Research ArticleRician Noise Removal via a Learned Dictionary
Jian Lu 1 Jiapeng Tian1 Lixin Shen2 Qingtang Jiang3 Xueying Zeng4 and Yuru Zou 1
1 Shenzhen Key Laboratory of Advanced Machine Learning and Applications College of Mathematics and StatisticsShenzhen University Shenzhen 518060 China
2Department of Mathematics Syracuse University Syracuse NY 13244 USA3Department of Mathematics and Computer Science University of Missouri-St Louis St Louis MO 63121 USA4School of Mathematical Sciences Ocean University of China Qingdao 266100 China
Correspondence should be addressed to Yuru Zou yrzou163com
Received 16 September 2018 Revised 9 January 2019 Accepted 3 February 2019 Published 18 February 2019
Academic Editor Elisa Francomano
Copyright copy 2019 Jian Lu et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
This paper proposes a new effective model for denoising images with Rician noise The sparse representations of images have beenshown to be efficient approaches for image processing Inspired by this we learn a dictionary from the noisy image and thencombine the MAP model with it for Rician noise removal For solving the proposed model the primal-dual algorithm is appliedand its convergence is studied The computational results show that the proposed method is promising in restoring images withRician noise
1 Introduction
Image denoising is one of the most fundamental issues inimage processing [1 2] In the past decades many methodsare proposed for removing Gaussian white noise [3 4]impulse noise [5ndash9] Poisson noise [10ndash12] andmultiplicativenoise [13ndash21] With the Magnetic Resonance Imaging (MRI)being widely used people are becoming gradually concernedwith another vital noise Rician noise In this paper wemainlystudy Rician noise and propose a newmodel that can bemoreefficient to remove it Mathematically the image 119910 degradedby Rician noise can be given by
119910 = radic(119909 + 1205781)2 + 12057822 (1)
where 119909 is the original image and 1205781 1205782 sim 119873(0 1205902) Our goalis to find the unknown true image 119909 from the degraded image119910 as well as possible
In recent years there have been many methods proposedfor denoising on the images corrupted by Rician noise suchas filtering methods wavelet methods nonconvex methodsand convex methods Based on the anisotropic diffusionprocess proposed by Perona and Malik [22] Gerig [23]
presented the nonlinear anisotropic filtering method forRician noise removal with the edge information preservedbut some other details may be lost Studying the nonlocalmeans algorithm in [24] Prima et al proposed a nonlocalmeans variants method for denoising the diffusion-weightedand diffusion tensor MR image And Manjn et al [25] alsoimproved a new filter (below we call it NLM model) whichis based on the nonlocal means to reduce the noise in MRimages Inspired by it Wiest-Daessle et al [26] adaptedthe nonlocal means filter to data corrupted by Rician noiseand presented nonlocal means filtering method for betterrespecting the image details and structures In [27] Nowakcame up with wavelet domain filtering method which adaptsto variations in both the signal and the noise From a singleRician-distributed image Foi [28] presented a stable and fastiterative procedure for robustly estimating the noise level andalso proposed a variance-stabilization method for efficientlyremoving Rician noise In [29] Wood and Johnson proposedwavelet packet denoising method for Rician noise removalat low SNR Wavelet methods have been efficient to removeRician noise with the image details and edge preservedbut the problem that small dots influence image analysisprocess remains unresolved In the mean time the maximum
HindawiMathematical Problems in EngineeringVolume 2019 Article ID 8535206 13 pageshttpsdoiorg10115520198535206
2 Mathematical Problems in Engineering
a posteriori (MAP) estimation model was proposed which isconsidered from the feature of noise-free image and includesthe data fidelity term TheMAP model is as follows
argmin119909
121205902 int
Ω1199092119889119905 minus int
Ωlog 1198680 (119909119910
1205902 ) 119889119905+ 120582int
Ω|119863119909| 119889119905
(2)
where 1198680 is the modified Bessel function of the first kind withorder zero [30] But it is a nonconvex function and leads to adifficult problem to solve In view of the MAP model GTVmodel [31] was put forward by Getreuer et al which is aconvex approximation of the MAP model and can be easilysolved but the fidelity item of GTV model is a complicatedpiecewise function Chen [32] proposed a new convex modelthat added a statistical property of Ricain noise into the MAPmodel leading to a new strictly convex model under mildcondition that can be easily solved by primal-dual algorithmand below we call it CZ model
In this paper we study the Rician noise and propose anew reasonable and efficient model for Rician noise removalAs we know natural images have a vital feature that issparseness and dictionary learning is being widely used forimage denoising Dictionary learning has been demonstratedto be efficient for various noise removal Aharon and Elad [3334] proposed the K-SVD algorithm for designing dictionarywith sparse representation and it is proven to be effectivefor additive white Gaussian noise removal Inspired by theK-SVD algorithm Huang [15] proposed a new model thatcombined the ldquoAArdquo model [35] and K-SVD algorithm toremove multiplication noise and also presented a log minus1198970 minimization approach to solve it Xiao [11] and Ma[10] also proposed new model via dictionary learning forPoisson noise removal and Poisson image deblurring Inaddition Liu et al [36] applied two-level Bregmanmethod todictionary updating and proposed an efficient algorithm forreconstructing MR images In [37] integrating total variation(TV) and dictionary learning Liu et al also proposed anovel gradient for image recovery Similarly integrating totalgeneralized variation and adaptive dictionary learning Luet al [38] presented a novel dictionary learning model forMRI reconstruction So we attempt to apply the sparserepresentation and dictionary learning into the MAP modelfor Rician noise removal Owing to the nonconvexity ofthe MAP model we add the sparse representation term toovercome the drawback so we can use the classical primal-dual algorithm to solve the model
The following is the outline of our paper In Section 2we first briefly introduce the dictionary learning and sparserepresentation and then we propose the new model thatcombines the MAP model with sparse representation termAlso we give and elaborate the two-step algorithm for solvingour model In Section 3 we demonstrate that our modeloutperforms the othermethods for Rician noise removal withnumerical results In the end we draw our conclusion inSection 4
2 Our Proposed Model
Generally speaking we consider that every signal instancefrom the family can be represented as a linear combination offew columns from a redundant dictionary For the degradedimage 119910 isin R
radic119873timesradic119873 regarding the image patch of size radic119899 timesradic119899 we order it as column vector Y isin R119899 lexicographicallyAnd we define a dictionary of size D isin R119899times119896 to simplyconstruct the Sparseland model where 119896 gt 119899 impliesthat the dictionary D is redundant Meanwhile we shouldalso make an assumption that the dictionary D is knownand fixed Then column vector Y can be sparsely linearlyrepresented by few atoms selected from the dictionary DThat is to say that there is a sparse solution of the followingproblem
= argmin120572
1205720 119904119906119887119895119890119888119905 119905119900 D120572 asymp Y (3)
That is 1205720 ≪ 119899 where 1205720 denotes the number of thenonzero entries in120572 And ∙0 is used to constrain the sparsityof representation
For simplicity we substitute Y minus D1205722 le 120598 for D120572 asymp YAlso replacing the constraint with a penalty term we can getthe equivalent problem of (3)
= argmin120572
Y minus D12057222 + 120583 1205720 (4)
and a suitable choice of 120583 can make problem (3) equivalent toproblem (4)
Now we consider the entire image 119910 that is to considerall the patches of image 119910 Then we can construct the sparserepresentation model for the noisy image 119910 First we indexthe image 119910 with Ω = 1 2 radic1198732 then the imagepatches of size radic119899 times radic119899 located in Ω can be indexed byΓ = 1 2 radic119873minusradic119899+12 From patches to image problem(4) becomes the following problem
119894119895 = argmin120572119894119895
sum(119894119895)isinΓ
10038171003817100381710038171003817R119894119895119910 minus D1205721198941198951003817100381710038171003817100381722 + sum(119894119895)isinΓ
whereR119894119895 is an 119899 times119873matrix that we can use to extract the (ij) patch from the image
Looking back at the assumption that the dictionaryD is known and fixed we have the following questionhow to choose and settle the dictionary In [33] Aharonput forward the K-SVD algorithm for designing dictionaryGiving the initial dictionary Aharon applied the singu-lar value decomposition (SVD) into updating dictionaryIn [34] Elad and Aharon applied the MAP estimator toproblem (5) and had a comparison of different dictionaries(overcomplete DCT global trained dictionary and adap-tive dictionary trained on patches from the noisy image)whose results show that the adaptive dictionary training isbest
After studying the sparse representation and dictionarylearning of K-SVD algorithm [33] it inspires us to apply thesparse representation to the MAP model (2) and thus we
Mathematical Problems in Engineering 3
propose a new model for Rician noise removal which is asfollows
where 119878(Ω) fl V isin 119861119881(Ω) 0 le V le 255 and intΩ|119863119909|119889119905
is the total variation (TV) of 119909 The first and second termsof our model stem from the MAP model which is data-fidelity caused by the statistical properties of Rician noiseand the third and forth terms are inspired by the sparserepresentation The last TV regularization term can make thedenoised image smooth
In model (6) there are three unknown variables thenoise-free image 119909 that we need to solve the dictionary Dand the sparse coefficients 120572119894119895 Similar to [33 34] in order tosolve problem (6) effectively and efficiently here we have thefollowing two-step algorithm
(1) Based on the degraded image 119910 we give the initialdictionary and get the sparse representation coefficients 120572119894119895then use 120572119894119895 to learn dictionary D and update correspondingcoefficients 120572119894119895
(2) Use the primal-dual algorithm to get the recoveredimage that we wanted
21 Dictionary Learning In the first step of our algorithmwe will use the degraded image 119910 to train a dictionary and allthe image patches can be sparsely represented by the traineddictionary with the corresponding coefficients 120572119894119895 The wholeprocess is just using the orthogonal matching pursuit (OMP)and K-SVD algorithms [33 34] and the procedure is to solvethe following optimization problem
D 119894119895 = arg minD120572119894119895
12 sum(119894119895)
10038171003817100381710038171003817R119894119895119909 minus D120572119894119895100381710038171003817100381710038172
For solving the difficult problem we have the followingspecific steps
(1) Initialization Set 119909 = 119910 D = overcomplete DCTdictionary
(2) Iteration 119865119900119903 119898 = 0 119905119900 119873 119889119900(119886) Given 119909 and D we get the sparse representationcoefficients 120572119894119895 through solving the following problem
119894119895 = argmin120572119894119895
12sum(119894119895)
10038171003817100381710038171003817R119894119895119909 minus D120572119894119895100381710038171003817100381710038172 + sum(119894119895)
and we can efficiently and effectively solve model (8) by usingthe orthogonal matching pursuit (OMP) method [33 34](119887) Given 119909 and 120572119894119895 we can update the dictionary D =[1198891 1198892 119889119896] column by column [33 34] For each column119889119897 119897 = 1 119896 we update it as follows(119894) For those patches represented by119889119897 wewrite down anddenote it by 120577119897 = (119894 119895) | 120572119894119895(119897) = 0
(119894119894) For each index (119894 119895) isin 120577119897 we compute the correspond-ing representation error through
119890119897119894119895 = R119894119895119909 minus sum119898 =119897
119889119898120572119894119895 (119898) (9)
and then we use the columns 119890119897119894119895(119894119895)isin120577119897 to define a matrix E119897(119894119894119894) For the matrix E119897 we apply the singular valuedecomposition (SVD) into E119897 and get E119897 = UΔVT
Let the first column of U be 119889119897 to update 119889119897 and multiplythe first column of V by Δ(1 1) to update 120572119894119895(119897)(119894119895)isin120577119897
End forHere the dictionary training and sparse representation
are completed
22 Primal-Dual Algorithm After the first step of our algo-rithm we get the spare dictionary representation D120572119894119895 fromeach patch R119894119895119910 and we now minimize (6) with respect to 119909that is
arg min119909isin119878(Ω)
12057321205902 int
Ω1199092119889119905 minus 120573int
Ωlog 1198680 (119910119909
1205902 )119889119905+ 1
2 sum10038171003817100381710038171003817D120572119894119895 minus R119894119895119909100381710038171003817100381710038172 + 120582intΩ|119863119909| 119889119905
(10)
Proposition 1 Let 119910 be a bounded function such thatinfΩ 119910 gt 0 then the objective function in (10) is strictly convexwith the constraint 1205904120573 gt (supΩ 119910)2Proof Using the notations 119882 = sum(119894119895)isinΓR
119879119894119895R119894119895 and 119872 =
sum(119894119895)isinΓR119879119894119895D120572119894119895 model (10) can be rewritten as
arg min119909isin119878(Ω)
12057321205902 ⟨119909 119909⟩ minus 120573⟨log 1198680 (119910119909
1205902 ) 1Ω⟩+ 1
2 ⟨119882119909 119909⟩ minus ⟨119872 119909⟩ + 120582 119909119879119881 (11)
Also we define 119892(119909) = (12057321205902)⟨119909 119909⟩minus120573⟨log 1198680(1199101199091205902) 1Ω⟩+(12)⟨119882119909 119909⟩ minus ⟨119872 119909⟩(119909 gt 0) and ℎ(119909) = log 1198680(119909)(119909 gt0) According to [32] we have 0 lt ℎ10158401015840(119909) = 1 minus(2119909)(1198681(119909)1198680(119909)) minus (1198681(119909)1198680(119909))2 lt 1 Then 11989210158401015840(119909) = 1205731205902 minus120573[log 1198680(1199101199091205902)]10158401015840 + 119882 gt 1205731205902 minus 120573(11991021205904) + 119882 Due to1 le 119882119894119895 le 119899 11989210158401015840(119909) gt 0 holds when 1205904120573 gt (supΩ 119910)2and 119892(119909) is strictly convex We know that 119909 997888rarr 119909119879119881 isconvex so the objective function (11) is strictly convex thatis function (10) is strictly convex
Referring to Proposition 1 we know that model (10) isconvex then we can apply the primal-dual algorithm forsolving the minimization problem which enjoys nice saddle-point structures and has good performance on nonsmooth
4 Mathematical Problems in Engineering
1 Let 120581 and 120591 be given Set 1199090 = 119910 1199090 = 119910 and 1199010 = (0 0)119879 isin R21198892 Update 119901119899 119909119899 119909119899 iteratively as follows
(a) 119901119899+1 = argmax119901isin119875
120582⟨119909119899 div 119901⟩ minus 12120581119901 minus 11990111989922
(b) 119909119899+1 = arg min119909isin119878(Ω)
119865(119909) minus 120582⟨119909 div 119901119899+1⟩ + 12120591119909 minus 11990911989922
(c) 119909119899+1 = 2119909119899+1 minus 119909119899until some stop criterion is satisfied
Algorithm 1
convex optimization [39ndash41] Moreover we discuss a biascorrection technique and the convergence of our algorithm
Briefly we define 119865(119909) fl (12057321205902)1199092 minus 120573⟨log 1198680(1199101199091205902) 1⟩ + (12)sum D120572119894119895 minus R1198941198951199092 and model (10) can betranslated to the following format
min0le119909le255
119865 (119909) + 120582 nabla1199091 (12)
Because the 119879119881-model has the duality property we canchange the primal-dual formulation of optimization problemof (12) into the following format
max119901isin119875
min0le119909le255
119865 (119909) minus 120582 ⟨119909 div 119901⟩ (13)
where 119875 = 119901 isin R2119889 max119894isin1119889|(1199012119894 + 1199012
119894+119889)12| le 1Here 119901 is the dual variable and div = minusnabla119879 The specific
algorithm is as in Algorithm 1In particular the solution of the dual problem 2(a) can be
where 1205871(119902119894) = 119902119894max(1 |119902119894|) 1205871(119902119889+119894) = 119902119889+119894max(1 |119902119894|)and |119902119894| = radic1199022119894 + 1199022119894+119889 119894 = 1 119889 Moreover the solution ofthe primal problem2(b) can be obtained byNewtonrsquosmethod
Inspired by [39] we provide a bias correction techniquefor (13) so that the mean of the restored image 119909lowast equals thatof the observed image 119910 In numerical practice the followingstep is implemented in order to preserve the mean of 119910
after updating 119909119899 by Newtonmethod which also ensures that119909119899 ge 0Referring to Theorem 1 in [42] we will get the conver-
gence properties of our algorithm as follows
Proposition 2 Suppose that 1205811205911205822nabla2 lt 1 en the iterates(119909119899 119901119899) of our algorithm converge to a saddle point of (13)
Inspired by [42] the condition of convergence of ouralgorithm can be simplified into 1205811205911205822 lt 18 based on thefact that nabla2 le 8 with the unit spacing size between pixelsFor simplicity in our numerical experiments we let 120581 = 8120582
and 120591 = 0015120582 and just adjust 120582 and the convergence of thealgorithm is satisfied
Before we present numerical results here we give thebasic property of model (10) rewritten in the continuoussettings that is
where (Ω119904)119904isin119868 is a finite set of small patches covering Ω andRΩ119904
is the restriction onΩ119904
Theorem 3 Let Ω be a bounded open subset of R2 withLipschitz boundary Let 119910 be a positive bounded function For119909 satisfying 119909 isin 119861119881(Ω) 119909 gt 0 let
where 120582 gt 0 Suppose that (Ω119904)119904isin119868 is a finite local coverage ofΩ andsuminfin
119904isin119868 1198631205721199042 lt +infin en 119864(119909) has a unique minimizerwhen 1205904120573 gt (supΩ 119910)2Proof Let119864(119909) = 120582119909119879119881+119869(119909) and 119869(119909) = (12057321205902) int
Ω1199092119889119905minus
120573intΩlog 1198680(1199101199091205902)119889119905 + (12)sum D120572119894119895 minus R1198941198951199092 By Proposi-
tion 1 we know that 119869(119909) is convex Thus 119864(119909) is convexand bounded from below Consider a minimizer sequence(119909119899) for 119864(119909) Then 119909119899119879119881 and 119869(119909119899) are bounded fromabove Hence (119909119899) is bounded in 119861119881(Ω) As 119861119881 is compactin 1198711(Ω) there is 1199090 isin 119861119881 such that a subsequence (119909119899119896)converges 1199090 in 1198711(Ω) and 119861119881(Ω) and we may assumethat 119909119899119896 997888rarr 1199090 pointwise almost everywhere As the 119861119881norm is lsc 1199090119879119881 le lim inf 119909119899119896119879119881 Since 119869(119909119899119896) isbounded from below by Fatoursquos Lemma we have 119869(1199090) lelim inf 119869(119909119899119896) Thus 119864(1199090) le lim inf 119864(119909119899119896) = inf119861119881(Ω)119864(119909)and 1199090 minimizes 119864(119909) Consequently 119864(119909) admits at leastone minimizer 1199090 119864(119909) is strictly convex function hence119864(119909) admits a unique minimizer
Mathematical Problems in Engineering 5
Table 1 Parameter 120582 values for all testing algorithms
In this section we will present our numerical experimentsto evaluate the approximation accuracy and computationalefficiency of our proposed algorithm We compare ourmethod for the denoising cases with the MAP model (2) theGTV model [31] the CZ model [32] and the NLM model[25] Here we have a brief overview of the models that wecompared
GTV model
argmin119909
120582intΩ119866120590 (119909 119910) + int
Ω|119863119909| 119889119905 (18)
119866120590 (119909 119910) = 119867120590 (119909) if 119909 ge 119888120590119867120590 (119888120590) + 1198671015840
120590 (119888120590) (119909 minus 119888120590) if 119909 le 1198881205901198671015840
where 119888 = 08426 1198681 is the modified Bessel function of thefirst kind with order one [30] Because it is an approximateequality we can not write down the final explicit restorationmodel
CZ model
arg min119909isin119878(Ω)
121205902 int
Ω1199092119889119905 minus int
Ωlog 1198680 (119909119910
1205902 )119889119905+ 1
120590 intΩ(radic119909 minus radic119910)2 119889119905 + 120582int
Ω|119863119909| 119889119905
(20)
where 119878(Ω) fl ] isin 119861119881(Ω) 0 le ] le 255 And because119861119881(Ω) [35 43] is the subspace of function 119909 isin 1198711(Ω) thefollowing quantity is finite
isin 119862infin0 (ΩR2) 10038171003817100381710038171205851003817100381710038171003817119871infin(ΩR2) le 1
(21)
NLMmodelBefore introducing the NLM model we have a descrip-
tion of the sign that will be used For a user-defined radius
119877119904119894119898 we define a square neighborhood window centeredaround pixel 119894 as 119873119894 And a Gaussian weighted Euclidiandistance of all the pixels of each neighborhood is defined as
119871 (119898 119899) = 1198661205881003817100381710038171003817119910 (119873119898) minus 119910 (119873119899)10038171003817100381710038172119877119904119894119898 (22)
where 119866120588 is a normalized Gaussian weighting functionwith zero mean and 120588 standard deviation (usually set to1) By giving more weight to pixels near the center wecan use 119866120588 to penalize pixels far from the center of theneighborhood window And based on the similarity betweenthe neighborhoods119873119898 and119873119899 of pixels119898 and 119899 we calculatethe similarity 119891(119898 119899) as
119862(119898) is the normalizing constant and ℎ is a exponentialdecay control parameter
Then given an image 119910 using the NLM method we cancalculate the filtered value at a point 119898 by the followingformula119873119871119872(119910 (119898)) = sum
forall119899isin119910
119891 (119898 119899) 119910 (119899)
0 le 119891 (119898 119899) le 1 sumforall119899isin119910
119891 (119898 119899) = 1 (24)
The parameter 120582 values of MAP model GTV modelCZ model and our proposed model are listed in Table 1In order to preserve the mean of the observed image thebias correction technique (15) is utilized for the CZ methodand our method All the experiments are performed underWindows 7 andMATLAB R2017a running on a PC equippedwith 290GHz CPU and 4G RAM
In our tests we choose images ldquoLenardquo and ldquoBarbarardquo withsize of 512 times 512 ldquoHouserdquo and ldquoMonarchrdquo with size of 256 times256 ldquoBrainrdquo with size of 181 times 217 and ldquoMouse intestinerdquo(briefly we call it ldquoMouserdquo throughout this paper) with sizeof 248 times 254 which are shown in Figure 1 We evaluate thequality of recovered images obtained from various denoisingalgorithms by using the structural similarity index (SSIM)[44] and the peal-signal-to-noise ratio (PSNR) [45] definedby
where 119909 and 119909 denote the original image and the recoveredimage respectively
First we choose the test image of ldquoLenardquo and degrade it byRician noise with120590 = 10 and use differentmethods to recoverthe noisy image The numerical results are enumerated inTable 2 where the first column gives the test image names thesecond column gives the method names (here Algorithm 1denotes our method presented in Section 2) and columns
3-5 (resp 6-8 and 9-11) are the PSNR (dB) SSIM and CPUtime (s) for 120590 = 10 (resp 120590 = 15 120590 = 20) We observethat the PSNR values of the restored images by Algorithm 1are higher than those of MAP GTV CZ and NLM methodMoreover you can easily find that the PSNR values of therestored images by Algorithm 1 are always the highest amongall those five methods for all images As far as ldquoBarbarardquo isconcerned the PSNR value of our method is more than 3
dB higher than CZ model at 120590 = 10 The characteristicsof SSIM values are almost consistent with those of PSNRvalues However the SSIM values are composed of luminancecomparison contrast comparison and structure comparisonwhich makes differences at some points
The images are corrupted respectively by Rician distri-bution with 120590 = 10 120590 = 15 and 120590 = 20 and the restoredimages of the above five algorithms for the images ldquoLenardquoldquoBarbarardquo ldquoHouserdquo and ldquoMonarchrdquo are shown inFigures 2ndash5respectively We clearly find that much noise remains in the
Figure 6 Results of ldquoBrainrdquo with 120590 = 10 by different methods (a) is the original ldquoBrainrdquo image and (g) is the degraded image with 120590 = 10(b)-(f) are the denoised images ofMAPmethod (120582 = 17) GTVmethod (120582 = 15) CZmethod (120582 = 0065) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
results of MAP GTV and CZmethod For example in Row 4of Figure 3 we can see from the background that ldquoBarbarardquoimage restored by our method is more clearer than thoserestored by other methods and textures of Barbararsquos trousersand scarf are kept better in the restored image by ourmethodThe restored images by our method also preserve moresignificant details thanMAPGTV CZ andNLMmethods onthe hat tidbits of the ldquoLenardquo image What is more the flowersof the lower left of ldquoMonarchrdquo also indicate that our method
is superior to other methods because our method is a patch-based method which is a different framework with TV-basedmethods (ie MAP method and CZ method) The imagesobtained from our method provide smoother regions andbetter shape preservation (eg the backgrounds in ldquoLenardquoand ldquoBarbarardquo)
Figures 6ndash11 are results of experiments on ldquoBrainrdquo andldquoMouserdquo images In particular for the ldquoBrainrdquo and ldquoMouserdquoimages we not only present the recovered images but also
10 Mathematical Problems in Engineering
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 7 Results of Brain with 120590 = 15 by different methods (a) is the original ldquoBrainrdquo image and (g) is the degraded image with 120590 = 15(b)-(f) are the denoised images of MAPmethod (120582 = 25) GTVmethod (120582 = 22) CZmethod (120582 = 005) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 8 Results of Brain with 120590 = 20 by different methods (a) is the original ldquoBrainrdquo image and (g) is the degraded image with 120590 = 20(b)-(f) are the denoised images ofMAPmethod (120582 = 35) GTVmethod (120582 = 22) CZmethod (120582 = 0045) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
use residuals to make a comparison with the original imagesThe first row is original image and recovered images ofMAP method GTV method CZ method NLM methodand our method The second row is degraded image withvarious 120590 and residuals of those methods respectively InFigures 6ndash8 the recovered images of of MAP method GTVmethod CZ method and NLM method are still unclear sowe can conclude that the recovered images by ourmethod arebest through our visions and residuals As for the ldquoMouserdquoimage the recovered results are very similar in this casewe use residuals to see which result is better In general theblurrier residual is the better according recovered image isWe can find that the outline of residual by our method isblurriest
In conclusion the results of our numerical experimentsdemonstrate that our proposed method performs better thanthe MAP method GTV method CZ method and NLMmethod
4 Conclusion
In this paper we proposed a new effective model via alearned dictionary for denoising images with Rician noiseMore specifically based on the dictionary training we adda dictionary penalty term to the original nonconvex MAPmodel to establish a new denoising model and develop atwo-step algorithm to solve our model Also we carry out
Mathematical Problems in Engineering 11
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 9 Results of ldquoMouserdquo with 120590 = 10 by different methods (a) is the original ldquoMouserdquo image and (g) is the degraded image with 120590 = 10(b)-(f) are the denoised images of MAPmethod (120582 = 17) GTVmethod (120582 = 17) CZmethod (120582 = 006) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 10 Results of ldquoMouserdquo with 120590 = 15 by different methods (a) is the original ldquoMouserdquo image and (g) is the degraded image with 120590 = 15(b)-(f) are the denoised images of MAPmethod (120582 = 24) GTV method (120582 = 24) CZ method (120582 = 005) NLM method and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
experiments on various images to demonstrate the effective-ness of our modelThe numerical experiments show that ourproposedmodel is promising in denoising imageswithRiciannoise
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interest
Acknowledgments
The authors would like to thank one of the authors of[32] for providing the source code of CZ method Thiswork was supported in part by the National Natural ScienceFoundation of China under Grants 11871348 61872429 and61373087 by the Natural Science Foundation of GuangdongChina under Grants 2015A030313550 and 2015A030313557by the HD Video RampD Platform for Intelligent Analy-sis and Processing in Guangdong Engineering TechnologyResearch Centre of Colleges and Universities (no GCZX-A1409) by Natural Science Foundation of Shenzhen underGrant JCYJ20170818091621856 and by the Guangdong Key
12 Mathematical Problems in Engineering
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 11 Results of ldquoMouserdquo with 120590 = 20 by different methods (a) is the original ldquoMouserdquo image and (g) is the degraded image with 120590 = 20(b)-(f) are the denoised images of MAPmethod (120582 = 31) GTV method (120582 = 32) CZ method (120582 = 0035) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
