Abstract The total variation model proposed by Rudin, Os-her, and Fatemi performs very well for removing noise whilepreserving edges. However, it favors a piecewise constantsolution in BV space which often leads to the staircase ef-fect, and small details such as textures are often filtered outwith noise in the process of denoising. In this paper, we pro-pose a fractional-order multi-scale variational model whichcan better preserve the textural information and eliminatethe staircase effect. This is accomplished by replacing thefirst-order derivative with the fractional-order derivative inthe regularization term, and substituting a kind of multi-scale norm in negative Sobolev space for the L2 norm inthe fidelity term of the ROF model. To improve the results,we propose an adaptive parameter selection method for theproposed model by using the local variance measures andthe wavelet based estimation of the singularity. Using theoperator splitting technique, we develop a simple alternat-ing projection algorithm to solve the new model. Numericalresults show that our method can not only remove noise andeliminate the staircase effect efficiently in the non-texturedregion, but also preserve the small details such as textureswell in the textured region. It is for this reason that our adap-tive method can improve the result both visually and in termsof the peak signal to noise ratio efficiently.
J. Zhang (�)School of Science, Nanjing University of Scienceand Technology, Nanjing 210094, Chinae-mail: [email protected]
Z. Wei · L. XiaoSchool of Computer Science and Technology, Nanjing Universityof Science and Technology, Nanjing 210094, China
1 Introduction and Motivations
An important task in image processing is the restoration orreconstruction of a true image u from an observation f .Given an image function f ∈ L2(�), with � ⊂ R2 an openand bounded domain, the problem is to extract u from f . Inthe denoising case, one of the most well known techniquesis by energy minimization and regularization. In this cate-gory, Rudin, Osher, and Fatemi have proposed the followingminimization problem in [1] (the ROF model):
minu∈BV(�)
{F(u) = J (u) + λ
2‖f − u‖2
L2
}, (1)
where J (u) = ∫�
|Du| denotes the total variation of u calledregularization term, BV(�) = {u|J (u) < +∞} is the spaceof functions of bounded variation, ‖f − u‖2
L2 is a fidelityterm, and λ > 0 is the regularization parameter.
The ROF model performs very well for removing noisewhile preserving edges, especially for cartoon like images.However, it favors a piecewise constant solution in BV spaceand therefore often causes the staircase effect. Meanwhile,small details such as textures are often filtered out with noisein the process of denoising.
To eliminate the undesirable staircase effect, high or-der PDEs (typically fourth-order PDEs) for image restora-tion have been introduced in [2–5]. Though these methodscan eliminate the staircase effect efficiently, they often leadto a speckle effect. Recently, two classes of fractional- or-der models for image denoising have been proposed in [6]and [7]. Numerical results show that the fractional-ordermodels can eliminate both staircase effect and speckle ef-fect when the parameters are chosen appropriately. More-over, they can preserve textures better than the ROF model.However, how to choose the parameters appropriately andadaptively is not discussed.
In order to preserve the textures, the L2 norm in the fi-delity term of the ROF model has been replaced by somedual norms in [8–11]. In this category, a kind of multi-scalenorm in negative Sobolev space H−s has been introducedin [11]. The numerical results show that this method canpreserve textures in different scales when the parameters arechosen appropriately. However, how to choose the parame-ters appropriately and adaptively is also not discussed.
In this paper, with the aim to better preserve the texturalinformation and eliminate the staircase effect in the processof denoising, we replace the first-order derivative with theGrünwald-Letnikov fractional-order derivative in the regu-larization term of the ROF model similar to [7], and substi-tute the multi-scale norm in negative Sobolev space H−s forthe L2 norm in the fidelity term of the ROF model similarto [11]. Thus we propose a fractional-order multi-scale vari-ational model for image denoising. To improve the results,we propose an adaptive parameter selection method for thenew model. To solve the proposed model, we develop analternating projection algorithm based on the operator split-ting technique.
The outline of this paper is given as follows. In Sect. 2,we describe the fractional-order multi-scale model and pro-pose a kind of adaptive parameter selection method for thenew model. In Sect. 3, we discuss some properties of thefractional-order total variation and its convex conjugate op-erator, and develop an alternating projection algorithm forsolving the new model. Numerical examples are presentedin Sect. 4. Finally, we conclude our paper in Sect. 5.
