A combined first and fractional order regularization method for mixed Poisson-white spike noisy image restoration

Suhua Wei; Linghai Kong

doi:10.3934/ipi.2023021

Article Contents

2024, Volume 18, Issue 1: 38-61. Doi: 10.3934/ipi.2023021

This issue Previous Article A multi-tasking novel variational model for image decolorization and denoising Next Article An inverse problem for a semi-linear wave equation: A numerical study

A combined first and fractional order regularization method for mixed Poisson-white spike noisy image restoration

Suhua Wei and
Linghai Kong^,

Institute of Applied Physics and Computational Mathematics, China

^*Corresponding author: Linghai Kong
^*Corresponding author: Linghai Kong

Received: August 2022

Revised: March 2023

Early access: May 2023

Published: February 2024

L. Kong and S. Wei are partially supported by National Science Foundation of China (NSFC) under Grant numbers 11571003, 12271053.

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Abstract

In this paper we study the problem of restoring images affected by mixed Poisson-Square Cauchy noise. A multi-convex variational model is derived by integrating infimal convolution likelihood into a process of joint maximum a posteriori estimation, where a modified four-directional fractional-order total variation is united with the first order total variation to characterize the image prior. To solve the proposed model numerically, a block coefficient descent based algorithm is derived, in which variable splitting and alternating direction minimization of multipliers are utilized along with anisotropic diffusion and additive operator splitting to gain efficiency and quality. The obtained numerical results are compared with results obtained from two other total variation regularized models with different fidelities. The numerical discuss confirms the flexibility and validity of the proposed model.

Keywords:

Square Cauchy distribution,
four directional fractional order total variation,
mixed Poisson-white spike noise,
image restoration,
hybrid regularizer.

Mathematics Subject Classification: Primary: 65K10, 65R32; Secondary: 49M25, 49M41.

Citation:

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Figure 1. Line slices of a NR data

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Figure 2. PDFs of standard Gaussian, Square-Cauchy, Cauchy, and Laplace distributions (log domain)

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Figure 3. Original images for numerical experiments

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Figure 4. Image denoising: (a) Noisy image, obtained by adding mixed Poisson-Cauchy noise to Fig. 2(a). (b) An output of Algorithm 2. (c)An output of Algorithm 3. (d) Restored image derived by Algorithm with $ \alpha = 2 $. (e) A restored version of (d) with $ \alpha<2 $

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Figure 5. Image denoising: (a) Noisy image, obtained by adding MPC noise to Fig. 3(b). (b) Output of Algorithm 2. (c) Recovered image obtained by Algorithm 3. (d) Recovered image obtained by Algorithm 1 with $ \alpha>2 $. (e) Recovered image obtained by Algorithm 1 with $ \alpha \leq 2 $

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Figure 6. Image denoising: (a) Noisy image, obtained by adding Poisson noise to Figure 3(c). (b) Restored image by Algorithm 3. (c), (d) Restored images derived by Algorithm 1 with different $ K_d $ values

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Figure 7. Image denoising: (a) Noisy image, obtained by adding truncated Gaussian noise to Figure 3(d). (b)An output of Algorithm 3. (c)A restored image derived by Algorithm 1

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Figure 8. Image denoising: (a) A synthesized image with unknown noise. (b) A restored image using Algorithm 2. (c) A restored image derived by Algorithm 1. (d) the noise map $ w $

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Algorithm 1. Hybrid regularization model for MPS noisy image restoration
Given positive parameters $ \alpha $, $ \beta $, $ \lambda $, $ \tau $, $ \sigma_0^2 $, $ K_d $, $ K_h $, tolerance error $ \epsilon $, and number of iterations $ N $. Initialize $ u^0 = f $, $ \eta^0 $, $ (\sigma^2)^0 $. Set $ w^0 = 0 $, $ \mu_d^0 = \mu_h^0 = 0 $, $ d^0 = 0 $, and $ h^0 = 0 $. For $ \upsilon = 0, 1, 2, \ldots, N $, repeat the following steps:
Step 1. Calculate $ u^{v+1} $ by formuation (53). If $ \\|u^{v+1}-u^v\\|_2<\epsilon \max(1, \\|u^v\\|_2) $, then end the algorithm, otherwise,
Step 2. Update $ \eta $, $ d $, and $ h $ by formulations (57), (60), and (61), respectively.
Step 3. Update $ w $, $ \sigma^2 $ by formulations (72) and (75), or (79) and (80), respectively.
Step 4. Calculate $ \mu^{v+1} $ by formulation (50).
Let $ u^{v} = u^{v+1} $, $ v = v+1 $, and go to the Step 1.

