1Faculty of Computer Science, National Research University Higher School of Economics, Moscow, Russia2The University of Da Nang - University of Science and Technology, Da Nang, Vietnam
3Institute of Applied Mathematics and Computer Science, Tula State University, Tula, Russia4The University of Da Nang - University of Economics, Da Nang, Vietnam
Received: 15.03.2018 • Accepted/Published Online: 22.08.2018 • Final Version: 29.11.2018
Abstract: We present here an effective scheme for image denoising based on total variation regularization. The proposedscheme allows to efficiently remove Poisson noise as well as Gaussian noise simultaneously with the help of a newkind of data fidelity term, suitable for the mixed Poisson–Gaussian noise model. The results show that the algorithmcorresponding to our new scheme outperforms the existing methods for mixed Poisson–Gaussian noise removal.
1. IntroductionThis paper addresses the image reconstruction problem. Let there be an image, considered as a matrix of pixelvalues, which has been corrupted by some random noise. Our task is to compute an estimation of what theoriginal image was. A wide range of image acquisition devices are subject to noise and therefore this problemhas many applications, e.g., in medical imagery, astronomy, microscopy, or even usual digital cameras. Thedistribution of the random noise is usually known in advance since it can be deduced from the physical processof image acquisition. However, this distribution generally has some parameters that cannot be computed apriori and therefore have to be estimated by the image reconstruction algorithm itself.
Two types of noise are commonly found in image acquisition applications: Gaussian noise and Poissonnoise. A large body of literature is devoted to the denoising problem for both types: see [1] and its referencelist for the Poisson case, and [2–5] for the Gaussian case. In [6], the authors proposed to model the noise witha mixture of Poisson and Gaussian distributions. The Gaussian part accounts for signal-independent sources ofnoise, such as thermal and electronic noise [7, 8]. The Poisson part accounts for the uncertainty intrinsic to thephoton-counting process used in detectors. This makes noise strongly signal-dependent, which is especially truefor electronic microscopy or astronomy, so the additive Gaussian noise model itself cannot provide sufficientaccuracy for further data analysis and interpretation. Various approaches have been investigated to denoiseimages with Poisson–Gaussian noise [9–11].
In general, image denoising is an ill-posed problem. Therefore, we have to use some a priori knowledge∗Correspondence: [email protected]
This work is licensed under a Creative Commons Attribution 4.0 International License.2831
about the original image to be able to reconstruct it. The well-known ROF model [12] is a common approachto do so: it prescribes that the reconstructed image u∗ is computed from the observed image f = f(x) (withx = (x1, x2) ∈ Ω , Ω ⊆ R2 being an open bounded domain) by the following formula:
u∗ = arg minu∈BV (Ω)
∫Ω
|∇u|dx+λ
2
∫Ω
(u− f)2dx, (1)
where λ > 0 is a regularization parameter, BV is the space of functions of bounded variation, and∫Ω|∇u|
stands for the total variation of u (see [13]). The definition of the |∇ · | operator is given later, cf. Eq. (4).The second term is a data fidelity term, which ensures that the reconstructed image should be close
enough to the observed image. The first term is a smoothness term, which ensures that the reconstructed imageis not noisy. This is where the a priori knowledge lies: we assume that the original image was smooth enough,in the sense that its total variation is low. Finally, λ is a parameter allowing to scale the relative importanceof these two requirements. Using Euler–Lagrange equations, we can rewrite Eq. (1) into a form that may benumerically solved using gradient descent or other techniques.
In [14], Eq. (1) was adapted to Poisson noise instead of Gaussian noise. The proposed method is tocompute the following:
u∗ = arg minu∈BV (Ω)
∫Ω
|∇u|dx+ β
∫Ω
(u− f logu)dx, (2)
where β > 0 is a parameter giving the relative weight of the two constraints, similarly to λ . The model ofEq. (2) is called the modified ROF model (M-ROF). However, the numerical algorithm proposed in [14] tocompute a solution to this equation has drawbacks; most notably, the intermediate solutions obtained duringthe execution of the algorithm may contain pixels with negative values. This causes problems with the logfunction when evaluating Eq. (2), and the algorithm may end up with a suboptimal result. This concern wasaddressed in [15], in which the authors gave a new numerical algorithm that avoids this problem.