2 Adaptive Fractional-Order Multi-scale Model
2.1 Description of the Fractional-order Multi-scale Model
The definition of fractional-order derivative is not uni-fied until now. The fractional-order derivatives used in [6]and [7] are defined by the Fourier transform pairs, whichcause high computational cost. In this paper, we use theGrünwald-Letnikov fractional-order derivative [12]. Asthe discrete version of Grünwald-Letnikov fractional-orderderivative, the fractional-order difference is defined as
�αg(x) =K−1∑k=0
(−1)kCαk g(x − k), (2)
where g(x) is a real function, Cαk = �(α+1)
�(k+1)�(α−k+1)denotes
the generalized binomial coefficient and �(x) is the Gammafunction. In (2), K ≥ 3 is an integer constant (we use K = 20in this paper). Specially, if α = 1, then C1
k = 0 for k ≥ 2 and(2) is the first-order forward difference as usual.
To simplify, our images will be 2-dimensional matricesof size N ×N . We denote by X the Euclidean space R
N×N .
For any u = (ui,j )Ni,j=1 ∈ X, we denote ui,j = 0 for i, j < 1
or i, j > N .For any u ∈ X, the discrete fractional-order gradient ∇αu
is a vector in Y = X × X given by
∇αu := [(∇αu)i,j ]Ni,j=1
= ([(�α1 u)i,j ]Ni,j=1, [(�α
2 u)i,j ]Ni,j=1) (3)
with
(�α1 u)i,j =
K−1∑k=0
(−1)kCαk ui−k,j ,
(�α2 u)i,j =
K−1∑k=0
(−1)kCαk ui,j−k, i, j = 1, . . . ,N. (4)
For any p = (p1,p2) ∈ Y , the discrete fractional-order di-vergence is defined as
divαp := [(divαp)i,j ]Ni,j=1 (5)
with
(divαp)i,j = (−1)αK−1∑k=0
(−1)kCαk p1
i+k,j + (−1)α
×K−1∑k=0
(−1)kCαk p2
i,j+k, i, j = 1, . . . ,N. (6)
One can check easily that (3) and (5) satisfy the property asfollows
〈p,∇αu〉Y = 〈(−1)αdivαp,u〉X, ∀u ∈ X,p ∈ Y, (7)
where 〈u,v〉X = ∑Ni,j=1 ui,j vi,j ,∀u,v ∈ X and 〈p,q〉Y =∑N
i,j=1(p1i,j q
1i,j + p2
i,j q2i,j ),∀p = (p1, p2), q = (q1, q2) ∈
Y are the inner products of X and Y respectively.The discrete fractional-order total variation of u ∈ X can
be defined as
Jα(u) :=N∑
i,j=1
|(∇αu)i,j |. (8)
Specially, J1(u) = ∑Ni,j=1 |(∇1u)i,j | is the discrete total
variation of u ∈ X as usual [17].For the regularization term of the ROF model, we replace
the total variation J (u) with the discrete fractional-order to-tal variation Jα(u), which is similar to [7]. For the fidelityterm, we substitute the L2 norm with the multi-scale normin negative Sobolev space, which is similar to [11]. Thus, wepropose a coupling fractional-order multi-scale variational
J Math Imaging Vis (2012) 43:39–49 41
model as follows
minu
{E(u) = Jα(u) + 1
2
L∑j=0
2−2jsj ‖[λ(f − u)]j‖2X,
1 ≤ α ≤ 2,0 ≤ sj ≤ 1
}, (9)
where [λ(f − u)]j , j = 0,1, . . . ,L are the components ofλ(f − u) = [λi,j (fi,j − ui,j )]Ni,j=1 in different scales, andthey can be obtained by orthogonal wavelet decompositionand reconstruction. Differing from the parameter λ in [11],λ in (9) is not a constant any longer but a matrix, namelyλ = (λi,j )
Ni,j=1.