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Table 1. Parameter values for our proposed algorithms

Paras	$\alpha$	$b$	$\lambda$	$\tau$	$\beta$	$\Delta t$	$(\sigma^2)^0$	$K_d$	$K_h$
Values	[1.5, 3)	$10^{-2}$	0.85	$10^{-2}$	0.15	0.025	0.9	$0.0375$	$10^{-4}$

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Table 2. Denoising performance

Image	Noise	SSIM-n	SSIM-t	SSIM-$p_1$	SSIM-$p_2$
Fig. 4	MPS	0.4453	0.8643 (0.8254)	0.9024	0.9790
Fig. 5	MPS	0.3684	0.6498 (0.6119)	0.6725	0.7223
Fig. 6	Poisson	0.5003	0.8633	0.9207	0.9406
Fig. 7	Gaussian	0.5918	0.9408	-	0.9477

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References

[1]	S. Abe and A. K. Rajagopal, Information theoretic approach to statistical properties of multivariate Cauchy-Lorentz distributions, J. Phys. A: Math. Gen., 34 (2001), 8727-8731. doi: 10.1088/0305-4470/34/42/301.
[2]	I. Abraham, R. Abraham, M. Bergounioux and G. Carlier, Tomogrpahic reconstruction from a few views: A multi-marginal optimal transport approach, Appl. Math. Optim., 75 (2017), 55-73. doi: 10.1007/s00245-015-9323-3.
[3]	R. Abraham, M. Bergounioux and E. Trelat, A penalization approach for tomographic reconstruction of binary axially symmetric objects, Appl. Math. Optim., 58 (2008), 345-371. doi: 10.1007/s00245-008-9039-8.
[4]	X. Ai, G. Ni and T. Zeng, Nonconvex regularization for blurred images with Cauchy noise, Inverse Problems and Imaging, 16 (2022), 625-646. doi: 10.3934/ipi.2021065.
[5]	N. E. Alexander and A. H. Lettington, Derivation of the Loretzian probability model for use in constrained image restoration, J. Modern Optics, 52 (2005), 1893-1903. doi: 10.1080/09500340500141854.
[6]	I. S. Anderson, R. L. McGreevy and H. Z. Bilheux, Neutron Imaging and Applications, Springer, 2009.
[7]	T. J. Asaki, R. Chartrand, K. R. Vixie and B. Wohlberg, Abel inversion using total-variation regularization, Inverse Problems, 21 (2005), 1895-1903. doi: 10.1088/0266-5611/21/6/006.
[8]	L. Azzari and A. Foi, Gaussian-Cauchy mixture modeling for robust signal-dependent noise estimation, IEEE International Conference on Acoustic, Speech and Signal Processing (ICASSP), 2014,978-1-4799-2893-4/14, 10.1109/ICASSP.2014.6854626. doi: 10.1109/ICASSP.2014.6854626.
[9]	J. Bai and X.-C. Feng, Fractional-order anisotropic diffusion for image denoising, IEEE Trans. Image Proc., 16 (2007), 2492-2502. doi: 10.1109/TIP.2007.904971.
[10]	B. Bajić, J. Lindblad and N. Sldoje, Blind restoration of images degraded with mixed Poisson-Gaussian noise with application in transmission electron microscopy, in Inter. Sympo. Biomedical Imaging, IEEE (2016), 123-127. doi: 10.1109/ISBI.2016.7493226.
[11]	H. H. Barrett and W. Swindell, Radiological Imaging: The Theory of Image Formation, Detection, and Processing, Academic Press, New York, 1981.
[12]	F. Benvenuto, A. L. Camera, C. Theys, A. Ferrari, H. Lantéri and M. Bertero, The study of an iterative method for the recontruction of images corrupted by Poisson and Gaussian noise, Inverse Problems, 24 (2008), 035016-1-03516-20. doi: 10.1088/0266-5611/24/3/035016.
[13]	M. Bergounioux, Mathematical analysis of a inf-convolution model for image processing, J. Optim. Theory Appl., 168 (2016), 1-21. doi: 10.1007/s10957-015-0734-8.
