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ОБРАБОТКА ИНФОРМАЦИИ И УПРАВЛЕНИЕ /
UDC 004.93 Articles
doi:10.31799/1684-8853-2021-2-20-32
Second-order total generalized variation based model for restoring images with mixed Poisson - Gaussian noise
Pham Cong Thanga, PhD, Lecturer, orcid.org/0000-0002-6428-102X, [email protected] Tran Thi Thu Thaob, M. Sc., Lecturer, orcid.org/0000-0001-7705-2405 Nguyen Thanh Conga, M. Sc., Specialist, orcid.org/0000-0002-8060-0238 Vo Duc Hoanga, PhD, Lecturer, orcid.org/0000-0002-6974-9023
aThe University of Danang — University of Science and Technology, 54 Nguyen Luong Bang Street, Danang 550000, Vietnam
bThe University of Danang — University of Economics, 71 Ngu Hanh Son Street, Danang 550000, Vietnam
Introduction: A common problem in image restoration is image denoising. Among many noise models, the mixed Poisson — Gaussian model has recently aroused considerable interest. Purpose: Development of a model for denoising images corrupted by mixed Poisson — Gaussian noise, along with an algorithm for solving the resulting minimization problem. Results: We proposed a new total variation model for restoring an image with mixed Poisson — Gaussian noise, based on second-order total generalized variation. In order to solve this problem, an efficient alternating minimization algorithm is used. To illustrate its comparison with related methods, experimental results are presented, demonstrating the high efficiency of the proposed approach. Practical relevance: The proposed model allows you to remove mixed Poisson — Gaussian noise in digital images, preserving the edges. The presented numerical results demonstrate the competitive features of the proposed model.
Keywords — image denoising, total variation, minimization, mixed Poisson — Gausian noise.
For citation: Pham C. T., Tran T. T. T., Nguyen T. C., Vo D. H. Second-order total generalized variation based model for restoring images with mixed Poisson — Gaussian noise. Informatsionno-upravliaiushchie sistemy [Information and Control Systems], 2021, no. 2, pp. 20-32. doi:10.31799/1684-8853-2021-2-20-32
Introduction
Image denoising is an important task in digital image processing. During the formation procedure, the image is usually degraded by noise. The denoising problem is to recover u from an observed image f with the size of M x N. In literature, many types of noise generated by different devices and processes have been considered, e. g., Gaussian [1], Poisson [2], as well as mixed noise, e. g., mixed Poisson — Gaussian [3]. In practical, the Poisson — Gaussian model can accurately describe the noise present in a number of imaging applications such as astronomy, medicine, biology, etc... [4, 5]. The Poisson component accounts for the signal-dependent uncertainty inherent to the photon counting process, and the additive white Gaussian noise component accounts for the other signal-independent noise sources, such as thermal noise [6].
As is well known, several approaches have been developed for recovering images corrupted by the mixed Poisson — Gaussian noise. Among them, one of popular approaches is perhaps total variation (TV) model for mixed Poisson — Gaussian noise removal (TVPG) [7, 8] using the TV norm as regularization term, formulated as follows:
ix +
: argmindx + ~Jq(u - f ) dx
+ PjQ(u -f log и )d x), (1)
where f is the observed image; Q c R2 be bounded open set and u must be positive almost everywhere over Q; X, p are positive regularization parameters.
In literature, we can find many efficient algorithms for solving the TV regularized mixed Poison — Gaussian denoising model (1), such as a primal-dual algorithm [9], an augmented Lagrangian method [10-12], the split Bregman method [13, 14], etc.
As is well known, the TV regularizer framework preserves edges well but has the transformation of smooth regions into piecewise constant regions. To avoid this problem, many regularization techniques for the denoising problem have been introduced, including non-local total variation [15], TV combined with higher-order term [16], Euler's elastic model [17], a mean curvature model [18, 19]. Reccently, a well-known method is the total generalized variation (TGV) introduced as penalty functional for image restoration [20, 21]. TGV includes higher-order derivatives of u. Image reconstructed by TGV regularization usually includes sharp edges and
piecewise polynomial intensities [22]. With simplicity and prominence, the second-order TGV with weight a TGV2^ based models have been widely researched recently, and achieved great successes in image processing [23-25]. Applied for image denoising, the resulting model is given by:
u = arcmin ItGV^ («) + — J (u - ff d x|. (2)
u | 2 ^ )
The model (2) was proposed in [23] for denoising image corrupted by Gaussian noise. Therefore, in case of mixed Poisson — Gaussian noise, the model itself cannot provide necessary accuracy for further data interpretation and analysis.