More recent work [16, 17] showed that Eqs. (1) and (2) can be combined to denoise an image corruptedby a mixture of Poisson and Gaussian noise. However, similar problems of negative values arise in the proposednumerical algorithms. In this paper, we use the ideas from [15] to design a new numerical scheme that overcomesthese problems in the mixed Poisson–Gaussian case. We show that our scheme only yields positive values for eachpixel in each intermediate image it computes, thus avoiding the aforementioned problems. We also determinebounds on the rate of convergence of our algorithm. Then we provide experimental results, comparing ourproposal to the ones from [12, 14, 15, 18], which are based on the ROF model, as well as BM3D [19] andPES-TV [20]. We use standard metrics for performance evaluation: the peak signal-to-noise ratio (PSNR),structural similarity index (SSIM) [21], and Pratt’s figure of merit (FOM) [22].
Other techniques have been investigated to denoise images in the Poisson–Gaussian case. The PURE-LETapproach [11] uses statistical analysis on the noise model (PURE stands for Poisson Unbiased Risk Estimator,which can be modified to work on a Poisson–Gaussian mixed distribution [23]) to estimate the noise. Thedenoising process is then parametrized linearly, using linear expansion of thresholds (LET), so that the optimalparameters are simply computed by solving a system of linear equations.
Another technique uses maximum a priori (MAP) formulation of the denoising problem and statisticalanalysis of the noise distribution to give a total variation formulation [9]. The initial reasoning behind this
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approach is similar to ours; however, both the formulation and the algorithm scheme of [9] are more complexthan ours. They get possibly better results, at the cost of trickier implementation.
In [24], the authors dealt with a slightly different case, where we have many acquisitions of the sameimage (in other terms, several realizations of the random variable “image + noise”). Their assumptions are thatthe original image is fixed, while the noises are independent between realizations. They use an expectation-maximization approach to solve the problem.
This paper is organized as follows: Section 2 presents some preliminaries about Poisson–Gaussian noise.In Section 3, which is the main part of our contribution, we discuss a new numerical scheme for Poisson–Gaussiandenoising. Section 4 consists of experiments and discussions of the obtained results. Finally, some limits andconclusions are described in Section 5.
2. PreliminariesIn what follows, f denotes the observed image. We assume that f is positive everywhere it is defined, andbounded. Moreover, we suppose that the total variation of f is also bounded. An efficient numerical approachto solve the minimization problem in the case of the M-ROF model (2) is given in [15], called the modifiedscheme for Poisson-modified total variation model (MS-ROF): start with u(0) = f , the observed image, andcompute successive values of u(n) with the following formula:
u(n+1) − u(n)
τ= div
(∇u(n)∣∣∇u(u)
∣∣)
− β
(1− f
u(n+1)
), (3)
where τ ∈ (0; 1] is the time-step parameter of the gradient descent, while β > 0 is a parameter weighting therelative importance of data fidelity and smoothness requirements. Keep computing u(n) for successive values ofn , until the point where the difference u(n+1)−u(n) becomes negligible. The final u(n) is then the reconstructedimage, u∗ .
The images that we are handling are discrete, i.e. matrices of pixel values rather than functions from R2
to R . Therefore, we have to choose a discretization scheme for numerical computations. If u is a image, wewrite uj,k for the pixel at coordinates (j, k) in u (j = 1, ..,M ; k = 1, .., N ). We define the following quantities:
u(1)j,k = uj+1,k − uj−1,k u
(2)j,k = uj,k+1 − uj,k−1
∇uj,k = (u(1)j,k, u
(2)j,k) |∇uj,k| =
√(u
(1)j,k)
2 + (u(2)j,k)
2 + ε2, (4)
where ε is a small positive quantity, added for considerations of numerical stability.
As suggested in [16, 17], we can combine the models of Eqs. (1) and (2) into a single one, aiming atremoving a mixture of Gaussian and Poisson noise:
u∗ = arg minu
∫|∇u|dx+
λ
2
∫(u− f)2dx+ β
∫(u− f logu)dx, (5)
where f is the observed image, while λ > 0 and β > 0 are parameters that give the relative weight of Gaussianand Poisson noise (those parameters will have to be determined experimentally). One drawback of this modelis that the variances of Poisson and Gaussian noise are assumed to be the same.
If we try to use a similar reasoning to [15] in order to derive a numerical approach solving Eq. (5), thenwe run into problems. Indeed, it is impossible to guarantee that u will always remain positive; we need thepositivity since we have to compute logu to evaluate Eq. (5). We propose another numerical scheme to solveEq. (5), which avoids these problems.