We notice that∑L
j=0[λ(f − u)]j = λ(f − u), and there-fore our proposed model includes the following specialcases:
(i) if α = 1.0, sj = 0 (j = 0,1, . . . ,L) and λi,j ≡ C
(i, j = 1, . . . ,N,C > 0 is a constant), then the pro-posed model is the same as the classical ROF model;
(ii) if α = 1.0,0 ≤ sj ≤ 1 (j = 0,1, . . . ,L) and λi,j ≡ C
(i, j = 1, . . . ,N,C > 0 is a constant), then the pro-posed model is the same as the multi-scale model pro-posed in [11];
(iii) if 1.0 ≤ α ≤ 2.0, sj = 0 (j = 0,1, . . . ,L) and λi,j ≡ C
(i, j = 1, . . . ,N,C > 0 is a constant), then the pro-posed model is similar to the fractional-order varia-tional model proposed in [7].
In order to improve the performance of noise removaland texture preserving, we will choose the parameters of ourmodel adaptively. These parameters are related to the imagecharacteristics such as regional properties (textured regionor non-textured region) and scale properties.
2.2 Parameter Selection
2.2.1 The Selection of λ
Some regularization parameter selection methods have beenintroduced in [13–16]. In [14] Gilboa et al. proposed to se-lect the regularization parameter of the ROF model related tothe “local variance” instead of the global variance of noise.This method was subsequently used in [15] and [16]. In thispaper, we will select the regularization parameter by usingthe local variance measures similar to [14–16].
The Euler-Lagrange equation of the fractional-ordermulti-scale variational model (9) is
∂Jα(u) − λ
L∑j=0
2−2jsj [λ(f − u)]j = 0, (10)
where ∂Jα(u) = (−1)αdivα( ∇αu|∇αu| ) is a subgradient of Jα
at u.
Using the orthogonal wavelet decomposition and recon-struction, we rewrite (10) as follows
L∑j=0
[∂Jα(u)]j =L∑
j=0
2−2jsj λ · [λ(f − u)]j , (11)
where [∂Jα(u)]j and [λ(f − u)]j , j = 0,1, . . . ,L are thecomponents of ∂Jα(u) and λ(f − u) in different scales re-spectively.
N .We propose to add a local adaptability behavior to Q fol-
lowing the idea in [14]. Let W be a normalized and radiallysymmetric Gaussian window of size M × M(M is odd), weassume Q as
Q = Q ∗ W,
where ∗ denotes convolution and Q ∈ X is an undeterminedmatrix. Notice that W is symmetric and W(i, j) = 0 fori, j < 1 or i, j > M , if we multiply (12) by (f − u) andsum over, then we can obtain
N∑k,l=0
[(f − u)
L∑j=0
22jsj [∂Jα(u)]j]
k,l
=N∑
k,l=0
(Q ∗ W)k,l(f − u)2k,l
=N∑
m,n=0
[(f − u)2 ∗ W ]m,nQm,n. (13)
A sufficient condition of (13) is
Qk,l = [(f − u)∑L
j=0 22jsj (∂Jα(u))j ]k,l
Pk,l
,
k, l = 1, . . . ,N, (14)
where P = (f − u)2 ∗ W .If the average of the residue v = f − u is zero, then P
is the discrete version of “local variance” of v proposedin [14]. Similar to [14], we use the local variance of theresidual image vROF = f − uROF to estimate P in (14),
42 J Math Imaging Vis (2012) 43:39–49
where uROF denotes the solution of the ROF model. We es-timate P approximately as follows
P ≈ σ 4
PvROF
, (15)
where σ 2 is the global variance of vROF and PvROF=
[vROF − mean(vROF )]2 ∗ W is the local variance of vROF .In [14], they estimate P as P ≈ σ 4/PvROF
, where σ 2 is thevariance of noise. Since σ 2 is often unknown in practice, wereplace σ 2 with σ 2 in this paper. Using (15), we estimate Q
as
Q ≈ (f − u)∑L
j=0 22jsj [∂Jα(u)]jσ 2
· PvROF
σ 2. (16)
We notice that W is a normalized Gaussian window, andtherefore we have
1
N2
N∑i,j=0
(PvROF)i,j = 1
N2
N∑i,j=0
[vROF − mean(vROF )]2i,j
= σ 2. (17)
The equation (17) implies that the average of PvROFis
σ 2. For a cartoon image where there are only piecewise con-stant components, vROF contains mainly noise, and there-fore σ 2 ≈ σ 2, (PvROF
)i,j ≈ σ 2 for i, j = 1, . . . ,N . For apartly textured image, vROF contains both textures andnoise. In the non-textured region, we have (PvROF
)i,j � σ 2.In the textured region, affected by the textures, we have(PvROF
)i,j ≥ σ 2. Compared with Q for cartoon images, thevalues of Q for partly textured images are increased in thetextured region and reduced in the non-textured region, and
therefore λ =√
Q ∗ W will have the same character.