Inspired by the advantages of TGV^ regularization, we propose an second-order TGV regularized model for the mixed Poisson — Gaussian noise removal problem as follows:
In this paper, we employ the the second-order TGV instead of the standard TV norm in the model (1) and propose the following optimization problem:
u* = argmin I TGVa2 (u)+^ J (u - ff d x +X 2 J (u - f log u)d A (3)
u I 2 ^ ^
where X1 and X2 are positive parameter.
Our main contributions in this paper are following. We introduce a new total variation model for restoring image with mixed Poisson — Gaussian on the basis of the TGV^. The second important advantage is to extend an efficient alternating minimization method for solving the proposed model. Furthermore, we provide experimental results to demonstrate the high efficiency of our algorithm for considered problem, in comparison with related methods.
Proposed method
The denoising model
In this paper, we consider the following optimization problem (3):
u* = argmin | TGV2 (u) +—J (u - ff d x + X2 J (u - f log u)d x).
u I 2 ^ ^
Referring [20, 24], we shortly review the concept of the second-order TGV. The definnitions can be found in Appendix.
Following the Refs. [7, 23-26], we have theorem (Theorem 1) for the considered model.
Theorem 1. The optimization problem (3) has a solution.
Proof: The proof will be given in the Appendix for completeness.
According to [20, 23-25], the discrete TGV2 regularization of u can be formulated as
TGV2 (u) = mina^lWu -w\ 1 + k(w)l1 ,
where w = (wp w2)T; s(w) = (l/2)(vw + VwT ).
The operators e(w) and Vu can be expressed as follows:
Vu =
V1 u V2 u
and e(w ) =
Viwi 2 (V2w1 + V1w2 )
1 (V2wl + Viw2 ) V2w2
w
where V = (V1; V2), V1 and V2 are derivative operators in the horizontal and vertical directions, respectively.
According to the version of TGVa , the discrete version of the minimization problem (3) is given by
= argmin (ai ||Vu - w| 1 +a21 |s(w + 2| |u -f| |2 +ß ^1, u - f log u^ I.
u, w
Computational method
In this section, we derive the numerical method for problem (4) in detail. By the classical augmented Lagrangian multiplier method [16, 17, 19-21], we introduce three new variables (d, g, z) and rewrite the equation (4) in the constrained optimization problem as follows:
min (ai lldl 1 +a21 Igl 1 +-I\z - f\2 +ß (l, z - f log z) I,
u, d,g,zv 1 1 2" 2 x ')
s.t. d = Vu - w, g = e(w), 2 = u
with
d =
and g =
gi g3 g 3 g2
(5)
The augmented Lagrangian functional for the constrained optimization problem (5) is defined as
C (u, w, d, g, z, 9!, |a2, §3 ) = j)i | dll +a2 I |g| 1 +2|z - f| + P( 1, z - f log z)-(e, d-Vu + w) + + |d - Vu + w||2 - (§, g - e(w)) +-y |g - e(w)||2 -(m z - u) + |z - u^ J,
(6)
where r^, r|2, -q3 — positive parameters; 6, Ç, p — with Lagrangian multipliers.
The discrete gradient Vz and the second-order derivatives V2u of an image u for the pixel location (i, j) in u(i = 1...M; j = 1...N) are defined like:
V1uij = ui+1j - uij, V2ui,j = ui+1j - uij \Ui,j = (Vxui,j, Vyui,j), \Vui,j j = ^(Viuy )2 +(V2ui,j )2 .
V
The minimization method to solve the problem (6) can be expressed as follows:
(fe+i) u ' = arg min
-/e«, d(fe) - Vu + w(fe)}+^-
d(fe) - Vu + w
(fe)
-/¿k ) z(fe )-u)
z(fe )- u
21
w
(+1) =
= arg min
w
- (e, d(fe) - Vu(fe+i) + w) + ^ d(fe) - Vu(fe+i) + w
- ^
«-B(w)\+M|g -
<w i2)
d(fe+i) = argmin
g(fe+i) = argmin
z(fe+i) = argmin
ai d 1-e, d -Vu(fe+i) +w(fe+i)+-^1| d-Vu(fe+i)+w(fe+1
«2||g|| i^, g-B^)|g-e(w(+i))H;
V
|z - f| 12 +ß (1, z - f log 4-(,« z - u(fe+1 ^+222.