Laboratory of Intelligent Information Processing ShenzhenUniversity China (518060)
References
[1] J-F Cai B Dong and Z Shen ldquoImage restoration A waveletframe based model for piecewise smooth functions andbeyondrdquoApplied andComputational Harmonic Analysis vol 41no 1 pp 94ndash138 2016
[2] R H Chan and J Ma ldquoA multiplicative iterative algorithm forbox-constrained penalized likelihood image restorationrdquo IEEETransactions on Image Processing vol 21 no 7 pp 3168ndash31812012
[3] C A Micchelli L Shen and Y Xu ldquoProximity algorithmsfor image models Denoisingrdquo Inverse Problems vol 27 no 4Article ID 045009 p 30 2011
[4] YWang J Yang W Yin and Y Zhang ldquoA new alternatingmin-imization algorithm for total variation image reconstructionrdquoSIAM Journal on Imaging Sciences vol 1 no 3 pp 248ndash2722008
[5] Y Dong M Hintermuller and M Neri ldquoAn efficient primal-dual method for l1-TV image restorationrdquo SIAM Journal onImaging Sciences vol 2 no 4 pp 1168ndash1189 2009
[6] J Lu K Qiao L Shen and Y Zou ldquoFixed-point algorithmsfor a TVL1 image restoration modelrdquo International Journal ofComputer Mathematics vol 95 no 9 pp 1829ndash1844 2018
[7] J Lu Z Ye and Y Zou ldquoHuber fractal image coding based ona fitting planerdquo IEEE Transactions on Image Processing vol 22no 1 pp 134ndash145 2013
[8] C A Micchelli L Shen Y Xu and X Zeng ldquoProximityalgorithms for the L1TV image denoising modelrdquo Advances inComputational Mathematics vol 38 no 2 pp 401ndash426 2013
[9] J Yang Y Zhang and W Yin ldquoAn efficient TVL1 algorithm fordeblurring multichannel images corrupted by impulsive noiserdquoSIAM Journal on Scientific Computing vol 31 no 4 pp 2842ndash2865 2009
[10] L Ma L Moisan J Yu and T Zeng ldquoA dictionary learningapproach for poisson image deblurringrdquo IEEE Transactions onMedical Imaging vol 32 no 7 pp 1277ndash1289 2013
[11] Y Xiao and T Zeng ldquoPoisson noise removal via learned dictio-naryrdquo in Proceedings of the 17th IEEE International Conferenceon Image Processing (ICIP) pp 1177ndash1180 2010
[12] H Zhang Y Dong and Q Fan ldquoWavelet frame based Poissonnoise removal and image deblurringrdquo Signal Processing vol 137pp 363ndash372 2017
[13] Y Dong and T Zeng ldquoA convex variational model for restoringblurred images with multiplicative noiserdquo SIAM Journal onImaging Sciences vol 6 no 3 pp 1598ndash1625 2013
[14] Y Huang M K Ng and YWen ldquoA new total variation methodfor multiplicative noise removalrdquo SIAM Journal on ImagingSciences vol 2 no 1 pp 20ndash40 2009
[15] Y-M Huang L Moisan M K Ng and T Zeng ldquoMultiplicativenoise removal via a learned dictionaryrdquo IEEE Transactions onImage Processing vol 21 no 11 pp 4534ndash4543 2012
[16] Z Jin and X Yang ldquoA variational model to remove the multi-plicative noise in ultrasound imagesrdquo Journal of MathematicalImaging and Vision vol 39 no 1 pp 62ndash74 2011
[17] M Kang S Yun and H Woo ldquoTwo-level convex relaxedvariational model for multiplicative denoisingrdquo SIAM Journalon Imaging Sciences vol 6 no 2 pp 875ndash903 2013
[18] J Lu L Shen C Xu and Y Xu ldquoMultiplicative noise removalin imaging An exp-model and its fixed-point proximity algo-rithmrdquo Applied and Computational Harmonic Analysis vol 41no 2 pp 518ndash539 2016
[19] J Lu L Shen C Xu and Y Xu ldquoMultiplicative noise removalwith a sparsity-aware optimization modelrdquo Inverse Problemsand Imaging vol 11 no 6 pp 949ndash974 2017
[20] J Lu Z Yang L Shen Z Lu H Yang and C Xu ldquoA frameletalgorithm for de-blurring images corrupted by multiplicativenoiserdquoApplied Mathematical Modelling vol 62 pp 51ndash61 2018
[21] J Lu Y Chen Y Zou and L Shen ldquoA new total variationmodelfor restoring blurred and speckle noisy imagesrdquo InternationalJournal of Wavelets Multiresolution and Information Processingvol 15 no 2 19 pages 2017
Mathematical Problems in Engineering 13
[22] P Perona and J Malik ldquoScale-space and edge detection usinganisotropic diffusionrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 12 no 7 pp 629ndash639 1990
[23] G Gerig O Kubler R Kikinis and F A Jolesz ldquoNonlinearanisotropic filtering ofMRI datardquo IEEE Transactions onMedicalImaging vol 11 no 2 pp 221ndash232 1992
[24] S Prima S P Morrissey and C Barillot ldquoNon-local meansvariants for denoising of diffusion-weighted and diffusiontensor MRIrdquo in Proceedings of the International Conference onMedical Image Computing and Computer-Assisted Interventionpp 344ndash351 Springer Berlin Germany 2007
[25] J V Manjon J Carbonell-Caballero J J Lull G Garcıa-MartiL Martı-Bonmati and M Robles ldquoMRI denoising using Non-LocalMeansrdquoMedical Image Analysis vol 12 no 4 pp 514ndash5232008
[26] N Wiest-Daessle S Prima P Coupe S P Morrissey and CBarillot ldquoRician noise removal by non-local means filteringfor low signal-to-noise ratio MRI Applications to DT-MRIrdquoMedical Image Computing and Computer-Assisted Interventionvol 5242 no 2 pp 171ndash179 2008
[27] R D Nowak ldquoWavelet-based Rician noise removal for mag-netic resonance imagingrdquo IEEE Transactions on Image Process-ing vol 8 no 10 pp 1408ndash1419 1999
[28] A Foi ldquoNoise estimation and removal in MR imaging Thevariance-stabilization approachrdquo in Proceedings of the 8th IEEEInternational Symposium on Biomedical Imaging From Nano toMacro (ISBI) pp 1809ndash1814 IEEE Chicago IL USA 2011
[29] J C Wood and K M Johnson ldquoWavelet packet denoising ofmagnetic resonance images Importance of Rician noise at lowSNRrdquo Magnetic Resonance in Medicine vol 41 no 3 pp 631ndash635 1999
[30] F Bowman Introduction to Bessel functions Dover PublicationsMineola NY USA 2012
[31] P Getreuer M Tong and L A Vese ldquoA variational model forthe restoration of MR images corrupted by blur and Riciannoiserdquo in Proceedings of the 7th international conference onAdvances in visual computing vol Part I of Lecture Notes inComputer Science pp 686ndash698 Springer Las Vegas NV USA2011
[32] L Chen and T Zeng ldquoA convex variational model for restoringblurred imageswith large Rician noiserdquo Journal ofMathematicalImaging and Vision vol 53 no 1 pp 92ndash111 2015
[33] M Aharon M Elad and A M Bruckstein ldquoK-SVD Analgorithm for designing overcomplete dictionaries for sparserepresentationrdquo IEEE Transactions on Signal Processing vol 54no 11 pp 4311ndash4322 2006
[34] M Elad and M Aharon ldquoImage denoising via sparse andredundant representations over learned dictionariesrdquo IEEETransactions on Image Processing vol 15 no 12 pp 3736ndash37452006
[35] G Aubert and J Aujol ldquoA variational approach to removingmultiplicative noiserdquo SIAM Journal on Applied Mathematicsvol 68 no 4 pp 925ndash946 2008
[36] Q Liu S Wang K Yang J Luo Y Zhu and D Liang ldquoHighlyundersampled magnetic resonance image reconstruction usingtwo-level Bregman method with dictionary updatingrdquo IEEETransactions on Medical Imaging vol 32 no 7 pp 1290ndash13012013
[37] Q Liu S Wang L Ying X Peng Y Zhu and D LiangldquoAdaptive dictionary learning in sparse gradient domain forimage recoveryrdquo IEEE Transactions on Image Processing vol 22no 12 pp 4652ndash4663 2013
[38] H Lu J Wei Q Liu Y Wang and X Deng ldquoA dictionarylearning method with total generalized variation for mri recon-structionrdquo International Journal of Biomedical Imaging vol2016 Article ID 7512471 13 pages 2016
[39] A Chambolle and T Pock ldquoA first-order primal-dual algorithmfor convex problems with applications to imagingrdquo Journal ofMathematical Imaging and Vision vol 40 no 1 pp 120ndash1452011
[40] E Esser X Zhang and T F Chan ldquoA general frameworkfor a class of first order primal-dual algorithms for convexoptimization in imaging sciencerdquo SIAM Journal on ImagingSciences vol 3 no 4 pp 1015ndash1046 2010
[41] N Komodakis and J-C Pesquet ldquoPlaying with duality Anoverview of recent primaldual approaches for solving large-scale optimization problemsrdquo IEEE Signal Processing Magazinevol 32 no 6 pp 31ndash54 2015
[42] A Chambolle ldquoAn algorithm for total variation minimizationand applicationsrdquo Journal of Mathematical Imaging and Visionvol 20 no 1-2 pp 89ndash97 2004
[43] L Ambrosio N Fusco and D Pallara Functions of BoundedVariation and Free Discontinuity Problem Oxford UniversityPress London 2000
[44] Z Wang A C Bovik H R Sheikh and E P SimoncellildquoImage quality assessment From error visibility to structuralsimilarityrdquo IEEE Transactions on Image Processing vol 13 no4 pp 600ndash612 2004
[45] J Gibson and A Bovik Handbook of Image and Video Process-ing Academic Press 2000
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2 Mathematical Problems in Engineering
a posteriori (MAP) estimation model was proposed which isconsidered from the feature of noise-free image and includesthe data fidelity term TheMAP model is as follows
argmin119909
121205902 int
Ω1199092119889119905 minus int
Ωlog 1198680 (119909119910
1205902 ) 119889119905+ 120582int
Ω|119863119909| 119889119905
(2)
where 1198680 is the modified Bessel function of the first kind withorder zero [30] But it is a nonconvex function and leads to adifficult problem to solve In view of the MAP model GTVmodel [31] was put forward by Getreuer et al which is aconvex approximation of the MAP model and can be easilysolved but the fidelity item of GTV model is a complicatedpiecewise function Chen [32] proposed a new convex modelthat added a statistical property of Ricain noise into the MAPmodel leading to a new strictly convex model under mildcondition that can be easily solved by primal-dual algorithmand below we call it CZ model
In this paper we study the Rician noise and propose anew reasonable and efficient model for Rician noise removalAs we know natural images have a vital feature that issparseness and dictionary learning is being widely used forimage denoising Dictionary learning has been demonstratedto be efficient for various noise removal Aharon and Elad [3334] proposed the K-SVD algorithm for designing dictionarywith sparse representation and it is proven to be effectivefor additive white Gaussian noise removal Inspired by theK-SVD algorithm Huang [15] proposed a new model thatcombined the ldquoAArdquo model [35] and K-SVD algorithm toremove multiplication noise and also presented a log minus1198970 minimization approach to solve it Xiao [11] and Ma[10] also proposed new model via dictionary learning forPoisson noise removal and Poisson image deblurring Inaddition Liu et al [36] applied two-level Bregmanmethod todictionary updating and proposed an efficient algorithm forreconstructing MR images In [37] integrating total variation(TV) and dictionary learning Liu et al also proposed anovel gradient for image recovery Similarly integrating totalgeneralized variation and adaptive dictionary learning Luet al [38] presented a novel dictionary learning model forMRI reconstruction So we attempt to apply the sparserepresentation and dictionary learning into the MAP modelfor Rician noise removal Owing to the nonconvexity ofthe MAP model we add the sparse representation term toovercome the drawback so we can use the classical primal-dual algorithm to solve the model
The following is the outline of our paper In Section 2we first briefly introduce the dictionary learning and sparserepresentation and then we propose the new model thatcombines the MAP model with sparse representation termAlso we give and elaborate the two-step algorithm for solvingour model In Section 3 we demonstrate that our modeloutperforms the othermethods for Rician noise removal withnumerical results In the end we draw our conclusion inSection 4
2 Our Proposed Model
Generally speaking we consider that every signal instancefrom the family can be represented as a linear combination offew columns from a redundant dictionary For the degradedimage 119910 isin R
radic119873timesradic119873 regarding the image patch of size radic119899 timesradic119899 we order it as column vector Y isin R119899 lexicographicallyAnd we define a dictionary of size D isin R119899times119896 to simplyconstruct the Sparseland model where 119896 gt 119899 impliesthat the dictionary D is redundant Meanwhile we shouldalso make an assumption that the dictionary D is knownand fixed Then column vector Y can be sparsely linearlyrepresented by few atoms selected from the dictionary DThat is to say that there is a sparse solution of the followingproblem
= argmin120572
1205720 119904119906119887119895119890119888119905 119905119900 D120572 asymp Y (3)
That is 1205720 ≪ 119899 where 1205720 denotes the number of thenonzero entries in120572 And ∙0 is used to constrain the sparsityof representation
For simplicity we substitute Y minus D1205722 le 120598 for D120572 asymp YAlso replacing the constraint with a penalty term we can getthe equivalent problem of (3)
= argmin120572
Y minus D12057222 + 120583 1205720 (4)
and a suitable choice of 120583 can make problem (3) equivalent toproblem (4)
Now we consider the entire image 119910 that is to considerall the patches of image 119910 Then we can construct the sparserepresentation model for the noisy image 119910 First we indexthe image 119910 with Ω = 1 2 radic1198732 then the imagepatches of size radic119899 times radic119899 located in Ω can be indexed byΓ = 1 2 radic119873minusradic119899+12 From patches to image problem(4) becomes the following problem
119894119895 = argmin120572119894119895
sum(119894119895)isinΓ
10038171003817100381710038171003817R119894119895119910 minus D1205721198941198951003817100381710038171003817100381722 + sum(119894119895)isinΓ
whereR119894119895 is an 119899 times119873matrix that we can use to extract the (ij) patch from the image
Looking back at the assumption that the dictionaryD is known and fixed we have the following questionhow to choose and settle the dictionary In [33] Aharonput forward the K-SVD algorithm for designing dictionaryGiving the initial dictionary Aharon applied the singu-lar value decomposition (SVD) into updating dictionaryIn [34] Elad and Aharon applied the MAP estimator toproblem (5) and had a comparison of different dictionaries(overcomplete DCT global trained dictionary and adap-tive dictionary trained on patches from the noisy image)whose results show that the adaptive dictionary training isbest
After studying the sparse representation and dictionarylearning of K-SVD algorithm [33] it inspires us to apply thesparse representation to the MAP model (2) and thus we
Mathematical Problems in Engineering 3
propose a new model for Rician noise removal which is asfollows
where 119878(Ω) fl V isin 119861119881(Ω) 0 le V le 255 and intΩ|119863119909|119889119905
is the total variation (TV) of 119909 The first and second termsof our model stem from the MAP model which is data-fidelity caused by the statistical properties of Rician noiseand the third and forth terms are inspired by the sparserepresentation The last TV regularization term can make thedenoised image smooth
In model (6) there are three unknown variables thenoise-free image 119909 that we need to solve the dictionary Dand the sparse coefficients 120572119894119895 Similar to [33 34] in order tosolve problem (6) effectively and efficiently here we have thefollowing two-step algorithm
(1) Based on the degraded image 119910 we give the initialdictionary and get the sparse representation coefficients 120572119894119895then use 120572119894119895 to learn dictionary D and update correspondingcoefficients 120572119894119895
(2) Use the primal-dual algorithm to get the recoveredimage that we wanted
21 Dictionary Learning In the first step of our algorithmwe will use the degraded image 119910 to train a dictionary and allthe image patches can be sparsely represented by the traineddictionary with the corresponding coefficients 120572119894119895 The wholeprocess is just using the orthogonal matching pursuit (OMP)and K-SVD algorithms [33 34] and the procedure is to solvethe following optimization problem
D 119894119895 = arg minD120572119894119895
12 sum(119894119895)
10038171003817100381710038171003817R119894119895119909 minus D120572119894119895100381710038171003817100381710038172
For solving the difficult problem we have the followingspecific steps
(1) Initialization Set 119909 = 119910 D = overcomplete DCTdictionary
(2) Iteration 119865119900119903 119898 = 0 119905119900 119873 119889119900(119886) Given 119909 and D we get the sparse representationcoefficients 120572119894119895 through solving the following problem
119894119895 = argmin120572119894119895
12sum(119894119895)
10038171003817100381710038171003817R119894119895119909 minus D120572119894119895100381710038171003817100381710038172 + sum(119894119895)
and we can efficiently and effectively solve model (8) by usingthe orthogonal matching pursuit (OMP) method [33 34](119887) Given 119909 and 120572119894119895 we can update the dictionary D =[1198891 1198892 119889119896] column by column [33 34] For each column119889119897 119897 = 1 119896 we update it as follows(119894) For those patches represented by119889119897 wewrite down anddenote it by 120577119897 = (119894 119895) | 120572119894119895(119897) = 0
(119894119894) For each index (119894 119895) isin 120577119897 we compute the correspond-ing representation error through
119890119897119894119895 = R119894119895119909 minus sum119898 =119897
119889119898120572119894119895 (119898) (9)
and then we use the columns 119890119897119894119895(119894119895)isin120577119897 to define a matrix E119897(119894119894119894) For the matrix E119897 we apply the singular valuedecomposition (SVD) into E119897 and get E119897 = UΔVT
Let the first column of U be 119889119897 to update 119889119897 and multiplythe first column of V by Δ(1 1) to update 120572119894119895(119897)(119894119895)isin120577119897
End forHere the dictionary training and sparse representation
are completed
22 Primal-Dual Algorithm After the first step of our algo-rithm we get the spare dictionary representation D120572119894119895 fromeach patch R119894119895119910 and we now minimize (6) with respect to 119909that is
arg min119909isin119878(Ω)
12057321205902 int
Ω1199092119889119905 minus 120573int
Ωlog 1198680 (119910119909
1205902 )119889119905+ 1
2 sum10038171003817100381710038171003817D120572119894119895 minus R119894119895119909100381710038171003817100381710038172 + 120582intΩ|119863119909| 119889119905
(10)
Proposition 1 Let 119910 be a bounded function such thatinfΩ 119910 gt 0 then the objective function in (10) is strictly convexwith the constraint 1205904120573 gt (supΩ 119910)2Proof Using the notations 119882 = sum(119894119895)isinΓR
119879119894119895R119894119895 and 119872 =
sum(119894119895)isinΓR119879119894119895D120572119894119895 model (10) can be rewritten as
arg min119909isin119878(Ω)
12057321205902 ⟨119909 119909⟩ minus 120573⟨log 1198680 (119910119909
1205902 ) 1Ω⟩+ 1
2 ⟨119882119909 119909⟩ minus ⟨119872 119909⟩ + 120582 119909119879119881 (11)
Also we define 119892(119909) = (12057321205902)⟨119909 119909⟩minus120573⟨log 1198680(1199101199091205902) 1Ω⟩+(12)⟨119882119909 119909⟩ minus ⟨119872 119909⟩(119909 gt 0) and ℎ(119909) = log 1198680(119909)(119909 gt0) According to [32] we have 0 lt ℎ10158401015840(119909) = 1 minus(2119909)(1198681(119909)1198680(119909)) minus (1198681(119909)1198680(119909))2 lt 1 Then 11989210158401015840(119909) = 1205731205902 minus120573[log 1198680(1199101199091205902)]10158401015840 + 119882 gt 1205731205902 minus 120573(11991021205904) + 119882 Due to1 le 119882119894119895 le 119899 11989210158401015840(119909) gt 0 holds when 1205904120573 gt (supΩ 119910)2and 119892(119909) is strictly convex We know that 119909 997888rarr 119909119879119881 isconvex so the objective function (11) is strictly convex thatis function (10) is strictly convex
Referring to Proposition 1 we know that model (10) isconvex then we can apply the primal-dual algorithm forsolving the minimization problem which enjoys nice saddle-point structures and has good performance on nonsmooth
4 Mathematical Problems in Engineering
1 Let 120581 and 120591 be given Set 1199090 = 119910 1199090 = 119910 and 1199010 = (0 0)119879 isin R21198892 Update 119901119899 119909119899 119909119899 iteratively as follows
(a) 119901119899+1 = argmax119901isin119875
120582⟨119909119899 div 119901⟩ minus 12120581119901 minus 11990111989922
(b) 119909119899+1 = arg min119909isin119878(Ω)
119865(119909) minus 120582⟨119909 div 119901119899+1⟩ + 12120591119909 minus 11990911989922
(c) 119909119899+1 = 2119909119899+1 minus 119909119899until some stop criterion is satisfied
Algorithm 1
convex optimization [39ndash41] Moreover we discuss a biascorrection technique and the convergence of our algorithm
Briefly we define 119865(119909) fl (12057321205902)1199092 minus 120573⟨log 1198680(1199101199091205902) 1⟩ + (12)sum D120572119894119895 minus R1198941198951199092 and model (10) can betranslated to the following format
min0le119909le255
119865 (119909) + 120582 nabla1199091 (12)
Because the 119879119881-model has the duality property we canchange the primal-dual formulation of optimization problemof (12) into the following format
max119901isin119875
min0le119909le255
119865 (119909) minus 120582 ⟨119909 div 119901⟩ (13)
where 119875 = 119901 isin R2119889 max119894isin1119889|(1199012119894 + 1199012
119894+119889)12| le 1Here 119901 is the dual variable and div = minusnabla119879 The specific
algorithm is as in Algorithm 1In particular the solution of the dual problem 2(a) can be
where 1205871(119902119894) = 119902119894max(1 |119902119894|) 1205871(119902119889+119894) = 119902119889+119894max(1 |119902119894|)and |119902119894| = radic1199022119894 + 1199022119894+119889 119894 = 1 119889 Moreover the solution ofthe primal problem2(b) can be obtained byNewtonrsquosmethod
Inspired by [39] we provide a bias correction techniquefor (13) so that the mean of the restored image 119909lowast equals thatof the observed image 119910 In numerical practice the followingstep is implemented in order to preserve the mean of 119910
after updating 119909119899 by Newtonmethod which also ensures that119909119899 ge 0Referring to Theorem 1 in [42] we will get the conver-
gence properties of our algorithm as follows
Proposition 2 Suppose that 1205811205911205822nabla2 lt 1 en the iterates(119909119899 119901119899) of our algorithm converge to a saddle point of (13)
Inspired by [42] the condition of convergence of ouralgorithm can be simplified into 1205811205911205822 lt 18 based on thefact that nabla2 le 8 with the unit spacing size between pixelsFor simplicity in our numerical experiments we let 120581 = 8120582
and 120591 = 0015120582 and just adjust 120582 and the convergence of thealgorithm is satisfied
Before we present numerical results here we give thebasic property of model (10) rewritten in the continuoussettings that is
where (Ω119904)119904isin119868 is a finite set of small patches covering Ω andRΩ119904
is the restriction onΩ119904
Theorem 3 Let Ω be a bounded open subset of R2 withLipschitz boundary Let 119910 be a positive bounded function For119909 satisfying 119909 isin 119861119881(Ω) 119909 gt 0 let
where 120582 gt 0 Suppose that (Ω119904)119904isin119868 is a finite local coverage ofΩ andsuminfin
119904isin119868 1198631205721199042 lt +infin en 119864(119909) has a unique minimizerwhen 1205904120573 gt (supΩ 119910)2Proof Let119864(119909) = 120582119909119879119881+119869(119909) and 119869(119909) = (12057321205902) int
Ω1199092119889119905minus
120573intΩlog 1198680(1199101199091205902)119889119905 + (12)sum D120572119894119895 minus R1198941198951199092 By Proposi-
tion 1 we know that 119869(119909) is convex Thus 119864(119909) is convexand bounded from below Consider a minimizer sequence(119909119899) for 119864(119909) Then 119909119899119879119881 and 119869(119909119899) are bounded fromabove Hence (119909119899) is bounded in 119861119881(Ω) As 119861119881 is compactin 1198711(Ω) there is 1199090 isin 119861119881 such that a subsequence (119909119899119896)converges 1199090 in 1198711(Ω) and 119861119881(Ω) and we may assumethat 119909119899119896 997888rarr 1199090 pointwise almost everywhere As the 119861119881norm is lsc 1199090119879119881 le lim inf 119909119899119896119879119881 Since 119869(119909119899119896) isbounded from below by Fatoursquos Lemma we have 119869(1199090) lelim inf 119869(119909119899119896) Thus 119864(1199090) le lim inf 119864(119909119899119896) = inf119861119881(Ω)119864(119909)and 1199090 minimizes 119864(119909) Consequently 119864(119909) admits at leastone minimizer 1199090 119864(119909) is strictly convex function hence119864(119909) admits a unique minimizer
Mathematical Problems in Engineering 5
Table 1 Parameter 120582 values for all testing algorithms
In this section we will present our numerical experimentsto evaluate the approximation accuracy and computationalefficiency of our proposed algorithm We compare ourmethod for the denoising cases with the MAP model (2) theGTV model [31] the CZ model [32] and the NLM model[25] Here we have a brief overview of the models that wecompared
GTV model
argmin119909
120582intΩ119866120590 (119909 119910) + int
Ω|119863119909| 119889119905 (18)
119866120590 (119909 119910) = 119867120590 (119909) if 119909 ge 119888120590119867120590 (119888120590) + 1198671015840
120590 (119888120590) (119909 minus 119888120590) if 119909 le 1198881205901198671015840
where 119888 = 08426 1198681 is the modified Bessel function of thefirst kind with order one [30] Because it is an approximateequality we can not write down the final explicit restorationmodel
CZ model
arg min119909isin119878(Ω)
121205902 int
Ω1199092119889119905 minus int
Ωlog 1198680 (119909119910
1205902 )119889119905+ 1
120590 intΩ(radic119909 minus radic119910)2 119889119905 + 120582int
Ω|119863119909| 119889119905
(20)
where 119878(Ω) fl ] isin 119861119881(Ω) 0 le ] le 255 And because119861119881(Ω) [35 43] is the subspace of function 119909 isin 1198711(Ω) thefollowing quantity is finite
isin 119862infin0 (ΩR2) 10038171003817100381710038171205851003817100381710038171003817119871infin(ΩR2) le 1
(21)
NLMmodelBefore introducing the NLM model we have a descrip-
tion of the sign that will be used For a user-defined radius
119877119904119894119898 we define a square neighborhood window centeredaround pixel 119894 as 119873119894 And a Gaussian weighted Euclidiandistance of all the pixels of each neighborhood is defined as
119871 (119898 119899) = 1198661205881003817100381710038171003817119910 (119873119898) minus 119910 (119873119899)10038171003817100381710038172119877119904119894119898 (22)
where 119866120588 is a normalized Gaussian weighting functionwith zero mean and 120588 standard deviation (usually set to1) By giving more weight to pixels near the center wecan use 119866120588 to penalize pixels far from the center of theneighborhood window And based on the similarity betweenthe neighborhoods119873119898 and119873119899 of pixels119898 and 119899 we calculatethe similarity 119891(119898 119899) as
119862(119898) is the normalizing constant and ℎ is a exponentialdecay control parameter
Then given an image 119910 using the NLM method we cancalculate the filtered value at a point 119898 by the followingformula119873119871119872(119910 (119898)) = sum
forall119899isin119910
119891 (119898 119899) 119910 (119899)
0 le 119891 (119898 119899) le 1 sumforall119899isin119910
119891 (119898 119899) = 1 (24)
The parameter 120582 values of MAP model GTV modelCZ model and our proposed model are listed in Table 1In order to preserve the mean of the observed image thebias correction technique (15) is utilized for the CZ methodand our method All the experiments are performed underWindows 7 andMATLAB R2017a running on a PC equippedwith 290GHz CPU and 4G RAM
In our tests we choose images ldquoLenardquo and ldquoBarbarardquo withsize of 512 times 512 ldquoHouserdquo and ldquoMonarchrdquo with size of 256 times256 ldquoBrainrdquo with size of 181 times 217 and ldquoMouse intestinerdquo(briefly we call it ldquoMouserdquo throughout this paper) with sizeof 248 times 254 which are shown in Figure 1 We evaluate thequality of recovered images obtained from various denoisingalgorithms by using the structural similarity index (SSIM)[44] and the peal-signal-to-noise ratio (PSNR) [45] definedby
where 119909 and 119909 denote the original image and the recoveredimage respectively
First we choose the test image of ldquoLenardquo and degrade it byRician noise with120590 = 10 and use differentmethods to recoverthe noisy image The numerical results are enumerated inTable 2 where the first column gives the test image names thesecond column gives the method names (here Algorithm 1denotes our method presented in Section 2) and columns
3-5 (resp 6-8 and 9-11) are the PSNR (dB) SSIM and CPUtime (s) for 120590 = 10 (resp 120590 = 15 120590 = 20) We observethat the PSNR values of the restored images by Algorithm 1are higher than those of MAP GTV CZ and NLM methodMoreover you can easily find that the PSNR values of therestored images by Algorithm 1 are always the highest amongall those five methods for all images As far as ldquoBarbarardquo isconcerned the PSNR value of our method is more than 3
dB higher than CZ model at 120590 = 10 The characteristicsof SSIM values are almost consistent with those of PSNRvalues However the SSIM values are composed of luminancecomparison contrast comparison and structure comparisonwhich makes differences at some points
The images are corrupted respectively by Rician distri-bution with 120590 = 10 120590 = 15 and 120590 = 20 and the restoredimages of the above five algorithms for the images ldquoLenardquoldquoBarbarardquo ldquoHouserdquo and ldquoMonarchrdquo are shown inFigures 2ndash5respectively We clearly find that much noise remains in the
Figure 6 Results of ldquoBrainrdquo with 120590 = 10 by different methods (a) is the original ldquoBrainrdquo image and (g) is the degraded image with 120590 = 10(b)-(f) are the denoised images ofMAPmethod (120582 = 17) GTVmethod (120582 = 15) CZmethod (120582 = 0065) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
