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ОБРАБОТКА ИНФОРМАЦИИ И УПРАВЛЕНИЕ /
UDC 004.93 Articles
doi:10.31799/1684-8853-2021-5-10-19
Combined total variation of first and fractional orders for Poisson noise removal in digital images
Pham Cong Thanga, PhD, Lecturer, orcid.org/0000-0002-6428-102X, pcthang@dut.udn.vn Tran Thi Thu Thaob, MSc., Lecturer, orcid.org/0000-0001-7705-2405 Pham Minh Tuana, PhD, Lecturer, orcid.org/0000-0001-9843-9676 Nguyen Thanh Conga, MSc., Specialist, orcid.org/0000-0002-8060-0238
aThe University of Danang — University of Science and Technology, 54, Nguyen Luong Bang St., 550000, Danang, Vietnam
bThe University of Danang — University of Economics, 71, Ngu Hanh Son St., 550000, Danang, Vietnam
Introduction: Many methods have been proposed to handle the image restoration problem with Poisson noise. A popular approach to Poissonian image reconstruction is the one based on total variation. This method can provide significantly sharp edges and visually fine images, but it results in piecewise-constant regions in the resulting images. Purpose: Developing an adaptive total variation-based model for the reconstruction of images contaminated by Poisson noise, and an algorithm for solving the optimization problem. Results: We proposed an effective way to restore images degraded by Poisson noise. Using the Bayesian framework, we proposed an adaptive model based on a combination of first-order total variation and fractional order total variation. The first-order total variation model is efficient for suppressing the noise and preserving the keen edges simultaneously. However, the first-order total variation method usually causes artifact problems in the obtained results. To avoid this drawback, we can use high-order total variation models, one of which is the fractional-order total variation-based model for image restoration. In the fractional-order total variation model, the derivatives have an order greater than or equal to one. It leads to the convenience of computation with a compact discrete form. However, methods based on the fractional-order total variation may cause image blurring. Thus, the proposed model incorporates the advantages of two total variation regularization models, having a significant effect on the edge-preserving image restoration. In order to solve the considered optimization problem, the Split Bregman method is used. Experimental results are provided, demonstrating the effectiveness of the proposed method. Practical relevance: The proposed method allows you to restore Poissonian images preserving their edges. The presented numerical simulation demonstrates the competitive performance of the model proposed for image reconstruction. Discussion: From the experimental results, we can see that the proposed algorithm is effective in suppressing noise and preserving the image edges. However, the weighted parameters in the proposed model were not automatically selected at each iteration of the proposed algorithm. This requires additional research.
Keywords — total variation, image restoration, Poisson noise, minimization method.
For citation: Pham C. T., Tran T. T. T., Pham M. T., Nguyen T. C. Combined total variation of first and fractional orders for Poisson noise removal in digital images. Informatsionno-upravliaiushchie sistemy [Information and Control Systems], 2021, no. 5, pp. 10-19. doi:10.31799/1684-8853-2021-5-10-19
Introduction
Poisson noise is known as photon noise or shot noise [1]. Poisson noise removal is an important task in various applications such as electronic microscopy [2, 3], tomography [4, 5], X-ray [6, 7], etc. In electronic microscopy imaging, the number of electrons collected to create an image pixel follows the Poisson distribution [8]. It means that parts of an image with higher intensity have a higher noise level than parts with low intensity. A low signal-to-noise ratio leads to a high level of noise, which affects detected edges and accuracy in measurement [9, 10]. With the electron tomography technique, data acquisition is the detection of scattered electron particles. The emission and subsequent detection of electrons follow a Poisson process [11]. These processes degrade data quality due to Poisson noise [12, 13]. Meanwhile, X-ray is a popular medical imaging technique. X-rays are produced using photons, which have very small wavelengths and high
penetration ability [14]. X-ray images are produced using a digital receptor. The photon scattering follows Poisson noise distribution which leads to degradation of the X-ray image quality and to the difficulty of diagnosis [15].