S- u(fe+i)
2 1
î
2 ) 21
(7)
2
u
d
g
z
ОБРАБОТКА ИНФОРМАЦИИ И УПРАВЛЕНИЕ
with update for |(г+1), „з^1 :
>+!) = e(k) + Л1 (W^1) - d(k+1) - w^);
x(k+1)=M(k ) The u subproblem in (7) is given by:
„ /=^/+^3 ((+1) - Z(k+1)).
= arg min
d(k)-Vu + w(k )
+(3 u 2 > = (1 d(k)-vu + w(k )-e(k ) 2 + (3 Ï )- u -„<* '
2 2 V 2 (1 2 2 (3
Thus, we get
^v1
Vu + d(k)- w(k) (1
+ (3
u +
z(k)
(3
Л
= 0.
(9)
We can rewrite the equation (9) as follows:
((V + Пз )u(k+1)=^VT
(k )
(1
+ (3
(k )
(3
(10)
It is obvious that system (10) is linear and symmetric positive definite, therefore z(k+1) can be efficiently solved by fast Fourier transform [18], under the periodic boundary conditions:
u(k+1)= F-1
^v1
V v
d(k )+ w(k)-e
(k)
(1
+ (3
z(k)-„
(k)
(3
//
(1F (vTv) + ^3
(11)
where F and F-1 are the forward and inverse Fourier transform operators. The w problem is
w(k+l) = arg min f _/0M, d(k)_Vu(k+1) +w) +
(1
d(k)-vu(k+1)
+ w
2-(çM, g«-s(w)} + ^|g-s(w)||2'
2
w +
d(k)-vu(k+1)-e
,(k )
(1
(2
:(w)
- >)+!
(k )
(2
Therefore, we get:
2
w
ni
Jk) V (k+i) + 01 ; ai '+ wi —1—
ni
ViWi - gi +
Si
(k)
n2
f i ^(k) + n2V| -(V2Wi +ViW2)g( + -3—
2 n2
= 0;
ni
a2k)-v2u(k+i)+ w2 2
(k)
+ n2V|
Ak)
V2W2 - g2 +"
ni
= 0.
+ n2VT
-(V2wi + Viw2 )- g( + 2 n2
(k 3
n2
(12)
We have:
nil + n2 VT Vi + n2 V2V2 J wi + n2 VT ViW2 = ni
(
Viu(k+i) - aik)+0i
,(k)
ni
+ n2Vi
gi
Si
(k)
ni
+ n2VT
g(
S(k)A
S3
ni
(
n2VTV2Wi +UiI + V2Vi +n2VTV2 |W2 =ni
V2u(k+i) - a2k) +—
(k)
ni
+ n2V2
g(
S3
n2
+ n2V2l
g2
S2 n2
(13)
From (13), we have a system of linear equations in two unknowns wl
(k+i) w(k+i).
a b c d
(k+i) ,,(k+i)
(14)
with
a = | nil + n2VTVi + 22VTV2 I; b = 22v£v-; c = 22V^;
d = | nil + yVTVi +n2ViV2 |;
s = ni
Viu(k+i) -di(k)+0i
(k )
A
ni
(
+ n2 V2
t = ni
f û(k) V2u(k+i) -d2k)+02-
ni
gi
Si
(k )
A
+ n2 V2
g(
ni
/
(k ^ n2
+ n2V2
+ n2V2
g2
S(k ^ ni
/
(k ^ n2
2
Similar to the u subproblem, we can solve problems (14) with fast Fourier transform, under the periodic boundary conditions:
w-
,(k+i)= F-i
F (sd - bt ) F (ad - cb)
w
,(k+i)= F-i
F(at - cs ) F (ad - cb)
(15)
ОБРАБОТКА ИНФОРМАЦИИ И УПРАВЛЕНИЕ
The d subproblem is given by:
= arg min I a,!