3. Modified numerical schemeWe want to solve Eq. (5) by gradient descent. We have:
δu(t)
δt= div
(∇u
|∇u|
)− β(1− f
u)− λ(u− f), (6)
where f is the observed image, and λ, β are regularization parameters.For numerical implementation of Eq. (6), the authors in [18] used the following modified scheme for the
mixed Poisson–Gaussian model (MPGS):
u(n+1) − u(n)
τ= div
(∇u(n)∣∣∇u(n)
∣∣)
− β(1− f
u(n))− λ
σ2(u(n) − f), (7)
where f is the observed image, σ > 0 is an estimation for variance noise by the method of Immerker [25], andλ, β are positive parameters such that λ+ β = 1 .
Based on the ideas in [15], we propose the following numerical scheme to implement the gradient descent:
u(n+1) − u(n)
τ= div
(∇u(n)∣∣∇u(n)
∣∣)
− β
(1− f
u(n+1)
)− λ
σ2
(u(n+1) − f
), (8)
where f is the observed image, τ ∈ (0, 1] is the time-step parameter, and λ, β are positive parameters suchthat λ+ β = 1 . σ is an estimation for variance noise computed as follows [25]:
σ =
√π
2
1
6(M − 2)(N − 2)
M∑j
N∑k
|uj,k ∗K| K =
1 −2 1−2 4 −21 −2 1
.
Parameters λ, β determine the relative importance of Gaussian and Poisson noise compensation in ourscheme. We will determine the values of those parameters in the experimentation phase. Let us group the
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similar terms to rewrite Eq. (8) as:
an
(u(n+1)
)2+ bnu
(n+1) + cn = 0, (9)
where
an = (1 + τλ
σ2) bn = −
(u(n) + τ div
(∇u(n)∣∣∇u(n)
∣∣)
− τβ + τλ
σ2f
)cn = −τβf.
The solution is given by:
u(n+1) =−bn +
√(−bn)2 − 4ancn2an
. (10)
Theorem 1 If f ∈ BV (Ω) we always receive a positive solution u(n+1) in Eq. (10).
ProofLet M = sup(f) and m = inf(f) . If f ∈ BV (Ω) , using the proof of Theorem 1 in [15], we
have log(f) ∈ L1(Ω) and E(u) bounded in BV (Ω) . Therefore, we have 0 ≤ m ≤ u ≤ f ≤ M andcn = −τβf < 0 (τ > 0, β > 0) . Since an = (1 + τ λ
σ2 ) > 0 with τ > 0 and λ > 0, σ > 0 , cn < 0 , wehave ancn < 0 and ∆n > 0 . Hence, we always receive a positive solution for Eq. (10). 2
The next theorem shows the theoretical rate of convergence for our numerical approach.
Theorem 2 If f ∈ BV (Ω) and fj,k−1 < 4τλ(1 + τλ) for each j, k , then
u(n) ≤ sn(f − r) + r,
where s = a−1n = (1 + τ λ
σ2 )−1 and r = t/(1− s) , with t = sτ(
√8− β) + τf(s λ
σ2 + 2β) .
Proof By definition of an and cn , the hypothesis on f implies that −4ancn > 1 ; moreover, −bn > 0 . Wecan use the bound
√x2 + y ≤ x+ y to rewrite Eq. (10) into the following:
u(n+1) ≤ −2bn − 4ancn2an
= − bnan
− 2cn.
Following [26], we have:∣∣∣∣div( ∇u(n)
|∇u(n)|
)∣∣∣∣ ≤ √8.
Plug in the definitions of an, bn, cn to get:
un+1 ≤ un
1 + τ λσ2
+τ(√8− β + λ
σ2 f)
1 + τ λσ2
+ 2τβf.
By definition of s and t , this rewrites into u(n+1) ≤ su(n)+t . A simple recurrence shows that the generalterm is indeed u(n+1) ≤ sn(u(0) − r) + r . Since u(0) = f , the theorem is proved. 2
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Algorithm 1 Modified numerical scheme for mixed Poisson–Gaussian denoising.1: Initialize: λ , β , u(0) = f , k = 0 ;2: While the stopping criterion is not satisfied do3: -k = k + 1 .4: -Compute u(k) by using Eqs. (9) and (10).5: End while6: Return u∗ = u(k) .
The proposed denoising algorithm is described as follows:We terminate the iteration if the following stopping condition is satisfied for some prescribed tolerance
ς :∥u(k) − u(k−1)∥2
∥u(k)∥2< ς, (11)
where ς is a small positive parameter.
4. Experiments4.1. Implementation issues
The original images are 8-bit gray-scale standard images shown1 in Figures 1a−1j. All experiments were run ona machine with Core i7-CPU 2.GHz, SDRAM 4GB-DDR III 2.00 GHz, Windows 10 (64-bit), and implementedin MATLAB. We present the denoising results of the test images degraded by Poisson noise with a peak intensityImax and Gaussian noise with a standard deviation σG .