2.2.2 The Selection of sj and α
In the proposed model (9), the parameter sj is a kind of in-dex represents the regularity of [λ(f − u)]j ∈ H−sj . Ac-cording to wavelet analysis, the regularity of [λ(f − u)]jcan be represented by the Lipchitz exponent γj also, and itcan be computed as follows
γj = 1
2
(1
2log2
(max |W2j |
max |W2j−1 |)
− 1
), j = 1,2, . . . ,L,
(18)
where W2j (j = 0,1, . . . ,L) are the wavelet coefficients ofλ(f − u) at scale 2j . In this paper, we use the Lipchitzexponent γj to estimate sj . Considering the range of sj
(0 ≤ sj ≤ 1), we estimate sj as follows
sj =⎧⎨⎩
0, if j = 0 or (−γj ) < 0,
1, if j > 0 and (−γj ) > 1,
−γj , if j > 0 and 0 < (−γj ) < 1,
j = 0,1,2, . . . ,L. (19)
The numerical results in [6] and [7] imply that the use ofthe fractional-order derivative with order α > 1 can elimi-nate the staircase effect efficiently. With the increase of α,the textures can be preserved better in the textured regionbut the speckle effect will appear in the non-textured region.To eliminate both staircase effect and speckle effect in thenon-textured region, we choose α slightly larger than 1.0,and therefore we limit 1.0 ≤ α ≤ 1.2 in the non-textured re-gion. To preserve the textures in the textured region, we limit1.2 ≤ α ≤ 1.6 in the textured region. Hence, we choose α
adaptively as follows
α =
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
1.0 + 25π
arctan(PvROF− min(PvROF
)),
if PvROF< σ 2.
1.2 + 45π
arctan(PvROF− σ 2)),
if PvROF≥ σ 2.
(20)
3 Description of the Adaptive Alternating ProjectionAlgorithm
In [17], Chambolle has proposed a fast projection algo-rithm for the ROF model and this algorithm has been ex-tended to many TV based problems. Inspiring from theseworks, we develop a projection algorithm for solving thenew fractional-order multi-scale model correspondingly.
For the discrete fractional-order total variation operatorJα defined by (8), we have
Lemma 3.1 For any u = (ui,j )Ni,j=1 ∈ X, if we denote
ui,j = 0 for i, j < 1 or i, j > N , then the discrete fractional-order total variation Jα(u) satisfies
Jα(u) = supp
〈p,∇αu〉Y , (21)
where p = (p1,p2) ∈ Y and |pi,j | =√
(p1i,j )
2 + (p2i,j )
2 ≤1 for i, j = 1,2, . . . ,N .
Proof For any p = (p1,p2) �= 0 ∈ Y and u = (ui,j )Ni,j=1 ∈
X, we have
〈p,∇αu〉Y =∑
1≤i,j≤N
[p1i,j · (�α
1 u)i,j + p2i,j · (�α
2 u)i,j ]
=∑
1≤i,j≤N
pi,j · (∇αu)i,j
J Math Imaging Vis (2012) 43:39–49 43
=∑
1≤i,j≤N
|(∇αu)i,j | · |pi,j | · cos θi,j ,
where pi,j = (p1i,j , p
2i,j ) and θi,j is the angle between two
vectors pi,j and (∇αu)i,j . Since |pi,j | ≤ 1, it is obviouslythat if and only if |pi,j | = 1 and cos θi,j = 1 for i, j =1,2, . . . ,N , then we can obtain the maximum of 〈p,∇αu〉Y .Therefore, we have
supp
〈p,∇αu〉Y =∑
1≤i,j≤N
|(∇αu)i,j | = Jα(u).