= arg min
d
a1 ||d|| 1 + — U II1 2
d -Vu(k+!)+ w(k+!)-e
(k)
H1
2 A
The solution of the d subproblem can readily be obtained by applying the soft thresholding operator [27]:
(k)
Vu(k+1)- w(k+1)+e
d (k+1) =_Hi
Vu(k+1)- w(k+1)-
H1
•max
Vu(k+1)- w(k+1 )+e
(k)
H1
-01, о H1
The g subproblem is given by:
g(k+1) = arg min
02|HI g-s(w(k+1))+^22 g-s(w(k+1))
= arg min
g
f
a2 i II H2 '«1+11
V
5(w(k+1))-
,(k)
H2
2 A
The solution of the g subproblem can be obtained by applying the soft thresholding operator too:
*(k+1)=.
(+1))-
,(k)
H2
(+1))-
,(k)
H2
• max
:(w (k+1))
(k)
H2
Л
-02, 0 H2
The z subproblem is given by:
z(k+l) = arg min
= argmin
z
+ p( 1, z- f logz)-(p« z-uM^
21 z - f 12+p <1, z - f log ^+H3-
z - fll
f
X
— |z -
2
V
z - u(k
- u(k+1)-P 3
(k)
H3
2 A
Therefore, we get
X(z-f ) + P[l-Z ) + * (z-,(fe+l))-p^fe)=0.
This equation can be rewritten as follows:
(X + ^a )z2 - z+ Ppt) -P + Xf ) - Pf = 0.
(16)
(17)
g
z
The solution of z(k+1) is the positive solution given by:
=
,3 U^1 )+p(k )-P + X/ ^ u(k+1 )+p3" )-P + ^f ] + 4 (T13 +*))/
2(î!3 + X)
The complete method is summarized in Algorithm 1. We need a stopping criterion for the iteration: we end the loop if the maximum number of allowed outer iterations N has been carried out (to guarantee an upper bound on running time) or the following condition is satisfied for some prescribed tolerance ct:
U(k)- U(k-1)
»
-<ç»
(19)
where ct is a small positive parameter.
Algorithm 1: Alternating minimization method for solving the model (5).
1. Initialize: z(0) = u(0) = f; d(0) = g<0) = 0; w(0) = 0; k = 0.
2. While Stopping condition is not satisfied do:
3. Compute u(k+1) according to (11).
4. Compute w(k+1) according to (15).
4. Compute d(k+1) according to (16).
5. Compute g(k+1) according to (17).
6. Compute z(k+1) according to (18).
Î+1 by (8).
7. Update 9(1k+1),
u(k+1) M-2 ,
10. k = k + 1.
11. Endwhile.
12. Return u.
2
2
Numerical experiments
In this section, we present some numerical results to illustrate the performance of the proposed model for MPGN removal. In order to prove the superiority of the proposed model, we compare our results with closely related approaches [8, 23]: the TVPG model (1) and TGV model (2). For compared models, the optimization problem are implemented by the state-of-the-art alternating minimization algorithm. The original test images are shown in Fig. 1, a-d.
All experiments were carried out in Windows 10 and Matlab running on a desktop equipped with an Intel Corei3, 2.1 GHz and 12 GB of RAM. To assess quality of the restoration results, we use peak signal-to-noise ratio (PSNR) defined as follows:
(
PSNR = 10log10
A
2552 • MN 2
2 y
where u, u* are the original image, the reconstructed or noisy image accordingly; M and N are the number of image pixels in rows and columns.
We also use other popular measure called SSIM (structural similarity index measure). The SSIM measure compares local patterns of pixel intensities normalized for luminance and contrast, and allows us to get more consistent with human visual characteristics [28]:
SSIM
(u, u* ) :
■ + c
1 )(2ctu,u* + c2
)
((+^u* + ci )(u +ctu* + c2)
where pu, pu* are the means of u, u* respectively; ctu, ctu* — their standard deviations; ctu u* — the covariance of two images u and u*; c1 = (K1L)2; c2 = (K2L)2, L is the dynamic range of the pixel values (255 for 8-bit grayscale images), and K1 << 1, K2 < 1 are small constants.
For our experiments, we set tolerance in (19): ct = 0.0001 and N = 200. The observed images in our experiments are simulated as follows. To test different noise levels, the noisy images are generated by Poisson noise with some fixed peak /max, and by Gaussian noise with standard deviation CTg. Empirically, all of the compared methods perform image denoising with their optimal parameters. All images are processed with the equivalent parameters X = 0.4, p = 0.6, which gave the best restoration results. For our models, we set r = 5, r|2 = 5 and la = 1.