results of MAP GTV and CZmethod For example in Row 4of Figure 3 we can see from the background that ldquoBarbarardquoimage restored by our method is more clearer than thoserestored by other methods and textures of Barbararsquos trousersand scarf are kept better in the restored image by ourmethodThe restored images by our method also preserve moresignificant details thanMAPGTV CZ andNLMmethods onthe hat tidbits of the ldquoLenardquo image What is more the flowersof the lower left of ldquoMonarchrdquo also indicate that our method
is superior to other methods because our method is a patch-based method which is a different framework with TV-basedmethods (ie MAP method and CZ method) The imagesobtained from our method provide smoother regions andbetter shape preservation (eg the backgrounds in ldquoLenardquoand ldquoBarbarardquo)
Figures 6ndash11 are results of experiments on ldquoBrainrdquo andldquoMouserdquo images In particular for the ldquoBrainrdquo and ldquoMouserdquoimages we not only present the recovered images but also
10 Mathematical Problems in Engineering
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 7 Results of Brain with 120590 = 15 by different methods (a) is the original ldquoBrainrdquo image and (g) is the degraded image with 120590 = 15(b)-(f) are the denoised images of MAPmethod (120582 = 25) GTVmethod (120582 = 22) CZmethod (120582 = 005) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 8 Results of Brain with 120590 = 20 by different methods (a) is the original ldquoBrainrdquo image and (g) is the degraded image with 120590 = 20(b)-(f) are the denoised images ofMAPmethod (120582 = 35) GTVmethod (120582 = 22) CZmethod (120582 = 0045) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
use residuals to make a comparison with the original imagesThe first row is original image and recovered images ofMAP method GTV method CZ method NLM methodand our method The second row is degraded image withvarious 120590 and residuals of those methods respectively InFigures 6ndash8 the recovered images of of MAP method GTVmethod CZ method and NLM method are still unclear sowe can conclude that the recovered images by ourmethod arebest through our visions and residuals As for the ldquoMouserdquoimage the recovered results are very similar in this casewe use residuals to see which result is better In general theblurrier residual is the better according recovered image isWe can find that the outline of residual by our method isblurriest
In conclusion the results of our numerical experimentsdemonstrate that our proposed method performs better thanthe MAP method GTV method CZ method and NLMmethod
4 Conclusion
In this paper we proposed a new effective model via alearned dictionary for denoising images with Rician noiseMore specifically based on the dictionary training we adda dictionary penalty term to the original nonconvex MAPmodel to establish a new denoising model and develop atwo-step algorithm to solve our model Also we carry out
Mathematical Problems in Engineering 11
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 9 Results of ldquoMouserdquo with 120590 = 10 by different methods (a) is the original ldquoMouserdquo image and (g) is the degraded image with 120590 = 10(b)-(f) are the denoised images of MAPmethod (120582 = 17) GTVmethod (120582 = 17) CZmethod (120582 = 006) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 10 Results of ldquoMouserdquo with 120590 = 15 by different methods (a) is the original ldquoMouserdquo image and (g) is the degraded image with 120590 = 15(b)-(f) are the denoised images of MAPmethod (120582 = 24) GTV method (120582 = 24) CZ method (120582 = 005) NLM method and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
experiments on various images to demonstrate the effective-ness of our modelThe numerical experiments show that ourproposedmodel is promising in denoising imageswithRiciannoise
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interest
Acknowledgments
The authors would like to thank one of the authors of[32] for providing the source code of CZ method Thiswork was supported in part by the National Natural ScienceFoundation of China under Grants 11871348 61872429 and61373087 by the Natural Science Foundation of GuangdongChina under Grants 2015A030313550 and 2015A030313557by the HD Video RampD Platform for Intelligent Analy-sis and Processing in Guangdong Engineering TechnologyResearch Centre of Colleges and Universities (no GCZX-A1409) by Natural Science Foundation of Shenzhen underGrant JCYJ20170818091621856 and by the Guangdong Key
12 Mathematical Problems in Engineering
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 11 Results of ldquoMouserdquo with 120590 = 20 by different methods (a) is the original ldquoMouserdquo image and (g) is the degraded image with 120590 = 20(b)-(f) are the denoised images of MAPmethod (120582 = 31) GTV method (120582 = 32) CZ method (120582 = 0035) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
Laboratory of Intelligent Information Processing ShenzhenUniversity China (518060)
References
[1] J-F Cai B Dong and Z Shen ldquoImage restoration A waveletframe based model for piecewise smooth functions andbeyondrdquoApplied andComputational Harmonic Analysis vol 41no 1 pp 94ndash138 2016
[2] R H Chan and J Ma ldquoA multiplicative iterative algorithm forbox-constrained penalized likelihood image restorationrdquo IEEETransactions on Image Processing vol 21 no 7 pp 3168ndash31812012
[3] C A Micchelli L Shen and Y Xu ldquoProximity algorithmsfor image models Denoisingrdquo Inverse Problems vol 27 no 4Article ID 045009 p 30 2011
[4] YWang J Yang W Yin and Y Zhang ldquoA new alternatingmin-imization algorithm for total variation image reconstructionrdquoSIAM Journal on Imaging Sciences vol 1 no 3 pp 248ndash2722008
[5] Y Dong M Hintermuller and M Neri ldquoAn efficient primal-dual method for l1-TV image restorationrdquo SIAM Journal onImaging Sciences vol 2 no 4 pp 1168ndash1189 2009
[6] J Lu K Qiao L Shen and Y Zou ldquoFixed-point algorithmsfor a TVL1 image restoration modelrdquo International Journal ofComputer Mathematics vol 95 no 9 pp 1829ndash1844 2018
[7] J Lu Z Ye and Y Zou ldquoHuber fractal image coding based ona fitting planerdquo IEEE Transactions on Image Processing vol 22no 1 pp 134ndash145 2013
[8] C A Micchelli L Shen Y Xu and X Zeng ldquoProximityalgorithms for the L1TV image denoising modelrdquo Advances inComputational Mathematics vol 38 no 2 pp 401ndash426 2013
[9] J Yang Y Zhang and W Yin ldquoAn efficient TVL1 algorithm fordeblurring multichannel images corrupted by impulsive noiserdquoSIAM Journal on Scientific Computing vol 31 no 4 pp 2842ndash2865 2009
[10] L Ma L Moisan J Yu and T Zeng ldquoA dictionary learningapproach for poisson image deblurringrdquo IEEE Transactions onMedical Imaging vol 32 no 7 pp 1277ndash1289 2013
[11] Y Xiao and T Zeng ldquoPoisson noise removal via learned dictio-naryrdquo in Proceedings of the 17th IEEE International Conferenceon Image Processing (ICIP) pp 1177ndash1180 2010
[12] H Zhang Y Dong and Q Fan ldquoWavelet frame based Poissonnoise removal and image deblurringrdquo Signal Processing vol 137pp 363ndash372 2017
[13] Y Dong and T Zeng ldquoA convex variational model for restoringblurred images with multiplicative noiserdquo SIAM Journal onImaging Sciences vol 6 no 3 pp 1598ndash1625 2013
[14] Y Huang M K Ng and YWen ldquoA new total variation methodfor multiplicative noise removalrdquo SIAM Journal on ImagingSciences vol 2 no 1 pp 20ndash40 2009
[15] Y-M Huang L Moisan M K Ng and T Zeng ldquoMultiplicativenoise removal via a learned dictionaryrdquo IEEE Transactions onImage Processing vol 21 no 11 pp 4534ndash4543 2012
[16] Z Jin and X Yang ldquoA variational model to remove the multi-plicative noise in ultrasound imagesrdquo Journal of MathematicalImaging and Vision vol 39 no 1 pp 62ndash74 2011
[17] M Kang S Yun and H Woo ldquoTwo-level convex relaxedvariational model for multiplicative denoisingrdquo SIAM Journalon Imaging Sciences vol 6 no 2 pp 875ndash903 2013
[18] J Lu L Shen C Xu and Y Xu ldquoMultiplicative noise removalin imaging An exp-model and its fixed-point proximity algo-rithmrdquo Applied and Computational Harmonic Analysis vol 41no 2 pp 518ndash539 2016
[19] J Lu L Shen C Xu and Y Xu ldquoMultiplicative noise removalwith a sparsity-aware optimization modelrdquo Inverse Problemsand Imaging vol 11 no 6 pp 949ndash974 2017
[20] J Lu Z Yang L Shen Z Lu H Yang and C Xu ldquoA frameletalgorithm for de-blurring images corrupted by multiplicativenoiserdquoApplied Mathematical Modelling vol 62 pp 51ndash61 2018
[21] J Lu Y Chen Y Zou and L Shen ldquoA new total variationmodelfor restoring blurred and speckle noisy imagesrdquo InternationalJournal of Wavelets Multiresolution and Information Processingvol 15 no 2 19 pages 2017
Mathematical Problems in Engineering 13
[22] P Perona and J Malik ldquoScale-space and edge detection usinganisotropic diffusionrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 12 no 7 pp 629ndash639 1990
[23] G Gerig O Kubler R Kikinis and F A Jolesz ldquoNonlinearanisotropic filtering ofMRI datardquo IEEE Transactions onMedicalImaging vol 11 no 2 pp 221ndash232 1992
[24] S Prima S P Morrissey and C Barillot ldquoNon-local meansvariants for denoising of diffusion-weighted and diffusiontensor MRIrdquo in Proceedings of the International Conference onMedical Image Computing and Computer-Assisted Interventionpp 344ndash351 Springer Berlin Germany 2007
[25] J V Manjon J Carbonell-Caballero J J Lull G Garcıa-MartiL Martı-Bonmati and M Robles ldquoMRI denoising using Non-LocalMeansrdquoMedical Image Analysis vol 12 no 4 pp 514ndash5232008
[26] N Wiest-Daessle S Prima P Coupe S P Morrissey and CBarillot ldquoRician noise removal by non-local means filteringfor low signal-to-noise ratio MRI Applications to DT-MRIrdquoMedical Image Computing and Computer-Assisted Interventionvol 5242 no 2 pp 171ndash179 2008
[27] R D Nowak ldquoWavelet-based Rician noise removal for mag-netic resonance imagingrdquo IEEE Transactions on Image Process-ing vol 8 no 10 pp 1408ndash1419 1999
[28] A Foi ldquoNoise estimation and removal in MR imaging Thevariance-stabilization approachrdquo in Proceedings of the 8th IEEEInternational Symposium on Biomedical Imaging From Nano toMacro (ISBI) pp 1809ndash1814 IEEE Chicago IL USA 2011
[29] J C Wood and K M Johnson ldquoWavelet packet denoising ofmagnetic resonance images Importance of Rician noise at lowSNRrdquo Magnetic Resonance in Medicine vol 41 no 3 pp 631ndash635 1999
[30] F Bowman Introduction to Bessel functions Dover PublicationsMineola NY USA 2012
[31] P Getreuer M Tong and L A Vese ldquoA variational model forthe restoration of MR images corrupted by blur and Riciannoiserdquo in Proceedings of the 7th international conference onAdvances in visual computing vol Part I of Lecture Notes inComputer Science pp 686ndash698 Springer Las Vegas NV USA2011
[32] L Chen and T Zeng ldquoA convex variational model for restoringblurred imageswith large Rician noiserdquo Journal ofMathematicalImaging and Vision vol 53 no 1 pp 92ndash111 2015
[33] M Aharon M Elad and A M Bruckstein ldquoK-SVD Analgorithm for designing overcomplete dictionaries for sparserepresentationrdquo IEEE Transactions on Signal Processing vol 54no 11 pp 4311ndash4322 2006
[34] M Elad and M Aharon ldquoImage denoising via sparse andredundant representations over learned dictionariesrdquo IEEETransactions on Image Processing vol 15 no 12 pp 3736ndash37452006
[35] G Aubert and J Aujol ldquoA variational approach to removingmultiplicative noiserdquo SIAM Journal on Applied Mathematicsvol 68 no 4 pp 925ndash946 2008
[36] Q Liu S Wang K Yang J Luo Y Zhu and D Liang ldquoHighlyundersampled magnetic resonance image reconstruction usingtwo-level Bregman method with dictionary updatingrdquo IEEETransactions on Medical Imaging vol 32 no 7 pp 1290ndash13012013
[37] Q Liu S Wang L Ying X Peng Y Zhu and D LiangldquoAdaptive dictionary learning in sparse gradient domain forimage recoveryrdquo IEEE Transactions on Image Processing vol 22no 12 pp 4652ndash4663 2013
[38] H Lu J Wei Q Liu Y Wang and X Deng ldquoA dictionarylearning method with total generalized variation for mri recon-structionrdquo International Journal of Biomedical Imaging vol2016 Article ID 7512471 13 pages 2016
[39] A Chambolle and T Pock ldquoA first-order primal-dual algorithmfor convex problems with applications to imagingrdquo Journal ofMathematical Imaging and Vision vol 40 no 1 pp 120ndash1452011
[40] E Esser X Zhang and T F Chan ldquoA general frameworkfor a class of first order primal-dual algorithms for convexoptimization in imaging sciencerdquo SIAM Journal on ImagingSciences vol 3 no 4 pp 1015ndash1046 2010
[41] N Komodakis and J-C Pesquet ldquoPlaying with duality Anoverview of recent primaldual approaches for solving large-scale optimization problemsrdquo IEEE Signal Processing Magazinevol 32 no 6 pp 31ndash54 2015
[42] A Chambolle ldquoAn algorithm for total variation minimizationand applicationsrdquo Journal of Mathematical Imaging and Visionvol 20 no 1-2 pp 89ndash97 2004
[43] L Ambrosio N Fusco and D Pallara Functions of BoundedVariation and Free Discontinuity Problem Oxford UniversityPress London 2000
[44] Z Wang A C Bovik H R Sheikh and E P SimoncellildquoImage quality assessment From error visibility to structuralsimilarityrdquo IEEE Transactions on Image Processing vol 13 no4 pp 600ndash612 2004
[45] J Gibson and A Bovik Handbook of Image and Video Process-ing Academic Press 2000
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
where 119878(Ω) fl V isin 119861119881(Ω) 0 le V le 255 and intΩ|119863119909|119889119905
is the total variation (TV) of 119909 The first and second termsof our model stem from the MAP model which is data-fidelity caused by the statistical properties of Rician noiseand the third and forth terms are inspired by the sparserepresentation The last TV regularization term can make thedenoised image smooth
In model (6) there are three unknown variables thenoise-free image 119909 that we need to solve the dictionary Dand the sparse coefficients 120572119894119895 Similar to [33 34] in order tosolve problem (6) effectively and efficiently here we have thefollowing two-step algorithm
(1) Based on the degraded image 119910 we give the initialdictionary and get the sparse representation coefficients 120572119894119895then use 120572119894119895 to learn dictionary D and update correspondingcoefficients 120572119894119895
(2) Use the primal-dual algorithm to get the recoveredimage that we wanted
21 Dictionary Learning In the first step of our algorithmwe will use the degraded image 119910 to train a dictionary and allthe image patches can be sparsely represented by the traineddictionary with the corresponding coefficients 120572119894119895 The wholeprocess is just using the orthogonal matching pursuit (OMP)and K-SVD algorithms [33 34] and the procedure is to solvethe following optimization problem
D 119894119895 = arg minD120572119894119895
12 sum(119894119895)
10038171003817100381710038171003817R119894119895119909 minus D120572119894119895100381710038171003817100381710038172
For solving the difficult problem we have the followingspecific steps
(1) Initialization Set 119909 = 119910 D = overcomplete DCTdictionary
(2) Iteration 119865119900119903 119898 = 0 119905119900 119873 119889119900(119886) Given 119909 and D we get the sparse representationcoefficients 120572119894119895 through solving the following problem
119894119895 = argmin120572119894119895
12sum(119894119895)
10038171003817100381710038171003817R119894119895119909 minus D120572119894119895100381710038171003817100381710038172 + sum(119894119895)
and we can efficiently and effectively solve model (8) by usingthe orthogonal matching pursuit (OMP) method [33 34](119887) Given 119909 and 120572119894119895 we can update the dictionary D =[1198891 1198892 119889119896] column by column [33 34] For each column119889119897 119897 = 1 119896 we update it as follows(119894) For those patches represented by119889119897 wewrite down anddenote it by 120577119897 = (119894 119895) | 120572119894119895(119897) = 0
(119894119894) For each index (119894 119895) isin 120577119897 we compute the correspond-ing representation error through
119890119897119894119895 = R119894119895119909 minus sum119898 =119897
119889119898120572119894119895 (119898) (9)
and then we use the columns 119890119897119894119895(119894119895)isin120577119897 to define a matrix E119897(119894119894119894) For the matrix E119897 we apply the singular valuedecomposition (SVD) into E119897 and get E119897 = UΔVT
Let the first column of U be 119889119897 to update 119889119897 and multiplythe first column of V by Δ(1 1) to update 120572119894119895(119897)(119894119895)isin120577119897
End forHere the dictionary training and sparse representation
are completed
22 Primal-Dual Algorithm After the first step of our algo-rithm we get the spare dictionary representation D120572119894119895 fromeach patch R119894119895119910 and we now minimize (6) with respect to 119909that is
arg min119909isin119878(Ω)
12057321205902 int
Ω1199092119889119905 minus 120573int
Ωlog 1198680 (119910119909
1205902 )119889119905+ 1
2 sum10038171003817100381710038171003817D120572119894119895 minus R119894119895119909100381710038171003817100381710038172 + 120582intΩ|119863119909| 119889119905
(10)
Proposition 1 Let 119910 be a bounded function such thatinfΩ 119910 gt 0 then the objective function in (10) is strictly convexwith the constraint 1205904120573 gt (supΩ 119910)2Proof Using the notations 119882 = sum(119894119895)isinΓR
119879119894119895R119894119895 and 119872 =
sum(119894119895)isinΓR119879119894119895D120572119894119895 model (10) can be rewritten as
arg min119909isin119878(Ω)
12057321205902 ⟨119909 119909⟩ minus 120573⟨log 1198680 (119910119909
1205902 ) 1Ω⟩+ 1
2 ⟨119882119909 119909⟩ minus ⟨119872 119909⟩ + 120582 119909119879119881 (11)
Also we define 119892(119909) = (12057321205902)⟨119909 119909⟩minus120573⟨log 1198680(1199101199091205902) 1Ω⟩+(12)⟨119882119909 119909⟩ minus ⟨119872 119909⟩(119909 gt 0) and ℎ(119909) = log 1198680(119909)(119909 gt0) According to [32] we have 0 lt ℎ10158401015840(119909) = 1 minus(2119909)(1198681(119909)1198680(119909)) minus (1198681(119909)1198680(119909))2 lt 1 Then 11989210158401015840(119909) = 1205731205902 minus120573[log 1198680(1199101199091205902)]10158401015840 + 119882 gt 1205731205902 minus 120573(11991021205904) + 119882 Due to1 le 119882119894119895 le 119899 11989210158401015840(119909) gt 0 holds when 1205904120573 gt (supΩ 119910)2and 119892(119909) is strictly convex We know that 119909 997888rarr 119909119879119881 isconvex so the objective function (11) is strictly convex thatis function (10) is strictly convex
Referring to Proposition 1 we know that model (10) isconvex then we can apply the primal-dual algorithm forsolving the minimization problem which enjoys nice saddle-point structures and has good performance on nonsmooth
4 Mathematical Problems in Engineering
1 Let 120581 and 120591 be given Set 1199090 = 119910 1199090 = 119910 and 1199010 = (0 0)119879 isin R21198892 Update 119901119899 119909119899 119909119899 iteratively as follows
(a) 119901119899+1 = argmax119901isin119875
120582⟨119909119899 div 119901⟩ minus 12120581119901 minus 11990111989922
(b) 119909119899+1 = arg min119909isin119878(Ω)
119865(119909) minus 120582⟨119909 div 119901119899+1⟩ + 12120591119909 minus 11990911989922
(c) 119909119899+1 = 2119909119899+1 minus 119909119899until some stop criterion is satisfied
Algorithm 1
convex optimization [39ndash41] Moreover we discuss a biascorrection technique and the convergence of our algorithm
Briefly we define 119865(119909) fl (12057321205902)1199092 minus 120573⟨log 1198680(1199101199091205902) 1⟩ + (12)sum D120572119894119895 minus R1198941198951199092 and model (10) can betranslated to the following format
min0le119909le255
119865 (119909) + 120582 nabla1199091 (12)
Because the 119879119881-model has the duality property we canchange the primal-dual formulation of optimization problemof (12) into the following format
max119901isin119875
min0le119909le255
119865 (119909) minus 120582 ⟨119909 div 119901⟩ (13)
where 119875 = 119901 isin R2119889 max119894isin1119889|(1199012119894 + 1199012
119894+119889)12| le 1Here 119901 is the dual variable and div = minusnabla119879 The specific
algorithm is as in Algorithm 1In particular the solution of the dual problem 2(a) can be
where 1205871(119902119894) = 119902119894max(1 |119902119894|) 1205871(119902119889+119894) = 119902119889+119894max(1 |119902119894|)and |119902119894| = radic1199022119894 + 1199022119894+119889 119894 = 1 119889 Moreover the solution ofthe primal problem2(b) can be obtained byNewtonrsquosmethod
Inspired by [39] we provide a bias correction techniquefor (13) so that the mean of the restored image 119909lowast equals thatof the observed image 119910 In numerical practice the followingstep is implemented in order to preserve the mean of 119910
after updating 119909119899 by Newtonmethod which also ensures that119909119899 ge 0Referring to Theorem 1 in [42] we will get the conver-
gence properties of our algorithm as follows
Proposition 2 Suppose that 1205811205911205822nabla2 lt 1 en the iterates(119909119899 119901119899) of our algorithm converge to a saddle point of (13)
Inspired by [42] the condition of convergence of ouralgorithm can be simplified into 1205811205911205822 lt 18 based on thefact that nabla2 le 8 with the unit spacing size between pixelsFor simplicity in our numerical experiments we let 120581 = 8120582
and 120591 = 0015120582 and just adjust 120582 and the convergence of thealgorithm is satisfied
Before we present numerical results here we give thebasic property of model (10) rewritten in the continuoussettings that is
where (Ω119904)119904isin119868 is a finite set of small patches covering Ω andRΩ119904
is the restriction onΩ119904
Theorem 3 Let Ω be a bounded open subset of R2 withLipschitz boundary Let 119910 be a positive bounded function For119909 satisfying 119909 isin 119861119881(Ω) 119909 gt 0 let
where 120582 gt 0 Suppose that (Ω119904)119904isin119868 is a finite local coverage ofΩ andsuminfin
119904isin119868 1198631205721199042 lt +infin en 119864(119909) has a unique minimizerwhen 1205904120573 gt (supΩ 119910)2Proof Let119864(119909) = 120582119909119879119881+119869(119909) and 119869(119909) = (12057321205902) int
Ω1199092119889119905minus
120573intΩlog 1198680(1199101199091205902)119889119905 + (12)sum D120572119894119895 minus R1198941198951199092 By Proposi-
tion 1 we know that 119869(119909) is convex Thus 119864(119909) is convexand bounded from below Consider a minimizer sequence(119909119899) for 119864(119909) Then 119909119899119879119881 and 119869(119909119899) are bounded fromabove Hence (119909119899) is bounded in 119861119881(Ω) As 119861119881 is compactin 1198711(Ω) there is 1199090 isin 119861119881 such that a subsequence (119909119899119896)converges 1199090 in 1198711(Ω) and 119861119881(Ω) and we may assumethat 119909119899119896 997888rarr 1199090 pointwise almost everywhere As the 119861119881norm is lsc 1199090119879119881 le lim inf 119909119899119896119879119881 Since 119869(119909119899119896) isbounded from below by Fatoursquos Lemma we have 119869(1199090) lelim inf 119869(119909119899119896) Thus 119864(1199090) le lim inf 119864(119909119899119896) = inf119861119881(Ω)119864(119909)and 1199090 minimizes 119864(119909) Consequently 119864(119909) admits at leastone minimizer 1199090 119864(119909) is strictly convex function hence119864(119909) admits a unique minimizer
Mathematical Problems in Engineering 5
Table 1 Parameter 120582 values for all testing algorithms
In this section we will present our numerical experimentsto evaluate the approximation accuracy and computationalefficiency of our proposed algorithm We compare ourmethod for the denoising cases with the MAP model (2) theGTV model [31] the CZ model [32] and the NLM model[25] Here we have a brief overview of the models that wecompared
GTV model
argmin119909
120582intΩ119866120590 (119909 119910) + int
Ω|119863119909| 119889119905 (18)
119866120590 (119909 119910) = 119867120590 (119909) if 119909 ge 119888120590119867120590 (119888120590) + 1198671015840
120590 (119888120590) (119909 minus 119888120590) if 119909 le 1198881205901198671015840
where 119888 = 08426 1198681 is the modified Bessel function of thefirst kind with order one [30] Because it is an approximateequality we can not write down the final explicit restorationmodel
CZ model
arg min119909isin119878(Ω)
121205902 int
Ω1199092119889119905 minus int
Ωlog 1198680 (119909119910
1205902 )119889119905+ 1
120590 intΩ(radic119909 minus radic119910)2 119889119905 + 120582int
Ω|119863119909| 119889119905
(20)
where 119878(Ω) fl ] isin 119861119881(Ω) 0 le ] le 255 And because119861119881(Ω) [35 43] is the subspace of function 119909 isin 1198711(Ω) thefollowing quantity is finite
isin 119862infin0 (ΩR2) 10038171003817100381710038171205851003817100381710038171003817119871infin(ΩR2) le 1
(21)
NLMmodelBefore introducing the NLM model we have a descrip-
tion of the sign that will be used For a user-defined radius
119877119904119894119898 we define a square neighborhood window centeredaround pixel 119894 as 119873119894 And a Gaussian weighted Euclidiandistance of all the pixels of each neighborhood is defined as
119871 (119898 119899) = 1198661205881003817100381710038171003817119910 (119873119898) minus 119910 (119873119899)10038171003817100381710038172119877119904119894119898 (22)
where 119866120588 is a normalized Gaussian weighting functionwith zero mean and 120588 standard deviation (usually set to1) By giving more weight to pixels near the center wecan use 119866120588 to penalize pixels far from the center of theneighborhood window And based on the similarity betweenthe neighborhoods119873119898 and119873119899 of pixels119898 and 119899 we calculatethe similarity 119891(119898 119899) as
119862(119898) is the normalizing constant and ℎ is a exponentialdecay control parameter
Then given an image 119910 using the NLM method we cancalculate the filtered value at a point 119898 by the followingformula119873119871119872(119910 (119898)) = sum
forall119899isin119910
119891 (119898 119899) 119910 (119899)
0 le 119891 (119898 119899) le 1 sumforall119899isin119910
119891 (119898 119899) = 1 (24)
The parameter 120582 values of MAP model GTV modelCZ model and our proposed model are listed in Table 1In order to preserve the mean of the observed image thebias correction technique (15) is utilized for the CZ methodand our method All the experiments are performed underWindows 7 andMATLAB R2017a running on a PC equippedwith 290GHz CPU and 4G RAM
In our tests we choose images ldquoLenardquo and ldquoBarbarardquo withsize of 512 times 512 ldquoHouserdquo and ldquoMonarchrdquo with size of 256 times256 ldquoBrainrdquo with size of 181 times 217 and ldquoMouse intestinerdquo(briefly we call it ldquoMouserdquo throughout this paper) with sizeof 248 times 254 which are shown in Figure 1 We evaluate thequality of recovered images obtained from various denoisingalgorithms by using the structural similarity index (SSIM)[44] and the peal-signal-to-noise ratio (PSNR) [45] definedby
where 119909 and 119909 denote the original image and the recoveredimage respectively
First we choose the test image of ldquoLenardquo and degrade it byRician noise with120590 = 10 and use differentmethods to recoverthe noisy image The numerical results are enumerated inTable 2 where the first column gives the test image names thesecond column gives the method names (here Algorithm 1denotes our method presented in Section 2) and columns
3-5 (resp 6-8 and 9-11) are the PSNR (dB) SSIM and CPUtime (s) for 120590 = 10 (resp 120590 = 15 120590 = 20) We observethat the PSNR values of the restored images by Algorithm 1are higher than those of MAP GTV CZ and NLM methodMoreover you can easily find that the PSNR values of therestored images by Algorithm 1 are always the highest amongall those five methods for all images As far as ldquoBarbarardquo isconcerned the PSNR value of our method is more than 3
dB higher than CZ model at 120590 = 10 The characteristicsof SSIM values are almost consistent with those of PSNRvalues However the SSIM values are composed of luminancecomparison contrast comparison and structure comparisonwhich makes differences at some points
The images are corrupted respectively by Rician distri-bution with 120590 = 10 120590 = 15 and 120590 = 20 and the restoredimages of the above five algorithms for the images ldquoLenardquoldquoBarbarardquo ldquoHouserdquo and ldquoMonarchrdquo are shown inFigures 2ndash5respectively We clearly find that much noise remains in the
Figure 6 Results of ldquoBrainrdquo with 120590 = 10 by different methods (a) is the original ldquoBrainrdquo image and (g) is the degraded image with 120590 = 10(b)-(f) are the denoised images ofMAPmethod (120582 = 17) GTVmethod (120582 = 15) CZmethod (120582 = 0065) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
results of MAP GTV and CZmethod For example in Row 4of Figure 3 we can see from the background that ldquoBarbarardquoimage restored by our method is more clearer than thoserestored by other methods and textures of Barbararsquos trousersand scarf are kept better in the restored image by ourmethodThe restored images by our method also preserve moresignificant details thanMAPGTV CZ andNLMmethods onthe hat tidbits of the ldquoLenardquo image What is more the flowersof the lower left of ldquoMonarchrdquo also indicate that our method
is superior to other methods because our method is a patch-based method which is a different framework with TV-basedmethods (ie MAP method and CZ method) The imagesobtained from our method provide smoother regions andbetter shape preservation (eg the backgrounds in ldquoLenardquoand ldquoBarbarardquo)
Figures 6ndash11 are results of experiments on ldquoBrainrdquo andldquoMouserdquo images In particular for the ldquoBrainrdquo and ldquoMouserdquoimages we not only present the recovered images but also
10 Mathematical Problems in Engineering