[14]	M. Bergounioux, I. Abraham, R. Abraham, G. Carlier, E. Le Pennec and E. Trélat, Variational methods for tomographic reconstruction with few views, Milan J. Math., 86 (2018), 157-200. doi: 10.1007/s00032-018-0285-1.
[15]	K. Bredies, K Kunisch and T. Pock, Total generalized variation, SIAM J. Imaging Sci., 3 (2010), 492-526. doi: 10.1137/090769521.
[16]	M. Burger, J. Müller, E. Papoutsellis and C. B. Schönlieb, Total variation regularization in measurement and image space for PET reconstruction, Inverse Problems, 30 (2014), 105003. doi: 10.1088/0266-5611/30/10/105003.
[17]	M. Burger and S. Osher, A guide to the TV Zoo, in Level Set and PDE Based Reconstruction Methods in Imaging (eds. M. Burger and S. Osher), Springer International Publishing House, Switzerland, 2013. doi: 10.1007/978-3-319-01712-9_1.
[18]	J.-F. Cai, R. H. Chan and M. Nikolova, Two-phase approach for deblurring images corrupted by impulse plus Gaussian noise, Inverse Problems and Imaging, 2 (2008), 187-204. doi: 10.3934/ipi.2008.2.187.
[19]	L. Calatroni, J. C. De Los Reyes and C.-B. Schönlieb, Infimal convolution of data discrepancies for mixed noise removal, SIAM J. Imaging Sci., 10 (2017), 1196-1233. doi: 10.1137/16M1101684.
[20]	L. Calatroni and K. Papafitsoros, Analysis and automatic parameter selection of a variational model for mixed Gaussian and Salt-and-Pepper noise removal, Inverse Problems, 35 (2019), 114001. doi: 10.1088/1361-6420/ab291a.
[21]	R. H. Chan, A. Lcnza, S. Morigi and F. Sgallari, An adaptive stragegy for the restoration fo textured images using fracitonal order regularization, Numer. Math. Theor. Meth. Appl., 6 (2013), 276-196. doi: 10.4208/nmtma.2013.mssvm15.
[22]	T. Chan, A. Marquina and P. Mulet, High-order total variation-based image restoration, SIAM J. Sci. Comput., 22 (2001), 503-516. doi: 10.1137/S1064827598344169.
[23]	Y. Chen, W. W. Hager, M Yashiti, X. Ye and H. Zhang, Bregman operator splitting with variable stepsize for total variation image reconstruction, Comput. Optim. Appl., 54 (2013), 317-342. doi: 10.1007/s10589-012-9519-2.
[24]	G. Chen and X. Liu, Cauchy pdf modelling and its application to SAR image despeckling, J. Syst. Engin. Elect., 19 (2008), 717-721. doi: 10.1016/S1004-4132(08)60144-9.
[25]	E. Chouzenoux, A. Jezierska, J.-C. Pesquet and H. Talbot, A convex approach for image restoration with exact Poisson-Gaussian likelihood, SIAM J. Imaging Sci., 8 (2015), 2662-2682. doi: 10.1137/15M1014395.
[26]	M. R. Chowdhury, J. Qin and Y. Lou, Non-blind and blind deconvolution under Poisson noise using fractional-order total variation, J. Math. Image Vis., 62 (2020), 1238-1255. doi: 10.1007/s10851-020-00987-0.
[27]	M. R. Chowdhury, J. Zhang, J. Qin and Y. Lou, Poisson image denoising based on fractional-order total variation, Inverse Prob. Imaging, 14 (2020), 77-96. doi: 10.3934/ipi.2019064.
[28]	S. Das, Functional Fractional Calculus, Springer-Verlag, Berlin Heidelberg, 2011. doi: 10.1007/978-3-642-20545-3.
[29]	J. D. Drummound, The adaptive optics Lorentzian point spread function, SPIE Proceedings, 3353 (1998), 1030-1037. doi: 10.1117/12.321648.
[30]	C. Ekdahl, Characterizing flash-radiography source spots, J. Opt. Soc. Amer. A, 28 (2011), 2501-2509. doi: 10.1364/JOSAA.28.002501.