In Figures 2, a-d and 3, a-d we exhibit the results of compared methods for noise levels 'max = 120, CTg = 5 and /max = 60, CTg = 5.
For a better visual comparison, we show some details of the restored images in Fig. 4 for noise levels /max = 120, CTg = 5, and in Fig. 5 for /max = 60, CTg = 5. In these Figures, we include details of the noisy and original images. It can be seen that our method gives even better visual improvement than the other two methods. For the comparison of the performance quantitatively, the measures of PSNR and SSIM values are reported in Tables 1 and 2. In each of the Tables, we include the PSNR and SSIM values for noisy images and recovered images, and the average results over test images for each method are shown. The better restored results are highlighted in bold.
In Figures 6, a-d and 7, a-d, we also show the results details of compared methods for noise levels 'max = 120, CTg = 10 and 'max = 60, CTg = 10, respectively. We report the PSNR and SSIM values for noisy images and recovered images in Tables 3 and 4. The average results over test images also appear in last row of each table. The better restored results are highlighted in bold.
From Figures, we can see that the images recovered by our proposed model are better quality than those of the compared approaches. Beside, the measurable comparisons reported in Tables 1-4, the our proposed approach gets higher PSNR, SSIM values than those of the TVPG and TGV approaches. It indicates the competitive performance of the proposed method for denoising image corrupted by MPGN.
A ftjlfc \t ' ' j, 'v /i
IF 4
M
■ Fig. 2. Recovered results for the test images with noise level /max = 120, ct g = 0.5: a — Noisy; b — TVPG; c — TGV; d —^urs g
■ Fig. 3. Recovered results for the test images with noise level /max = 60, ct g = 0.5: a — Noisy; b — TVPG; c — TGV; d — Ours g
■ Fig. 4. The zoom-in part of the recovered images in first row and in second row of Fig. 2: a — details of original images; b — details of noisy images; c — details of restored images by TVPG; d — details of restored images by TGV; e — details of restored images by our approach
■ Fig. 5. The zoom-in part of the recovered images in third row and in second row of Fig. 3: a — details of original images; b — details of noisy images; c — details of restored images by TVPG; d — details of restored images by TGV; e — details of restored images by our approach
■ Table 1. PSNR and SSIM values for noisy images and restored images with noise level Imax = 120, ct„ = 5
Image PSNR SSIM
Noisy TGV TVPG Ours Noisy TGV TVPG Ours
Board 20.5670 26.9777 27.1435 27.5823 0.5482 0.7688 0.7749 0.7812
Clock 15.3632 24.2404 25.9160 26.4658 0.36742 0.8856 0.8884 0.8956
Lake 18.6823 24.7286 24.7002 25.7141 0.61996 0.7649 0.7779 0.7864
Head 20.7322 26.9048 27.9500 28.8874 0.60745 0.8624 0.8657 0.8739
Average 18.8362 25.7129 26.4274 27.1624 0.5358 0.8204 0.8267 0.8343
■ Table 2. PSNR and SSIM values for noisy images and restored images with noise level Imax = 60, ct
g
5
Image PSNR SSIM
Noisy TGV TVPG Ours Noisy TGV TVPG Ours
Board 18.6799 24.0460 24.7064 25.1713 0.3871 0.6701 0.6818 0.6931
Clock 13.0537 24.3635 24.4234 25.5345 0.2600 0.8409 0.8423 0.8587
Lake 16.339 22.0954 22.4670 22.8379 0.4735 0.6762 0.6877 0.6920
Head 16.7107 25.2752 25.6161 26.4411 0.5736 0.7724 0.7923 0.8087
Average 16.1958 23.9450 24.3032 24.9962 0.4235 0.7399 0.7510 0.7631
■ Fig. 6. Recovered results for the test images with noise level Imax = 120, CTg = 10: a — Noisy; b — TVPG; c — TGV; d — Ours