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 7 Results of Brain with 120590 = 15 by different methods (a) is the original ldquoBrainrdquo image and (g) is the degraded image with 120590 = 15(b)-(f) are the denoised images of MAPmethod (120582 = 25) GTVmethod (120582 = 22) CZmethod (120582 = 005) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 8 Results of Brain with 120590 = 20 by different methods (a) is the original ldquoBrainrdquo image and (g) is the degraded image with 120590 = 20(b)-(f) are the denoised images ofMAPmethod (120582 = 35) GTVmethod (120582 = 22) CZmethod (120582 = 0045) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
use residuals to make a comparison with the original imagesThe first row is original image and recovered images ofMAP method GTV method CZ method NLM methodand our method The second row is degraded image withvarious 120590 and residuals of those methods respectively InFigures 6ndash8 the recovered images of of MAP method GTVmethod CZ method and NLM method are still unclear sowe can conclude that the recovered images by ourmethod arebest through our visions and residuals As for the ldquoMouserdquoimage the recovered results are very similar in this casewe use residuals to see which result is better In general theblurrier residual is the better according recovered image isWe can find that the outline of residual by our method isblurriest
In conclusion the results of our numerical experimentsdemonstrate that our proposed method performs better thanthe MAP method GTV method CZ method and NLMmethod
4 Conclusion
In this paper we proposed a new effective model via alearned dictionary for denoising images with Rician noiseMore specifically based on the dictionary training we adda dictionary penalty term to the original nonconvex MAPmodel to establish a new denoising model and develop atwo-step algorithm to solve our model Also we carry out
Mathematical Problems in Engineering 11
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 9 Results of ldquoMouserdquo with 120590 = 10 by different methods (a) is the original ldquoMouserdquo image and (g) is the degraded image with 120590 = 10(b)-(f) are the denoised images of MAPmethod (120582 = 17) GTVmethod (120582 = 17) CZmethod (120582 = 006) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 10 Results of ldquoMouserdquo with 120590 = 15 by different methods (a) is the original ldquoMouserdquo image and (g) is the degraded image with 120590 = 15(b)-(f) are the denoised images of MAPmethod (120582 = 24) GTV method (120582 = 24) CZ method (120582 = 005) NLM method and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
experiments on various images to demonstrate the effective-ness of our modelThe numerical experiments show that ourproposedmodel is promising in denoising imageswithRiciannoise
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interest
Acknowledgments
The authors would like to thank one of the authors of[32] for providing the source code of CZ method Thiswork was supported in part by the National Natural ScienceFoundation of China under Grants 11871348 61872429 and61373087 by the Natural Science Foundation of GuangdongChina under Grants 2015A030313550 and 2015A030313557by the HD Video RampD Platform for Intelligent Analy-sis and Processing in Guangdong Engineering TechnologyResearch Centre of Colleges and Universities (no GCZX-A1409) by Natural Science Foundation of Shenzhen underGrant JCYJ20170818091621856 and by the Guangdong Key
12 Mathematical Problems in Engineering
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 11 Results of ldquoMouserdquo with 120590 = 20 by different methods (a) is the original ldquoMouserdquo image and (g) is the degraded image with 120590 = 20(b)-(f) are the denoised images of MAPmethod (120582 = 31) GTV method (120582 = 32) CZ method (120582 = 0035) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
Laboratory of Intelligent Information Processing ShenzhenUniversity China (518060)
References
[1] J-F Cai B Dong and Z Shen ldquoImage restoration A waveletframe based model for piecewise smooth functions andbeyondrdquoApplied andComputational Harmonic Analysis vol 41no 1 pp 94ndash138 2016
[2] R H Chan and J Ma ldquoA multiplicative iterative algorithm forbox-constrained penalized likelihood image restorationrdquo IEEETransactions on Image Processing vol 21 no 7 pp 3168ndash31812012
[3] C A Micchelli L Shen and Y Xu ldquoProximity algorithmsfor image models Denoisingrdquo Inverse Problems vol 27 no 4Article ID 045009 p 30 2011
[4] YWang J Yang W Yin and Y Zhang ldquoA new alternatingmin-imization algorithm for total variation image reconstructionrdquoSIAM Journal on Imaging Sciences vol 1 no 3 pp 248ndash2722008
[5] Y Dong M Hintermuller and M Neri ldquoAn efficient primal-dual method for l1-TV image restorationrdquo SIAM Journal onImaging Sciences vol 2 no 4 pp 1168ndash1189 2009
[6] J Lu K Qiao L Shen and Y Zou ldquoFixed-point algorithmsfor a TVL1 image restoration modelrdquo International Journal ofComputer Mathematics vol 95 no 9 pp 1829ndash1844 2018
[7] J Lu Z Ye and Y Zou ldquoHuber fractal image coding based ona fitting planerdquo IEEE Transactions on Image Processing vol 22no 1 pp 134ndash145 2013
[8] C A Micchelli L Shen Y Xu and X Zeng ldquoProximityalgorithms for the L1TV image denoising modelrdquo Advances inComputational Mathematics vol 38 no 2 pp 401ndash426 2013
[9] J Yang Y Zhang and W Yin ldquoAn efficient TVL1 algorithm fordeblurring multichannel images corrupted by impulsive noiserdquoSIAM Journal on Scientific Computing vol 31 no 4 pp 2842ndash2865 2009
[10] L Ma L Moisan J Yu and T Zeng ldquoA dictionary learningapproach for poisson image deblurringrdquo IEEE Transactions onMedical Imaging vol 32 no 7 pp 1277ndash1289 2013
[11] Y Xiao and T Zeng ldquoPoisson noise removal via learned dictio-naryrdquo in Proceedings of the 17th IEEE International Conferenceon Image Processing (ICIP) pp 1177ndash1180 2010
[12] H Zhang Y Dong and Q Fan ldquoWavelet frame based Poissonnoise removal and image deblurringrdquo Signal Processing vol 137pp 363ndash372 2017
[13] Y Dong and T Zeng ldquoA convex variational model for restoringblurred images with multiplicative noiserdquo SIAM Journal onImaging Sciences vol 6 no 3 pp 1598ndash1625 2013
[14] Y Huang M K Ng and YWen ldquoA new total variation methodfor multiplicative noise removalrdquo SIAM Journal on ImagingSciences vol 2 no 1 pp 20ndash40 2009
[15] Y-M Huang L Moisan M K Ng and T Zeng ldquoMultiplicativenoise removal via a learned dictionaryrdquo IEEE Transactions onImage Processing vol 21 no 11 pp 4534ndash4543 2012
[16] Z Jin and X Yang ldquoA variational model to remove the multi-plicative noise in ultrasound imagesrdquo Journal of MathematicalImaging and Vision vol 39 no 1 pp 62ndash74 2011
[17] M Kang S Yun and H Woo ldquoTwo-level convex relaxedvariational model for multiplicative denoisingrdquo SIAM Journalon Imaging Sciences vol 6 no 2 pp 875ndash903 2013
[18] J Lu L Shen C Xu and Y Xu ldquoMultiplicative noise removalin imaging An exp-model and its fixed-point proximity algo-rithmrdquo Applied and Computational Harmonic Analysis vol 41no 2 pp 518ndash539 2016
[19] J Lu L Shen C Xu and Y Xu ldquoMultiplicative noise removalwith a sparsity-aware optimization modelrdquo Inverse Problemsand Imaging vol 11 no 6 pp 949ndash974 2017
[20] J Lu Z Yang L Shen Z Lu H Yang and C Xu ldquoA frameletalgorithm for de-blurring images corrupted by multiplicativenoiserdquoApplied Mathematical Modelling vol 62 pp 51ndash61 2018
[21] J Lu Y Chen Y Zou and L Shen ldquoA new total variationmodelfor restoring blurred and speckle noisy imagesrdquo InternationalJournal of Wavelets Multiresolution and Information Processingvol 15 no 2 19 pages 2017
Mathematical Problems in Engineering 13
[22] P Perona and J Malik ldquoScale-space and edge detection usinganisotropic diffusionrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 12 no 7 pp 629ndash639 1990
[23] G Gerig O Kubler R Kikinis and F A Jolesz ldquoNonlinearanisotropic filtering ofMRI datardquo IEEE Transactions onMedicalImaging vol 11 no 2 pp 221ndash232 1992
[24] S Prima S P Morrissey and C Barillot ldquoNon-local meansvariants for denoising of diffusion-weighted and diffusiontensor MRIrdquo in Proceedings of the International Conference onMedical Image Computing and Computer-Assisted Interventionpp 344ndash351 Springer Berlin Germany 2007
[25] J V Manjon J Carbonell-Caballero J J Lull G Garcıa-MartiL Martı-Bonmati and M Robles ldquoMRI denoising using Non-LocalMeansrdquoMedical Image Analysis vol 12 no 4 pp 514ndash5232008
[26] N Wiest-Daessle S Prima P Coupe S P Morrissey and CBarillot ldquoRician noise removal by non-local means filteringfor low signal-to-noise ratio MRI Applications to DT-MRIrdquoMedical Image Computing and Computer-Assisted Interventionvol 5242 no 2 pp 171ndash179 2008
[27] R D Nowak ldquoWavelet-based Rician noise removal for mag-netic resonance imagingrdquo IEEE Transactions on Image Process-ing vol 8 no 10 pp 1408ndash1419 1999
[28] A Foi ldquoNoise estimation and removal in MR imaging Thevariance-stabilization approachrdquo in Proceedings of the 8th IEEEInternational Symposium on Biomedical Imaging From Nano toMacro (ISBI) pp 1809ndash1814 IEEE Chicago IL USA 2011
[29] J C Wood and K M Johnson ldquoWavelet packet denoising ofmagnetic resonance images Importance of Rician noise at lowSNRrdquo Magnetic Resonance in Medicine vol 41 no 3 pp 631ndash635 1999
[30] F Bowman Introduction to Bessel functions Dover PublicationsMineola NY USA 2012
[31] P Getreuer M Tong and L A Vese ldquoA variational model forthe restoration of MR images corrupted by blur and Riciannoiserdquo in Proceedings of the 7th international conference onAdvances in visual computing vol Part I of Lecture Notes inComputer Science pp 686ndash698 Springer Las Vegas NV USA2011
[32] L Chen and T Zeng ldquoA convex variational model for restoringblurred imageswith large Rician noiserdquo Journal ofMathematicalImaging and Vision vol 53 no 1 pp 92ndash111 2015
[33] M Aharon M Elad and A M Bruckstein ldquoK-SVD Analgorithm for designing overcomplete dictionaries for sparserepresentationrdquo IEEE Transactions on Signal Processing vol 54no 11 pp 4311ndash4322 2006
[34] M Elad and M Aharon ldquoImage denoising via sparse andredundant representations over learned dictionariesrdquo IEEETransactions on Image Processing vol 15 no 12 pp 3736ndash37452006
[35] G Aubert and J Aujol ldquoA variational approach to removingmultiplicative noiserdquo SIAM Journal on Applied Mathematicsvol 68 no 4 pp 925ndash946 2008
[36] Q Liu S Wang K Yang J Luo Y Zhu and D Liang ldquoHighlyundersampled magnetic resonance image reconstruction usingtwo-level Bregman method with dictionary updatingrdquo IEEETransactions on Medical Imaging vol 32 no 7 pp 1290ndash13012013
[37] Q Liu S Wang L Ying X Peng Y Zhu and D LiangldquoAdaptive dictionary learning in sparse gradient domain forimage recoveryrdquo IEEE Transactions on Image Processing vol 22no 12 pp 4652ndash4663 2013
[38] H Lu J Wei Q Liu Y Wang and X Deng ldquoA dictionarylearning method with total generalized variation for mri recon-structionrdquo International Journal of Biomedical Imaging vol2016 Article ID 7512471 13 pages 2016
[39] A Chambolle and T Pock ldquoA first-order primal-dual algorithmfor convex problems with applications to imagingrdquo Journal ofMathematical Imaging and Vision vol 40 no 1 pp 120ndash1452011
[40] E Esser X Zhang and T F Chan ldquoA general frameworkfor a class of first order primal-dual algorithms for convexoptimization in imaging sciencerdquo SIAM Journal on ImagingSciences vol 3 no 4 pp 1015ndash1046 2010
[41] N Komodakis and J-C Pesquet ldquoPlaying with duality Anoverview of recent primaldual approaches for solving large-scale optimization problemsrdquo IEEE Signal Processing Magazinevol 32 no 6 pp 31ndash54 2015
[42] A Chambolle ldquoAn algorithm for total variation minimizationand applicationsrdquo Journal of Mathematical Imaging and Visionvol 20 no 1-2 pp 89ndash97 2004
[43] L Ambrosio N Fusco and D Pallara Functions of BoundedVariation and Free Discontinuity Problem Oxford UniversityPress London 2000
[44] Z Wang A C Bovik H R Sheikh and E P SimoncellildquoImage quality assessment From error visibility to structuralsimilarityrdquo IEEE Transactions on Image Processing vol 13 no4 pp 600ndash612 2004
[45] J Gibson and A Bovik Handbook of Image and Video Process-ing Academic Press 2000
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4 Mathematical Problems in Engineering
1 Let 120581 and 120591 be given Set 1199090 = 119910 1199090 = 119910 and 1199010 = (0 0)119879 isin R21198892 Update 119901119899 119909119899 119909119899 iteratively as follows
(a) 119901119899+1 = argmax119901isin119875
120582⟨119909119899 div 119901⟩ minus 12120581119901 minus 11990111989922
(b) 119909119899+1 = arg min119909isin119878(Ω)
119865(119909) minus 120582⟨119909 div 119901119899+1⟩ + 12120591119909 minus 11990911989922
(c) 119909119899+1 = 2119909119899+1 minus 119909119899until some stop criterion is satisfied
Algorithm 1
convex optimization [39ndash41] Moreover we discuss a biascorrection technique and the convergence of our algorithm
Briefly we define 119865(119909) fl (12057321205902)1199092 minus 120573⟨log 1198680(1199101199091205902) 1⟩ + (12)sum D120572119894119895 minus R1198941198951199092 and model (10) can betranslated to the following format
min0le119909le255
119865 (119909) + 120582 nabla1199091 (12)
Because the 119879119881-model has the duality property we canchange the primal-dual formulation of optimization problemof (12) into the following format
max119901isin119875
min0le119909le255
119865 (119909) minus 120582 ⟨119909 div 119901⟩ (13)
where 119875 = 119901 isin R2119889 max119894isin1119889|(1199012119894 + 1199012
119894+119889)12| le 1Here 119901 is the dual variable and div = minusnabla119879 The specific
algorithm is as in Algorithm 1In particular the solution of the dual problem 2(a) can be
where 1205871(119902119894) = 119902119894max(1 |119902119894|) 1205871(119902119889+119894) = 119902119889+119894max(1 |119902119894|)and |119902119894| = radic1199022119894 + 1199022119894+119889 119894 = 1 119889 Moreover the solution ofthe primal problem2(b) can be obtained byNewtonrsquosmethod
Inspired by [39] we provide a bias correction techniquefor (13) so that the mean of the restored image 119909lowast equals thatof the observed image 119910 In numerical practice the followingstep is implemented in order to preserve the mean of 119910
after updating 119909119899 by Newtonmethod which also ensures that119909119899 ge 0Referring to Theorem 1 in [42] we will get the conver-
gence properties of our algorithm as follows
Proposition 2 Suppose that 1205811205911205822nabla2 lt 1 en the iterates(119909119899 119901119899) of our algorithm converge to a saddle point of (13)
Inspired by [42] the condition of convergence of ouralgorithm can be simplified into 1205811205911205822 lt 18 based on thefact that nabla2 le 8 with the unit spacing size between pixelsFor simplicity in our numerical experiments we let 120581 = 8120582
and 120591 = 0015120582 and just adjust 120582 and the convergence of thealgorithm is satisfied
Before we present numerical results here we give thebasic property of model (10) rewritten in the continuoussettings that is
where (Ω119904)119904isin119868 is a finite set of small patches covering Ω andRΩ119904
is the restriction onΩ119904
Theorem 3 Let Ω be a bounded open subset of R2 withLipschitz boundary Let 119910 be a positive bounded function For119909 satisfying 119909 isin 119861119881(Ω) 119909 gt 0 let
where 120582 gt 0 Suppose that (Ω119904)119904isin119868 is a finite local coverage ofΩ andsuminfin
119904isin119868 1198631205721199042 lt +infin en 119864(119909) has a unique minimizerwhen 1205904120573 gt (supΩ 119910)2Proof Let119864(119909) = 120582119909119879119881+119869(119909) and 119869(119909) = (12057321205902) int
Ω1199092119889119905minus
120573intΩlog 1198680(1199101199091205902)119889119905 + (12)sum D120572119894119895 minus R1198941198951199092 By Proposi-
tion 1 we know that 119869(119909) is convex Thus 119864(119909) is convexand bounded from below Consider a minimizer sequence(119909119899) for 119864(119909) Then 119909119899119879119881 and 119869(119909119899) are bounded fromabove Hence (119909119899) is bounded in 119861119881(Ω) As 119861119881 is compactin 1198711(Ω) there is 1199090 isin 119861119881 such that a subsequence (119909119899119896)converges 1199090 in 1198711(Ω) and 119861119881(Ω) and we may assumethat 119909119899119896 997888rarr 1199090 pointwise almost everywhere As the 119861119881norm is lsc 1199090119879119881 le lim inf 119909119899119896119879119881 Since 119869(119909119899119896) isbounded from below by Fatoursquos Lemma we have 119869(1199090) lelim inf 119869(119909119899119896) Thus 119864(1199090) le lim inf 119864(119909119899119896) = inf119861119881(Ω)119864(119909)and 1199090 minimizes 119864(119909) Consequently 119864(119909) admits at leastone minimizer 1199090 119864(119909) is strictly convex function hence119864(119909) admits a unique minimizer
Mathematical Problems in Engineering 5
Table 1 Parameter 120582 values for all testing algorithms
In this section we will present our numerical experimentsto evaluate the approximation accuracy and computationalefficiency of our proposed algorithm We compare ourmethod for the denoising cases with the MAP model (2) theGTV model [31] the CZ model [32] and the NLM model[25] Here we have a brief overview of the models that wecompared
GTV model
argmin119909
120582intΩ119866120590 (119909 119910) + int
Ω|119863119909| 119889119905 (18)
119866120590 (119909 119910) = 119867120590 (119909) if 119909 ge 119888120590119867120590 (119888120590) + 1198671015840
120590 (119888120590) (119909 minus 119888120590) if 119909 le 1198881205901198671015840
where 119888 = 08426 1198681 is the modified Bessel function of thefirst kind with order one [30] Because it is an approximateequality we can not write down the final explicit restorationmodel
CZ model
arg min119909isin119878(Ω)
121205902 int
Ω1199092119889119905 minus int
Ωlog 1198680 (119909119910
1205902 )119889119905+ 1
120590 intΩ(radic119909 minus radic119910)2 119889119905 + 120582int
Ω|119863119909| 119889119905
(20)
where 119878(Ω) fl ] isin 119861119881(Ω) 0 le ] le 255 And because119861119881(Ω) [35 43] is the subspace of function 119909 isin 1198711(Ω) thefollowing quantity is finite
isin 119862infin0 (ΩR2) 10038171003817100381710038171205851003817100381710038171003817119871infin(ΩR2) le 1
(21)
NLMmodelBefore introducing the NLM model we have a descrip-
tion of the sign that will be used For a user-defined radius
119877119904119894119898 we define a square neighborhood window centeredaround pixel 119894 as 119873119894 And a Gaussian weighted Euclidiandistance of all the pixels of each neighborhood is defined as
119871 (119898 119899) = 1198661205881003817100381710038171003817119910 (119873119898) minus 119910 (119873119899)10038171003817100381710038172119877119904119894119898 (22)
where 119866120588 is a normalized Gaussian weighting functionwith zero mean and 120588 standard deviation (usually set to1) By giving more weight to pixels near the center wecan use 119866120588 to penalize pixels far from the center of theneighborhood window And based on the similarity betweenthe neighborhoods119873119898 and119873119899 of pixels119898 and 119899 we calculatethe similarity 119891(119898 119899) as
119862(119898) is the normalizing constant and ℎ is a exponentialdecay control parameter
Then given an image 119910 using the NLM method we cancalculate the filtered value at a point 119898 by the followingformula119873119871119872(119910 (119898)) = sum
forall119899isin119910
119891 (119898 119899) 119910 (119899)
0 le 119891 (119898 119899) le 1 sumforall119899isin119910
119891 (119898 119899) = 1 (24)
The parameter 120582 values of MAP model GTV modelCZ model and our proposed model are listed in Table 1In order to preserve the mean of the observed image thebias correction technique (15) is utilized for the CZ methodand our method All the experiments are performed underWindows 7 andMATLAB R2017a running on a PC equippedwith 290GHz CPU and 4G RAM
In our tests we choose images ldquoLenardquo and ldquoBarbarardquo withsize of 512 times 512 ldquoHouserdquo and ldquoMonarchrdquo with size of 256 times256 ldquoBrainrdquo with size of 181 times 217 and ldquoMouse intestinerdquo(briefly we call it ldquoMouserdquo throughout this paper) with sizeof 248 times 254 which are shown in Figure 1 We evaluate thequality of recovered images obtained from various denoisingalgorithms by using the structural similarity index (SSIM)[44] and the peal-signal-to-noise ratio (PSNR) [45] definedby
where 119909 and 119909 denote the original image and the recoveredimage respectively
First we choose the test image of ldquoLenardquo and degrade it byRician noise with120590 = 10 and use differentmethods to recoverthe noisy image The numerical results are enumerated inTable 2 where the first column gives the test image names thesecond column gives the method names (here Algorithm 1denotes our method presented in Section 2) and columns
3-5 (resp 6-8 and 9-11) are the PSNR (dB) SSIM and CPUtime (s) for 120590 = 10 (resp 120590 = 15 120590 = 20) We observethat the PSNR values of the restored images by Algorithm 1are higher than those of MAP GTV CZ and NLM methodMoreover you can easily find that the PSNR values of therestored images by Algorithm 1 are always the highest amongall those five methods for all images As far as ldquoBarbarardquo isconcerned the PSNR value of our method is more than 3
dB higher than CZ model at 120590 = 10 The characteristicsof SSIM values are almost consistent with those of PSNRvalues However the SSIM values are composed of luminancecomparison contrast comparison and structure comparisonwhich makes differences at some points
The images are corrupted respectively by Rician distri-bution with 120590 = 10 120590 = 15 and 120590 = 20 and the restoredimages of the above five algorithms for the images ldquoLenardquoldquoBarbarardquo ldquoHouserdquo and ldquoMonarchrdquo are shown inFigures 2ndash5respectively We clearly find that much noise remains in the
Figure 6 Results of ldquoBrainrdquo with 120590 = 10 by different methods (a) is the original ldquoBrainrdquo image and (g) is the degraded image with 120590 = 10(b)-(f) are the denoised images ofMAPmethod (120582 = 17) GTVmethod (120582 = 15) CZmethod (120582 = 0065) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
results of MAP GTV and CZmethod For example in Row 4of Figure 3 we can see from the background that ldquoBarbarardquoimage restored by our method is more clearer than thoserestored by other methods and textures of Barbararsquos trousersand scarf are kept better in the restored image by ourmethodThe restored images by our method also preserve moresignificant details thanMAPGTV CZ andNLMmethods onthe hat tidbits of the ldquoLenardquo image What is more the flowersof the lower left of ldquoMonarchrdquo also indicate that our method
is superior to other methods because our method is a patch-based method which is a different framework with TV-basedmethods (ie MAP method and CZ method) The imagesobtained from our method provide smoother regions andbetter shape preservation (eg the backgrounds in ldquoLenardquoand ldquoBarbarardquo)
Figures 6ndash11 are results of experiments on ldquoBrainrdquo andldquoMouserdquo images In particular for the ldquoBrainrdquo and ldquoMouserdquoimages we not only present the recovered images but also
10 Mathematical Problems in Engineering
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 7 Results of Brain with 120590 = 15 by different methods (a) is the original ldquoBrainrdquo image and (g) is the degraded image with 120590 = 15(b)-(f) are the denoised images of MAPmethod (120582 = 25) GTVmethod (120582 = 22) CZmethod (120582 = 005) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 8 Results of Brain with 120590 = 20 by different methods (a) is the original ldquoBrainrdquo image and (g) is the degraded image with 120590 = 20(b)-(f) are the denoised images ofMAPmethod (120582 = 35) GTVmethod (120582 = 22) CZmethod (120582 = 0045) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
use residuals to make a comparison with the original imagesThe first row is original image and recovered images ofMAP method GTV method CZ method NLM methodand our method The second row is degraded image withvarious 120590 and residuals of those methods respectively InFigures 6ndash8 the recovered images of of MAP method GTVmethod CZ method and NLM method are still unclear sowe can conclude that the recovered images by ourmethod arebest through our visions and residuals As for the ldquoMouserdquoimage the recovered results are very similar in this casewe use residuals to see which result is better In general theblurrier residual is the better according recovered image isWe can find that the outline of residual by our method isblurriest
In conclusion the results of our numerical experimentsdemonstrate that our proposed method performs better thanthe MAP method GTV method CZ method and NLMmethod
4 Conclusion
In this paper we proposed a new effective model via alearned dictionary for denoising images with Rician noiseMore specifically based on the dictionary training we adda dictionary penalty term to the original nonconvex MAPmodel to establish a new denoising model and develop atwo-step algorithm to solve our model Also we carry out
Mathematical Problems in Engineering 11
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 9 Results of ldquoMouserdquo with 120590 = 10 by different methods (a) is the original ldquoMouserdquo image and (g) is the degraded image with 120590 = 10(b)-(f) are the denoised images of MAPmethod (120582 = 17) GTVmethod (120582 = 17) CZmethod (120582 = 006) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 10 Results of ldquoMouserdquo with 120590 = 15 by different methods (a) is the original ldquoMouserdquo image and (g) is the degraded image with 120590 = 15(b)-(f) are the denoised images of MAPmethod (120582 = 24) GTV method (120582 = 24) CZ method (120582 = 005) NLM method and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
experiments on various images to demonstrate the effective-ness of our modelThe numerical experiments show that ourproposedmodel is promising in denoising imageswithRiciannoise
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interest
Acknowledgments
The authors would like to thank one of the authors of[32] for providing the source code of CZ method Thiswork was supported in part by the National Natural ScienceFoundation of China under Grants 11871348 61872429 and61373087 by the Natural Science Foundation of GuangdongChina under Grants 2015A030313550 and 2015A030313557by the HD Video RampD Platform for Intelligent Analy-sis and Processing in Guangdong Engineering TechnologyResearch Centre of Colleges and Universities (no GCZX-A1409) by Natural Science Foundation of Shenzhen underGrant JCYJ20170818091621856 and by the Guangdong Key
12 Mathematical Problems in Engineering
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 11 Results of ldquoMouserdquo with 120590 = 20 by different methods (a) is the original ldquoMouserdquo image and (g) is the degraded image with 120590 = 20(b)-(f) are the denoised images of MAPmethod (120582 = 31) GTV method (120582 = 32) CZ method (120582 = 0035) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
Laboratory of Intelligent Information Processing ShenzhenUniversity China (518060)
References
[1] J-F Cai B Dong and Z Shen ldquoImage restoration A waveletframe based model for piecewise smooth functions andbeyondrdquoApplied andComputational Harmonic Analysis vol 41no 1 pp 94ndash138 2016
[2] R H Chan and J Ma ldquoA multiplicative iterative algorithm forbox-constrained penalized likelihood image restorationrdquo IEEETransactions on Image Processing vol 21 no 7 pp 3168ndash31812012
[3] C A Micchelli L Shen and Y Xu ldquoProximity algorithmsfor image models Denoisingrdquo Inverse Problems vol 27 no 4Article ID 045009 p 30 2011
[4] YWang J Yang W Yin and Y Zhang ldquoA new alternatingmin-imization algorithm for total variation image reconstructionrdquoSIAM Journal on Imaging Sciences vol 1 no 3 pp 248ndash2722008
[5] Y Dong M Hintermuller and M Neri ldquoAn efficient primal-dual method for l1-TV image restorationrdquo SIAM Journal onImaging Sciences vol 2 no 4 pp 1168ndash1189 2009
[6] J Lu K Qiao L Shen and Y Zou ldquoFixed-point algorithmsfor a TVL1 image restoration modelrdquo International Journal ofComputer Mathematics vol 95 no 9 pp 1829ndash1844 2018
[7] J Lu Z Ye and Y Zou ldquoHuber fractal image coding based ona fitting planerdquo IEEE Transactions on Image Processing vol 22no 1 pp 134ndash145 2013
[8] C A Micchelli L Shen Y Xu and X Zeng ldquoProximityalgorithms for the L1TV image denoising modelrdquo Advances inComputational Mathematics vol 38 no 2 pp 401ndash426 2013
[9] J Yang Y Zhang and W Yin ldquoAn efficient TVL1 algorithm fordeblurring multichannel images corrupted by impulsive noiserdquoSIAM Journal on Scientific Computing vol 31 no 4 pp 2842ndash2865 2009
[10] L Ma L Moisan J Yu and T Zeng ldquoA dictionary learningapproach for poisson image deblurringrdquo IEEE Transactions onMedical Imaging vol 32 no 7 pp 1277ndash1289 2013
[11] Y Xiao and T Zeng ldquoPoisson noise removal via learned dictio-naryrdquo in Proceedings of the 17th IEEE International Conferenceon Image Processing (ICIP) pp 1177ndash1180 2010
[12] H Zhang Y Dong and Q Fan ldquoWavelet frame based Poissonnoise removal and image deblurringrdquo Signal Processing vol 137pp 363ndash372 2017
[13] Y Dong and T Zeng ldquoA convex variational model for restoringblurred images with multiplicative noiserdquo SIAM Journal onImaging Sciences vol 6 no 3 pp 1598ndash1625 2013
[14] Y Huang M K Ng and YWen ldquoA new total variation methodfor multiplicative noise removalrdquo SIAM Journal on ImagingSciences vol 2 no 1 pp 20ndash40 2009