For many years, many variational methods have been proposed to handle the restoration problem with Poisson noise [16-23]. One of variational models for Poissonian image reconstruction is the one based on the total variation (TV) norm as regularization term [24]:
min(|| ^+P<1, Kz- f logKz)), (1)
zeQ
where p is a positive regularization parameter; z must be positive in Q; K is a blurring operator, the operator ||z||TV stands for the TV of u. Authors in [25] replaced the TV regularization term in (1)
X 2
by || z\\tv +—1| z |2 and proposed bounded varia-
tion-based Poissonian images restoration model as follows (TV-model):
z* = argmin ||| z | \TV +— 11 z \ || +P<1, Kz-flog Kz) ]. (2)
zeQ V 2 )
The model (2) performs very well for preserving edges while removing noise. However, it often causes undesired artifact effects in smooth regions. To overcome these effects, some high-order models have been introduced for restoring blurred images corrupted by Poisson noise. The authors in [26] replaced the term \\z\\TV in (2) with higher-order ||z||_HW and proposed following model (HTV-model):
*
z = arg min
X,
zeQ
lz I \HTV + 2
|z\\2 +p<1, Z-flogZ>|. (3)
Recently, fractional-order derivatives are widely applied in image processing [27-31]. The works have reflected the good performance of the fractional-order derivative in image denois-ing with edge-preserving. Following [29], the fractional-order TV model for Poissonian image denoising and deblurring is as follows (FTV-model):
z = argmin(\\z \\tv + p<1, Kz -f log Kz)), (4)
where p is positive parameter; !\z\Tv stands for fractional-order TV defined in Eq. (10).
Motivated by the above studies, we introduce an adaptive TV based optimization problem as follows:
*
z = argminE(z), (5)
zeS(Q)
z = argmin(^i \ \z \ \v + \ \z \ Tva +
zeQ
+ -2 \\z \ 22 +p<1, Kz - f log Kz)),
where p1, p2 and p are positive paramters,
P2 e (0, 1).
Motivated by the previous works, we propose a Poisson noise removal model that can substantially reduce artifact effects while preserving edges in the restored images. The proposed model is designed by combining advantages of the first order TV and fractional order TV. We extend the split Bregman method for solving the optimization problem. Furthermore, we provide experimental results to demonstrate the efficiency of our algorithm for the considered problem, in comparison with state-of-the-art methods.
Preliminaries
We recall the principle behind Eq. (5). We aim at reconstructing the original image u with the known noisy image f. Our strategy is to find the image u which maximizes the conditional probability P(u\f). Bayes's rule gives
P(z | f ) =
P(f I z)P(z) P(f) '
(6)
The probability density function of the observed image f corrupted by Poisson noise is:
P(f I z) =
zf ex
p(-z )
f!
Suppose that f has size M x N, and let J = {1, „., M} x {1, ..., N} denote the domain of f. For i e J, we write ft the pixel of f at position i (and similarly ui the pixel of u at position i) [32]. Then:
P(f I z) = n
ieI
(zi )fi e~zi]
fi!
Maximizing P(z\f) is equivalent to minimizing -log(P(z|f)), so let us compute the quantity -log(P(f|z)):
£zi - f. log(Zi ) + logfi!).
ieI
(7)
Since f is constant, we can ignore the term log(fj!). Now we assume that P(z) follows a choice of the prior:
P(z) = exp| --(^(z)
(8)
where x is a normalization factor being positive and constant.
In this work, we assumpt that
^(z) = Ml \\z \ TV +H-2 \\z \ \TV \\z \2 .
a 2
The assumption on P(u) means that each pixel depends (weakly) on the neighbouring pixels only, so we do not lose much by assuming independence.
We now have all the ingredients to maximize P(z\f). By Eq. (6), this amounts to minimize the expression -log(P(f\z)) - log(P(z)), so we can plug in Equations (7) and (8) to get:
z* = argmin^f 11^(z;) + (z; - fi log(z;)) 1
z ieIVx )
u
and we can view this expression as a discrete approximation of the functional E(-) defined as
E(z) = [Vi 11 z | |tv + ^2 11 z | |\Va+f |\ z 12 + p<l, z-flogz)
where p = x is positive and constant parameter.