■ Fig. 7. Recovered results for the test images with noise level Imax = 60, ct g = 10: a — Noisy; b — TVPG; c — TGV; d — Ours
■ Table 3. PSNR and SSIM values for noisy images and restored images with noise level I x = 120, ct„ = 10
Image PSNR SSIM
Noisy TGV TVPG Ours Noisy TGV TVPG Ours
Board 19.7547 24.7675 25.9887 26.1733 0.4376 0.7255 0.7218 0.7316
Clock 14.5413 22.4326 24.6121 25.9687 0.2980 0.8571 0.8421 0.8749
Lake 17.805 23.3247 23.7328 24.2787 0.5191 0.7249 0.7270 0.7396
Head 16.031 26.3812 26.5097 27.0119 0.6075 0.8154 0.8292 0.8358
Average 17.0330 24.2265 25.2108 25.8582 0.4655 0.78073 0.7800 0.7955
■ Table 4. PSNR and SSIM values for noisy images and restored images with noise level Im = 60, ct„ = 10
Image PSNR SSIM
Noisy TGV TVPG Ours Noisy TGV TVPG Ours
Board 17.5737 23.3837 23.5885 23.8189 0.2566 0.6060 0.6054 0.6215
Clock 12.2833 24.3595 24.2930 24.4320 0.1793 0.7965 0.7726 0.8150
Lake 14.6131 20.8641 20.8629 21.5523 0.3230 0.6018 0.6097 0.6207
Head 14.1531 23.4717 24.2758 24.6904 0.4588 0.7304 0.7386 0.7496
Average 14.6558 23.0198 23.2551 23.6234 0.3044 0.6837 0.6816 0.7017
Conclusions
In this paper, we have investigated a second-order TGV^ based model for denoising image corrupted by MPGN. Computationally, an alternating
minimization algorithm is employed for solving the proposed optimization problem. Finally, compared with several existing state-of-the-art approaches, the experiments demonstrate competitive performance of the proposed method.
Financial support
This work was supported by The University of Danang, University of Science and Technology, code number of Project T2020-02-33.
Appendix
Definition 1 [20, 23-25]. Let Q c R2 be a bound domain, k > 1 and a = (a0, a1) > 0.
Then the total generalized variation of order k with weight a for u £ L1(Q) is defined as the value of the functional:
TGVa2 (u) = sup jjQudiv2Mx|S e C% (q, Sdxd ),
||$|| <an, ||div$|| <a1 L ii iiro u II II^ 1 j
where d denotes the dimension of images, 0% (q, is the space of compactly supported
symmetric d x d matrix fields, Sdxd is the set of all symmetric d x d matrices,
(divS).=V*^, (divV) = X* .
V h dx. V Ii =1,7=1 dx,dxj
d2§i;
The infinite norms of 6 and div6 are given by
IIC=sup (X h.=1 ^
xeQ V
1 |2 ^2
References
1. Pham Cong Thang, Andrei V. Kopylov. Tree-serial parametric dynamic programming with flexible prior model for image denoising. Computer Optics, 2018, vol. 42(5), pp. 838-845.
2. Le T., Chartrand R., Asaki T. J. A variational approach to reconstructing images corrupted by Poisson noise. Journal of Mathematical Imaging and Vision, 2007, vol. 27, pp. 257-263.
3. Li J., Shen Z., Yin R., Zhang X. A reweighted method for image restoration with Poisson and mixed Pois-son-Gaussian noise. Inverse Problems & Imaging, 2015, vol. 9 (3), pp. 875-894.
4. Chouzenoux E., Jezierska A., Pesquet J. C., Talbot H. A convex approach for image restoration with exact Poisson-Gaussian likelihood. SIAM Journal on Imaging Sciences, 2015, vol. 8(4), pp. 2662-2682.
5. Benvenuto F., Camera A. L., Theys C., Ferrari A., Lanteri H., Bertero M. The study of an iterative method for the reconstruction of images corrupted by Poisson and Gaussian noise. Inverse Problems, 2008, vol. 24(3), pp. 35016.
divôL= SUP X 1=1 (divS); (x)
xeQ ' 1 '
2
Definition 2 [20, 23-25]. The space of functions of bounded generalized variation (BGV) is defined as follows:
BGV2 (Q) = ju e L1 (Q) TGV^ (u) < <»},
2 =||u| 1 + TGv2 (u).
Wbgv
BGV2(Q) is a Banach space independent of the weight vector a, TGVa is a seminorm and a convex function in BGV2(Q). Subsequently, we denote the spaces U = C% (Q,M), V = C% (q, M2) and G = 0% (Q, §2x2).