[31]	L. C. Evans, Partial Differential Equations, AMS, Rhode Island, 1998. doi: 10.1090/gsm/019.
[32]	D. Gabay, Applications of the method of multipliers to variational inequalities, in Augmented Lagrange Methods: Applications to the Solution of Boundary-Valued Problems (eds. M. Fortin, R. Glowinski), North Holland, Amsterdam, 1983,299-331. doi: 10.1016/S0168-2024(08)70034-1.
[33]	M. Ghulyani and M. Arigovindan, Fast roughness minimizing image restoration under mixed Poisson-Gaussian noise, IEEE Trans. Image Proc., 30 (2021), 134-149. doi: 10.1109/TIP.2020.3032036.
[34]	F. K. Golbaghi, M. Rezghi and M. R Eslahchi, A hybrid image denoising method based on integer and fractional-order total variation, Iran J. Sci. Technol. Trans. Sci., 44 (2020), 1803-1814. doi: 10.1007/s40995-020-00977-2.
[35]	Z. Gong, Z. Shen and K.-C. Toh, Image restoration with mixed or unknown noises, MultiScale Model. Simul., 12 (2014), 458-487. doi: 10.1137/130904533.
[36]	K. M. Hanson, Introduction to Bayesian image analysis, in Image Processing (ed. M. H. Loew), Proc. SPIE 1898, (1993), 716-731.
[37]	A. A. Harms and D. R. Wyman, Mathematics and Physics of Neutron Radiography, Springer, Dordrecht, 1986. doi: 10.1007/978-94-015-6937-8.
[38]	A. K. Heller and J. S. Brenizer, Neutron radiography, in Neutron Imaging and Applications, (eds. I.S. Anderson, R. L. McGreevy, H. Z. Bilheux), Springer, New York, 2009. doi: 10.1007/978-0-387-78693-3_5.
[39]	M. Holler and K. Kunisch, On infimal convolution of TV type functionals and applications to video and image reconstruction, SIAM J. Imaging Sci., 7 (2014), 2258-2300. doi: 10.1137/130948793.
[40]	D. S. Hussey, K. J. Coakley, E. baltic and D. L. Jacobson, Improving quantitative neutron radiography through image restoration, Nuc. Inst. Meth. Phys. Res. A, 729 (2013), 316-321. doi: 10.1016/j.nima.2013.07.013.
[41]	M. Idan and J. L. Speyer, Cauchy estimation for linear scalar systems, IEEE Trans. Automat. Control., 55 (2010), 1329-1342. doi: 10.1109/TAC.2010.2042009.
[42]	A. Ito, Recognition of sounds using square Cauchy mixture distribution, in IEEE Int. Conf. Signal and Image Proc. (ICSIP), 2016. doi: 10.1109/SIPROCESS.2016.7888359.
[43]	E. Jonsson, C.-S. Huang and T. Chan, Total Variation Regularization in Positron Emission Tomography, CAM Report 98-48, UCLA, 1998.
[44]	Y. Kaganovsky, S. Han, S. Degirmenci, D. G. Politte, D. J. Brady, J. A. O'Sullivan and L. Carin, Alteranting minimization algorithm with automatic relevance determination for transmission tomography under Poisson noise, SIAM J. Imaging Sci., 8 (2015), 2087-2132. doi: 10.1137/141000038.
[45]	S. H. Kayyar and P. Jidesh, Non-local total variation regularization approach for image restoration under a Poisson degradation, J. Mode. Optics, 65 (2018), 2265-2276. doi: 10.1080/09500340.2018.1506058.
[46]	L. Kong and S. Wei, A variational method for Abel inversion Tomography with mixed Poisson-Laplace-Gaussian noise, Inverse Problems and Imaging, 16 (2022), 967-995. doi: 10.3934/ipi.2022007.
[47]	F. Laus, F. Pierre and G. Steidl, Nonlocal myriad filters for Cauchy noise removal, J. Math. Imaging Vis., 60 (2018), 1324-1354. doi: 10.1007/s10851-018-0816-y.
[48]	T. Le, R. Chartrand and T. J. Asaki, A variational approach to reconstructing images corrupted by Poisson noise, J. Math. Imaging Vis., 27 (2007), 257-263. doi: 10.1007/s10851-007-0652-y.