The lemma is proved. �
If we denote by J ∗α the convex conjugate operator of Jα ,
then it can be defined as
J ∗α (v) = sup
u∈X
{〈u,v〉X − Jα(u)}, v ∈ X,
and J ∗α has the following property:
Theorem 3.1 If we define Kα as the closure of the set
{(−1)αdivαp : p ∈ Y, |pi,j | ≤ 1, i, j = 1,2, . . . ,N}, (22)
then the convex conjugate of Jα(u) satisfies
J ∗α (v) =
{0, if v ∈ Kα.
+∞, ifv /∈ Kα.(23)
Proof Denote L(u, v) := 〈u,v〉X − Jα(u), then we haveJ ∗
α (v) = supu∈X L(u, v). According to Lemma 3.1 and (7),we have
L(u, v) = 〈u,v〉X − supp∈Y,|pi,j |≤1
〈p,∇αu〉Y
= 〈u,v〉X − supp∈Y,|pi,j |≤1
〈u, (−1)αdivαp〉X.
(i) if v ∈ Kα , then we have L(u, v) = 〈u,v〉X −supp∈Y,|pi,j |≤1〈u, (−1)αdivαp〉X ≤ 0. On the other hand, wenotice that L(0, v) = 0, and therefore J ∗
α (v) =supu L(u, v) ≥ 0. Hence, we can obtain J ∗
α (v) = 0 for anyv ∈ Kα .
(ii) if v /∈ Kα , then we need to prove J ∗α (v) = +∞. De-
note
g(ε) := L(u + εw,v)
= 〈u + εw,v〉 − Jα(u + εw), u, v,w ∈ X, ε ∈ R.
If J ∗α (v) < +∞, then there exist u ∈ X s.t. g′(0) = 0 for any
w ∈ X. Let g′(0) = 0, we have
g′(0) =⟨v − (−1)αdivα
( ∇αu
|∇αu|)
,w
⟩X
= 0, ∀w ∈ X.
Hence, v = (−1)αdivα( ∇αu|∇αu| ) ∈ Kα . It conflicts with v /∈
Kα , and therefore J ∗α (v) = +∞ for any v /∈ Kα . The the-
orem is proved. �
Using the operator splitting technique [18, 19], werewrite the Euler-Lagrange equation (10) as follows
{g = u + λ
∑Lj=0 2−2jsj [λ(f − u)]j ,
∂Jα(u) − (g − u) = 0.(24)
To solve this problem, we consider an alternating schemeas follows:
(a) Let u(n) be fixed, we compute
g(n) = u(n) + λ
L∑j=0
2−2jsj [λ(f − u(n))]j .
(b) Let g(n) be fixed, we solve
∂Jα(u) − (g(n) − u) = 0. (25)
If we denote w = g(n) − u, then we can obtain the dualequation of (25) as follows
∂J ∗α (w) − (g(n) − w) = 0,
and it can be considered as the Euler-Lagrange equation ofthe following minimization problem:
minw
J ∗α (w) + 1
2‖g(n) − w‖2
X. (26)
According to Theorem 3.1, if we denote w = (−1)αdivαp,then (26) is equivalent to finding the solution of the follow-ing problem:
minp
{∥∥(−1)αdivαp − g(n)∥∥
X: p ∈ Y, |pi,j |2 ≤ 1,
i, j = 1,2, . . . ,N}. (27)
Similar to the algorithm in [17], we propose an iterationscheme for solving (27) as follows:
Giving τ > 0 and initial value p(0), we compute
p(k+1)i,j = p
(k)i,j − τ(∇α((−1)αdivαp(k) − g(n)))i,j
1 + τ |(∇α((−1)αdivαp(k) − g(n)))i,j |,
k = 0,1,2,3, . . . (28)
for i, j = 1,2, . . . ,N .If the iteration scheme (28) is convergent, namely if there
is p∗ ∈ Y , s.t limk→+∞ p(k) = p∗, then we can obtain thesolution of (25) as u(n+1) = g(n) − (−1)αdivαp∗. In prac-tice, we just need an approximation of p∗ denoted by p(n+1)
and obtain u(n+1) = g(n) − (−1)αdivαp(n+1) approximately.