Proof for Theorem 1.
Let u(k) be a bounded minimizing sequence. By the compactness property in the space of bound variation BV(Q), there exists u* £ BV(Q), such that u(k) converges weakly to u* £ BV(Q) and u(k) converges strongly to u* in L1(Q). According to [7, 23-26], we know that the functions TGVa2 (u) and data fidelity term are all lower semi-continuous, proper and convex; and according to Fatou's lemma [29], we have
E(u) > E(u*).
Thus, u* is a minimizer of the optimization problem (4).
6. Lanza A., Morigi S., Sgallari F., Wen Y. W. Image restoration with Poisson-Gaussian mixed noise. Computer Methods in Biomechanics and Biomedical Engineering: Imaging and Visualization, 2014, vol. 2(1), pp. 12-24.
7. Calatroni L., De Los Reyes J., and Schronlieb C. Infi-mal convolution of data discrepancies for mixed noise removal. S/AM Journal on Imaging Sciences, 2017, vol. 10(3), pp. 1196-1233.
8. Pham C. T., Gamard G., Kopylov A., Tran T. T. T. An algorithm for image restoration with mixed noise using total variation regularization. Turkish Journal of Electrical Engineering and Computer Sciences, 2018, vol. 26(6), pp. 2831-2845.
9. Chambolle A. An algorithm for total variation minimization and applications. Journal of Mathematical /maging and Vision, 2004, vol. 20, pp. 89-97.
10. He C., Hu C., Zhang W., Shi B. A fast adaptive parameter estimation for total variation image restoration. /EEE Transactions on /mage Processing, 2014, vol. 23(12), pp. 4954-4967.
11. Huang Y. M., Ng M. K., Wen Y. W. A fast total variation minimization method for image restoration.
Multiscale Modeling and Simulation, 2008, vol. 7(2), pp. 774-795.
12. Wang Y., Yang J., Yin W., Zhang Y. A New alternating minimization algorithm for total variation image reconstruction. SIAM Journal on Imaging Sciences, 2008, vol. 1 (3), pp. 248-272.
13. Goldstein T., Osher S. The split Bregman method for Ll-regularized problems. SIAM Journal on Imaging Sciences, 2009, vol. 2(2), pp. 89-97.
14. Chen H., Wang C., Song Y., and Li Z. Split bregman-ized anisotropic total variationmodel for image de-blurring. Journal of Visual Communication and Image Representation, 2015, vol. 31, pp. 282-293.
15. Kayyar S. H., Jidesh P. Non-local total variation reg-ularization approach for image restoration under a Poisson degradation. Journal of Modern Optics, 2018, vol. 65, pp. 2231-2242.
16. Lysaker M., Lundervold A., Tai X.-C. Noise removal using fourth order partial differential equation with applications to medical magnetic resonance images in space and time. IEEE Transactions on Image Processing, 2003, vol. 12 , pp. 1579-1590.
17. Zhang J., Chen R., Deng C., and Wang S. Fast linearized augmented Lagrangian method for Euler's elastica model. Numerical Mathematics: Theory, Methods and Applications, 2017, vol. 10, pp. 98115.
18. Zhu W., Tai X. C., Chan T. A fast algorithm for a mean curvature based image denoising model using augmented Lagrangian method. In: Efficient Algorithms for Global Optimization Methods in Computer Vision. Lecture Notes in Computer Science. Springer, Berlin, Heidelberg, 2014. Vol. 8293. Pp. 104118.
19. Myllykoski M., Glowinski R., Karkkainen T., Rossi T. A new augmented Lagrangian approach for -mean curvature image denoising. SIAM Journal on Imaging Sciences, 2015, vol. 8(1), pp. 95-125.
20. Bredies K., Kunisch K., Pock T. Total generalized variation. SIAM Journal on Imaging Sciences, 2010, vol. 3(3), pp. 492-526.
21. Bredies L. K, Dong Y., Hintermller M. Spatially dependent regularization parameter selection in total generalized variation models for image restoration. International Journal of Computer Mathematics,
2013, vol. 90(1), pp. 109-123.