In case of the blur effect, we can generalize the model (5) for restoring a blurred image corrupted by Poisson noise as follows:
E(z) = (^1 Ilz I TV +^2 Ilz I Tvf
+a 112 112 + P<i,Kz - f log Kz)
where p is positive and constant parameter.
The discrete gradients of an image u for the pixel location (i, j) in z (i = 1..M; j = 1..N) are defined like [33-35]:
| |z | \tv =V(Viz)2 + (V2z)2, (9)
V1zi,j = zi+1,j - zi,j' V2zi,j = zi,j+i - zi,j, V11zi,j = zi+1,j - 2zi,j + zi-1,j' V22zi,j = zi,j+1 - 2zi,j + zi,j-1' V12zi,j = V21zi,j = zi,j - zi,j-1 - zi-1,j + zi-1,j-1'
| |z | IhTV ^(Vnz)2 + (V12z)2 + (V21z)2 + (V22z)2 .
Due to the convenience in numerical implementation, from Greunwald — Letnikov (GL) fractional-order derivative, the discrete fractional-order TV | | z | \V of z e Q is defined as follows [27, 31]:
I ITV„
^(Vfz)2 + (Vf z)2
(10)
where the discrete gradients Vfz and Vfz are defined as follows:
L-1 L-1
(V1 z)i,j — X Cf zi-k,j , (V2 z)i,j — X Cf Zi,j-k • k—0 k—0
Parameter L is the number of neighboring pixels that are used to compute the fractional-order derivative at each pixel; the coefficients {Cf} are defined as follows [27]:
Ck — :
T(a +1)
r(k + 1)r(a +1 - k)
k — 1.. L -1.
Computational method
In this section, we derive the numerical method for the problem (5) in detail. There are many meth-
ods which can be employed to obtain the solution of the optimization problem. In this article, we decide to employ the split Bregman method for solving the optimization problem.
The split Bregman method performs break the minimization problem down into easy subproblems [36-38]. Subproblems can be directly solved with tools like fast Fourier transform (FFT), shrinkage operator that makes the optimization algorithm rather fast. We have a scalar y and two convex func-tionals T(-) and G(-); and that we need to solve the following constrained optimization problem:
argmin(|H|+yG(z)), s.t.w = T(z). (11)
z,w
We convert (11) into an unconstrained problem: argmin I 11 w | |+yG(z) + - |\d-T (z) - b 12 I, (12)
z,w V 2 J
where w — splitting variable; p is positive constant parameters; b — variable of the Bregman iterations.
The solution to problem (12) can be approximated by the Split Bregman method:
z(k+1) — argmin^yG(z) + 2 IIw(k) -Y(z)-b(k) ||| J; — argminI I WI1 +— IW- T(z(k++-b(k) || |,
w V 2 J
b(k+1)— b(k) +¥(z(k+1)) -w(k+1) •
We return to the problem (5). By introducing three auxiliary variables u, p and q, Eq. (5) is equivalent to the constrained optimization problem:
mm
p,q,z,u
MI pI I + mI I qI K-1 Iz 12 + p<1, u - f log u> J
s.t. p — Vz, q — Vaz, u — Kz, (13)
where Vz and V*z are defined in (9) and (10) respectively.
We convert the problem (5) to the unconstrained one as follows:
min
p,q,z,u
11 p 11 + H-211 qI I + 2 I Iz 112 +
+ P<1, u-f log u> + —1 11 p-Vz 112 +
+ £2 11 q - Vaz 12 +P3 I Iu - Kz 12
(14)
where p2, p and y are positive parameters; p1, p2 are Lagrangian multipliers.