44 J Math Imaging Vis (2012) 43:39–49
Fig. 1 Original test images.(a) Barbara image with pixels256 × 256; (b) Synthetic imagewith pixels 256 × 256;(c) Barbara image with pixels512 × 512
To improve the performance, the parameters are re-freshed adaptively in the iterative process. To summarize,we propose the adaptive alternating projection algorithm asfollows:
Algorithm:
1. Preprocessing:1.1 For a given noisy image f of size N × N , we ob-
tain the denoised image uROF and the residual im-age vROF = f − uROF by using Chambolle’s algo-rithm in [17] for the ROF model.
1.2 Compute the global variance σ 2 and the local vari-ance PvROF
of the residual image vROF respectively.Compute α by using (20).
τ > 0.3. Iteration: For n = 0,1,2, . . .: Compute u(n+1) by the fol-
lowing steps:3.1 Compute
g(n) = u(n) + λ(n)L∑
j=0
2−2js(n)j [λ(n)(f − u(n))]j .
3.2 Compute
p(n+1)i,j = p
(n)i,j − τ(∇α((−1)αdivαp(n) − g(n)))i,j
1 + τ |(∇α((−1)αdivαp(n) − g(n)))i,j |,
i, j = 1,2, . . . ,N.
3.3 Compute u(n+1) = g(n)−(−1)αdivαp(n+1). If u(n+1)
satisfies the given condition, then we terminate theiteration and output u(n+1); otherwise, go to step 4.
4. Parameter updating:4.1 Compute
λ(n+1) =√√√√√
[(f − u(n+1))
L∑j=0
22js(n)j [∂Jα(u(n+1))]j · PvROF
σ 4
]∗ W.
4.2 Refresh s(n+1)j , j = 0,1,2, . . . ,L by using (18) and
(19). Denote n := n + 1 and go to step 3.
4 Numerical Experiments
In this section, we present numerical results for image de-noising obtained by our method. As an objective evaluationindex of denoising performance, we use the peak signal tonoise ratio (PSNR) defined as
PSNR = 10 lgmax1≤i,j≤N |(uo)i,j |2
1N2
∑Ni=1
∑Nj=1[(uo)i,j − ui,j ]2
,
where uo is the original image without noise and u is thedenoised image.
In our experiments, we use the Daubechies wavelet of thefourth order (Db4) for the wavelet decomposition and recon-struction. The size of the normalized and symmetric Gaus-sian window is 9 × 9. The iteration termination condition ofthe proposed algorithm is
mean(|u(n) − u(n−1)|) < 0.1,
where mean(|u(n) − u(n−1)|) is the average of the absolutevalues of u(n) − u(n−1).
The original test images without noise are showed inFig. 1. In these images, there are high presences of texturescombined with non-textured parts.
Experiment 1 Our method is an adaptive method for imagedenoising. In this experiment, we will compare our methodwith two famous adaptive methods for image denoising: thescale TV and the adaptive TV. These two methods were pro-posed for the ROF model by Gilboa et al. in [14]. In the scaleTV, λ is selected adaptively as a constant in each iterationloop. If we fix α ≡ 1, sj ≡ 0, j = 0,1, . . . ,L and replace σ 2
in (16) with the variance of noise σ 2, then our model (9) isthe same as the ROF model and our algorithm is the same asthe adaptive TV proposed in [14].
We add white Gaussian noise with standard deviation 10,20 and 30 to Fig. 1(a) and Fig. 1(b) respectively and com-pare our results with that obtained by the scale TV and theadaptive TV.