[15] Y-M Huang L Moisan M K Ng and T Zeng ldquoMultiplicativenoise removal via a learned dictionaryrdquo IEEE Transactions onImage Processing vol 21 no 11 pp 4534ndash4543 2012
[16] Z Jin and X Yang ldquoA variational model to remove the multi-plicative noise in ultrasound imagesrdquo Journal of MathematicalImaging and Vision vol 39 no 1 pp 62ndash74 2011
[17] M Kang S Yun and H Woo ldquoTwo-level convex relaxedvariational model for multiplicative denoisingrdquo SIAM Journalon Imaging Sciences vol 6 no 2 pp 875ndash903 2013
[18] J Lu L Shen C Xu and Y Xu ldquoMultiplicative noise removalin imaging An exp-model and its fixed-point proximity algo-rithmrdquo Applied and Computational Harmonic Analysis vol 41no 2 pp 518ndash539 2016
[19] J Lu L Shen C Xu and Y Xu ldquoMultiplicative noise removalwith a sparsity-aware optimization modelrdquo Inverse Problemsand Imaging vol 11 no 6 pp 949ndash974 2017
[20] J Lu Z Yang L Shen Z Lu H Yang and C Xu ldquoA frameletalgorithm for de-blurring images corrupted by multiplicativenoiserdquoApplied Mathematical Modelling vol 62 pp 51ndash61 2018
[21] J Lu Y Chen Y Zou and L Shen ldquoA new total variationmodelfor restoring blurred and speckle noisy imagesrdquo InternationalJournal of Wavelets Multiresolution and Information Processingvol 15 no 2 19 pages 2017
Mathematical Problems in Engineering 13
[22] P Perona and J Malik ldquoScale-space and edge detection usinganisotropic diffusionrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 12 no 7 pp 629ndash639 1990
[23] G Gerig O Kubler R Kikinis and F A Jolesz ldquoNonlinearanisotropic filtering ofMRI datardquo IEEE Transactions onMedicalImaging vol 11 no 2 pp 221ndash232 1992
[24] S Prima S P Morrissey and C Barillot ldquoNon-local meansvariants for denoising of diffusion-weighted and diffusiontensor MRIrdquo in Proceedings of the International Conference onMedical Image Computing and Computer-Assisted Interventionpp 344ndash351 Springer Berlin Germany 2007
[25] J V Manjon J Carbonell-Caballero J J Lull G Garcıa-MartiL Martı-Bonmati and M Robles ldquoMRI denoising using Non-LocalMeansrdquoMedical Image Analysis vol 12 no 4 pp 514ndash5232008
[26] N Wiest-Daessle S Prima P Coupe S P Morrissey and CBarillot ldquoRician noise removal by non-local means filteringfor low signal-to-noise ratio MRI Applications to DT-MRIrdquoMedical Image Computing and Computer-Assisted Interventionvol 5242 no 2 pp 171ndash179 2008
[27] R D Nowak ldquoWavelet-based Rician noise removal for mag-netic resonance imagingrdquo IEEE Transactions on Image Process-ing vol 8 no 10 pp 1408ndash1419 1999
[28] A Foi ldquoNoise estimation and removal in MR imaging Thevariance-stabilization approachrdquo in Proceedings of the 8th IEEEInternational Symposium on Biomedical Imaging From Nano toMacro (ISBI) pp 1809ndash1814 IEEE Chicago IL USA 2011
[29] J C Wood and K M Johnson ldquoWavelet packet denoising ofmagnetic resonance images Importance of Rician noise at lowSNRrdquo Magnetic Resonance in Medicine vol 41 no 3 pp 631ndash635 1999
[30] F Bowman Introduction to Bessel functions Dover PublicationsMineola NY USA 2012
[31] P Getreuer M Tong and L A Vese ldquoA variational model forthe restoration of MR images corrupted by blur and Riciannoiserdquo in Proceedings of the 7th international conference onAdvances in visual computing vol Part I of Lecture Notes inComputer Science pp 686ndash698 Springer Las Vegas NV USA2011
[32] L Chen and T Zeng ldquoA convex variational model for restoringblurred imageswith large Rician noiserdquo Journal ofMathematicalImaging and Vision vol 53 no 1 pp 92ndash111 2015
[33] M Aharon M Elad and A M Bruckstein ldquoK-SVD Analgorithm for designing overcomplete dictionaries for sparserepresentationrdquo IEEE Transactions on Signal Processing vol 54no 11 pp 4311ndash4322 2006
[34] M Elad and M Aharon ldquoImage denoising via sparse andredundant representations over learned dictionariesrdquo IEEETransactions on Image Processing vol 15 no 12 pp 3736ndash37452006
[35] G Aubert and J Aujol ldquoA variational approach to removingmultiplicative noiserdquo SIAM Journal on Applied Mathematicsvol 68 no 4 pp 925ndash946 2008
[36] Q Liu S Wang K Yang J Luo Y Zhu and D Liang ldquoHighlyundersampled magnetic resonance image reconstruction usingtwo-level Bregman method with dictionary updatingrdquo IEEETransactions on Medical Imaging vol 32 no 7 pp 1290ndash13012013
[37] Q Liu S Wang L Ying X Peng Y Zhu and D LiangldquoAdaptive dictionary learning in sparse gradient domain forimage recoveryrdquo IEEE Transactions on Image Processing vol 22no 12 pp 4652ndash4663 2013
[38] H Lu J Wei Q Liu Y Wang and X Deng ldquoA dictionarylearning method with total generalized variation for mri recon-structionrdquo International Journal of Biomedical Imaging vol2016 Article ID 7512471 13 pages 2016
[39] A Chambolle and T Pock ldquoA first-order primal-dual algorithmfor convex problems with applications to imagingrdquo Journal ofMathematical Imaging and Vision vol 40 no 1 pp 120ndash1452011
[40] E Esser X Zhang and T F Chan ldquoA general frameworkfor a class of first order primal-dual algorithms for convexoptimization in imaging sciencerdquo SIAM Journal on ImagingSciences vol 3 no 4 pp 1015ndash1046 2010
[41] N Komodakis and J-C Pesquet ldquoPlaying with duality Anoverview of recent primaldual approaches for solving large-scale optimization problemsrdquo IEEE Signal Processing Magazinevol 32 no 6 pp 31ndash54 2015
[42] A Chambolle ldquoAn algorithm for total variation minimizationand applicationsrdquo Journal of Mathematical Imaging and Visionvol 20 no 1-2 pp 89ndash97 2004
[43] L Ambrosio N Fusco and D Pallara Functions of BoundedVariation and Free Discontinuity Problem Oxford UniversityPress London 2000
[44] Z Wang A C Bovik H R Sheikh and E P SimoncellildquoImage quality assessment From error visibility to structuralsimilarityrdquo IEEE Transactions on Image Processing vol 13 no4 pp 600ndash612 2004
[45] J Gibson and A Bovik Handbook of Image and Video Process-ing Academic Press 2000
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
In this section we will present our numerical experimentsto evaluate the approximation accuracy and computationalefficiency of our proposed algorithm We compare ourmethod for the denoising cases with the MAP model (2) theGTV model [31] the CZ model [32] and the NLM model[25] Here we have a brief overview of the models that wecompared
GTV model
argmin119909
120582intΩ119866120590 (119909 119910) + int
Ω|119863119909| 119889119905 (18)
119866120590 (119909 119910) = 119867120590 (119909) if 119909 ge 119888120590119867120590 (119888120590) + 1198671015840
120590 (119888120590) (119909 minus 119888120590) if 119909 le 1198881205901198671015840
where 119888 = 08426 1198681 is the modified Bessel function of thefirst kind with order one [30] Because it is an approximateequality we can not write down the final explicit restorationmodel
CZ model
arg min119909isin119878(Ω)
121205902 int
Ω1199092119889119905 minus int
Ωlog 1198680 (119909119910
1205902 )119889119905+ 1
120590 intΩ(radic119909 minus radic119910)2 119889119905 + 120582int
Ω|119863119909| 119889119905
(20)
where 119878(Ω) fl ] isin 119861119881(Ω) 0 le ] le 255 And because119861119881(Ω) [35 43] is the subspace of function 119909 isin 1198711(Ω) thefollowing quantity is finite
isin 119862infin0 (ΩR2) 10038171003817100381710038171205851003817100381710038171003817119871infin(ΩR2) le 1
(21)
NLMmodelBefore introducing the NLM model we have a descrip-
tion of the sign that will be used For a user-defined radius
119877119904119894119898 we define a square neighborhood window centeredaround pixel 119894 as 119873119894 And a Gaussian weighted Euclidiandistance of all the pixels of each neighborhood is defined as
119871 (119898 119899) = 1198661205881003817100381710038171003817119910 (119873119898) minus 119910 (119873119899)10038171003817100381710038172119877119904119894119898 (22)
where 119866120588 is a normalized Gaussian weighting functionwith zero mean and 120588 standard deviation (usually set to1) By giving more weight to pixels near the center wecan use 119866120588 to penalize pixels far from the center of theneighborhood window And based on the similarity betweenthe neighborhoods119873119898 and119873119899 of pixels119898 and 119899 we calculatethe similarity 119891(119898 119899) as
119862(119898) is the normalizing constant and ℎ is a exponentialdecay control parameter
Then given an image 119910 using the NLM method we cancalculate the filtered value at a point 119898 by the followingformula119873119871119872(119910 (119898)) = sum
forall119899isin119910
119891 (119898 119899) 119910 (119899)
0 le 119891 (119898 119899) le 1 sumforall119899isin119910
119891 (119898 119899) = 1 (24)
The parameter 120582 values of MAP model GTV modelCZ model and our proposed model are listed in Table 1In order to preserve the mean of the observed image thebias correction technique (15) is utilized for the CZ methodand our method All the experiments are performed underWindows 7 andMATLAB R2017a running on a PC equippedwith 290GHz CPU and 4G RAM
In our tests we choose images ldquoLenardquo and ldquoBarbarardquo withsize of 512 times 512 ldquoHouserdquo and ldquoMonarchrdquo with size of 256 times256 ldquoBrainrdquo with size of 181 times 217 and ldquoMouse intestinerdquo(briefly we call it ldquoMouserdquo throughout this paper) with sizeof 248 times 254 which are shown in Figure 1 We evaluate thequality of recovered images obtained from various denoisingalgorithms by using the structural similarity index (SSIM)[44] and the peal-signal-to-noise ratio (PSNR) [45] definedby
where 119909 and 119909 denote the original image and the recoveredimage respectively
First we choose the test image of ldquoLenardquo and degrade it byRician noise with120590 = 10 and use differentmethods to recoverthe noisy image The numerical results are enumerated inTable 2 where the first column gives the test image names thesecond column gives the method names (here Algorithm 1denotes our method presented in Section 2) and columns
3-5 (resp 6-8 and 9-11) are the PSNR (dB) SSIM and CPUtime (s) for 120590 = 10 (resp 120590 = 15 120590 = 20) We observethat the PSNR values of the restored images by Algorithm 1are higher than those of MAP GTV CZ and NLM methodMoreover you can easily find that the PSNR values of therestored images by Algorithm 1 are always the highest amongall those five methods for all images As far as ldquoBarbarardquo isconcerned the PSNR value of our method is more than 3
dB higher than CZ model at 120590 = 10 The characteristicsof SSIM values are almost consistent with those of PSNRvalues However the SSIM values are composed of luminancecomparison contrast comparison and structure comparisonwhich makes differences at some points
The images are corrupted respectively by Rician distri-bution with 120590 = 10 120590 = 15 and 120590 = 20 and the restoredimages of the above five algorithms for the images ldquoLenardquoldquoBarbarardquo ldquoHouserdquo and ldquoMonarchrdquo are shown inFigures 2ndash5respectively We clearly find that much noise remains in the
Figure 6 Results of ldquoBrainrdquo with 120590 = 10 by different methods (a) is the original ldquoBrainrdquo image and (g) is the degraded image with 120590 = 10(b)-(f) are the denoised images ofMAPmethod (120582 = 17) GTVmethod (120582 = 15) CZmethod (120582 = 0065) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
results of MAP GTV and CZmethod For example in Row 4of Figure 3 we can see from the background that ldquoBarbarardquoimage restored by our method is more clearer than thoserestored by other methods and textures of Barbararsquos trousersand scarf are kept better in the restored image by ourmethodThe restored images by our method also preserve moresignificant details thanMAPGTV CZ andNLMmethods onthe hat tidbits of the ldquoLenardquo image What is more the flowersof the lower left of ldquoMonarchrdquo also indicate that our method
is superior to other methods because our method is a patch-based method which is a different framework with TV-basedmethods (ie MAP method and CZ method) The imagesobtained from our method provide smoother regions andbetter shape preservation (eg the backgrounds in ldquoLenardquoand ldquoBarbarardquo)
Figures 6ndash11 are results of experiments on ldquoBrainrdquo andldquoMouserdquo images In particular for the ldquoBrainrdquo and ldquoMouserdquoimages we not only present the recovered images but also
10 Mathematical Problems in Engineering
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 7 Results of Brain with 120590 = 15 by different methods (a) is the original ldquoBrainrdquo image and (g) is the degraded image with 120590 = 15(b)-(f) are the denoised images of MAPmethod (120582 = 25) GTVmethod (120582 = 22) CZmethod (120582 = 005) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 8 Results of Brain with 120590 = 20 by different methods (a) is the original ldquoBrainrdquo image and (g) is the degraded image with 120590 = 20(b)-(f) are the denoised images ofMAPmethod (120582 = 35) GTVmethod (120582 = 22) CZmethod (120582 = 0045) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
use residuals to make a comparison with the original imagesThe first row is original image and recovered images ofMAP method GTV method CZ method NLM methodand our method The second row is degraded image withvarious 120590 and residuals of those methods respectively InFigures 6ndash8 the recovered images of of MAP method GTVmethod CZ method and NLM method are still unclear sowe can conclude that the recovered images by ourmethod arebest through our visions and residuals As for the ldquoMouserdquoimage the recovered results are very similar in this casewe use residuals to see which result is better In general theblurrier residual is the better according recovered image isWe can find that the outline of residual by our method isblurriest
In conclusion the results of our numerical experimentsdemonstrate that our proposed method performs better thanthe MAP method GTV method CZ method and NLMmethod
4 Conclusion
In this paper we proposed a new effective model via alearned dictionary for denoising images with Rician noiseMore specifically based on the dictionary training we adda dictionary penalty term to the original nonconvex MAPmodel to establish a new denoising model and develop atwo-step algorithm to solve our model Also we carry out
Mathematical Problems in Engineering 11
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 9 Results of ldquoMouserdquo with 120590 = 10 by different methods (a) is the original ldquoMouserdquo image and (g) is the degraded image with 120590 = 10(b)-(f) are the denoised images of MAPmethod (120582 = 17) GTVmethod (120582 = 17) CZmethod (120582 = 006) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 10 Results of ldquoMouserdquo with 120590 = 15 by different methods (a) is the original ldquoMouserdquo image and (g) is the degraded image with 120590 = 15(b)-(f) are the denoised images of MAPmethod (120582 = 24) GTV method (120582 = 24) CZ method (120582 = 005) NLM method and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
experiments on various images to demonstrate the effective-ness of our modelThe numerical experiments show that ourproposedmodel is promising in denoising imageswithRiciannoise
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interest
Acknowledgments
The authors would like to thank one of the authors of[32] for providing the source code of CZ method Thiswork was supported in part by the National Natural ScienceFoundation of China under Grants 11871348 61872429 and61373087 by the Natural Science Foundation of GuangdongChina under Grants 2015A030313550 and 2015A030313557by the HD Video RampD Platform for Intelligent Analy-sis and Processing in Guangdong Engineering TechnologyResearch Centre of Colleges and Universities (no GCZX-A1409) by Natural Science Foundation of Shenzhen underGrant JCYJ20170818091621856 and by the Guangdong Key
12 Mathematical Problems in Engineering
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 11 Results of ldquoMouserdquo with 120590 = 20 by different methods (a) is the original ldquoMouserdquo image and (g) is the degraded image with 120590 = 20(b)-(f) are the denoised images of MAPmethod (120582 = 31) GTV method (120582 = 32) CZ method (120582 = 0035) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
Laboratory of Intelligent Information Processing ShenzhenUniversity China (518060)
References
[1] J-F Cai B Dong and Z Shen ldquoImage restoration A waveletframe based model for piecewise smooth functions andbeyondrdquoApplied andComputational Harmonic Analysis vol 41no 1 pp 94ndash138 2016
[2] R H Chan and J Ma ldquoA multiplicative iterative algorithm forbox-constrained penalized likelihood image restorationrdquo IEEETransactions on Image Processing vol 21 no 7 pp 3168ndash31812012
[3] C A Micchelli L Shen and Y Xu ldquoProximity algorithmsfor image models Denoisingrdquo Inverse Problems vol 27 no 4Article ID 045009 p 30 2011
[4] YWang J Yang W Yin and Y Zhang ldquoA new alternatingmin-imization algorithm for total variation image reconstructionrdquoSIAM Journal on Imaging Sciences vol 1 no 3 pp 248ndash2722008
[5] Y Dong M Hintermuller and M Neri ldquoAn efficient primal-dual method for l1-TV image restorationrdquo SIAM Journal onImaging Sciences vol 2 no 4 pp 1168ndash1189 2009
[6] J Lu K Qiao L Shen and Y Zou ldquoFixed-point algorithmsfor a TVL1 image restoration modelrdquo International Journal ofComputer Mathematics vol 95 no 9 pp 1829ndash1844 2018
[7] J Lu Z Ye and Y Zou ldquoHuber fractal image coding based ona fitting planerdquo IEEE Transactions on Image Processing vol 22no 1 pp 134ndash145 2013
[8] C A Micchelli L Shen Y Xu and X Zeng ldquoProximityalgorithms for the L1TV image denoising modelrdquo Advances inComputational Mathematics vol 38 no 2 pp 401ndash426 2013
[9] J Yang Y Zhang and W Yin ldquoAn efficient TVL1 algorithm fordeblurring multichannel images corrupted by impulsive noiserdquoSIAM Journal on Scientific Computing vol 31 no 4 pp 2842ndash2865 2009
[10] L Ma L Moisan J Yu and T Zeng ldquoA dictionary learningapproach for poisson image deblurringrdquo IEEE Transactions onMedical Imaging vol 32 no 7 pp 1277ndash1289 2013
[11] Y Xiao and T Zeng ldquoPoisson noise removal via learned dictio-naryrdquo in Proceedings of the 17th IEEE International Conferenceon Image Processing (ICIP) pp 1177ndash1180 2010
[12] H Zhang Y Dong and Q Fan ldquoWavelet frame based Poissonnoise removal and image deblurringrdquo Signal Processing vol 137pp 363ndash372 2017
[13] Y Dong and T Zeng ldquoA convex variational model for restoringblurred images with multiplicative noiserdquo SIAM Journal onImaging Sciences vol 6 no 3 pp 1598ndash1625 2013
[14] Y Huang M K Ng and YWen ldquoA new total variation methodfor multiplicative noise removalrdquo SIAM Journal on ImagingSciences vol 2 no 1 pp 20ndash40 2009
[15] Y-M Huang L Moisan M K Ng and T Zeng ldquoMultiplicativenoise removal via a learned dictionaryrdquo IEEE Transactions onImage Processing vol 21 no 11 pp 4534ndash4543 2012
[16] Z Jin and X Yang ldquoA variational model to remove the multi-plicative noise in ultrasound imagesrdquo Journal of MathematicalImaging and Vision vol 39 no 1 pp 62ndash74 2011
[17] M Kang S Yun and H Woo ldquoTwo-level convex relaxedvariational model for multiplicative denoisingrdquo SIAM Journalon Imaging Sciences vol 6 no 2 pp 875ndash903 2013
[18] J Lu L Shen C Xu and Y Xu ldquoMultiplicative noise removalin imaging An exp-model and its fixed-point proximity algo-rithmrdquo Applied and Computational Harmonic Analysis vol 41no 2 pp 518ndash539 2016
[19] J Lu L Shen C Xu and Y Xu ldquoMultiplicative noise removalwith a sparsity-aware optimization modelrdquo Inverse Problemsand Imaging vol 11 no 6 pp 949ndash974 2017
[20] J Lu Z Yang L Shen Z Lu H Yang and C Xu ldquoA frameletalgorithm for de-blurring images corrupted by multiplicativenoiserdquoApplied Mathematical Modelling vol 62 pp 51ndash61 2018
[21] J Lu Y Chen Y Zou and L Shen ldquoA new total variationmodelfor restoring blurred and speckle noisy imagesrdquo InternationalJournal of Wavelets Multiresolution and Information Processingvol 15 no 2 19 pages 2017
Mathematical Problems in Engineering 13
[22] P Perona and J Malik ldquoScale-space and edge detection usinganisotropic diffusionrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 12 no 7 pp 629ndash639 1990
[23] G Gerig O Kubler R Kikinis and F A Jolesz ldquoNonlinearanisotropic filtering ofMRI datardquo IEEE Transactions onMedicalImaging vol 11 no 2 pp 221ndash232 1992
[24] S Prima S P Morrissey and C Barillot ldquoNon-local meansvariants for denoising of diffusion-weighted and diffusiontensor MRIrdquo in Proceedings of the International Conference onMedical Image Computing and Computer-Assisted Interventionpp 344ndash351 Springer Berlin Germany 2007
[25] J V Manjon J Carbonell-Caballero J J Lull G Garcıa-MartiL Martı-Bonmati and M Robles ldquoMRI denoising using Non-LocalMeansrdquoMedical Image Analysis vol 12 no 4 pp 514ndash5232008
[26] N Wiest-Daessle S Prima P Coupe S P Morrissey and CBarillot ldquoRician noise removal by non-local means filteringfor low signal-to-noise ratio MRI Applications to DT-MRIrdquoMedical Image Computing and Computer-Assisted Interventionvol 5242 no 2 pp 171ndash179 2008
[27] R D Nowak ldquoWavelet-based Rician noise removal for mag-netic resonance imagingrdquo IEEE Transactions on Image Process-ing vol 8 no 10 pp 1408ndash1419 1999
[28] A Foi ldquoNoise estimation and removal in MR imaging Thevariance-stabilization approachrdquo in Proceedings of the 8th IEEEInternational Symposium on Biomedical Imaging From Nano toMacro (ISBI) pp 1809ndash1814 IEEE Chicago IL USA 2011
[29] J C Wood and K M Johnson ldquoWavelet packet denoising ofmagnetic resonance images Importance of Rician noise at lowSNRrdquo Magnetic Resonance in Medicine vol 41 no 3 pp 631ndash635 1999
[30] F Bowman Introduction to Bessel functions Dover PublicationsMineola NY USA 2012
[31] P Getreuer M Tong and L A Vese ldquoA variational model forthe restoration of MR images corrupted by blur and Riciannoiserdquo in Proceedings of the 7th international conference onAdvances in visual computing vol Part I of Lecture Notes inComputer Science pp 686ndash698 Springer Las Vegas NV USA2011
[32] L Chen and T Zeng ldquoA convex variational model for restoringblurred imageswith large Rician noiserdquo Journal ofMathematicalImaging and Vision vol 53 no 1 pp 92ndash111 2015
[33] M Aharon M Elad and A M Bruckstein ldquoK-SVD Analgorithm for designing overcomplete dictionaries for sparserepresentationrdquo IEEE Transactions on Signal Processing vol 54no 11 pp 4311ndash4322 2006
[34] M Elad and M Aharon ldquoImage denoising via sparse andredundant representations over learned dictionariesrdquo IEEETransactions on Image Processing vol 15 no 12 pp 3736ndash37452006
[35] G Aubert and J Aujol ldquoA variational approach to removingmultiplicative noiserdquo SIAM Journal on Applied Mathematicsvol 68 no 4 pp 925ndash946 2008
[36] Q Liu S Wang K Yang J Luo Y Zhu and D Liang ldquoHighlyundersampled magnetic resonance image reconstruction usingtwo-level Bregman method with dictionary updatingrdquo IEEETransactions on Medical Imaging vol 32 no 7 pp 1290ndash13012013
[37] Q Liu S Wang L Ying X Peng Y Zhu and D LiangldquoAdaptive dictionary learning in sparse gradient domain forimage recoveryrdquo IEEE Transactions on Image Processing vol 22no 12 pp 4652ndash4663 2013
[38] H Lu J Wei Q Liu Y Wang and X Deng ldquoA dictionarylearning method with total generalized variation for mri recon-structionrdquo International Journal of Biomedical Imaging vol2016 Article ID 7512471 13 pages 2016
[39] A Chambolle and T Pock ldquoA first-order primal-dual algorithmfor convex problems with applications to imagingrdquo Journal ofMathematical Imaging and Vision vol 40 no 1 pp 120ndash1452011
[40] E Esser X Zhang and T F Chan ldquoA general frameworkfor a class of first order primal-dual algorithms for convexoptimization in imaging sciencerdquo SIAM Journal on ImagingSciences vol 3 no 4 pp 1015ndash1046 2010
[41] N Komodakis and J-C Pesquet ldquoPlaying with duality Anoverview of recent primaldual approaches for solving large-scale optimization problemsrdquo IEEE Signal Processing Magazinevol 32 no 6 pp 31ndash54 2015
[42] A Chambolle ldquoAn algorithm for total variation minimizationand applicationsrdquo Journal of Mathematical Imaging and Visionvol 20 no 1-2 pp 89ndash97 2004
[43] L Ambrosio N Fusco and D Pallara Functions of BoundedVariation and Free Discontinuity Problem Oxford UniversityPress London 2000
[44] Z Wang A C Bovik H R Sheikh and E P SimoncellildquoImage quality assessment From error visibility to structuralsimilarityrdquo IEEE Transactions on Image Processing vol 13 no4 pp 600ndash612 2004
[45] J Gibson and A Bovik Handbook of Image and Video Process-ing Academic Press 2000
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
where 119909 and 119909 denote the original image and the recoveredimage respectively
First we choose the test image of ldquoLenardquo and degrade it byRician noise with120590 = 10 and use differentmethods to recoverthe noisy image The numerical results are enumerated inTable 2 where the first column gives the test image names thesecond column gives the method names (here Algorithm 1denotes our method presented in Section 2) and columns
3-5 (resp 6-8 and 9-11) are the PSNR (dB) SSIM and CPUtime (s) for 120590 = 10 (resp 120590 = 15 120590 = 20) We observethat the PSNR values of the restored images by Algorithm 1are higher than those of MAP GTV CZ and NLM methodMoreover you can easily find that the PSNR values of therestored images by Algorithm 1 are always the highest amongall those five methods for all images As far as ldquoBarbarardquo isconcerned the PSNR value of our method is more than 3
dB higher than CZ model at 120590 = 10 The characteristicsof SSIM values are almost consistent with those of PSNRvalues However the SSIM values are composed of luminancecomparison contrast comparison and structure comparisonwhich makes differences at some points
The images are corrupted respectively by Rician distri-bution with 120590 = 10 120590 = 15 and 120590 = 20 and the restoredimages of the above five algorithms for the images ldquoLenardquoldquoBarbarardquo ldquoHouserdquo and ldquoMonarchrdquo are shown inFigures 2ndash5respectively We clearly find that much noise remains in the
Figure 6 Results of ldquoBrainrdquo with 120590 = 10 by different methods (a) is the original ldquoBrainrdquo image and (g) is the degraded image with 120590 = 10(b)-(f) are the denoised images ofMAPmethod (120582 = 17) GTVmethod (120582 = 15) CZmethod (120582 = 0065) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
results of MAP GTV and CZmethod For example in Row 4of Figure 3 we can see from the background that ldquoBarbarardquoimage restored by our method is more clearer than thoserestored by other methods and textures of Barbararsquos trousersand scarf are kept better in the restored image by ourmethodThe restored images by our method also preserve moresignificant details thanMAPGTV CZ andNLMmethods onthe hat tidbits of the ldquoLenardquo image What is more the flowersof the lower left of ldquoMonarchrdquo also indicate that our method
is superior to other methods because our method is a patch-based method which is a different framework with TV-basedmethods (ie MAP method and CZ method) The imagesobtained from our method provide smoother regions andbetter shape preservation (eg the backgrounds in ldquoLenardquoand ldquoBarbarardquo)
Figures 6ndash11 are results of experiments on ldquoBrainrdquo andldquoMouserdquo images In particular for the ldquoBrainrdquo and ldquoMouserdquoimages we not only present the recovered images but also
10 Mathematical Problems in Engineering
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 7 Results of Brain with 120590 = 15 by different methods (a) is the original ldquoBrainrdquo image and (g) is the degraded image with 120590 = 15(b)-(f) are the denoised images of MAPmethod (120582 = 25) GTVmethod (120582 = 22) CZmethod (120582 = 005) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 8 Results of Brain with 120590 = 20 by different methods (a) is the original ldquoBrainrdquo image and (g) is the degraded image with 120590 = 20(b)-(f) are the denoised images ofMAPmethod (120582 = 35) GTVmethod (120582 = 22) CZmethod (120582 = 0045) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
use residuals to make a comparison with the original imagesThe first row is original image and recovered images ofMAP method GTV method CZ method NLM methodand our method The second row is degraded image withvarious 120590 and residuals of those methods respectively InFigures 6ndash8 the recovered images of of MAP method GTVmethod CZ method and NLM method are still unclear sowe can conclude that the recovered images by ourmethod arebest through our visions and residuals As for the ldquoMouserdquoimage the recovered results are very similar in this casewe use residuals to see which result is better In general theblurrier residual is the better according recovered image isWe can find that the outline of residual by our method isblurriest
In conclusion the results of our numerical experimentsdemonstrate that our proposed method performs better thanthe MAP method GTV method CZ method and NLMmethod
4 Conclusion
In this paper we proposed a new effective model via alearned dictionary for denoising images with Rician noiseMore specifically based on the dictionary training we adda dictionary penalty term to the original nonconvex MAPmodel to establish a new denoising model and develop atwo-step algorithm to solve our model Also we carry out
Mathematical Problems in Engineering 11
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 9 Results of ldquoMouserdquo with 120590 = 10 by different methods (a) is the original ldquoMouserdquo image and (g) is the degraded image with 120590 = 10(b)-(f) are the denoised images of MAPmethod (120582 = 17) GTVmethod (120582 = 17) CZmethod (120582 = 006) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 10 Results of ldquoMouserdquo with 120590 = 15 by different methods (a) is the original ldquoMouserdquo image and (g) is the degraded image with 120590 = 15(b)-(f) are the denoised images of MAPmethod (120582 = 24) GTV method (120582 = 24) CZ method (120582 = 005) NLM method and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