[49]	A. H. Lettington and Q. H. Hong, Image restoration using a Lorentzian probability model, J. Modern Optics, 42 (1995), 1367-1376. doi: 10.1080/09500349514551201.
[50]	A. H. Lettington, M. P. Rollason, S. Tzimoplulou and E. Boukouvala, Image restoration using a two-dimensional Lorentzian probability model, J. Modern Optics, 47 (2000), 931-938. doi: 10.1080/09500340008235101.
[51]	A. Levin, Y. Weiss, F. Durand and W. T. Freeman, Understanding and evaluating blind deconvolution algorithms, in IEEE Conf. Cmputer Vis. Pattern Recog., 2009, 1964-1971. doi: 10.1109/CVPR.2009.5206815.
[52]	F. Li, C. Shen, J. Fan and C. Shen, Image restoration combining a total variational filter and a fourth-order filter, J. Vis. Commun. Image R., 18 (2007), 322-330. doi: 10.1016/j.jvcir.2007.04.005.
[53]	J. Li, Z. Shen, R. Yin and X. Zhang, A reweighted $\ell^2$ method for image restoration with Poisson and mixed Possion-Gaussian noise, Inverse Problems and Imaging, 9 (2015), 875-894. doi: 10.3934/ipi.2015.9.875.
[54]	Q. Liu and S. Gao, Directional fractional-order total variation hybrid regularization for image deblurring, J. Electron. Imaging, 29 (2020), 033001. doi: 10.1117/1.JEI.29.3.033001.
[55]	X. Liu and L. Huang, Toatal variational variation-based Poissonian images recovery by split Bregman iteration, Math. Meth. Appl. Sci., 35 (2012), 520-529. doi: 10.1002/mma.1588.
[56]	J. Liu, H. Huang, Z. Huan and H. Zhang, Adaptive variational method for restoring color images with high density impulsive noise, Int. J. Comput. Vision, 90 (2010), 131-149. doi: 10.1007/s11263-010-0351-9.
[57]	J. Liu, X.-C. Tai, H. Huang and Z. Huan, A weighted dictionary learning model for denoising images corrupted by mixed noise, IEEE Trans. Image Proc., 22 (2013), 1108-1120. doi: 10.1109/TIP.2012.2227766.
[58]	J. Liu and H. Zhang, Image segmentation using a local GMM in a variational framework, J. Math. Imaging Vis., 46 (2013), 161-176. doi: 10.1007/s10851-012-0376-5.
[59]	F. Luisier, T. Blu and M. Unser, Image denoising in mixed Poisson-Gaussian noise, IEEE Trans. Image Proc., 20 (2011), 696-708. doi: 10.1109/TIP.2010.2073477.
[60]	M. Lysaker, A. Lundervold and X.-C. Tai, Noise removal using fourth-order partial differential equation with application to medical magnetic resonance images in space and time, IEEE Trans. on Image Proc., 12 (2009), 1579-1590. doi: 10.1109/TIP.2003.819229.
[61]	M. Matsubayashi, A. Tsuruno, T. Kodaira and H. Kobayashi, High resolution static imaging system using a cooled CCD camera, Nuclear Instruments and Methods in Physics Research A, 377 (1996), 107-110. doi: 10.1016/0168-9002(96)00126-X.
[62]	G. Mclachlan and D. Peel, Finite Mxiture Models, Wiley, New York, 2000. doi: 10.1002/0471721182.
[63]	J.-J. Mei, Y. Dong, T.-Z. Huang and W. Yin, Cauchy noise removal by nonconvex ADMM with convergence guarantees, J. Sci. Comput., 74 (2018), 743-766. doi: 10.1007/s10915-017-0460-5.
[64]	M. Nikolova, A variational approach to remove outliers and impulse noise, J. Math. Imaging Vis., 20 (2004), 99-120.
[65]	S. Oh, H. Woo, S. Yun and M. Kang, Non-convex hybrid total variation for image denoising, J. Vis. Commun. Image R., 24 (2013), 322-334. doi: 10.1016/j.jvcir.2013.01.010.