The extended split Bregman iterative method employed for solving the problem (5) can be described as follows:
f 11 *i 2I I p(k) -vz - ц +л
(k+1) „. • z ' = argmin
+ <2 | |q(k)-Va z - b^I |22 + + ||u(k) - Kz - bZk)| |2
(k+l) „. • u ' = arg min
P(1, u - f log u} +
Л
+ <2 11 u - ^z(k+1) - bZk)| 22
V 2
^ = argmin L 11 p| | +| | p - Vz(k+1) - b(pk) | Ц
p
q(k+1) = argmin ^2 | |q| | + f | к -V« z(k+1) - | Ц
with update for b(pk+1), b^k+1), b^k+1) :
bpk+1)= b(pk) + Vz(k+1) - p(k+1) bqk+1) = bqk) + Vaz(k+1)-q(k+1). bZk+1)= bzk) + Kz(k+1) - u(k+1)
Hence, we have three subproblems to solve: z, p and q.
Subproblem z. For the z subproblem, optimality condition reads:
9z + Pl (V)T (Vz + bp) - p(k)) + + P2(Va)T(Vaz + bf -q(k)) = 0.
Therefore, we have
(6 + pi(V)T V + pn(Va )T V« )z = = Pl(V)T(p(k)+ p2(Va)T(q(k) -bqk)). (15)
The Eq. (15) can be solved efficiently with one Fourier transform operation and one inverse FFT operation as follows:
z(k+i) = F -l
F(pi(V)T(p(k) - b(pk)) + + P2(Va )T (q(k) -bqk))) 6 + piF((V)T V) + P2 F((Va )T Va )
(16)
where F and F"1 are the forward and inverse Fourier transform operators.
Subproblem u. For the u subproblem, optimality condition reads:
P+ P3(u-Kz(k+1) -bzk)) = 0. u
At the (k + 1)-th iteration, we compute u by discretization scheme:
,(k+i)
- f
,(k)
+ p3(u(k+1) -Kz(k+1) -bzk)) = 0.
Therefore, we have
u(k+i) = Pf + p3U(k)(Kz(k+1) + bzk)) P + p3U(k) .
Subproblems p and q. The solution of the p subproblem can readily be obtained by applying the soft thresholding operator:
,(k+1) = shrinkVz(k+1) + b(nk), К-Л
p2
Vz(k+1) + bpk) | Vz(k+1) + b(pk)| '
| Vz(k+1) + b(pk) | , 0
pl
. (18)
The solution of the q subproblem can also be obtained by applying the soft thresholding operator:
q(k+1) = shrinkfVau(h+1) + b(pk), '
V p2 J
Vau(k+1) + b(pk) |Vau(k+1) + b(k)\
Vau(k+1) + , 0 . (19)
p2 J
The complete method is summarized in Algorithm 1.
Algorithm 1: Adaptive split Bregman method for solving the problem (5)
1. Initialize: z(°) = f; p(0) = q(0) = 0; b(„0) = b{n0) =
= b(V; k = 1 " 9
II ,,(k) - „(k-1) M
2. while (M---^ <E) II (k <Nmax) do
II u 12
3. Calculate z(k+1) using (16)
4. Calculate u(k+1) using (17)
5. Calculate p(k+1) using (18)
6. Calculate q(k+1) using (19)
7. bpk+1)= bk + Vz(k+1) -p(k+1)
8. b(k+1) = b(k) +Vaz(k+1) (k+1)
z
u
9. bk + Kz(k+1) - uk1)
10. k = k + 1
11. endwhile
12. return z
Experimental results
In this section, we present some numerical results to illustrate the performance of the proposed model for Poisson noise removal. In order to prove the efficiency of the proposed model, we compare our reconstruction results with those of the mentioned models: TV-model, HTV-model and FTV-model. The compared models are implemented by the split Bregman method. We performed all experiments under MATLAB and Windows 10 on a PC with an Intel Core (TM) i5 CPU at 2.4 GHz and 8 GB of RAM. Empirically, all images are processed with the equivalent parameters a = 1.5, = 0.6, p2 = 0.4, = 0.01, p2 = 0.01, p3 = 0.01, 9 = 0.0001. We set the stopping condition for Algorithm 1: e = 0.00004 and N = 500. The observed images in our experiments are simulated as follows. Poisson noise is data dependent, the noise level of the observed images depends on the pixel intensity value. To test different noise levels, the noisy image is
simulated by adding Poisson with some fixed value Peak. The test images are shown in Fig. 1.