experiments on various images to demonstrate the effective-ness of our modelThe numerical experiments show that ourproposedmodel is promising in denoising imageswithRiciannoise
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interest
Acknowledgments
The authors would like to thank one of the authors of[32] for providing the source code of CZ method Thiswork was supported in part by the National Natural ScienceFoundation of China under Grants 11871348 61872429 and61373087 by the Natural Science Foundation of GuangdongChina under Grants 2015A030313550 and 2015A030313557by the HD Video RampD Platform for Intelligent Analy-sis and Processing in Guangdong Engineering TechnologyResearch Centre of Colleges and Universities (no GCZX-A1409) by Natural Science Foundation of Shenzhen underGrant JCYJ20170818091621856 and by the Guangdong Key
12 Mathematical Problems in Engineering
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 11 Results of ldquoMouserdquo with 120590 = 20 by different methods (a) is the original ldquoMouserdquo image and (g) is the degraded image with 120590 = 20(b)-(f) are the denoised images of MAPmethod (120582 = 31) GTV method (120582 = 32) CZ method (120582 = 0035) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
Laboratory of Intelligent Information Processing ShenzhenUniversity China (518060)
References
[1] J-F Cai B Dong and Z Shen ldquoImage restoration A waveletframe based model for piecewise smooth functions andbeyondrdquoApplied andComputational Harmonic Analysis vol 41no 1 pp 94ndash138 2016
[2] R H Chan and J Ma ldquoA multiplicative iterative algorithm forbox-constrained penalized likelihood image restorationrdquo IEEETransactions on Image Processing vol 21 no 7 pp 3168ndash31812012
[3] C A Micchelli L Shen and Y Xu ldquoProximity algorithmsfor image models Denoisingrdquo Inverse Problems vol 27 no 4Article ID 045009 p 30 2011
[4] YWang J Yang W Yin and Y Zhang ldquoA new alternatingmin-imization algorithm for total variation image reconstructionrdquoSIAM Journal on Imaging Sciences vol 1 no 3 pp 248ndash2722008
[5] Y Dong M Hintermuller and M Neri ldquoAn efficient primal-dual method for l1-TV image restorationrdquo SIAM Journal onImaging Sciences vol 2 no 4 pp 1168ndash1189 2009
[6] J Lu K Qiao L Shen and Y Zou ldquoFixed-point algorithmsfor a TVL1 image restoration modelrdquo International Journal ofComputer Mathematics vol 95 no 9 pp 1829ndash1844 2018
[7] J Lu Z Ye and Y Zou ldquoHuber fractal image coding based ona fitting planerdquo IEEE Transactions on Image Processing vol 22no 1 pp 134ndash145 2013
[8] C A Micchelli L Shen Y Xu and X Zeng ldquoProximityalgorithms for the L1TV image denoising modelrdquo Advances inComputational Mathematics vol 38 no 2 pp 401ndash426 2013
[9] J Yang Y Zhang and W Yin ldquoAn efficient TVL1 algorithm fordeblurring multichannel images corrupted by impulsive noiserdquoSIAM Journal on Scientific Computing vol 31 no 4 pp 2842ndash2865 2009
[10] L Ma L Moisan J Yu and T Zeng ldquoA dictionary learningapproach for poisson image deblurringrdquo IEEE Transactions onMedical Imaging vol 32 no 7 pp 1277ndash1289 2013
[11] Y Xiao and T Zeng ldquoPoisson noise removal via learned dictio-naryrdquo in Proceedings of the 17th IEEE International Conferenceon Image Processing (ICIP) pp 1177ndash1180 2010
[12] H Zhang Y Dong and Q Fan ldquoWavelet frame based Poissonnoise removal and image deblurringrdquo Signal Processing vol 137pp 363ndash372 2017
[13] Y Dong and T Zeng ldquoA convex variational model for restoringblurred images with multiplicative noiserdquo SIAM Journal onImaging Sciences vol 6 no 3 pp 1598ndash1625 2013
[14] Y Huang M K Ng and YWen ldquoA new total variation methodfor multiplicative noise removalrdquo SIAM Journal on ImagingSciences vol 2 no 1 pp 20ndash40 2009
[15] Y-M Huang L Moisan M K Ng and T Zeng ldquoMultiplicativenoise removal via a learned dictionaryrdquo IEEE Transactions onImage Processing vol 21 no 11 pp 4534ndash4543 2012
[16] Z Jin and X Yang ldquoA variational model to remove the multi-plicative noise in ultrasound imagesrdquo Journal of MathematicalImaging and Vision vol 39 no 1 pp 62ndash74 2011
[17] M Kang S Yun and H Woo ldquoTwo-level convex relaxedvariational model for multiplicative denoisingrdquo SIAM Journalon Imaging Sciences vol 6 no 2 pp 875ndash903 2013
[18] J Lu L Shen C Xu and Y Xu ldquoMultiplicative noise removalin imaging An exp-model and its fixed-point proximity algo-rithmrdquo Applied and Computational Harmonic Analysis vol 41no 2 pp 518ndash539 2016
[19] J Lu L Shen C Xu and Y Xu ldquoMultiplicative noise removalwith a sparsity-aware optimization modelrdquo Inverse Problemsand Imaging vol 11 no 6 pp 949ndash974 2017
[20] J Lu Z Yang L Shen Z Lu H Yang and C Xu ldquoA frameletalgorithm for de-blurring images corrupted by multiplicativenoiserdquoApplied Mathematical Modelling vol 62 pp 51ndash61 2018
[21] J Lu Y Chen Y Zou and L Shen ldquoA new total variationmodelfor restoring blurred and speckle noisy imagesrdquo InternationalJournal of Wavelets Multiresolution and Information Processingvol 15 no 2 19 pages 2017
Mathematical Problems in Engineering 13
[22] P Perona and J Malik ldquoScale-space and edge detection usinganisotropic diffusionrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 12 no 7 pp 629ndash639 1990
[23] G Gerig O Kubler R Kikinis and F A Jolesz ldquoNonlinearanisotropic filtering ofMRI datardquo IEEE Transactions onMedicalImaging vol 11 no 2 pp 221ndash232 1992
[24] S Prima S P Morrissey and C Barillot ldquoNon-local meansvariants for denoising of diffusion-weighted and diffusiontensor MRIrdquo in Proceedings of the International Conference onMedical Image Computing and Computer-Assisted Interventionpp 344ndash351 Springer Berlin Germany 2007
[25] J V Manjon J Carbonell-Caballero J J Lull G Garcıa-MartiL Martı-Bonmati and M Robles ldquoMRI denoising using Non-LocalMeansrdquoMedical Image Analysis vol 12 no 4 pp 514ndash5232008
[26] N Wiest-Daessle S Prima P Coupe S P Morrissey and CBarillot ldquoRician noise removal by non-local means filteringfor low signal-to-noise ratio MRI Applications to DT-MRIrdquoMedical Image Computing and Computer-Assisted Interventionvol 5242 no 2 pp 171ndash179 2008
[27] R D Nowak ldquoWavelet-based Rician noise removal for mag-netic resonance imagingrdquo IEEE Transactions on Image Process-ing vol 8 no 10 pp 1408ndash1419 1999
[28] A Foi ldquoNoise estimation and removal in MR imaging Thevariance-stabilization approachrdquo in Proceedings of the 8th IEEEInternational Symposium on Biomedical Imaging From Nano toMacro (ISBI) pp 1809ndash1814 IEEE Chicago IL USA 2011
[29] J C Wood and K M Johnson ldquoWavelet packet denoising ofmagnetic resonance images Importance of Rician noise at lowSNRrdquo Magnetic Resonance in Medicine vol 41 no 3 pp 631ndash635 1999
[30] F Bowman Introduction to Bessel functions Dover PublicationsMineola NY USA 2012
[31] P Getreuer M Tong and L A Vese ldquoA variational model forthe restoration of MR images corrupted by blur and Riciannoiserdquo in Proceedings of the 7th international conference onAdvances in visual computing vol Part I of Lecture Notes inComputer Science pp 686ndash698 Springer Las Vegas NV USA2011
[32] L Chen and T Zeng ldquoA convex variational model for restoringblurred imageswith large Rician noiserdquo Journal ofMathematicalImaging and Vision vol 53 no 1 pp 92ndash111 2015
[33] M Aharon M Elad and A M Bruckstein ldquoK-SVD Analgorithm for designing overcomplete dictionaries for sparserepresentationrdquo IEEE Transactions on Signal Processing vol 54no 11 pp 4311ndash4322 2006
[34] M Elad and M Aharon ldquoImage denoising via sparse andredundant representations over learned dictionariesrdquo IEEETransactions on Image Processing vol 15 no 12 pp 3736ndash37452006
[35] G Aubert and J Aujol ldquoA variational approach to removingmultiplicative noiserdquo SIAM Journal on Applied Mathematicsvol 68 no 4 pp 925ndash946 2008
[36] Q Liu S Wang K Yang J Luo Y Zhu and D Liang ldquoHighlyundersampled magnetic resonance image reconstruction usingtwo-level Bregman method with dictionary updatingrdquo IEEETransactions on Medical Imaging vol 32 no 7 pp 1290ndash13012013
[37] Q Liu S Wang L Ying X Peng Y Zhu and D LiangldquoAdaptive dictionary learning in sparse gradient domain forimage recoveryrdquo IEEE Transactions on Image Processing vol 22no 12 pp 4652ndash4663 2013
[38] H Lu J Wei Q Liu Y Wang and X Deng ldquoA dictionarylearning method with total generalized variation for mri recon-structionrdquo International Journal of Biomedical Imaging vol2016 Article ID 7512471 13 pages 2016
[39] A Chambolle and T Pock ldquoA first-order primal-dual algorithmfor convex problems with applications to imagingrdquo Journal ofMathematical Imaging and Vision vol 40 no 1 pp 120ndash1452011
[40] E Esser X Zhang and T F Chan ldquoA general frameworkfor a class of first order primal-dual algorithms for convexoptimization in imaging sciencerdquo SIAM Journal on ImagingSciences vol 3 no 4 pp 1015ndash1046 2010
[41] N Komodakis and J-C Pesquet ldquoPlaying with duality Anoverview of recent primaldual approaches for solving large-scale optimization problemsrdquo IEEE Signal Processing Magazinevol 32 no 6 pp 31ndash54 2015
[42] A Chambolle ldquoAn algorithm for total variation minimizationand applicationsrdquo Journal of Mathematical Imaging and Visionvol 20 no 1-2 pp 89ndash97 2004
[43] L Ambrosio N Fusco and D Pallara Functions of BoundedVariation and Free Discontinuity Problem Oxford UniversityPress London 2000
[44] Z Wang A C Bovik H R Sheikh and E P SimoncellildquoImage quality assessment From error visibility to structuralsimilarityrdquo IEEE Transactions on Image Processing vol 13 no4 pp 600ndash612 2004
[45] J Gibson and A Bovik Handbook of Image and Video Process-ing Academic Press 2000
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
dB higher than CZ model at 120590 = 10 The characteristicsof SSIM values are almost consistent with those of PSNRvalues However the SSIM values are composed of luminancecomparison contrast comparison and structure comparisonwhich makes differences at some points
The images are corrupted respectively by Rician distri-bution with 120590 = 10 120590 = 15 and 120590 = 20 and the restoredimages of the above five algorithms for the images ldquoLenardquoldquoBarbarardquo ldquoHouserdquo and ldquoMonarchrdquo are shown inFigures 2ndash5respectively We clearly find that much noise remains in the
Figure 6 Results of ldquoBrainrdquo with 120590 = 10 by different methods (a) is the original ldquoBrainrdquo image and (g) is the degraded image with 120590 = 10(b)-(f) are the denoised images ofMAPmethod (120582 = 17) GTVmethod (120582 = 15) CZmethod (120582 = 0065) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
results of MAP GTV and CZmethod For example in Row 4of Figure 3 we can see from the background that ldquoBarbarardquoimage restored by our method is more clearer than thoserestored by other methods and textures of Barbararsquos trousersand scarf are kept better in the restored image by ourmethodThe restored images by our method also preserve moresignificant details thanMAPGTV CZ andNLMmethods onthe hat tidbits of the ldquoLenardquo image What is more the flowersof the lower left of ldquoMonarchrdquo also indicate that our method
is superior to other methods because our method is a patch-based method which is a different framework with TV-basedmethods (ie MAP method and CZ method) The imagesobtained from our method provide smoother regions andbetter shape preservation (eg the backgrounds in ldquoLenardquoand ldquoBarbarardquo)
Figures 6ndash11 are results of experiments on ldquoBrainrdquo andldquoMouserdquo images In particular for the ldquoBrainrdquo and ldquoMouserdquoimages we not only present the recovered images but also
10 Mathematical Problems in Engineering
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 7 Results of Brain with 120590 = 15 by different methods (a) is the original ldquoBrainrdquo image and (g) is the degraded image with 120590 = 15(b)-(f) are the denoised images of MAPmethod (120582 = 25) GTVmethod (120582 = 22) CZmethod (120582 = 005) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 8 Results of Brain with 120590 = 20 by different methods (a) is the original ldquoBrainrdquo image and (g) is the degraded image with 120590 = 20(b)-(f) are the denoised images ofMAPmethod (120582 = 35) GTVmethod (120582 = 22) CZmethod (120582 = 0045) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
use residuals to make a comparison with the original imagesThe first row is original image and recovered images ofMAP method GTV method CZ method NLM methodand our method The second row is degraded image withvarious 120590 and residuals of those methods respectively InFigures 6ndash8 the recovered images of of MAP method GTVmethod CZ method and NLM method are still unclear sowe can conclude that the recovered images by ourmethod arebest through our visions and residuals As for the ldquoMouserdquoimage the recovered results are very similar in this casewe use residuals to see which result is better In general theblurrier residual is the better according recovered image isWe can find that the outline of residual by our method isblurriest
In conclusion the results of our numerical experimentsdemonstrate that our proposed method performs better thanthe MAP method GTV method CZ method and NLMmethod
4 Conclusion
In this paper we proposed a new effective model via alearned dictionary for denoising images with Rician noiseMore specifically based on the dictionary training we adda dictionary penalty term to the original nonconvex MAPmodel to establish a new denoising model and develop atwo-step algorithm to solve our model Also we carry out
Mathematical Problems in Engineering 11
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 9 Results of ldquoMouserdquo with 120590 = 10 by different methods (a) is the original ldquoMouserdquo image and (g) is the degraded image with 120590 = 10(b)-(f) are the denoised images of MAPmethod (120582 = 17) GTVmethod (120582 = 17) CZmethod (120582 = 006) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 10 Results of ldquoMouserdquo with 120590 = 15 by different methods (a) is the original ldquoMouserdquo image and (g) is the degraded image with 120590 = 15(b)-(f) are the denoised images of MAPmethod (120582 = 24) GTV method (120582 = 24) CZ method (120582 = 005) NLM method and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
experiments on various images to demonstrate the effective-ness of our modelThe numerical experiments show that ourproposedmodel is promising in denoising imageswithRiciannoise
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interest
Acknowledgments
The authors would like to thank one of the authors of[32] for providing the source code of CZ method Thiswork was supported in part by the National Natural ScienceFoundation of China under Grants 11871348 61872429 and61373087 by the Natural Science Foundation of GuangdongChina under Grants 2015A030313550 and 2015A030313557by the HD Video RampD Platform for Intelligent Analy-sis and Processing in Guangdong Engineering TechnologyResearch Centre of Colleges and Universities (no GCZX-A1409) by Natural Science Foundation of Shenzhen underGrant JCYJ20170818091621856 and by the Guangdong Key
12 Mathematical Problems in Engineering
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 11 Results of ldquoMouserdquo with 120590 = 20 by different methods (a) is the original ldquoMouserdquo image and (g) is the degraded image with 120590 = 20(b)-(f) are the denoised images of MAPmethod (120582 = 31) GTV method (120582 = 32) CZ method (120582 = 0035) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
Laboratory of Intelligent Information Processing ShenzhenUniversity China (518060)
References
[1] J-F Cai B Dong and Z Shen ldquoImage restoration A waveletframe based model for piecewise smooth functions andbeyondrdquoApplied andComputational Harmonic Analysis vol 41no 1 pp 94ndash138 2016
[2] R H Chan and J Ma ldquoA multiplicative iterative algorithm forbox-constrained penalized likelihood image restorationrdquo IEEETransactions on Image Processing vol 21 no 7 pp 3168ndash31812012
[3] C A Micchelli L Shen and Y Xu ldquoProximity algorithmsfor image models Denoisingrdquo Inverse Problems vol 27 no 4Article ID 045009 p 30 2011
[4] YWang J Yang W Yin and Y Zhang ldquoA new alternatingmin-imization algorithm for total variation image reconstructionrdquoSIAM Journal on Imaging Sciences vol 1 no 3 pp 248ndash2722008
[5] Y Dong M Hintermuller and M Neri ldquoAn efficient primal-dual method for l1-TV image restorationrdquo SIAM Journal onImaging Sciences vol 2 no 4 pp 1168ndash1189 2009
[6] J Lu K Qiao L Shen and Y Zou ldquoFixed-point algorithmsfor a TVL1 image restoration modelrdquo International Journal ofComputer Mathematics vol 95 no 9 pp 1829ndash1844 2018
[7] J Lu Z Ye and Y Zou ldquoHuber fractal image coding based ona fitting planerdquo IEEE Transactions on Image Processing vol 22no 1 pp 134ndash145 2013
[8] C A Micchelli L Shen Y Xu and X Zeng ldquoProximityalgorithms for the L1TV image denoising modelrdquo Advances inComputational Mathematics vol 38 no 2 pp 401ndash426 2013
[9] J Yang Y Zhang and W Yin ldquoAn efficient TVL1 algorithm fordeblurring multichannel images corrupted by impulsive noiserdquoSIAM Journal on Scientific Computing vol 31 no 4 pp 2842ndash2865 2009
[10] L Ma L Moisan J Yu and T Zeng ldquoA dictionary learningapproach for poisson image deblurringrdquo IEEE Transactions onMedical Imaging vol 32 no 7 pp 1277ndash1289 2013
[11] Y Xiao and T Zeng ldquoPoisson noise removal via learned dictio-naryrdquo in Proceedings of the 17th IEEE International Conferenceon Image Processing (ICIP) pp 1177ndash1180 2010
[12] H Zhang Y Dong and Q Fan ldquoWavelet frame based Poissonnoise removal and image deblurringrdquo Signal Processing vol 137pp 363ndash372 2017
[13] Y Dong and T Zeng ldquoA convex variational model for restoringblurred images with multiplicative noiserdquo SIAM Journal onImaging Sciences vol 6 no 3 pp 1598ndash1625 2013
[14] Y Huang M K Ng and YWen ldquoA new total variation methodfor multiplicative noise removalrdquo SIAM Journal on ImagingSciences vol 2 no 1 pp 20ndash40 2009
[15] Y-M Huang L Moisan M K Ng and T Zeng ldquoMultiplicativenoise removal via a learned dictionaryrdquo IEEE Transactions onImage Processing vol 21 no 11 pp 4534ndash4543 2012
[16] Z Jin and X Yang ldquoA variational model to remove the multi-plicative noise in ultrasound imagesrdquo Journal of MathematicalImaging and Vision vol 39 no 1 pp 62ndash74 2011
[17] M Kang S Yun and H Woo ldquoTwo-level convex relaxedvariational model for multiplicative denoisingrdquo SIAM Journalon Imaging Sciences vol 6 no 2 pp 875ndash903 2013
[18] J Lu L Shen C Xu and Y Xu ldquoMultiplicative noise removalin imaging An exp-model and its fixed-point proximity algo-rithmrdquo Applied and Computational Harmonic Analysis vol 41no 2 pp 518ndash539 2016
[19] J Lu L Shen C Xu and Y Xu ldquoMultiplicative noise removalwith a sparsity-aware optimization modelrdquo Inverse Problemsand Imaging vol 11 no 6 pp 949ndash974 2017
[20] J Lu Z Yang L Shen Z Lu H Yang and C Xu ldquoA frameletalgorithm for de-blurring images corrupted by multiplicativenoiserdquoApplied Mathematical Modelling vol 62 pp 51ndash61 2018
[21] J Lu Y Chen Y Zou and L Shen ldquoA new total variationmodelfor restoring blurred and speckle noisy imagesrdquo InternationalJournal of Wavelets Multiresolution and Information Processingvol 15 no 2 19 pages 2017
Mathematical Problems in Engineering 13
[22] P Perona and J Malik ldquoScale-space and edge detection usinganisotropic diffusionrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 12 no 7 pp 629ndash639 1990
[23] G Gerig O Kubler R Kikinis and F A Jolesz ldquoNonlinearanisotropic filtering ofMRI datardquo IEEE Transactions onMedicalImaging vol 11 no 2 pp 221ndash232 1992
[24] S Prima S P Morrissey and C Barillot ldquoNon-local meansvariants for denoising of diffusion-weighted and diffusiontensor MRIrdquo in Proceedings of the International Conference onMedical Image Computing and Computer-Assisted Interventionpp 344ndash351 Springer Berlin Germany 2007
[25] J V Manjon J Carbonell-Caballero J J Lull G Garcıa-MartiL Martı-Bonmati and M Robles ldquoMRI denoising using Non-LocalMeansrdquoMedical Image Analysis vol 12 no 4 pp 514ndash5232008
[26] N Wiest-Daessle S Prima P Coupe S P Morrissey and CBarillot ldquoRician noise removal by non-local means filteringfor low signal-to-noise ratio MRI Applications to DT-MRIrdquoMedical Image Computing and Computer-Assisted Interventionvol 5242 no 2 pp 171ndash179 2008
[27] R D Nowak ldquoWavelet-based Rician noise removal for mag-netic resonance imagingrdquo IEEE Transactions on Image Process-ing vol 8 no 10 pp 1408ndash1419 1999
[28] A Foi ldquoNoise estimation and removal in MR imaging Thevariance-stabilization approachrdquo in Proceedings of the 8th IEEEInternational Symposium on Biomedical Imaging From Nano toMacro (ISBI) pp 1809ndash1814 IEEE Chicago IL USA 2011
[29] J C Wood and K M Johnson ldquoWavelet packet denoising ofmagnetic resonance images Importance of Rician noise at lowSNRrdquo Magnetic Resonance in Medicine vol 41 no 3 pp 631ndash635 1999
[30] F Bowman Introduction to Bessel functions Dover PublicationsMineola NY USA 2012
[31] P Getreuer M Tong and L A Vese ldquoA variational model forthe restoration of MR images corrupted by blur and Riciannoiserdquo in Proceedings of the 7th international conference onAdvances in visual computing vol Part I of Lecture Notes inComputer Science pp 686ndash698 Springer Las Vegas NV USA2011
[32] L Chen and T Zeng ldquoA convex variational model for restoringblurred imageswith large Rician noiserdquo Journal ofMathematicalImaging and Vision vol 53 no 1 pp 92ndash111 2015
[33] M Aharon M Elad and A M Bruckstein ldquoK-SVD Analgorithm for designing overcomplete dictionaries for sparserepresentationrdquo IEEE Transactions on Signal Processing vol 54no 11 pp 4311ndash4322 2006
[34] M Elad and M Aharon ldquoImage denoising via sparse andredundant representations over learned dictionariesrdquo IEEETransactions on Image Processing vol 15 no 12 pp 3736ndash37452006
[35] G Aubert and J Aujol ldquoA variational approach to removingmultiplicative noiserdquo SIAM Journal on Applied Mathematicsvol 68 no 4 pp 925ndash946 2008
[36] Q Liu S Wang K Yang J Luo Y Zhu and D Liang ldquoHighlyundersampled magnetic resonance image reconstruction usingtwo-level Bregman method with dictionary updatingrdquo IEEETransactions on Medical Imaging vol 32 no 7 pp 1290ndash13012013
[37] Q Liu S Wang L Ying X Peng Y Zhu and D LiangldquoAdaptive dictionary learning in sparse gradient domain forimage recoveryrdquo IEEE Transactions on Image Processing vol 22no 12 pp 4652ndash4663 2013
[38] H Lu J Wei Q Liu Y Wang and X Deng ldquoA dictionarylearning method with total generalized variation for mri recon-structionrdquo International Journal of Biomedical Imaging vol2016 Article ID 7512471 13 pages 2016
[39] A Chambolle and T Pock ldquoA first-order primal-dual algorithmfor convex problems with applications to imagingrdquo Journal ofMathematical Imaging and Vision vol 40 no 1 pp 120ndash1452011
[40] E Esser X Zhang and T F Chan ldquoA general frameworkfor a class of first order primal-dual algorithms for convexoptimization in imaging sciencerdquo SIAM Journal on ImagingSciences vol 3 no 4 pp 1015ndash1046 2010
[41] N Komodakis and J-C Pesquet ldquoPlaying with duality Anoverview of recent primaldual approaches for solving large-scale optimization problemsrdquo IEEE Signal Processing Magazinevol 32 no 6 pp 31ndash54 2015
[42] A Chambolle ldquoAn algorithm for total variation minimizationand applicationsrdquo Journal of Mathematical Imaging and Visionvol 20 no 1-2 pp 89ndash97 2004
[43] L Ambrosio N Fusco and D Pallara Functions of BoundedVariation and Free Discontinuity Problem Oxford UniversityPress London 2000
[44] Z Wang A C Bovik H R Sheikh and E P SimoncellildquoImage quality assessment From error visibility to structuralsimilarityrdquo IEEE Transactions on Image Processing vol 13 no4 pp 600ndash612 2004
[45] J Gibson and A Bovik Handbook of Image and Video Process-ing Academic Press 2000
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Figure 6 Results of ldquoBrainrdquo with 120590 = 10 by different methods (a) is the original ldquoBrainrdquo image and (g) is the degraded image with 120590 = 10(b)-(f) are the denoised images ofMAPmethod (120582 = 17) GTVmethod (120582 = 15) CZmethod (120582 = 0065) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
results of MAP GTV and CZmethod For example in Row 4of Figure 3 we can see from the background that ldquoBarbarardquoimage restored by our method is more clearer than thoserestored by other methods and textures of Barbararsquos trousersand scarf are kept better in the restored image by ourmethodThe restored images by our method also preserve moresignificant details thanMAPGTV CZ andNLMmethods onthe hat tidbits of the ldquoLenardquo image What is more the flowersof the lower left of ldquoMonarchrdquo also indicate that our method
is superior to other methods because our method is a patch-based method which is a different framework with TV-basedmethods (ie MAP method and CZ method) The imagesobtained from our method provide smoother regions andbetter shape preservation (eg the backgrounds in ldquoLenardquoand ldquoBarbarardquo)
Figures 6ndash11 are results of experiments on ldquoBrainrdquo andldquoMouserdquo images In particular for the ldquoBrainrdquo and ldquoMouserdquoimages we not only present the recovered images but also
10 Mathematical Problems in Engineering
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 7 Results of Brain with 120590 = 15 by different methods (a) is the original ldquoBrainrdquo image and (g) is the degraded image with 120590 = 15(b)-(f) are the denoised images of MAPmethod (120582 = 25) GTVmethod (120582 = 22) CZmethod (120582 = 005) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 8 Results of Brain with 120590 = 20 by different methods (a) is the original ldquoBrainrdquo image and (g) is the degraded image with 120590 = 20(b)-(f) are the denoised images ofMAPmethod (120582 = 35) GTVmethod (120582 = 22) CZmethod (120582 = 0045) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
use residuals to make a comparison with the original imagesThe first row is original image and recovered images ofMAP method GTV method CZ method NLM methodand our method The second row is degraded image withvarious 120590 and residuals of those methods respectively InFigures 6ndash8 the recovered images of of MAP method GTVmethod CZ method and NLM method are still unclear sowe can conclude that the recovered images by ourmethod arebest through our visions and residuals As for the ldquoMouserdquoimage the recovered results are very similar in this casewe use residuals to see which result is better In general theblurrier residual is the better according recovered image isWe can find that the outline of residual by our method isblurriest
In conclusion the results of our numerical experimentsdemonstrate that our proposed method performs better thanthe MAP method GTV method CZ method and NLMmethod
4 Conclusion
In this paper we proposed a new effective model via alearned dictionary for denoising images with Rician noiseMore specifically based on the dictionary training we adda dictionary penalty term to the original nonconvex MAPmodel to establish a new denoising model and develop atwo-step algorithm to solve our model Also we carry out
Mathematical Problems in Engineering 11
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 9 Results of ldquoMouserdquo with 120590 = 10 by different methods (a) is the original ldquoMouserdquo image and (g) is the degraded image with 120590 = 10(b)-(f) are the denoised images of MAPmethod (120582 = 17) GTVmethod (120582 = 17) CZmethod (120582 = 006) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 10 Results of ldquoMouserdquo with 120590 = 15 by different methods (a) is the original ldquoMouserdquo image and (g) is the degraded image with 120590 = 15(b)-(f) are the denoised images of MAPmethod (120582 = 24) GTV method (120582 = 24) CZ method (120582 = 005) NLM method and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
experiments on various images to demonstrate the effective-ness of our modelThe numerical experiments show that ourproposedmodel is promising in denoising imageswithRiciannoise
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interest
Acknowledgments
The authors would like to thank one of the authors of[32] for providing the source code of CZ method Thiswork was supported in part by the National Natural ScienceFoundation of China under Grants 11871348 61872429 and61373087 by the Natural Science Foundation of GuangdongChina under Grants 2015A030313550 and 2015A030313557by the HD Video RampD Platform for Intelligent Analy-sis and Processing in Guangdong Engineering TechnologyResearch Centre of Colleges and Universities (no GCZX-A1409) by Natural Science Foundation of Shenzhen underGrant JCYJ20170818091621856 and by the Guangdong Key
12 Mathematical Problems in Engineering
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 11 Results of ldquoMouserdquo with 120590 = 20 by different methods (a) is the original ldquoMouserdquo image and (g) is the degraded image with 120590 = 20(b)-(f) are the denoised images of MAPmethod (120582 = 31) GTV method (120582 = 32) CZ method (120582 = 0035) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
Laboratory of Intelligent Information Processing ShenzhenUniversity China (518060)
References
[1] J-F Cai B Dong and Z Shen ldquoImage restoration A waveletframe based model for piecewise smooth functions andbeyondrdquoApplied andComputational Harmonic Analysis vol 41no 1 pp 94ndash138 2016