[66]	K. Papafitsoros and C. B. Schönlieb, A combined first and second order variational approach for image reconstruction, J. Math. Imaging Vis., 48 (2014), 308-338. doi: 10.1007/s10851-013-0445-4.
[67]	Y. Peng, J. Chen, X. Xu and F. Pu, SAR images statistical modeling and classification based on the mixture of alpha-stable distributions, Remote Sens., 5 (2013), 2145-2163. doi: 10.3390/rs5052145.
[68]	D. Perrone, R. Disthelm and P. Favaro, Blind Deconvolution via lower-bounded logarithmic image priors, in EMMCVPR 2015 (eds. X.-C. Tai), Springer International Publishing, (2015), 112-125. doi: 10.1007/978-3-319-14612-6_9.
[69]	D. Perrone and P. Favaro, A logarithmic image prior for blind deconvolution, Int. J. Comput. Vis., 117 (2016), 159-172. doi: 10.1007/s11263-015-0857-2.
[70]	I. Podlubny, Fractional Differential Equations, Academic Press, New York, 1999.
[71]	H. Rabbani, M. Vafadust, S. Gazor and I. Selesnick, Image denoising exploying a bivariate Cauchy distribution with local variance in complex wavelet domain, in IEEE 12th Digital Signal Processing Workshop, (2006), 203-208. doi: 10.1109/DSPWS.2006.265407.
[72]	A. Rehman, Z. Wang, D. Brunet and E. R. Vrscay, SSIM-inspried image denoising using saprse representations, IEEE ICASSP, (2011), 1121-1124. doi: 10.1109/ICASSP.2011.5946605.
[73]	L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F.
[74]	A. Sawatzky, (Nonlocal) Total Variation in Medical Imaging, Ph. D. thesis, University of Muenster, Germany, 2011.
[75]	A. Sawatzky, C. Brune, T. Kösters, F. Wübbeling and M. Burger, EM-TV methods for Inverse problems with Poisson noise, in Level Set and PDE Based Reconstruction Methods in Imaging (eds. M. Burger and S. Osher), Springer International Publishing House, Switzerland, 2013. doi: 10.1007/978-3-319-01712-9_2.
[76]	F. Sciacchitano, Y. Dong and T. Zeng, Variational approach for restoring blurred images with Cauchy noise, SIAM J. Imaging Sci., 8 (2015), 1894-1922. doi: 10.1137/140997816.
[77]	J. Shao, H. Lu and H. Cai, A point spread function model for X-ray imaging (in Chinese), Acta Opt. Sinica, 25 (2005), 1148-1152.
[78]	X. Shu and N. Ahuja, Hybrid compressive sampling via a new total variation TVL1, Lect. Notes Comput. Sci., 6316 (2010), 393-404. doi: 10.1007/978-3-642-15567-3_29.
[79]	D. L. Snyder, A. M. Hammoud and R. L. White, Image recovery from data acquired with a charge-coupled-device camera, J. Opt. Soc. Am. A, 10 (1993), 1014-1023. doi: 10.1364/JOSAA.10.001014.
[80]	A. Staglianò, P. Boccacci and M. Bertero, Analysis of an approximate model for Poisson data reconstruction and a related discrepancy principle, Inverse Problems, 27 (2017), 125003, 20 pp. doi: 10.1088/0266-5611/27/12/125003.
[81]	G. Steidl, Combined first and second order variational approaches for image processing, Jahresber Dtsch Math-Ver, 117 (2015), 133-160. doi: 10.1365/s13291-015-0113-2.
[82]	W. Treimer, Neutron tomography, in Neutron Imaging and Applications: A Reference for the Imaging Community, (eds. I.S. Andnerson, R. L. McGreevy, H. Z. Biheux), Springer, New York, 2009. doi: 10.1007/978-0-387-78693-3_6.
[83]	Z. Wang, A. C. Bovik, H. R. Sheikh and E. P. Simoncelli, Image quality assessment: From error visibility to structured similarity, IEEE Trans. Image Process., 13 (2004), 600-612. doi: 10.1109/TIP.2003.819861.
