American Institute of Mathematical Sciences

Advanced
February  2020, 14(1): 77-96. doi: 10.3934/ipi.2019064

Poisson image denoising based on fractional-order total variation

Mujibur Rahman Chowdhury 1, , Jun Zhang 2, , Jing Qin 3, and  Yifei Lou 1,,

1. 

Department of Mathematical Sciences, The University of Texas at Dallas, Richardson, TX 75080, USA
2. 

Jiangxi Province Key Laboratory of Water Information Cooperative Sensing and Intelligent Processing/College of Science, Nanchang Institute of Technology, Nanchang 330099, China
3. 

Department of Mathematics, University of Kentucky, Lexington, KY 40506, USA

* Corresponding author: Yifei Lou

Received  March 2019 Revised  August 2019 Published  November 2019

Poisson noise is an important type of electronic noise that is present in a variety of photon-limited imaging systems. Different from the Gaussian noise, Poisson noise depends on the image intensity, which makes image restoration very challenging. Moreover, complex geometry of images desires a regularization that is capable of preserving piecewise smoothness. In this paper, we propose a Poisson denoising model based on the fractional-order total variation (FOTV). The existence and uniqueness of a solution to the model are established. To solve the problem efficiently, we propose three numerical algorithms based on the Chambolle-Pock primal-dual method, a forward-backward splitting scheme, and the alternating direction method of multipliers (ADMM), each with guaranteed convergence. Various experimental results are provided to demonstrate the effectiveness and efficiency of our proposed methods over the state-of-the-art in Poisson denoising.

Keywords: Poisson noise, expectation-maximization, fractional-order total variation.
Mathematics Subject Classification: Primary: 65F22, 65Y04; Secondary: 52A41, 49N45.
Citation: Mujibur Rahman Chowdhury, Jun Zhang, Jing Qin, Yifei Lou. Poisson image denoising based on fractional-order total variation. Inverse Problems & Imaging, 2020, 14 (1) : 77-96. doi: 10.3934/ipi.2019064
References:
[1]

J. Bai and X. C. Feng, Fractional-order anisotropic diffusion for image denoising, IEEE Trans. Image Process., 16 (2007), 2492-2502.  doi: 10.1109/TIP.2007.904971.  Google Scholar

[2]

S. Bonettini and V. Ruggiero, An alternating extragradient method for total variation-based image restoration from Poisson data, Inverse Problems, 27 (2011), 26pp. doi: 10.1088/0266-5611/27/9/095001.  Google Scholar

[3]

S. Boyd, N. Parikh, E. Chu, B. Peleato and J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers, Found. Trends Mach. Learn., 3 (2011), 1-122.  doi: 10.1561/2200000016.  Google Scholar

[4]

K. Bredies, K. Kunisch and T. Pock, Total generalized variation, SIAM J. Imaging Sci., 3 (2010), 492-526.  doi: 10.1137/090769521.  Google Scholar

[5]

A. Chambolle, An algorithm for total variation minimization and applications, J. Math. Imaging Vision, 20 (2004), 89-97.  doi: 10.1023/B:JMIV.0000011325.36760.1e.  Google Scholar

[6]

A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, J. Math. Imaging Vision, 40 (2011), 120-145.  doi: 10.1007/s10851-010-0251-1.  Google Scholar

[7]

R. Chan, A. Lanza, S. Morigi and F. Sgallari, An adaptive strategy for the restoration of textured images using fractional order regularization, Numer. Math. Theory Methods Appl., 6 (2013), 276-296.  doi: 10.4208/nmtma.2013.mssvm15.  Google Scholar

[8]

T. Chan, A. Marquina and P. Mulet, High-order total variation-based image restoration, SIAM J. Sci. Comput., 22 (2000), 503-516.  doi: 10.1137/S1064827598344169.  Google Scholar

[9]

H. Chang, Y. Lou, M. K. Ng and T. Zeng, Phase retrieval from incomplete magnitude information via total variation regularization, SIAM J. Sci. Comput., 38 (2016), A3672–A3695. doi: 10.1137/15M1029357.  Google Scholar

[10]