[2] R H Chan and J Ma ldquoA multiplicative iterative algorithm forbox-constrained penalized likelihood image restorationrdquo IEEETransactions on Image Processing vol 21 no 7 pp 3168ndash31812012
[3] C A Micchelli L Shen and Y Xu ldquoProximity algorithmsfor image models Denoisingrdquo Inverse Problems vol 27 no 4Article ID 045009 p 30 2011
[4] YWang J Yang W Yin and Y Zhang ldquoA new alternatingmin-imization algorithm for total variation image reconstructionrdquoSIAM Journal on Imaging Sciences vol 1 no 3 pp 248ndash2722008
[5] Y Dong M Hintermuller and M Neri ldquoAn efficient primal-dual method for l1-TV image restorationrdquo SIAM Journal onImaging Sciences vol 2 no 4 pp 1168ndash1189 2009
[6] J Lu K Qiao L Shen and Y Zou ldquoFixed-point algorithmsfor a TVL1 image restoration modelrdquo International Journal ofComputer Mathematics vol 95 no 9 pp 1829ndash1844 2018
[7] J Lu Z Ye and Y Zou ldquoHuber fractal image coding based ona fitting planerdquo IEEE Transactions on Image Processing vol 22no 1 pp 134ndash145 2013
[8] C A Micchelli L Shen Y Xu and X Zeng ldquoProximityalgorithms for the L1TV image denoising modelrdquo Advances inComputational Mathematics vol 38 no 2 pp 401ndash426 2013
[9] J Yang Y Zhang and W Yin ldquoAn efficient TVL1 algorithm fordeblurring multichannel images corrupted by impulsive noiserdquoSIAM Journal on Scientific Computing vol 31 no 4 pp 2842ndash2865 2009
[10] L Ma L Moisan J Yu and T Zeng ldquoA dictionary learningapproach for poisson image deblurringrdquo IEEE Transactions onMedical Imaging vol 32 no 7 pp 1277ndash1289 2013
[11] Y Xiao and T Zeng ldquoPoisson noise removal via learned dictio-naryrdquo in Proceedings of the 17th IEEE International Conferenceon Image Processing (ICIP) pp 1177ndash1180 2010
[12] H Zhang Y Dong and Q Fan ldquoWavelet frame based Poissonnoise removal and image deblurringrdquo Signal Processing vol 137pp 363ndash372 2017
[13] Y Dong and T Zeng ldquoA convex variational model for restoringblurred images with multiplicative noiserdquo SIAM Journal onImaging Sciences vol 6 no 3 pp 1598ndash1625 2013
[14] Y Huang M K Ng and YWen ldquoA new total variation methodfor multiplicative noise removalrdquo SIAM Journal on ImagingSciences vol 2 no 1 pp 20ndash40 2009
[15] Y-M Huang L Moisan M K Ng and T Zeng ldquoMultiplicativenoise removal via a learned dictionaryrdquo IEEE Transactions onImage Processing vol 21 no 11 pp 4534ndash4543 2012
[16] Z Jin and X Yang ldquoA variational model to remove the multi-plicative noise in ultrasound imagesrdquo Journal of MathematicalImaging and Vision vol 39 no 1 pp 62ndash74 2011
[17] M Kang S Yun and H Woo ldquoTwo-level convex relaxedvariational model for multiplicative denoisingrdquo SIAM Journalon Imaging Sciences vol 6 no 2 pp 875ndash903 2013
[18] J Lu L Shen C Xu and Y Xu ldquoMultiplicative noise removalin imaging An exp-model and its fixed-point proximity algo-rithmrdquo Applied and Computational Harmonic Analysis vol 41no 2 pp 518ndash539 2016
[19] J Lu L Shen C Xu and Y Xu ldquoMultiplicative noise removalwith a sparsity-aware optimization modelrdquo Inverse Problemsand Imaging vol 11 no 6 pp 949ndash974 2017
[20] J Lu Z Yang L Shen Z Lu H Yang and C Xu ldquoA frameletalgorithm for de-blurring images corrupted by multiplicativenoiserdquoApplied Mathematical Modelling vol 62 pp 51ndash61 2018
[21] J Lu Y Chen Y Zou and L Shen ldquoA new total variationmodelfor restoring blurred and speckle noisy imagesrdquo InternationalJournal of Wavelets Multiresolution and Information Processingvol 15 no 2 19 pages 2017
Mathematical Problems in Engineering 13
[22] P Perona and J Malik ldquoScale-space and edge detection usinganisotropic diffusionrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 12 no 7 pp 629ndash639 1990
[23] G Gerig O Kubler R Kikinis and F A Jolesz ldquoNonlinearanisotropic filtering ofMRI datardquo IEEE Transactions onMedicalImaging vol 11 no 2 pp 221ndash232 1992
[24] S Prima S P Morrissey and C Barillot ldquoNon-local meansvariants for denoising of diffusion-weighted and diffusiontensor MRIrdquo in Proceedings of the International Conference onMedical Image Computing and Computer-Assisted Interventionpp 344ndash351 Springer Berlin Germany 2007
[25] J V Manjon J Carbonell-Caballero J J Lull G Garcıa-MartiL Martı-Bonmati and M Robles ldquoMRI denoising using Non-LocalMeansrdquoMedical Image Analysis vol 12 no 4 pp 514ndash5232008
[26] N Wiest-Daessle S Prima P Coupe S P Morrissey and CBarillot ldquoRician noise removal by non-local means filteringfor low signal-to-noise ratio MRI Applications to DT-MRIrdquoMedical Image Computing and Computer-Assisted Interventionvol 5242 no 2 pp 171ndash179 2008
[27] R D Nowak ldquoWavelet-based Rician noise removal for mag-netic resonance imagingrdquo IEEE Transactions on Image Process-ing vol 8 no 10 pp 1408ndash1419 1999
[28] A Foi ldquoNoise estimation and removal in MR imaging Thevariance-stabilization approachrdquo in Proceedings of the 8th IEEEInternational Symposium on Biomedical Imaging From Nano toMacro (ISBI) pp 1809ndash1814 IEEE Chicago IL USA 2011
[29] J C Wood and K M Johnson ldquoWavelet packet denoising ofmagnetic resonance images Importance of Rician noise at lowSNRrdquo Magnetic Resonance in Medicine vol 41 no 3 pp 631ndash635 1999
[30] F Bowman Introduction to Bessel functions Dover PublicationsMineola NY USA 2012
[31] P Getreuer M Tong and L A Vese ldquoA variational model forthe restoration of MR images corrupted by blur and Riciannoiserdquo in Proceedings of the 7th international conference onAdvances in visual computing vol Part I of Lecture Notes inComputer Science pp 686ndash698 Springer Las Vegas NV USA2011
[32] L Chen and T Zeng ldquoA convex variational model for restoringblurred imageswith large Rician noiserdquo Journal ofMathematicalImaging and Vision vol 53 no 1 pp 92ndash111 2015
[33] M Aharon M Elad and A M Bruckstein ldquoK-SVD Analgorithm for designing overcomplete dictionaries for sparserepresentationrdquo IEEE Transactions on Signal Processing vol 54no 11 pp 4311ndash4322 2006
[34] M Elad and M Aharon ldquoImage denoising via sparse andredundant representations over learned dictionariesrdquo IEEETransactions on Image Processing vol 15 no 12 pp 3736ndash37452006
[35] G Aubert and J Aujol ldquoA variational approach to removingmultiplicative noiserdquo SIAM Journal on Applied Mathematicsvol 68 no 4 pp 925ndash946 2008
[36] Q Liu S Wang K Yang J Luo Y Zhu and D Liang ldquoHighlyundersampled magnetic resonance image reconstruction usingtwo-level Bregman method with dictionary updatingrdquo IEEETransactions on Medical Imaging vol 32 no 7 pp 1290ndash13012013
[37] Q Liu S Wang L Ying X Peng Y Zhu and D LiangldquoAdaptive dictionary learning in sparse gradient domain forimage recoveryrdquo IEEE Transactions on Image Processing vol 22no 12 pp 4652ndash4663 2013
[38] H Lu J Wei Q Liu Y Wang and X Deng ldquoA dictionarylearning method with total generalized variation for mri recon-structionrdquo International Journal of Biomedical Imaging vol2016 Article ID 7512471 13 pages 2016
[39] A Chambolle and T Pock ldquoA first-order primal-dual algorithmfor convex problems with applications to imagingrdquo Journal ofMathematical Imaging and Vision vol 40 no 1 pp 120ndash1452011
[40] E Esser X Zhang and T F Chan ldquoA general frameworkfor a class of first order primal-dual algorithms for convexoptimization in imaging sciencerdquo SIAM Journal on ImagingSciences vol 3 no 4 pp 1015ndash1046 2010
[41] N Komodakis and J-C Pesquet ldquoPlaying with duality Anoverview of recent primaldual approaches for solving large-scale optimization problemsrdquo IEEE Signal Processing Magazinevol 32 no 6 pp 31ndash54 2015
[42] A Chambolle ldquoAn algorithm for total variation minimizationand applicationsrdquo Journal of Mathematical Imaging and Visionvol 20 no 1-2 pp 89ndash97 2004
[43] L Ambrosio N Fusco and D Pallara Functions of BoundedVariation and Free Discontinuity Problem Oxford UniversityPress London 2000
[44] Z Wang A C Bovik H R Sheikh and E P SimoncellildquoImage quality assessment From error visibility to structuralsimilarityrdquo IEEE Transactions on Image Processing vol 13 no4 pp 600ndash612 2004
[45] J Gibson and A Bovik Handbook of Image and Video Process-ing Academic Press 2000
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Figure 6 Results of ldquoBrainrdquo with 120590 = 10 by different methods (a) is the original ldquoBrainrdquo image and (g) is the degraded image with 120590 = 10(b)-(f) are the denoised images ofMAPmethod (120582 = 17) GTVmethod (120582 = 15) CZmethod (120582 = 0065) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
results of MAP GTV and CZmethod For example in Row 4of Figure 3 we can see from the background that ldquoBarbarardquoimage restored by our method is more clearer than thoserestored by other methods and textures of Barbararsquos trousersand scarf are kept better in the restored image by ourmethodThe restored images by our method also preserve moresignificant details thanMAPGTV CZ andNLMmethods onthe hat tidbits of the ldquoLenardquo image What is more the flowersof the lower left of ldquoMonarchrdquo also indicate that our method
is superior to other methods because our method is a patch-based method which is a different framework with TV-basedmethods (ie MAP method and CZ method) The imagesobtained from our method provide smoother regions andbetter shape preservation (eg the backgrounds in ldquoLenardquoand ldquoBarbarardquo)
Figures 6ndash11 are results of experiments on ldquoBrainrdquo andldquoMouserdquo images In particular for the ldquoBrainrdquo and ldquoMouserdquoimages we not only present the recovered images but also
10 Mathematical Problems in Engineering
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 7 Results of Brain with 120590 = 15 by different methods (a) is the original ldquoBrainrdquo image and (g) is the degraded image with 120590 = 15(b)-(f) are the denoised images of MAPmethod (120582 = 25) GTVmethod (120582 = 22) CZmethod (120582 = 005) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 8 Results of Brain with 120590 = 20 by different methods (a) is the original ldquoBrainrdquo image and (g) is the degraded image with 120590 = 20(b)-(f) are the denoised images ofMAPmethod (120582 = 35) GTVmethod (120582 = 22) CZmethod (120582 = 0045) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
use residuals to make a comparison with the original imagesThe first row is original image and recovered images ofMAP method GTV method CZ method NLM methodand our method The second row is degraded image withvarious 120590 and residuals of those methods respectively InFigures 6ndash8 the recovered images of of MAP method GTVmethod CZ method and NLM method are still unclear sowe can conclude that the recovered images by ourmethod arebest through our visions and residuals As for the ldquoMouserdquoimage the recovered results are very similar in this casewe use residuals to see which result is better In general theblurrier residual is the better according recovered image isWe can find that the outline of residual by our method isblurriest
In conclusion the results of our numerical experimentsdemonstrate that our proposed method performs better thanthe MAP method GTV method CZ method and NLMmethod
4 Conclusion
In this paper we proposed a new effective model via alearned dictionary for denoising images with Rician noiseMore specifically based on the dictionary training we adda dictionary penalty term to the original nonconvex MAPmodel to establish a new denoising model and develop atwo-step algorithm to solve our model Also we carry out
Mathematical Problems in Engineering 11
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 9 Results of ldquoMouserdquo with 120590 = 10 by different methods (a) is the original ldquoMouserdquo image and (g) is the degraded image with 120590 = 10(b)-(f) are the denoised images of MAPmethod (120582 = 17) GTVmethod (120582 = 17) CZmethod (120582 = 006) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 10 Results of ldquoMouserdquo with 120590 = 15 by different methods (a) is the original ldquoMouserdquo image and (g) is the degraded image with 120590 = 15(b)-(f) are the denoised images of MAPmethod (120582 = 24) GTV method (120582 = 24) CZ method (120582 = 005) NLM method and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
experiments on various images to demonstrate the effective-ness of our modelThe numerical experiments show that ourproposedmodel is promising in denoising imageswithRiciannoise
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interest
Acknowledgments
The authors would like to thank one of the authors of[32] for providing the source code of CZ method Thiswork was supported in part by the National Natural ScienceFoundation of China under Grants 11871348 61872429 and61373087 by the Natural Science Foundation of GuangdongChina under Grants 2015A030313550 and 2015A030313557by the HD Video RampD Platform for Intelligent Analy-sis and Processing in Guangdong Engineering TechnologyResearch Centre of Colleges and Universities (no GCZX-A1409) by Natural Science Foundation of Shenzhen underGrant JCYJ20170818091621856 and by the Guangdong Key
12 Mathematical Problems in Engineering
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 11 Results of ldquoMouserdquo with 120590 = 20 by different methods (a) is the original ldquoMouserdquo image and (g) is the degraded image with 120590 = 20(b)-(f) are the denoised images of MAPmethod (120582 = 31) GTV method (120582 = 32) CZ method (120582 = 0035) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
Laboratory of Intelligent Information Processing ShenzhenUniversity China (518060)
References
[1] J-F Cai B Dong and Z Shen ldquoImage restoration A waveletframe based model for piecewise smooth functions andbeyondrdquoApplied andComputational Harmonic Analysis vol 41no 1 pp 94ndash138 2016
[2] R H Chan and J Ma ldquoA multiplicative iterative algorithm forbox-constrained penalized likelihood image restorationrdquo IEEETransactions on Image Processing vol 21 no 7 pp 3168ndash31812012
[3] C A Micchelli L Shen and Y Xu ldquoProximity algorithmsfor image models Denoisingrdquo Inverse Problems vol 27 no 4Article ID 045009 p 30 2011
[4] YWang J Yang W Yin and Y Zhang ldquoA new alternatingmin-imization algorithm for total variation image reconstructionrdquoSIAM Journal on Imaging Sciences vol 1 no 3 pp 248ndash2722008
[5] Y Dong M Hintermuller and M Neri ldquoAn efficient primal-dual method for l1-TV image restorationrdquo SIAM Journal onImaging Sciences vol 2 no 4 pp 1168ndash1189 2009
[6] J Lu K Qiao L Shen and Y Zou ldquoFixed-point algorithmsfor a TVL1 image restoration modelrdquo International Journal ofComputer Mathematics vol 95 no 9 pp 1829ndash1844 2018
[7] J Lu Z Ye and Y Zou ldquoHuber fractal image coding based ona fitting planerdquo IEEE Transactions on Image Processing vol 22no 1 pp 134ndash145 2013
[8] C A Micchelli L Shen Y Xu and X Zeng ldquoProximityalgorithms for the L1TV image denoising modelrdquo Advances inComputational Mathematics vol 38 no 2 pp 401ndash426 2013
[9] J Yang Y Zhang and W Yin ldquoAn efficient TVL1 algorithm fordeblurring multichannel images corrupted by impulsive noiserdquoSIAM Journal on Scientific Computing vol 31 no 4 pp 2842ndash2865 2009
[10] L Ma L Moisan J Yu and T Zeng ldquoA dictionary learningapproach for poisson image deblurringrdquo IEEE Transactions onMedical Imaging vol 32 no 7 pp 1277ndash1289 2013
[11] Y Xiao and T Zeng ldquoPoisson noise removal via learned dictio-naryrdquo in Proceedings of the 17th IEEE International Conferenceon Image Processing (ICIP) pp 1177ndash1180 2010
[12] H Zhang Y Dong and Q Fan ldquoWavelet frame based Poissonnoise removal and image deblurringrdquo Signal Processing vol 137pp 363ndash372 2017
[13] Y Dong and T Zeng ldquoA convex variational model for restoringblurred images with multiplicative noiserdquo SIAM Journal onImaging Sciences vol 6 no 3 pp 1598ndash1625 2013
[14] Y Huang M K Ng and YWen ldquoA new total variation methodfor multiplicative noise removalrdquo SIAM Journal on ImagingSciences vol 2 no 1 pp 20ndash40 2009
[15] Y-M Huang L Moisan M K Ng and T Zeng ldquoMultiplicativenoise removal via a learned dictionaryrdquo IEEE Transactions onImage Processing vol 21 no 11 pp 4534ndash4543 2012
[16] Z Jin and X Yang ldquoA variational model to remove the multi-plicative noise in ultrasound imagesrdquo Journal of MathematicalImaging and Vision vol 39 no 1 pp 62ndash74 2011
[17] M Kang S Yun and H Woo ldquoTwo-level convex relaxedvariational model for multiplicative denoisingrdquo SIAM Journalon Imaging Sciences vol 6 no 2 pp 875ndash903 2013
[18] J Lu L Shen C Xu and Y Xu ldquoMultiplicative noise removalin imaging An exp-model and its fixed-point proximity algo-rithmrdquo Applied and Computational Harmonic Analysis vol 41no 2 pp 518ndash539 2016
[19] J Lu L Shen C Xu and Y Xu ldquoMultiplicative noise removalwith a sparsity-aware optimization modelrdquo Inverse Problemsand Imaging vol 11 no 6 pp 949ndash974 2017
[20] J Lu Z Yang L Shen Z Lu H Yang and C Xu ldquoA frameletalgorithm for de-blurring images corrupted by multiplicativenoiserdquoApplied Mathematical Modelling vol 62 pp 51ndash61 2018
[21] J Lu Y Chen Y Zou and L Shen ldquoA new total variationmodelfor restoring blurred and speckle noisy imagesrdquo InternationalJournal of Wavelets Multiresolution and Information Processingvol 15 no 2 19 pages 2017
Mathematical Problems in Engineering 13
[22] P Perona and J Malik ldquoScale-space and edge detection usinganisotropic diffusionrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 12 no 7 pp 629ndash639 1990
[23] G Gerig O Kubler R Kikinis and F A Jolesz ldquoNonlinearanisotropic filtering ofMRI datardquo IEEE Transactions onMedicalImaging vol 11 no 2 pp 221ndash232 1992
[24] S Prima S P Morrissey and C Barillot ldquoNon-local meansvariants for denoising of diffusion-weighted and diffusiontensor MRIrdquo in Proceedings of the International Conference onMedical Image Computing and Computer-Assisted Interventionpp 344ndash351 Springer Berlin Germany 2007
[25] J V Manjon J Carbonell-Caballero J J Lull G Garcıa-MartiL Martı-Bonmati and M Robles ldquoMRI denoising using Non-LocalMeansrdquoMedical Image Analysis vol 12 no 4 pp 514ndash5232008
[26] N Wiest-Daessle S Prima P Coupe S P Morrissey and CBarillot ldquoRician noise removal by non-local means filteringfor low signal-to-noise ratio MRI Applications to DT-MRIrdquoMedical Image Computing and Computer-Assisted Interventionvol 5242 no 2 pp 171ndash179 2008
[27] R D Nowak ldquoWavelet-based Rician noise removal for mag-netic resonance imagingrdquo IEEE Transactions on Image Process-ing vol 8 no 10 pp 1408ndash1419 1999
[28] A Foi ldquoNoise estimation and removal in MR imaging Thevariance-stabilization approachrdquo in Proceedings of the 8th IEEEInternational Symposium on Biomedical Imaging From Nano toMacro (ISBI) pp 1809ndash1814 IEEE Chicago IL USA 2011
[29] J C Wood and K M Johnson ldquoWavelet packet denoising ofmagnetic resonance images Importance of Rician noise at lowSNRrdquo Magnetic Resonance in Medicine vol 41 no 3 pp 631ndash635 1999
[30] F Bowman Introduction to Bessel functions Dover PublicationsMineola NY USA 2012
[31] P Getreuer M Tong and L A Vese ldquoA variational model forthe restoration of MR images corrupted by blur and Riciannoiserdquo in Proceedings of the 7th international conference onAdvances in visual computing vol Part I of Lecture Notes inComputer Science pp 686ndash698 Springer Las Vegas NV USA2011
[32] L Chen and T Zeng ldquoA convex variational model for restoringblurred imageswith large Rician noiserdquo Journal ofMathematicalImaging and Vision vol 53 no 1 pp 92ndash111 2015
[33] M Aharon M Elad and A M Bruckstein ldquoK-SVD Analgorithm for designing overcomplete dictionaries for sparserepresentationrdquo IEEE Transactions on Signal Processing vol 54no 11 pp 4311ndash4322 2006
[34] M Elad and M Aharon ldquoImage denoising via sparse andredundant representations over learned dictionariesrdquo IEEETransactions on Image Processing vol 15 no 12 pp 3736ndash37452006
[35] G Aubert and J Aujol ldquoA variational approach to removingmultiplicative noiserdquo SIAM Journal on Applied Mathematicsvol 68 no 4 pp 925ndash946 2008
[36] Q Liu S Wang K Yang J Luo Y Zhu and D Liang ldquoHighlyundersampled magnetic resonance image reconstruction usingtwo-level Bregman method with dictionary updatingrdquo IEEETransactions on Medical Imaging vol 32 no 7 pp 1290ndash13012013
[37] Q Liu S Wang L Ying X Peng Y Zhu and D LiangldquoAdaptive dictionary learning in sparse gradient domain forimage recoveryrdquo IEEE Transactions on Image Processing vol 22no 12 pp 4652ndash4663 2013
[38] H Lu J Wei Q Liu Y Wang and X Deng ldquoA dictionarylearning method with total generalized variation for mri recon-structionrdquo International Journal of Biomedical Imaging vol2016 Article ID 7512471 13 pages 2016
[39] A Chambolle and T Pock ldquoA first-order primal-dual algorithmfor convex problems with applications to imagingrdquo Journal ofMathematical Imaging and Vision vol 40 no 1 pp 120ndash1452011
[40] E Esser X Zhang and T F Chan ldquoA general frameworkfor a class of first order primal-dual algorithms for convexoptimization in imaging sciencerdquo SIAM Journal on ImagingSciences vol 3 no 4 pp 1015ndash1046 2010
[41] N Komodakis and J-C Pesquet ldquoPlaying with duality Anoverview of recent primaldual approaches for solving large-scale optimization problemsrdquo IEEE Signal Processing Magazinevol 32 no 6 pp 31ndash54 2015
[42] A Chambolle ldquoAn algorithm for total variation minimizationand applicationsrdquo Journal of Mathematical Imaging and Visionvol 20 no 1-2 pp 89ndash97 2004
[43] L Ambrosio N Fusco and D Pallara Functions of BoundedVariation and Free Discontinuity Problem Oxford UniversityPress London 2000
[44] Z Wang A C Bovik H R Sheikh and E P SimoncellildquoImage quality assessment From error visibility to structuralsimilarityrdquo IEEE Transactions on Image Processing vol 13 no4 pp 600ndash612 2004
[45] J Gibson and A Bovik Handbook of Image and Video Process-ing Academic Press 2000
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
10 Mathematical Problems in Engineering
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 7 Results of Brain with 120590 = 15 by different methods (a) is the original ldquoBrainrdquo image and (g) is the degraded image with 120590 = 15(b)-(f) are the denoised images of MAPmethod (120582 = 25) GTVmethod (120582 = 22) CZmethod (120582 = 005) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 8 Results of Brain with 120590 = 20 by different methods (a) is the original ldquoBrainrdquo image and (g) is the degraded image with 120590 = 20(b)-(f) are the denoised images ofMAPmethod (120582 = 35) GTVmethod (120582 = 22) CZmethod (120582 = 0045) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
use residuals to make a comparison with the original imagesThe first row is original image and recovered images ofMAP method GTV method CZ method NLM methodand our method The second row is degraded image withvarious 120590 and residuals of those methods respectively InFigures 6ndash8 the recovered images of of MAP method GTVmethod CZ method and NLM method are still unclear sowe can conclude that the recovered images by ourmethod arebest through our visions and residuals As for the ldquoMouserdquoimage the recovered results are very similar in this casewe use residuals to see which result is better In general theblurrier residual is the better according recovered image isWe can find that the outline of residual by our method isblurriest
In conclusion the results of our numerical experimentsdemonstrate that our proposed method performs better thanthe MAP method GTV method CZ method and NLMmethod
4 Conclusion
In this paper we proposed a new effective model via alearned dictionary for denoising images with Rician noiseMore specifically based on the dictionary training we adda dictionary penalty term to the original nonconvex MAPmodel to establish a new denoising model and develop atwo-step algorithm to solve our model Also we carry out
Mathematical Problems in Engineering 11
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 9 Results of ldquoMouserdquo with 120590 = 10 by different methods (a) is the original ldquoMouserdquo image and (g) is the degraded image with 120590 = 10(b)-(f) are the denoised images of MAPmethod (120582 = 17) GTVmethod (120582 = 17) CZmethod (120582 = 006) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 10 Results of ldquoMouserdquo with 120590 = 15 by different methods (a) is the original ldquoMouserdquo image and (g) is the degraded image with 120590 = 15(b)-(f) are the denoised images of MAPmethod (120582 = 24) GTV method (120582 = 24) CZ method (120582 = 005) NLM method and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
experiments on various images to demonstrate the effective-ness of our modelThe numerical experiments show that ourproposedmodel is promising in denoising imageswithRiciannoise
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interest
Acknowledgments
The authors would like to thank one of the authors of[32] for providing the source code of CZ method Thiswork was supported in part by the National Natural ScienceFoundation of China under Grants 11871348 61872429 and61373087 by the Natural Science Foundation of GuangdongChina under Grants 2015A030313550 and 2015A030313557by the HD Video RampD Platform for Intelligent Analy-sis and Processing in Guangdong Engineering TechnologyResearch Centre of Colleges and Universities (no GCZX-A1409) by Natural Science Foundation of Shenzhen underGrant JCYJ20170818091621856 and by the Guangdong Key
12 Mathematical Problems in Engineering
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 11 Results of ldquoMouserdquo with 120590 = 20 by different methods (a) is the original ldquoMouserdquo image and (g) is the degraded image with 120590 = 20(b)-(f) are the denoised images of MAPmethod (120582 = 31) GTV method (120582 = 32) CZ method (120582 = 0035) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
Laboratory of Intelligent Information Processing ShenzhenUniversity China (518060)
References
[1] J-F Cai B Dong and Z Shen ldquoImage restoration A waveletframe based model for piecewise smooth functions andbeyondrdquoApplied andComputational Harmonic Analysis vol 41no 1 pp 94ndash138 2016
[2] R H Chan and J Ma ldquoA multiplicative iterative algorithm forbox-constrained penalized likelihood image restorationrdquo IEEETransactions on Image Processing vol 21 no 7 pp 3168ndash31812012
[3] C A Micchelli L Shen and Y Xu ldquoProximity algorithmsfor image models Denoisingrdquo Inverse Problems vol 27 no 4Article ID 045009 p 30 2011
[4] YWang J Yang W Yin and Y Zhang ldquoA new alternatingmin-imization algorithm for total variation image reconstructionrdquoSIAM Journal on Imaging Sciences vol 1 no 3 pp 248ndash2722008
[5] Y Dong M Hintermuller and M Neri ldquoAn efficient primal-dual method for l1-TV image restorationrdquo SIAM Journal onImaging Sciences vol 2 no 4 pp 1168ndash1189 2009
[6] J Lu K Qiao L Shen and Y Zou ldquoFixed-point algorithmsfor a TVL1 image restoration modelrdquo International Journal ofComputer Mathematics vol 95 no 9 pp 1829ndash1844 2018
[7] J Lu Z Ye and Y Zou ldquoHuber fractal image coding based ona fitting planerdquo IEEE Transactions on Image Processing vol 22no 1 pp 134ndash145 2013
[8] C A Micchelli L Shen Y Xu and X Zeng ldquoProximityalgorithms for the L1TV image denoising modelrdquo Advances inComputational Mathematics vol 38 no 2 pp 401ndash426 2013
[9] J Yang Y Zhang and W Yin ldquoAn efficient TVL1 algorithm fordeblurring multichannel images corrupted by impulsive noiserdquoSIAM Journal on Scientific Computing vol 31 no 4 pp 2842ndash2865 2009
[10] L Ma L Moisan J Yu and T Zeng ldquoA dictionary learningapproach for poisson image deblurringrdquo IEEE Transactions onMedical Imaging vol 32 no 7 pp 1277ndash1289 2013
[11] Y Xiao and T Zeng ldquoPoisson noise removal via learned dictio-naryrdquo in Proceedings of the 17th IEEE International Conferenceon Image Processing (ICIP) pp 1177ndash1180 2010
[12] H Zhang Y Dong and Q Fan ldquoWavelet frame based Poissonnoise removal and image deblurringrdquo Signal Processing vol 137pp 363ndash372 2017
[13] Y Dong and T Zeng ldquoA convex variational model for restoringblurred images with multiplicative noiserdquo SIAM Journal onImaging Sciences vol 6 no 3 pp 1598ndash1625 2013
[14] Y Huang M K Ng and YWen ldquoA new total variation methodfor multiplicative noise removalrdquo SIAM Journal on ImagingSciences vol 2 no 1 pp 20ndash40 2009
[15] Y-M Huang L Moisan M K Ng and T Zeng ldquoMultiplicativenoise removal via a learned dictionaryrdquo IEEE Transactions onImage Processing vol 21 no 11 pp 4534ndash4543 2012
[16] Z Jin and X Yang ldquoA variational model to remove the multi-plicative noise in ultrasound imagesrdquo Journal of MathematicalImaging and Vision vol 39 no 1 pp 62ndash74 2011
[17] M Kang S Yun and H Woo ldquoTwo-level convex relaxedvariational model for multiplicative denoisingrdquo SIAM Journalon Imaging Sciences vol 6 no 2 pp 875ndash903 2013
[18] J Lu L Shen C Xu and Y Xu ldquoMultiplicative noise removalin imaging An exp-model and its fixed-point proximity algo-rithmrdquo Applied and Computational Harmonic Analysis vol 41no 2 pp 518ndash539 2016
[19] J Lu L Shen C Xu and Y Xu ldquoMultiplicative noise removalwith a sparsity-aware optimization modelrdquo Inverse Problemsand Imaging vol 11 no 6 pp 949ndash974 2017
[20] J Lu Z Yang L Shen Z Lu H Yang and C Xu ldquoA frameletalgorithm for de-blurring images corrupted by multiplicativenoiserdquoApplied Mathematical Modelling vol 62 pp 51ndash61 2018
[21] J Lu Y Chen Y Zou and L Shen ldquoA new total variationmodelfor restoring blurred and speckle noisy imagesrdquo InternationalJournal of Wavelets Multiresolution and Information Processingvol 15 no 2 19 pages 2017
Mathematical Problems in Engineering 13
[22] P Perona and J Malik ldquoScale-space and edge detection usinganisotropic diffusionrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 12 no 7 pp 629ndash639 1990