The peak signal-to-noise ratio (PSNR) used in comparison are defined as follows:
(
PSNR = 10log
10
A
2552 • MN 2~~
2 y
u - u
where M and N are the number of image pixels in rows and columns; u, u* are the original image, the reconstructed or noisy image accordingly. We also use other popular measure called structural similarity index measure (SSIM) which allows us to get more consistent with human visual characteristics [39]:
SSIM(u, u ) =
(2^u* + c1)(2CTu,u* + c2 ) (M>u + + c1)(CTu +al* + c2)
where |u, |u* are the means of u, u* respectively; au, ctu* — their standard deviations; au u* — the covar-iance of two images u and u*; cv c2 are positive constants.
We first deal with Image denoising. In this case, K is an identity matrix. In Figs. 2 and 3, we aim
■ Fig. 1. Test images: a — Lena; b — Man; c — Aerial; d — MRI
■ Fig. 2. Image "Lena". Recovered images of different methods for image denoising with Peak = 255: a — original image; b — noisy image; c — TV; d — HTV; e — FTV; f — ours
■ Fig. 3. Image "Lena". Recovered images of different methods for image denoising with Peak = 100: a — original image; b — noisy image; c — TV; d — HTV; e — FTV; f — ours
OEPAEOTKA MHOOPMAUMM M YnPABAEHME
to restore the image "Lena" corrupted by Poisson noise with Peak = 255 and Peak = 100, respectively. In these figures, we also present the zoom-in on small details of the recovered images.
We see that TV-model is efficient for removing noise and simultaneously preserving the edges. However, the numerous artifact exists in result image recovered by TV-model. Meanwhile, the HTV-model and FTV-model can lead to edge blurring when image denoising. The images recovered by our model are more smooth and distinct than those of another three approaches.
In Figs. 4, 6 and 8, we show the results of compared methods for noise levels Peak = 100. In Figs. 4, b, 6, b, and 8, b, we represent the noisy images. In the others, Figs. 4, c-f, 6, c-f, 8, c-f, we show respectively the reconstructions given by TV, HTV, FTV and our proposed approach. In Figs. 5, 7 and 9, we show the zoomed details of the original images, observed images and the zoomed details of the re-
c)
■ Fig. 4. Image "Aerial". Recovered images of different methods for image denoising with Peak = 100: a — original image; b — noisy image; c — TV; d — HTV; e — FTV; f — ours
■ Fig. 5. Image "Aerial". The zoomed-in details of the recovered images in Fig. 4: a — original image; b — noisy image; c — TV; d — HTV; e — FTV; f — ours
stored images respectively in Figs. 4, 6 and 8. From the details in Figs. 5, 7 and 9, we can see that the our model can get better visual improvement than the others. In Tables 1 and 2, we show the comparison results in terms of SSIM and PSNR (the best result is highlighted in bold). We can clearly see that our method outperforms the other relative methods for restoring images damaged by Poisson noise.
In the case of image deblurring and denoising, we consider blurred images degraded by Poisson noise. For simulation, we use the Gaussian blur with a window size 5 x 5 and standard deviation of 1. After the blurring operation, we degrade the images by Poission noise with Peak = 100. In Fig. 10, we perform simultaneously image deblurring and denoising on image "Lena". Fig. 10, b denotes corrupted image. In Fig. 10, c-f, we show respectively the reconstructions given by TV, HTV, FTV and our approach.