In Table 1, we list the PSNR of denoised images at dif-ferent noise levels. In Fig. 2 and Fig. 3, we show the de-noised images and the residual images of the test images
J Math Imaging Vis (2012) 43:39–49 45
Table 1 Comparison of PSNRof denoised images at differentnoise levels
Test image / noise level Noisy image Scale TV Adaptive TV Our method
Barbara (256 × 256)/σ = 10 27.5156 25.8181 28.0770 28.4021
Barbara (256 × 256)/σ = 20 21.5660 25.4841 26.3588 26.6484
Barbara (256 × 256)/σ = 30 18.1419 24.5723 25.0569 25.1322
Fig. 2 Comparison of thedenoised images u and theresidual imagesv = f − u + 100 of the Barbaraimage. (a) the denoised imageobtained by the Scale TV;(b) the denoised image obtainedby the Adaptive TV; (c) thedenoised image obtained by ourmethod; (d) the residual imageobtained by the Scale TV;(e) the residual image obtainedby the Adaptive TV; (f) theresidual image obtained by ourmethod
Fig. 3 Comparison of thedenoised images u and theresidual imagesv = f − u + 100 of theSynthetic image. (a) thedenoised image obtained by theScale TV; (b) the denoisedimage obtained by the adaptiveTV; (c) the denoised imageobtained by our method; (d) theresidual image obtained by thescale TV; (e) the residual imageobtained by the adaptive TV;(f) the residual image obtainedby our method
46 J Math Imaging Vis (2012) 43:39–49
Table 2 Comparison of PSNR and Error of denoised images
ROF (α = 1.0, s = 0.0) α = 1.0 s = 0.0 Our adaptive method
Scale TV Adaptive TV s = 0.3 s = 0.5 s = 0.8 α = 1.2 α = 1.4 α = 1.6
with white Gaussian noise of standard deviation σ = 20. Inorder to show the residual images clearly, we enhance themby adding their brightness.
In the scale TV, though λ is selected adaptively as adifferent constant in different iteration loop, the processis smoothing both textures and noise with the same λ ineach iteration loop. Therefore, textured regions are over-smoothed, whereas non-textured regions are not sufficientlydenoised, which are clearly demonstrated by Fig. 2(a),Fig. 2(d), Fig. 3(a) and Fig. 3(d). It is for this reason thatthe PSNR of the denoised image obtained by the scale TV iseven lower than the PSNR of the noisy Barbara image withpixels 256 × 256 and σ = 10.
The adaptive TV applies different levels of denoising indifferent regions. This improves the result both visually (thetextures are better preserved, and smooth regions are betterdenoised) and in terms of PSNR. Our method takes advan-tage of both the fractional-order differential regularizationand the multi-scale fidelity term. The results in Figs. 2 and 3show that our method not only can preserve textures betterthan the adaptive TV in the textured region, but also can re-move noise efficiently in the non-textured region. It is forthis reason that our method can improve the PSNR more ef-ficiently than the adaptive TV.
Experiment 2 Our model is a coupling model between thefractional-order model proposed in [7] and the multi-scalemodel proposed in [11]. The main improvement is that theparameters in our model are selected adaptively. Moreover,the algorithms proposed in [7] and [11] are based on the gra-dient descent method, and therefore the convergence speedsare very slow. Our algorithm is based on the operator split-ting method and projection method, the convergence speedis much faster than that of the algorithms proposed in [7]and [11].
In this experiment, we add white Gaussian noise withstandard deviation σ = 20 to the Barbara image with pix-els 512 × 512 showed in Fig. 1(c) and compare our resultswith that obtained by the following special cases:
(1) Let α = 1.0 and sj = s ≡ 0.0 (j = 0,1,2, . . . ,L), ourmodel is the same as the ROF model, and we can solveit by the scale TV and the adaptive TV proposed in [14];
(2) Let α = 1.0 and sj = s (j = 0,1,2, . . . ,L, s ∈ (0,1) isa constant), our model is the same as the model pro-
posed in [11]. We choose s = 0.3,0.5 and 0.8 respec-tively in this experiment;
(3) Let sj = s = 0.0 (j = 0,1,2, . . . ,L) and α be a con-stant, our model is similar to the model proposed in [7].We choose α = 1.2,1.4 and 1.6 respectively in this ex-periment.
In this experiment, the regularization parameters of thesemodels except the scale TV are computed adaptively by us-ing our method.
In Table 2, we compare the PSNR and Error of the de-noised images. The Error is defined as
Error = mean(|u0 − u(n)|),where u0 is the original image and u(n) is the denoised im-age.