[23] G Gerig O Kubler R Kikinis and F A Jolesz ldquoNonlinearanisotropic filtering ofMRI datardquo IEEE Transactions onMedicalImaging vol 11 no 2 pp 221ndash232 1992
[24] S Prima S P Morrissey and C Barillot ldquoNon-local meansvariants for denoising of diffusion-weighted and diffusiontensor MRIrdquo in Proceedings of the International Conference onMedical Image Computing and Computer-Assisted Interventionpp 344ndash351 Springer Berlin Germany 2007
[25] J V Manjon J Carbonell-Caballero J J Lull G Garcıa-MartiL Martı-Bonmati and M Robles ldquoMRI denoising using Non-LocalMeansrdquoMedical Image Analysis vol 12 no 4 pp 514ndash5232008
[26] N Wiest-Daessle S Prima P Coupe S P Morrissey and CBarillot ldquoRician noise removal by non-local means filteringfor low signal-to-noise ratio MRI Applications to DT-MRIrdquoMedical Image Computing and Computer-Assisted Interventionvol 5242 no 2 pp 171ndash179 2008
[27] R D Nowak ldquoWavelet-based Rician noise removal for mag-netic resonance imagingrdquo IEEE Transactions on Image Process-ing vol 8 no 10 pp 1408ndash1419 1999
[28] A Foi ldquoNoise estimation and removal in MR imaging Thevariance-stabilization approachrdquo in Proceedings of the 8th IEEEInternational Symposium on Biomedical Imaging From Nano toMacro (ISBI) pp 1809ndash1814 IEEE Chicago IL USA 2011
[29] J C Wood and K M Johnson ldquoWavelet packet denoising ofmagnetic resonance images Importance of Rician noise at lowSNRrdquo Magnetic Resonance in Medicine vol 41 no 3 pp 631ndash635 1999
[30] F Bowman Introduction to Bessel functions Dover PublicationsMineola NY USA 2012
[31] P Getreuer M Tong and L A Vese ldquoA variational model forthe restoration of MR images corrupted by blur and Riciannoiserdquo in Proceedings of the 7th international conference onAdvances in visual computing vol Part I of Lecture Notes inComputer Science pp 686ndash698 Springer Las Vegas NV USA2011
[32] L Chen and T Zeng ldquoA convex variational model for restoringblurred imageswith large Rician noiserdquo Journal ofMathematicalImaging and Vision vol 53 no 1 pp 92ndash111 2015
[33] M Aharon M Elad and A M Bruckstein ldquoK-SVD Analgorithm for designing overcomplete dictionaries for sparserepresentationrdquo IEEE Transactions on Signal Processing vol 54no 11 pp 4311ndash4322 2006
[34] M Elad and M Aharon ldquoImage denoising via sparse andredundant representations over learned dictionariesrdquo IEEETransactions on Image Processing vol 15 no 12 pp 3736ndash37452006
[35] G Aubert and J Aujol ldquoA variational approach to removingmultiplicative noiserdquo SIAM Journal on Applied Mathematicsvol 68 no 4 pp 925ndash946 2008
[36] Q Liu S Wang K Yang J Luo Y Zhu and D Liang ldquoHighlyundersampled magnetic resonance image reconstruction usingtwo-level Bregman method with dictionary updatingrdquo IEEETransactions on Medical Imaging vol 32 no 7 pp 1290ndash13012013
[37] Q Liu S Wang L Ying X Peng Y Zhu and D LiangldquoAdaptive dictionary learning in sparse gradient domain forimage recoveryrdquo IEEE Transactions on Image Processing vol 22no 12 pp 4652ndash4663 2013
[38] H Lu J Wei Q Liu Y Wang and X Deng ldquoA dictionarylearning method with total generalized variation for mri recon-structionrdquo International Journal of Biomedical Imaging vol2016 Article ID 7512471 13 pages 2016
[39] A Chambolle and T Pock ldquoA first-order primal-dual algorithmfor convex problems with applications to imagingrdquo Journal ofMathematical Imaging and Vision vol 40 no 1 pp 120ndash1452011
[40] E Esser X Zhang and T F Chan ldquoA general frameworkfor a class of first order primal-dual algorithms for convexoptimization in imaging sciencerdquo SIAM Journal on ImagingSciences vol 3 no 4 pp 1015ndash1046 2010
[41] N Komodakis and J-C Pesquet ldquoPlaying with duality Anoverview of recent primaldual approaches for solving large-scale optimization problemsrdquo IEEE Signal Processing Magazinevol 32 no 6 pp 31ndash54 2015
[42] A Chambolle ldquoAn algorithm for total variation minimizationand applicationsrdquo Journal of Mathematical Imaging and Visionvol 20 no 1-2 pp 89ndash97 2004
[43] L Ambrosio N Fusco and D Pallara Functions of BoundedVariation and Free Discontinuity Problem Oxford UniversityPress London 2000
[44] Z Wang A C Bovik H R Sheikh and E P SimoncellildquoImage quality assessment From error visibility to structuralsimilarityrdquo IEEE Transactions on Image Processing vol 13 no4 pp 600ndash612 2004
[45] J Gibson and A Bovik Handbook of Image and Video Process-ing Academic Press 2000
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Mathematical Problems in Engineering 11
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 9 Results of ldquoMouserdquo with 120590 = 10 by different methods (a) is the original ldquoMouserdquo image and (g) is the degraded image with 120590 = 10(b)-(f) are the denoised images of MAPmethod (120582 = 17) GTVmethod (120582 = 17) CZmethod (120582 = 006) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 10 Results of ldquoMouserdquo with 120590 = 15 by different methods (a) is the original ldquoMouserdquo image and (g) is the degraded image with 120590 = 15(b)-(f) are the denoised images of MAPmethod (120582 = 24) GTV method (120582 = 24) CZ method (120582 = 005) NLM method and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
experiments on various images to demonstrate the effective-ness of our modelThe numerical experiments show that ourproposedmodel is promising in denoising imageswithRiciannoise
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request
Conflicts of Interest
The authors declare that there are no conflicts of interest
Acknowledgments
The authors would like to thank one of the authors of[32] for providing the source code of CZ method Thiswork was supported in part by the National Natural ScienceFoundation of China under Grants 11871348 61872429 and61373087 by the Natural Science Foundation of GuangdongChina under Grants 2015A030313550 and 2015A030313557by the HD Video RampD Platform for Intelligent Analy-sis and Processing in Guangdong Engineering TechnologyResearch Centre of Colleges and Universities (no GCZX-A1409) by Natural Science Foundation of Shenzhen underGrant JCYJ20170818091621856 and by the Guangdong Key
12 Mathematical Problems in Engineering
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 11 Results of ldquoMouserdquo with 120590 = 20 by different methods (a) is the original ldquoMouserdquo image and (g) is the degraded image with 120590 = 20(b)-(f) are the denoised images of MAPmethod (120582 = 31) GTV method (120582 = 32) CZ method (120582 = 0035) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
Laboratory of Intelligent Information Processing ShenzhenUniversity China (518060)
References
[1] J-F Cai B Dong and Z Shen ldquoImage restoration A waveletframe based model for piecewise smooth functions andbeyondrdquoApplied andComputational Harmonic Analysis vol 41no 1 pp 94ndash138 2016
[2] R H Chan and J Ma ldquoA multiplicative iterative algorithm forbox-constrained penalized likelihood image restorationrdquo IEEETransactions on Image Processing vol 21 no 7 pp 3168ndash31812012
[3] C A Micchelli L Shen and Y Xu ldquoProximity algorithmsfor image models Denoisingrdquo Inverse Problems vol 27 no 4Article ID 045009 p 30 2011
[4] YWang J Yang W Yin and Y Zhang ldquoA new alternatingmin-imization algorithm for total variation image reconstructionrdquoSIAM Journal on Imaging Sciences vol 1 no 3 pp 248ndash2722008
[5] Y Dong M Hintermuller and M Neri ldquoAn efficient primal-dual method for l1-TV image restorationrdquo SIAM Journal onImaging Sciences vol 2 no 4 pp 1168ndash1189 2009
[6] J Lu K Qiao L Shen and Y Zou ldquoFixed-point algorithmsfor a TVL1 image restoration modelrdquo International Journal ofComputer Mathematics vol 95 no 9 pp 1829ndash1844 2018
[7] J Lu Z Ye and Y Zou ldquoHuber fractal image coding based ona fitting planerdquo IEEE Transactions on Image Processing vol 22no 1 pp 134ndash145 2013
[8] C A Micchelli L Shen Y Xu and X Zeng ldquoProximityalgorithms for the L1TV image denoising modelrdquo Advances inComputational Mathematics vol 38 no 2 pp 401ndash426 2013
[9] J Yang Y Zhang and W Yin ldquoAn efficient TVL1 algorithm fordeblurring multichannel images corrupted by impulsive noiserdquoSIAM Journal on Scientific Computing vol 31 no 4 pp 2842ndash2865 2009
[10] L Ma L Moisan J Yu and T Zeng ldquoA dictionary learningapproach for poisson image deblurringrdquo IEEE Transactions onMedical Imaging vol 32 no 7 pp 1277ndash1289 2013
[11] Y Xiao and T Zeng ldquoPoisson noise removal via learned dictio-naryrdquo in Proceedings of the 17th IEEE International Conferenceon Image Processing (ICIP) pp 1177ndash1180 2010
[12] H Zhang Y Dong and Q Fan ldquoWavelet frame based Poissonnoise removal and image deblurringrdquo Signal Processing vol 137pp 363ndash372 2017
[13] Y Dong and T Zeng ldquoA convex variational model for restoringblurred images with multiplicative noiserdquo SIAM Journal onImaging Sciences vol 6 no 3 pp 1598ndash1625 2013
[14] Y Huang M K Ng and YWen ldquoA new total variation methodfor multiplicative noise removalrdquo SIAM Journal on ImagingSciences vol 2 no 1 pp 20ndash40 2009
[15] Y-M Huang L Moisan M K Ng and T Zeng ldquoMultiplicativenoise removal via a learned dictionaryrdquo IEEE Transactions onImage Processing vol 21 no 11 pp 4534ndash4543 2012
[16] Z Jin and X Yang ldquoA variational model to remove the multi-plicative noise in ultrasound imagesrdquo Journal of MathematicalImaging and Vision vol 39 no 1 pp 62ndash74 2011
[17] M Kang S Yun and H Woo ldquoTwo-level convex relaxedvariational model for multiplicative denoisingrdquo SIAM Journalon Imaging Sciences vol 6 no 2 pp 875ndash903 2013
[18] J Lu L Shen C Xu and Y Xu ldquoMultiplicative noise removalin imaging An exp-model and its fixed-point proximity algo-rithmrdquo Applied and Computational Harmonic Analysis vol 41no 2 pp 518ndash539 2016
[19] J Lu L Shen C Xu and Y Xu ldquoMultiplicative noise removalwith a sparsity-aware optimization modelrdquo Inverse Problemsand Imaging vol 11 no 6 pp 949ndash974 2017
[20] J Lu Z Yang L Shen Z Lu H Yang and C Xu ldquoA frameletalgorithm for de-blurring images corrupted by multiplicativenoiserdquoApplied Mathematical Modelling vol 62 pp 51ndash61 2018
[21] J Lu Y Chen Y Zou and L Shen ldquoA new total variationmodelfor restoring blurred and speckle noisy imagesrdquo InternationalJournal of Wavelets Multiresolution and Information Processingvol 15 no 2 19 pages 2017
Mathematical Problems in Engineering 13
[22] P Perona and J Malik ldquoScale-space and edge detection usinganisotropic diffusionrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 12 no 7 pp 629ndash639 1990
[23] G Gerig O Kubler R Kikinis and F A Jolesz ldquoNonlinearanisotropic filtering ofMRI datardquo IEEE Transactions onMedicalImaging vol 11 no 2 pp 221ndash232 1992
[24] S Prima S P Morrissey and C Barillot ldquoNon-local meansvariants for denoising of diffusion-weighted and diffusiontensor MRIrdquo in Proceedings of the International Conference onMedical Image Computing and Computer-Assisted Interventionpp 344ndash351 Springer Berlin Germany 2007
[25] J V Manjon J Carbonell-Caballero J J Lull G Garcıa-MartiL Martı-Bonmati and M Robles ldquoMRI denoising using Non-LocalMeansrdquoMedical Image Analysis vol 12 no 4 pp 514ndash5232008
[26] N Wiest-Daessle S Prima P Coupe S P Morrissey and CBarillot ldquoRician noise removal by non-local means filteringfor low signal-to-noise ratio MRI Applications to DT-MRIrdquoMedical Image Computing and Computer-Assisted Interventionvol 5242 no 2 pp 171ndash179 2008
[27] R D Nowak ldquoWavelet-based Rician noise removal for mag-netic resonance imagingrdquo IEEE Transactions on Image Process-ing vol 8 no 10 pp 1408ndash1419 1999
[28] A Foi ldquoNoise estimation and removal in MR imaging Thevariance-stabilization approachrdquo in Proceedings of the 8th IEEEInternational Symposium on Biomedical Imaging From Nano toMacro (ISBI) pp 1809ndash1814 IEEE Chicago IL USA 2011
[29] J C Wood and K M Johnson ldquoWavelet packet denoising ofmagnetic resonance images Importance of Rician noise at lowSNRrdquo Magnetic Resonance in Medicine vol 41 no 3 pp 631ndash635 1999
[30] F Bowman Introduction to Bessel functions Dover PublicationsMineola NY USA 2012
[31] P Getreuer M Tong and L A Vese ldquoA variational model forthe restoration of MR images corrupted by blur and Riciannoiserdquo in Proceedings of the 7th international conference onAdvances in visual computing vol Part I of Lecture Notes inComputer Science pp 686ndash698 Springer Las Vegas NV USA2011
[32] L Chen and T Zeng ldquoA convex variational model for restoringblurred imageswith large Rician noiserdquo Journal ofMathematicalImaging and Vision vol 53 no 1 pp 92ndash111 2015
[33] M Aharon M Elad and A M Bruckstein ldquoK-SVD Analgorithm for designing overcomplete dictionaries for sparserepresentationrdquo IEEE Transactions on Signal Processing vol 54no 11 pp 4311ndash4322 2006
[34] M Elad and M Aharon ldquoImage denoising via sparse andredundant representations over learned dictionariesrdquo IEEETransactions on Image Processing vol 15 no 12 pp 3736ndash37452006
[35] G Aubert and J Aujol ldquoA variational approach to removingmultiplicative noiserdquo SIAM Journal on Applied Mathematicsvol 68 no 4 pp 925ndash946 2008
[36] Q Liu S Wang K Yang J Luo Y Zhu and D Liang ldquoHighlyundersampled magnetic resonance image reconstruction usingtwo-level Bregman method with dictionary updatingrdquo IEEETransactions on Medical Imaging vol 32 no 7 pp 1290ndash13012013
[37] Q Liu S Wang L Ying X Peng Y Zhu and D LiangldquoAdaptive dictionary learning in sparse gradient domain forimage recoveryrdquo IEEE Transactions on Image Processing vol 22no 12 pp 4652ndash4663 2013
[38] H Lu J Wei Q Liu Y Wang and X Deng ldquoA dictionarylearning method with total generalized variation for mri recon-structionrdquo International Journal of Biomedical Imaging vol2016 Article ID 7512471 13 pages 2016
[39] A Chambolle and T Pock ldquoA first-order primal-dual algorithmfor convex problems with applications to imagingrdquo Journal ofMathematical Imaging and Vision vol 40 no 1 pp 120ndash1452011
[40] E Esser X Zhang and T F Chan ldquoA general frameworkfor a class of first order primal-dual algorithms for convexoptimization in imaging sciencerdquo SIAM Journal on ImagingSciences vol 3 no 4 pp 1015ndash1046 2010
[41] N Komodakis and J-C Pesquet ldquoPlaying with duality Anoverview of recent primaldual approaches for solving large-scale optimization problemsrdquo IEEE Signal Processing Magazinevol 32 no 6 pp 31ndash54 2015
[42] A Chambolle ldquoAn algorithm for total variation minimizationand applicationsrdquo Journal of Mathematical Imaging and Visionvol 20 no 1-2 pp 89ndash97 2004
[43] L Ambrosio N Fusco and D Pallara Functions of BoundedVariation and Free Discontinuity Problem Oxford UniversityPress London 2000
[44] Z Wang A C Bovik H R Sheikh and E P SimoncellildquoImage quality assessment From error visibility to structuralsimilarityrdquo IEEE Transactions on Image Processing vol 13 no4 pp 600ndash612 2004
[45] J Gibson and A Bovik Handbook of Image and Video Process-ing Academic Press 2000
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
12 Mathematical Problems in Engineering
(a) (b) (c) (d) (e) (f)
(g) (h) (i) (j) (k) (l)
Figure 11 Results of ldquoMouserdquo with 120590 = 20 by different methods (a) is the original ldquoMouserdquo image and (g) is the degraded image with 120590 = 20(b)-(f) are the denoised images of MAPmethod (120582 = 31) GTV method (120582 = 32) CZ method (120582 = 0035) NLMmethod and ours (120582 = 001)and (h)-(l) are residuals of those methods respectively
Laboratory of Intelligent Information Processing ShenzhenUniversity China (518060)
References
[1] J-F Cai B Dong and Z Shen ldquoImage restoration A waveletframe based model for piecewise smooth functions andbeyondrdquoApplied andComputational Harmonic Analysis vol 41no 1 pp 94ndash138 2016
[2] R H Chan and J Ma ldquoA multiplicative iterative algorithm forbox-constrained penalized likelihood image restorationrdquo IEEETransactions on Image Processing vol 21 no 7 pp 3168ndash31812012
[3] C A Micchelli L Shen and Y Xu ldquoProximity algorithmsfor image models Denoisingrdquo Inverse Problems vol 27 no 4Article ID 045009 p 30 2011
[4] YWang J Yang W Yin and Y Zhang ldquoA new alternatingmin-imization algorithm for total variation image reconstructionrdquoSIAM Journal on Imaging Sciences vol 1 no 3 pp 248ndash2722008
[5] Y Dong M Hintermuller and M Neri ldquoAn efficient primal-dual method for l1-TV image restorationrdquo SIAM Journal onImaging Sciences vol 2 no 4 pp 1168ndash1189 2009
[6] J Lu K Qiao L Shen and Y Zou ldquoFixed-point algorithmsfor a TVL1 image restoration modelrdquo International Journal ofComputer Mathematics vol 95 no 9 pp 1829ndash1844 2018
[7] J Lu Z Ye and Y Zou ldquoHuber fractal image coding based ona fitting planerdquo IEEE Transactions on Image Processing vol 22no 1 pp 134ndash145 2013
[8] C A Micchelli L Shen Y Xu and X Zeng ldquoProximityalgorithms for the L1TV image denoising modelrdquo Advances inComputational Mathematics vol 38 no 2 pp 401ndash426 2013
[9] J Yang Y Zhang and W Yin ldquoAn efficient TVL1 algorithm fordeblurring multichannel images corrupted by impulsive noiserdquoSIAM Journal on Scientific Computing vol 31 no 4 pp 2842ndash2865 2009
[10] L Ma L Moisan J Yu and T Zeng ldquoA dictionary learningapproach for poisson image deblurringrdquo IEEE Transactions onMedical Imaging vol 32 no 7 pp 1277ndash1289 2013
[11] Y Xiao and T Zeng ldquoPoisson noise removal via learned dictio-naryrdquo in Proceedings of the 17th IEEE International Conferenceon Image Processing (ICIP) pp 1177ndash1180 2010
[12] H Zhang Y Dong and Q Fan ldquoWavelet frame based Poissonnoise removal and image deblurringrdquo Signal Processing vol 137pp 363ndash372 2017
[13] Y Dong and T Zeng ldquoA convex variational model for restoringblurred images with multiplicative noiserdquo SIAM Journal onImaging Sciences vol 6 no 3 pp 1598ndash1625 2013
[14] Y Huang M K Ng and YWen ldquoA new total variation methodfor multiplicative noise removalrdquo SIAM Journal on ImagingSciences vol 2 no 1 pp 20ndash40 2009
[15] Y-M Huang L Moisan M K Ng and T Zeng ldquoMultiplicativenoise removal via a learned dictionaryrdquo IEEE Transactions onImage Processing vol 21 no 11 pp 4534ndash4543 2012
[16] Z Jin and X Yang ldquoA variational model to remove the multi-plicative noise in ultrasound imagesrdquo Journal of MathematicalImaging and Vision vol 39 no 1 pp 62ndash74 2011
[17] M Kang S Yun and H Woo ldquoTwo-level convex relaxedvariational model for multiplicative denoisingrdquo SIAM Journalon Imaging Sciences vol 6 no 2 pp 875ndash903 2013
[18] J Lu L Shen C Xu and Y Xu ldquoMultiplicative noise removalin imaging An exp-model and its fixed-point proximity algo-rithmrdquo Applied and Computational Harmonic Analysis vol 41no 2 pp 518ndash539 2016
[19] J Lu L Shen C Xu and Y Xu ldquoMultiplicative noise removalwith a sparsity-aware optimization modelrdquo Inverse Problemsand Imaging vol 11 no 6 pp 949ndash974 2017
[20] J Lu Z Yang L Shen Z Lu H Yang and C Xu ldquoA frameletalgorithm for de-blurring images corrupted by multiplicativenoiserdquoApplied Mathematical Modelling vol 62 pp 51ndash61 2018
[21] J Lu Y Chen Y Zou and L Shen ldquoA new total variationmodelfor restoring blurred and speckle noisy imagesrdquo InternationalJournal of Wavelets Multiresolution and Information Processingvol 15 no 2 19 pages 2017
Mathematical Problems in Engineering 13
[22] P Perona and J Malik ldquoScale-space and edge detection usinganisotropic diffusionrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 12 no 7 pp 629ndash639 1990
[23] G Gerig O Kubler R Kikinis and F A Jolesz ldquoNonlinearanisotropic filtering ofMRI datardquo IEEE Transactions onMedicalImaging vol 11 no 2 pp 221ndash232 1992
[24] S Prima S P Morrissey and C Barillot ldquoNon-local meansvariants for denoising of diffusion-weighted and diffusiontensor MRIrdquo in Proceedings of the International Conference onMedical Image Computing and Computer-Assisted Interventionpp 344ndash351 Springer Berlin Germany 2007
[25] J V Manjon J Carbonell-Caballero J J Lull G Garcıa-MartiL Martı-Bonmati and M Robles ldquoMRI denoising using Non-LocalMeansrdquoMedical Image Analysis vol 12 no 4 pp 514ndash5232008
[26] N Wiest-Daessle S Prima P Coupe S P Morrissey and CBarillot ldquoRician noise removal by non-local means filteringfor low signal-to-noise ratio MRI Applications to DT-MRIrdquoMedical Image Computing and Computer-Assisted Interventionvol 5242 no 2 pp 171ndash179 2008
[27] R D Nowak ldquoWavelet-based Rician noise removal for mag-netic resonance imagingrdquo IEEE Transactions on Image Process-ing vol 8 no 10 pp 1408ndash1419 1999
[28] A Foi ldquoNoise estimation and removal in MR imaging Thevariance-stabilization approachrdquo in Proceedings of the 8th IEEEInternational Symposium on Biomedical Imaging From Nano toMacro (ISBI) pp 1809ndash1814 IEEE Chicago IL USA 2011
[29] J C Wood and K M Johnson ldquoWavelet packet denoising ofmagnetic resonance images Importance of Rician noise at lowSNRrdquo Magnetic Resonance in Medicine vol 41 no 3 pp 631ndash635 1999
[30] F Bowman Introduction to Bessel functions Dover PublicationsMineola NY USA 2012
[31] P Getreuer M Tong and L A Vese ldquoA variational model forthe restoration of MR images corrupted by blur and Riciannoiserdquo in Proceedings of the 7th international conference onAdvances in visual computing vol Part I of Lecture Notes inComputer Science pp 686ndash698 Springer Las Vegas NV USA2011
[32] L Chen and T Zeng ldquoA convex variational model for restoringblurred imageswith large Rician noiserdquo Journal ofMathematicalImaging and Vision vol 53 no 1 pp 92ndash111 2015
[33] M Aharon M Elad and A M Bruckstein ldquoK-SVD Analgorithm for designing overcomplete dictionaries for sparserepresentationrdquo IEEE Transactions on Signal Processing vol 54no 11 pp 4311ndash4322 2006
[34] M Elad and M Aharon ldquoImage denoising via sparse andredundant representations over learned dictionariesrdquo IEEETransactions on Image Processing vol 15 no 12 pp 3736ndash37452006
[35] G Aubert and J Aujol ldquoA variational approach to removingmultiplicative noiserdquo SIAM Journal on Applied Mathematicsvol 68 no 4 pp 925ndash946 2008
[36] Q Liu S Wang K Yang J Luo Y Zhu and D Liang ldquoHighlyundersampled magnetic resonance image reconstruction usingtwo-level Bregman method with dictionary updatingrdquo IEEETransactions on Medical Imaging vol 32 no 7 pp 1290ndash13012013
[37] Q Liu S Wang L Ying X Peng Y Zhu and D LiangldquoAdaptive dictionary learning in sparse gradient domain forimage recoveryrdquo IEEE Transactions on Image Processing vol 22no 12 pp 4652ndash4663 2013
[38] H Lu J Wei Q Liu Y Wang and X Deng ldquoA dictionarylearning method with total generalized variation for mri recon-structionrdquo International Journal of Biomedical Imaging vol2016 Article ID 7512471 13 pages 2016
[39] A Chambolle and T Pock ldquoA first-order primal-dual algorithmfor convex problems with applications to imagingrdquo Journal ofMathematical Imaging and Vision vol 40 no 1 pp 120ndash1452011
[40] E Esser X Zhang and T F Chan ldquoA general frameworkfor a class of first order primal-dual algorithms for convexoptimization in imaging sciencerdquo SIAM Journal on ImagingSciences vol 3 no 4 pp 1015ndash1046 2010
[41] N Komodakis and J-C Pesquet ldquoPlaying with duality Anoverview of recent primaldual approaches for solving large-scale optimization problemsrdquo IEEE Signal Processing Magazinevol 32 no 6 pp 31ndash54 2015
[42] A Chambolle ldquoAn algorithm for total variation minimizationand applicationsrdquo Journal of Mathematical Imaging and Visionvol 20 no 1-2 pp 89ndash97 2004
[43] L Ambrosio N Fusco and D Pallara Functions of BoundedVariation and Free Discontinuity Problem Oxford UniversityPress London 2000
[44] Z Wang A C Bovik H R Sheikh and E P SimoncellildquoImage quality assessment From error visibility to structuralsimilarityrdquo IEEE Transactions on Image Processing vol 13 no4 pp 600ndash612 2004
[45] J Gibson and A Bovik Handbook of Image and Video Process-ing Academic Press 2000
Hindawiwwwhindawicom Volume 2018
MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Mathematical Problems in Engineering
Applied MathematicsJournal of
Hindawiwwwhindawicom Volume 2018
Probability and StatisticsHindawiwwwhindawicom Volume 2018
Journal of
Hindawiwwwhindawicom Volume 2018
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawiwwwhindawicom Volume 2018
OptimizationJournal of
Hindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom Volume 2018
Engineering Mathematics
International Journal of
Hindawiwwwhindawicom Volume 2018
Operations ResearchAdvances in
Journal of
Hindawiwwwhindawicom Volume 2018
Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018
International Journal of Mathematics and Mathematical Sciences
Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in
Nature and SocietyHindawiwwwhindawicom Volume 2018
Hindawiwwwhindawicom
Dierential EquationsInternational Journal of
Volume 2018
Hindawiwwwhindawicom Volume 2018
Decision SciencesAdvances in
Hindawiwwwhindawicom Volume 2018
AnalysisInternational Journal of
Hindawiwwwhindawicom Volume 2018
Stochastic AnalysisInternational Journal of
Submit your manuscripts atwwwhindawicom
Mathematical Problems in Engineering 13
[22] P Perona and J Malik ldquoScale-space and edge detection usinganisotropic diffusionrdquo IEEE Transactions on Pattern Analysisand Machine Intelligence vol 12 no 7 pp 629ndash639 1990
[23] G Gerig O Kubler R Kikinis and F A Jolesz ldquoNonlinearanisotropic filtering ofMRI datardquo IEEE Transactions onMedicalImaging vol 11 no 2 pp 221ndash232 1992
[24] S Prima S P Morrissey and C Barillot ldquoNon-local meansvariants for denoising of diffusion-weighted and diffusiontensor MRIrdquo in Proceedings of the International Conference onMedical Image Computing and Computer-Assisted Interventionpp 344ndash351 Springer Berlin Germany 2007
[25] J V Manjon J Carbonell-Caballero J J Lull G Garcıa-MartiL Martı-Bonmati and M Robles ldquoMRI denoising using Non-LocalMeansrdquoMedical Image Analysis vol 12 no 4 pp 514ndash5232008
[26] N Wiest-Daessle S Prima P Coupe S P Morrissey and CBarillot ldquoRician noise removal by non-local means filteringfor low signal-to-noise ratio MRI Applications to DT-MRIrdquoMedical Image Computing and Computer-Assisted Interventionvol 5242 no 2 pp 171ndash179 2008
[27] R D Nowak ldquoWavelet-based Rician noise removal for mag-netic resonance imagingrdquo IEEE Transactions on Image Process-ing vol 8 no 10 pp 1408ndash1419 1999
[28] A Foi ldquoNoise estimation and removal in MR imaging Thevariance-stabilization approachrdquo in Proceedings of the 8th IEEEInternational Symposium on Biomedical Imaging From Nano toMacro (ISBI) pp 1809ndash1814 IEEE Chicago IL USA 2011
[29] J C Wood and K M Johnson ldquoWavelet packet denoising ofmagnetic resonance images Importance of Rician noise at lowSNRrdquo Magnetic Resonance in Medicine vol 41 no 3 pp 631ndash635 1999
[30] F Bowman Introduction to Bessel functions Dover PublicationsMineola NY USA 2012
[31] P Getreuer M Tong and L A Vese ldquoA variational model forthe restoration of MR images corrupted by blur and Riciannoiserdquo in Proceedings of the 7th international conference onAdvances in visual computing vol Part I of Lecture Notes inComputer Science pp 686ndash698 Springer Las Vegas NV USA2011
[32] L Chen and T Zeng ldquoA convex variational model for restoringblurred imageswith large Rician noiserdquo Journal ofMathematicalImaging and Vision vol 53 no 1 pp 92ndash111 2015
[33] M Aharon M Elad and A M Bruckstein ldquoK-SVD Analgorithm for designing overcomplete dictionaries for sparserepresentationrdquo IEEE Transactions on Signal Processing vol 54no 11 pp 4311ndash4322 2006
[34] M Elad and M Aharon ldquoImage denoising via sparse andredundant representations over learned dictionariesrdquo IEEETransactions on Image Processing vol 15 no 12 pp 3736ndash37452006
[35] G Aubert and J Aujol ldquoA variational approach to removingmultiplicative noiserdquo SIAM Journal on Applied Mathematicsvol 68 no 4 pp 925ndash946 2008
[36] Q Liu S Wang K Yang J Luo Y Zhu and D Liang ldquoHighlyundersampled magnetic resonance image reconstruction usingtwo-level Bregman method with dictionary updatingrdquo IEEETransactions on Medical Imaging vol 32 no 7 pp 1290ndash13012013
[37] Q Liu S Wang L Ying X Peng Y Zhu and D LiangldquoAdaptive dictionary learning in sparse gradient domain forimage recoveryrdquo IEEE Transactions on Image Processing vol 22no 12 pp 4652ndash4663 2013
[38] H Lu J Wei Q Liu Y Wang and X Deng ldquoA dictionarylearning method with total generalized variation for mri recon-structionrdquo International Journal of Biomedical Imaging vol2016 Article ID 7512471 13 pages 2016
[39] A Chambolle and T Pock ldquoA first-order primal-dual algorithmfor convex problems with applications to imagingrdquo Journal ofMathematical Imaging and Vision vol 40 no 1 pp 120ndash1452011
[40] E Esser X Zhang and T F Chan ldquoA general frameworkfor a class of first order primal-dual algorithms for convexoptimization in imaging sciencerdquo SIAM Journal on ImagingSciences vol 3 no 4 pp 1015ndash1046 2010
[41] N Komodakis and J-C Pesquet ldquoPlaying with duality Anoverview of recent primaldual approaches for solving large-scale optimization problemsrdquo IEEE Signal Processing Magazinevol 32 no 6 pp 31ndash54 2015
[42] A Chambolle ldquoAn algorithm for total variation minimizationand applicationsrdquo Journal of Mathematical Imaging and Visionvol 20 no 1-2 pp 89ndash97 2004
[43] L Ambrosio N Fusco and D Pallara Functions of BoundedVariation and Free Discontinuity Problem Oxford UniversityPress London 2000
[44] Z Wang A C Bovik H R Sheikh and E P SimoncellildquoImage quality assessment From error visibility to structuralsimilarityrdquo IEEE Transactions on Image Processing vol 13 no4 pp 600ndash612 2004
[45] J Gibson and A Bovik Handbook of Image and Video Process-ing Academic Press 2000
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