■ Fig. 6. Image "Man". Recovered images of different methods for image denoising with Peak = 100: a — original image; b — noisy image; c — TV; d — HTV; e — FTV; f — ours
■ Fig. 7. Image "Man". The zoomed-in details of the recovered images in Fig. 6: a — original image; b — noisy image; c — TV; d — HTV; e — FTV; f — ours
■ Fig. 8. Image "MRI". Recovered images of different methods for image denoising with Peak = 100: a — original image; b — noisy image; c — TV; d — HTV; e — FTV; f — ours
■ Fig. 9. Image "MRI". The zoomed-in details of the recovered images in Fig. (8): a — original image; b — noisy image; c — TV; d — HTV; e — FTV; f — ours
■ Table 1. PSNR values for recovered images given by the compared methods with various noisy levels
■ Table 2. SSIM values for recovered images given by the compared methods with various noisy levels
Noise level Peak PSNR
Noisy TV HTV FTV Ours
Lena
255 26.9200 31.1786 30.9261 31.0504 31.8640
100 22.8909 29.1995 28.8624 29.2134 29.4694
Man
255 27.6436 29.6582 29.5528 29.8153 30.1140
100 23.5718 26.5633 26.2971 26.7400 26.8960
Aerial
255 26.7701 28.4530 28.2417 28.4966 29.006
100 22.6521 25.8654 25.7517 25.6930 26.2966
MRI
255 30.1130 30.7451 30.7111 30.8487 30.9878
100 26.9175 28.9371 28.9151 28.9780 29.2772
Noise level Peak SSIM
Noisy TV HTV FTV Ours
Lena
255 0.6721 0.8794 0.8812 0.8780 0.8891
100 0.5234 0.8368 0.8423 0.8411 0.8505
Man
255 0.8028 0.8770 0.8731 0.8713 0.8835
100 0.6578 0.7830 0.7784 0.7862 0.7986
Aerial
255 0.8277 0.8649 0.8585 0.8645 0.8805
100 0.7154 0.7787 0.7639 0.7647 0.7882
MRI
255 0.9025 0.9401 0.9427 0.9344 0.9484
100 0.8416 0.9107 0.9114 0.9095 0.9272
■ Fig. 10. Image "Lena". Recovered results for the test images: a — original image; b — noisy image; c — TV; d — HTV; e — FTV; f — ours
■ Fig. 11. Image "Lena". The zoomed-in details of the recovered images in Fig. 10: a — original image; b — noisy image; c — TV; d — HTV; e — FTV; f — ours
■ Table 3. PSNR values for recovering blurred images with noisy level Peak = 100
Image PSNR
Noisy TV HTV FTV Ours
Lena 20.6275 22.7049 22.5994 22.8684 23.3133
Man 19.4880 20.5501 20.0390 20.4229 20.6916
Aerial 18.3362 18.7176 18.9762 18.8378 19.4537
MRI 24.3167 24.9252 24.231 24.6373 25.0166
■ Table 4. SSIM values for recovering blurred images with noisy level Peak = 100
Image SSIM
Noisy TV HTV FTV Ours
Lena 0.4354 0.7503 0.7190 0.7647 0.7739
Man 0.4748 0.5702 0.5432 0.5853 0.5994
Aerial 0.5220 0.6047 0.5430 0.6074 0.6214
MRI 0.7956 0.8642 0.8527 0.8690 0.8760
Meanwhile, Fig. 11 shows enlarged images recovered by four methods in Fig. 10. In Tables 3 and 4, we report the quantitative measures of PSNR and SSIM values for different images and compared methods. The Figures and Tables demonstrate again the effectiveness of our proposed method for image reconstruction under Poisson noise even in presence of blur.
Conclusions
In this paper, we have researched the hybrid regularizers model, combining the fractional-order and first-order TV for denoising images corrupted by Poisson noise. Computationally, an extended split Bregman method is employed for solving the proposed optimization problem. Finally, compared with the existing state-of-the-art models, the ex-
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Acknowledgments
Pham Cong Thang (corresponding author) would like to thank his colleagues at IT Faculty, DUT, for their helpful comments. The authors also thank reviewers and Editors for their insightful comments and suggestions.