To compare the efficiency of the algorithms, we use the PSNR, SSIM [21], and Pratt’s FOM [22]. Thefirst metric, PSNR (db), is defined by:
PSNR = 10 log10
(MNI2max
∥u∗ − u∥22
),
where u, u∗ are the original image and the reconstructed or noisy image accordingly, Imax is the maximumintensity of the original image, and M and N are the number of image pixels in rows and columns.
The second metric, SSIM, is defined by:
SSIM(u, u∗) =(2µuµu∗ + c1)(2σu,u∗ + c2)
(µ2u + µ2
u∗ + c1)(σ2u + σ2
u∗ + c2),
where µu, µu∗ are the means of images; σu, σu∗ are the standard deviations (the square root of variance) ofimages; σu,u∗ is the covariance of the two images u and u∗ ; c1 = (K1L)
2 , c2 = (K2L)2 ; L is the dynamic
range of the pixel values (255 for 8-bit gray-scale images); and K1 ≪ 1 , K2 ≪ 1 are small constants.The last metric, FOM, is given by:
FOM =1
max(Ea, Ed)
Ed∑i=1
1
1 + γd2i,
1Coming from http://www.imageprocessingplace.com and http://www.cs.tut.fi/~foi/GCF-BM3D/BM3D_images.zip, both ac-cessed 20/02/2018.
where Ea and Ed are the numbers of actual (ground truth) and detected edge points, respectively; di is theith detected edge pixel’s deviation or error distance; and γ stands for a positive scaling parameter. The scalingfactor γ is a constant typically set to 1/9 (see [22]). In experiments, the edge detector for ground truth andthe denoised image is chosen to be the Sobel edge detector as a built-in function in MATLAB.
In order to show the potentiality of our method, we compare the mixed Poisson–Gaussian denoisingresults of the algorithm corresponding to our scheme with other related methods, like the ROF model [12] usingthe model of Eq. (1), the modified ROF (M-ROF) [14] using the model of Eq. (2), the modified scheme for thePoisson-modified total variation model (MS-ROF) [15] using the scheme of Eq. (3) for the model of Eq. (2),and the modified scheme for the mixed Poisson–Gaussian model (MPGS) [18] using the scheme of Eq. (7) forthe model of Eq. (5). For these methods, we employ the Euler–Lagrange equation to solve the minimizationproblem (see [13] for more details). Furthermore, we also compare our method with some methods in imagedenoising, such as PES-TV [20] and BM3D [19].
For the ROF model, we set λ = 0.8 , and for M-ROF, we set β = 0.25 (see [14]). For MS-ROF, we setβ = 0.1 (see [15]). For MPGS, we set β = 0.2 , and λ = 0.8 (see [18]). For our method, the regularizationparameters λ and β are determined empirically: we set β = 0.6 and λ = 0.4 , which gave the best results. Thetolerance parameter ς in the stopping condition of Eq. (11) is set to ς = 5× 10−4 in the experiments.
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4.2. Numerical results and discussion
4.2.1. First experiment We compare the mixed Poisson–Gaussian denoising results of our algorithm withother related methods, like ROF, M-ROF, MS-ROF, and MPGS. Noisy observations are generated by Poissonnoise with some fixed peak Imax , and by Gaussian noise with standard deviation σG = Imax/10 to the testimages.
Figures 2a−2f show the results of the compared methods for the “Lake” image corrupted by noiseparameters Imax = 120 and σG = 12 . Table 1 shows the PSNR, SSIM, FOM, number of iterations, andcomputational time for each of the compared methods.
Similarly, Figures 3a−3f show the results of compared methods for the “Peppers” image corrupted bynoise parameters Imax = 60 and σG = 6 . Table 2 shows the numerical results.
Finally, Table 3 shows the average FOM, PSNR, and SSIM over all 10 test images for each methodfor Imax = 120, 60, 30 and σG = Imax/10 . Our experimental results show that the proposed algorithm hasapproximately the same computational time as the other methods. Our proposed method outperforms theother relative methods for mixed Poisson–Gaussian noise removal.
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Table 1. Comparison of measurements for Figure 2.
- b c d e fPSNR 21.8958 22.3070 22.0505 22.3733 22.7176SSIM 0.6485 0.6722 0.6566 0.6755 0.6909FOM 0.8616 0.8755 0.8703 0.8800 0.8883Number of iterations 215 200 212 201 192CPU time (s) 8.05 6.86 7.55 7.61 7.19Time in one iteration (s) 0.037 0.034 0.036 0.038 0.038
We compare the mixed Poisson–Gaussian denoising results of our algorithm with other related methods (thesame as in the previous experiment), keeping either Imax or σG constant and varying the other parameter.