[84]	Y. Wang, Q. Li, N. Chen, J.-M. Cheng, Y.-T. Xie, Y.-L. Liu and Q.-H. Long, Spot size measurement of a flash-radiography source using the pinhole imaging method, Chinese Phys. C, 40 (2016), 076202. doi: 10.1088/1674-1137/40/7/076202.
[85]	X. Wang, R. Song, C. Song and J. Tao, The NSCT-HMT model of remote sensing image based on Gaussian-Cauchy mixture distribution, IEEE Access, 6 (2018), 66007-66019. doi: 10.1109/ACCESS.2018.2876447.
[86]	Y. Wang, J. Yang, W. Yin and Y. Zhang, A new alternating minimization algorithm for total variation image reconstruction, SIAM J. Imaging Sci., 1 (2008), 248-272. doi: 10.1137/080724265.
[87]	J. Weickert and G. Kühne, Fast methods for implicit active contour models, in Geometic Level Set Methods in Imaging, Vision, and Graphics (eds. S. Osher and N. Paragios), Springer-Verlag, New York, (2003), 43-57. doi: 10.1007/0-387-21810-6_3.
[88]	R. Wituła and D. Słota, Cardano's formula, square roots, Chebyshev polynominals and radicals, J. Math. Anal. Appl., 363 (2010), 639-647. doi: 10.1016/j.jmaa.2009.09.056.
[89]	C. Wu and X.-C. Tai, Augmented Lagrangian Method, Dual Methods, and Split Bregman iteration for ROF, vectorial TV and high order models, SIAM J. Imaging Sci., 3 (2010), 303-329. doi: 10.1137/090767558.
[90]	Y. Xiao, T. Zeng, J. Yu and M. K. Ng, Rstoration of images corrupted by mixed Gaussian-impulse noise via $l_1$-$l_0$ minimization, Pattern Recognition, 44 (2011), 1708-1720. doi: 10.1016/j.patcog.2011.02.002.
[91]	Y. Xu, Block Coordinate Descent for Regularized Multi-Convex Optimization, MA thesis, Rice University, Housten, Texas, 2013.
[92]	Y. Xu and W. Yin, A block coordinate descent method for regularized multiconvex optimization with applications to nonnegative tensor factorization and completion, SIAM J. Imaging Sci., 6 (2013), 1758-1789. doi: 10.1137/120887795.
[93]	X. Zhang, M. Burger and S. Osher, A unified primal-dual algorithm framework based on Bregman iteration, J. Sci. Comput., 46 (2011), 20-46. doi: 10.1007/s10915-010-9408-8.
[94]	J. Zhang and K. Chen, A total fractional-order variation model for image restoration with nonhomogeneous boundary conditions and its numerical solution, SIAM J. Imaging Sciences, 8 (2015), 2487-2518. doi: 10.1137/14097121X.
[95]	J. Zhang and Z. Wei, A class of fractional-order multi-scale variational models and alternating projection algorithm for image denoising, App. Math. Mod., 35 (2011), 2516-2528. doi: 10.1016/j.apm.2010.11.049.
[96]	J. Zhang, Z. Wei and L. Xiao, Fractional-order iterative regularization method for total variation based image denoising, J. Elect. Imaging, 21 (2012), 043005-1. doi: 10.1117/1.JEI.21.4.043005.
[97]	X. Zhang, M. K. Ng and M. Bai, A fast algorithm for deconvolution and Poisson noise removal, J. Sci. Comput., 75 (2018), 1535-1554. doi: 10.1007/s10915-017-0597-2.
[98]	Y. Zhang, W. Zhang, Y. Lei and J. Zhou, Few-view image reconstruction with fractional-order total variation, J. Opt. Soc. Am. A, 31 (2014), 981-995. doi: 10.1364/JOSAA.31.000981.
[99]	H. Zhu, Study on the Image Restoration of $\gamma$-ray Pinhole Imaging System (in Chinese), En.D thesis, Tsinghua University in Beijing, 2008.