Financial support
This work is supported by The Murata Science Foundation and The University of Danang, University of Science and Technology, code number of Project T2020-02-07MSF.
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УДК 004.93
doi:10.31799/1684-8853-2021-5-10-19
Комбинированная полная вариация первого и дробного порядка для удаления пуассоновского шума на цифровых изображениях
Фам Конг Тханга, PhD, преподаватель, orcid.org/0000-0002-6428-102X, pcthang@dut.udn.vn Чан Тхи Тху Тхаоб, магистр, преподаватель, orcid.org/0000-0001-7705-2405 Фам Минь Туана, PhD, преподаватель, orcid.org/0000-0001-9843-9676 Нгуен Тхань Конга, магистр, специалист, orcid.org/0000-0002-8060-0238
^Университет Дананга — Университет науки и техники, Нгуен Лыонг Банг, 54, Дананг, 550000, Вьетнам
бУниверситет Дананга — Университет экономики, Нгу Хань Шон, 71, Дананг, 550000, Вьетнам
Введение: известно множество методов для решения проблемы восстановления изображений с помощью пуассоновского шума. Одним из популярных подходов к реконструкции пуассоновского изображения является метод, основанный на полной вариации. C помощью этого метода можно получить весьма резкие границы и визуально четкие изображения, но он приводит к образованию кусочно-постоянных областей в результирующих изображениях. Цель: разработка адаптивной модели на основе полных вариаций для реконструкции изображений, искаженных пуассоновским шумом, и алгоритма решения задачи оптимизации. Результаты: предложен эффективный метод для восстановления изображений, искаженных пуассоновским шумом. На базе байесовской структуры предложена адаптивная модель, основанная на комбинации полной вариации первого порядка и полной вариации дробного порядка. Восстановление изображения на основе модели полной вариации первого порядка эффективно для шумоподавления и одновременно сохранения острых границ. Однако метод полной вариации первого порядка обычно вызывает проблемы с артефактами в полученных результатах. Чтобы избежать этого недостатка, использованы модели полной вариации высокого порядка, одна из которых является основанной на полной вариации дробного порядка для восстановления изображений. В модели полной вариации дробного порядка производные имеют порядок больше или равный единице. Это приводит к удобству вычислений с компактной дискретной формой. Но методы, основанные на полной вариации дробного порядка, могут вызвать размытие изображения. Таким образом, предложенная модель включает в себя преимущества двух моделей регуляризации полной вариации и существенно влияет на восстановление изображения с сохранением границ. Для решения рассматриваемой оптимизационной задачи используется метод Split Bregman. Приведены экспериментальные результаты, демонстрирующие эффективность предложенного метода. Практическая значимость: предлагаемый метод позволяет восстанавливать пуассоновские изображения с сохранением границ. Представленное численное моделирование наглядно демонстрирует конкурентоспособность предложенной модели для восстановления изображений, деградированных пуассоновским шумом. Обсуждение: из результатов экспериментов видно, что предложенный алгоритм эффективен в шумоподавлении и сохранении границ изображения. Однако взвешенные параметры, существующие в предложенной модели, не выбирались автоматически на каждой итерации предложенного алгоритма, что требует дополнительных исследований.
Ключевые слова — полная вариация, восстановление изображения, пуассоновский шум, метод минимизации.
Для цитирования: Pham C. T., Tran T. T. T., Pham M. T., Nguyen T. C. Combined total variation of first and fractional orders for Poisson noise removal in digital images. Информационно-управляющие системы, 2021, № 5, с. 10-19. doi:10.31799/1684-8853-2021-5-10-19
For citation: Pham C. T., Tran T. T. T., Pham M. T., Nguyen T. C. Combined total variation of first and fractional orders for Poisson noise removal in digital images. Informatsionno-upravliaiushchie sistemy [Information and Control Systems], 2021, no. 5, pp. 10-19. doi:10.31799/1684-8853-2021-5-10-19