Figures 4a−4f show the results of the compared methods for the “Boat” image corrupted by Poissonnoise with peak Imax = 120 and Gaussian white noise with standard deviation σG = 10 . Figures 5a−5f showthe results of the compared methods for the “Cameraman” image corrupted by noise with parameters Imax = 60
and σG = 10 .
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Table 2. Comparison of measurements for Figure 3.
- b c d e fPSNR 23.7174 24.5055 24.1221 24.8585 25.2848SSIm 0.6870 0.7197 0.7026 0.7345 0.7505FOM 0.7889 0.8371 0.8110 0.8411 0.8683Number of iterations 184 160 171 148 139CPU time (s) 6.78 5.76 5.98 5.34 5.15Time in one iteration (s) 0.037 0.036 0.035 0.036 0.037
Table 3. Mean results of the methods with σG = Imax/10 .
We compare the mixed Poisson–Gaussian denoising results in two cases: first, Poisson noise with peakImax = 120, 60, 30 and Gaussian white noise with constant standard deviation σG = 10 , and then Poisson noisewith constant peak Imax = 120 and Gaussian white noise with standard deviation σG = 5, 10, 15 . The averageresults over all images for each method appear in Table 4. Our proposed method gets better results than otherrelative methods in the vast majority of cases.
4.2.2. Third experiment
We compare the mixed Poisson–Gaussian denoising results of our algorithm with some well-known methods,such as PES-TV [20] and BM3D [19]. Figures 6a−6l show the results of the compared methods for the “House”images corrupted by Poisson noise with peak Imax = 120 and Gaussian white noise with standard deviationσG = 10, 15, 20 .
We compare mixed Poisson–Gaussian denoising results of our proposed method with PES-TV and BM3Dfor Poisson noise with peak Imax = 60, 120 and Gaussian noise with standard deviation σG = 10, 15, 20 . Table 5shows the mean results over all images for the tested methods.
Figure 6. House (256×256). Recovered images of different approaches for removing mixed Poisson–Gaussian noise. Firstrow from left to right: noisy images with PSNRNoisy = 19.4321 (a), PSNRNoisy = 17.0552 (b), PSNRNoisy = 15.0527(c); second row: restored images by PES-TV (d–f); third row: restored images by BM3D (g–i); fourth row: restoredimages by our proposed method (j–l).
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Table 4. Mean results of the methods with various values of Imax and σG .
PES-TV and the BM3D are well-known techniques for image denoising. They can allow good results forremoving mixed Poisson–Gaussian noise. However, our method gets similar results in some cases. Particularly,in the case of a high level of mixed Poisson–Gaussian noise, our method outperforms PES-TV and the BM3Din terms of PSNR and SSIM. Please see the Appendix for more detailed results.
5. ConclusionWe proposed a new computational scheme based on total variation regularization for mixed Poisson–Gaussiannoise. Our proposed scheme allows us to avoid sign-changing of the solution during the optimization processand guarantees the reconstructed image to be positive in the image domain. Therefore, the proposed algorithmcan lead to mixed Poisson–Gaussian noise removal with good results. The comparison of the results achieved bythe proposed algorithm with the other methods has shown that the proposed algorithm perceptibly improvesthe quality of the denoised images with mixed Poisson–Gaussian noise. Moreover, it has been shown that theproposed scheme has approximately the same computational time as the other methods.
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Table 5. The mean results of the methods for all test images.
Our algorithm has the advantage of being rather easy to implement and offers a good compromise betweenperformance and quality of results. Like similar algorithms, no previous training is required (in contrast to deeplearning techniques), and only one observation of the image is needed (in contrast to the EM approaches [24]).The user can tweak parameters λ and β if she precisely knows the noise distribution. One disadvantageintrinsic to our approach is that the Poisson and Gaussian noise are assumed to have the same variance. Othermodels could be designed to overcome this obstacle, probably at the cost of ease of implementation. Moreover,parameter-tweaking might be required in some specific cases.
Acknowledgments
The authors would like to thank the anonymous referees for their helpful remarks and suggestions. The authorswere supported by Russian Academic Excellence Project ‘5-100’ at the National Research University HigherSchool of Economics (HSE), Moscow, Russia.
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Appendix: Detailed values of experimental results.
Appendix A: Detailed values used to compute Table 3 with σG = Imax/10 .