The results in Table 2 show that our adaptive method per-forms better for improving the PSNR than the methods pro-posed in [7, 11] and [14].
In Fig. 4, we compare the performance of preserving tex-tures and eliminating the staircase effect. The results implythat the scale TV removes most textures and has seriousstaircase effect. The adaptive TV can preserve textures wellin the textured region and remove the noise effectively in thenon-textured region. When α = 1.0, our model is the sameas the model proposed in [11] and can preserve textures indifferent scales. Specially, when s = 0.5, it preserves tex-tures better than the adaptive TV. When s = 0, our model issimilar to the model proposed in [7] and can prevent stair-case effect better than the adaptive TV. When all parametersare computed adaptively, our coupling model can not onlyremove noise and eliminate the staircase effect well in thenon-textured region, but also preserve textures well in thetextured region. It is for this reason that our adaptive methodcan improve the result both visually and in terms of PSNRefficiently.
5 Conclusion
In this paper, we propose a fractional-order multi-scalemodel for image denoising. We choose the parameters ofour model adaptively by using the local variance measuresand the wavelet based estimation of the singularity. We de-velop a simple alternating projection algorithm for solving
J Math Imaging Vis (2012) 43:39–49 47
Fig. 4 Comparison of thedenoised images, non-texturedregions and textured regions.The images in the first row areobtained by the scale TV; theimages in the second row areobtained by the adaptive TV; theimages in the third row areobtained by our method withα = 1.0 and s = 0.5; the imagesin the forth row are obtained byour method with α = 1.2 ands = 0.0; the images in the fifthrow are obtained by our adaptivemethod
48 J Math Imaging Vis (2012) 43:39–49
the new model. Numerical results show that the proposedmethod can not only remove noise and eliminate the stair-case effect well in the non-textured region, but also preservetextures well in the textured region. Thus, our method andcan improve the result visually and in terms of PSNR effi-ciently.
We are currently working on extending our approach toproblems involving non-additive and non-Gaussian noisemodels. Moreover, we consider the local variance measuresin a normalized and radially symmetric Gaussian windowand the size of the window is fixed in this paper. In the futurework, we will consider replacing the fixed Gaussian windowwith an adaptive anisotropic window, which can better dealwith the textures and edges in different directions.
Acknowledgements This work was supported by Specialized Re-search Fund for the Doctoral Program of Higher Education(No.200802880018); National Natural Science Foundation of China(No.60802039 and 61071146); NJUST Research Funding(No.2010ZYTS070 and 2010ZDJH07).
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Jun Zhang received B.Sc., M.Sc.degrees in computational math-ematics from Wuhan University,Wuhan, Hubei, China, in 1999,2002 respectively and the Ph.D.degree in pattern recognition andintelligent systems from NanjingUniversity of Science and Tech-nology, Nanjing, Jiangsu, China, in2010. He is currently a lecturer ofNanjing University of Science andTechnology. His research interestsare in mathematical image process-ing. His current research is focusedon the partial differential equation
based image modeling and algorithms in image denoising, inpainting,segmentation, and super-resolution.
Zhihui Wei received B.Sc., M.Sc.,and Ph.D. degrees from South EastUniversity, Nanjing, Jiangsu, China,in 1983, 1986, and 2003 respec-tively. He is currently the professorand doctoral supervisor of NanjingUniversity of Science and Tech-nology. His main research interestsare mathematical image process-ing, image modeling, multiscaleanalysis, video and image codingand compressing, watermarking andsteganography, and speech and au-dio processing. His current researchis focused on the theory of image
Liang Xiao received B.Sc., de-gree in Applied Mathematics andPh.D. degree in Computer Sciencefrom Nanjing University of Scienceand Technology (NUST), Nanjing,Jiangsu, China, in 1999 and 2004respectively. From 2006 to 2008,he was a Post-doctor Research Fel-low at the Pattern Recognition Lab-oratory of the NUST. From 2009–2010, he was post-doctor at Rens-selaer Polytechnic Institute (RPI),USA. He is currently an AssociateProfessor at the School of Com-puter Science of NUST. His main
research areas include inverse problems in image processing, scientificcomputing, data mining, and pattern recognition.