22. He C., Hu C., Yang X., He H., Zhang Qi. An adaptive total generalized variation model with augmented Lagrangian method for image denoising. Mathematical Problems in Engineering, 2014, vol. 2014, Article ID 157893, 11 p.
23. Knoll F., Bredies K., Pock T., Stollberger R. Second order total generalized variation (TGV) for MRI. Magnetic Resonance in Medicine, 2011, vol. 65(2), pp.480-491.
24. Guo W., Qin J., Yin W. A new detail-preserving regularity scheme. SIAM Journal on Imaging Sciences,
2014, vol. 7(2), pp. 1309-1334.
25. Liu X. Augmented Lagrangian method for total generalized variation based Poissonian image restoration. Computers & Mathematics with Applications, 2016, vol. 71(8), pp. 1694-1705.
26. Pham C. T., Tran T. T. T., Gamard G. An efficient total variation minimization method for image restoration. Informatica, 2020, vol. 31(3), pp. 539-560.
27. Micchelli C. A., Shen L., Xu Y. Proximity algorithms for image models: denoising. Inverse Problem, 2011, vol. 27(4), 045009.
28. Wang Z., Bovik A. C., Sheikh H. R., Simoncelli E. P. Image quality assessment: From error visibility to structural similarity. IEEE Transactions on Image Processing, 2004, vol. 13, pp. 600-612.
29. Feinberg E. A., Kasyanov P. O., Zadoianchuk N. V. Fatou's lemma for weakly converging probabilities. Theory of Probability & its Applications, 2014, vol. 58(4), pp. 683-689.
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УДК 004.93
doi:10.31799/1684-8853-2021-2-20-32
Модель на основе полной обобщенной вариации второго порядка для восстановления изображений со смешанным пуассоновско-гауссовским шумом
Фам Конг Тханга, PhD, преподаватель, orcid.org/0000-0002-6428-102X, [email protected]
Чан Тхи Тху Тхаоб, магистр, преподаватель, orcid.org/0000-0001-7705-2405
Нгуен Тхань Конга, магистр, специалист, orcid.org/0000-0002-8060-0238
Во Дык Хоанга, PhD, преподаватель, orcid.org/0000-0002-6974-9023
^Университет науки и техники, Нгуэн Лунг Банг, 54, Дананг, 550000, Вьетнам
бУниверситет экономики, Нгу Ханх Сон, 71, Дананг, 550000, Вьетнам
Введение: восстановление изображений играет важную роль в обработке цифровых изображений. Распространенной проблемой восстановления изображений является шумоподавление. В области шумоподавления изображений существует множество моделей шума, одной из них можно назвать модель смешанного пуассоновско-гауссовского шума, которая с недавнего времени вызывает большой интерес. Цель: разработка модели шумоподавления изображений, искаженных смешанным пуассоновско-га-уссовским шумом, и алгоритма для решения результирующей задачи минимизации. Результаты: предложена новая модель полной вариации для восстановления изображения со смешанным пуассоновско-гауссовским шумом на основе полной обобщенной
вариации второго порядка. Для решения рассматриваемой задачи оптимизации применяется эффективный алгоритм чередующейся минимизации. В качестве иллюстрации, в сравнение с родственными методами, представлены экспериментальные результаты, свидетельствующие о высокой эффективности предлагаемого подхода. Практическая значимость: разработанная модель позволяет удалить смешанныый пуассоновско-гауссовский шум на цифровых изображениях с сохранением границ. Приведенные численные результаты демонстрируют конкурентоспособные характеристики предложенной модели для шумоподавления изображений, искаженных смешанным пуассоновско-гауссовским шумом.
Ключевые слова — шумоподавление изображения, полная вариация, минимизация, смешанный пуассоновско-гауссовский шум.
Для цитирования: Pham C. T., Tran T. T. T., Nguyen T. C., Vo D. H. Second-order total generalized variation based model for restoring images with mixed Poisson — Gaussian noise. Информационно-управляющие системы, 2021, № 2, с. 20-32. doi:10.31799/1684-8853-2021-2-20-32
For citation: Pham C. T., Tran T. T. T., Nguyen T. C., Vo D. H. Second-order total generalized variation based model for restoring images with mixed Poisson — Gaussian noise. Informatsionno-upravliaiushchie sistemy [Information and Control Systems], 2021, no. 2, pp. 20-32. doi:10.31799/1684-8853-2021-2-20-32
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