D. Chen, Y. Q. Chen and D. Xue, Fractional-order total variation image denoising based on proximity algorithm, Appl. Math. Comput., 257 (2015), 537-545.  doi: 10.1016/j.amc.2015.01.012.  Google Scholar

[11]

H. Chen, J. Song and X. C. Tai, A dual algorithm for minimization of the LLT model, Adv. Comput. Math., 31 (2009), 115-130.  doi: 10.1007/s10444-008-9097-0.  Google Scholar

[12]

K. Dabov, A. Foi, V. Katkovnik and K. Egiazarian, Image denoising by sparse 3D transform-domain collaborative filtering, IEEE Trans. Image Process., 16 (2007), 2080-2095.  doi: 10.1109/TIP.2007.901238.  Google Scholar

[13]

D. di Serafino, G. Landi and M. Viola, ACQUIRE: An inexact iteratively reweighted norm approach for TV-based Poisson image restoration, Appl. Math. Comput., 364 (2020), 23pp. doi: 10.1016/j.amc.2019.124678.  Google Scholar

[14]

F. Dong and Y. Chen, A fractional-order derivative based variational framework for image denoising, Inverse Probl. Imaging, 10 (2016), 27-50.  doi: 10.3934/ipi.2016.10.27.  Google Scholar

[15]

Y. Duan, Y. Wang and J. Hahn, A fast augmented Lagrangian method for Euler's elastica models, Numer. Math. Theory Methods Appl., 6 (2013), 47-71.  doi: 10.4208/nmtma.2013.mssvm03.  Google Scholar

[16]

J. Eckstein and D. Bertsekas, On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators, Math. Programming, 55 (1992), 293-318.  doi: 10.1007/BF01581204.  Google Scholar

[17]

M. Figueiredo and J. Bioucas-Dias, Restoration of Poissonian images using alternating direction optimization, IEEE Trans. Image Process., 19 (2010), 3133-3145.  doi: 10.1109/TIP.2010.2053941.  Google Scholar

[18]

M. Figueiredo and J. M. Bioucas-Dias, Deconvolution of Poissonian images using variable splitting and augmented Lagrangian optimization, preprint, arXiv: math/0904.4868. doi: 10.1109/TIP.2010.2045029.  Google Scholar

[19]

W. Guo, J. Qin and W. Yin, A new detail-preserving regularization scheme, SIAM J. Imaging Sci., 7 (2014), 1309-1334.  doi: 10.1137/120904263.  Google Scholar

[20]

L. Jiang, J. Huang, X. G. Lv and J. Liu, Alternating direction method for the high-order total variation-based Poisson noise removal problem, Numer. Algorithms, 69 (2015), 495-516.  doi: 10.1007/s11075-014-9908-y.  Google Scholar

[21]

S. H. Kayyar and P. Jidesh, Non-local total variation regularization approach for image restoration under a Poisson degradation, J. Modern Optics, 65 (2018), 2231-2242.  doi: 10.1080/09500340.2018.1506058.  Google Scholar

[22]

G. Landi and E. L. Piccolomini, NPTool: A MATLAB software for nonnegative image restoration with Newton projection methods, Numer. Algorithms, 62 (2013), 487-504.  doi: 10.1007/s11075-012-9602-x.  Google Scholar

[23]

H. Lantéri and C. Theys, Restoration of astrophysical images: The case of Poisson data with additive Gaussian noise, EURASIP J. Adv. Signal Process., 2005 (2005), 2500-2513.  doi: 10.1155/ASP.2005.2500.  Google Scholar

[24]

T. Le, R. Chartrand and T. J. Asaki, A variational approach to reconstructing images corrupted by Poisson noise, J. Math. Imaging Vision, 27 (2007), 257-263.  doi: 10.1007/s10851-007-0652-y.  Google Scholar

[25]

D. Li, X. Tian, Q. Jin and K. Hirasawa, Adaptive fractional-order total variation image restoration with split Bregman iteration, ISA Transactions, 82 (2017), 210-222.  doi: 10.1016/j.isatra.2017.08.014.  Google Scholar

[26]

H. Li, J. Wang and H. Dou, Second-order TGV model for Poisson noise image restoration, SpringerPlus, 5 (2016). doi: 10.1186/s40064-016-2929-3.  Google Scholar

[27]

Y. Li, J. Qin, Y. Hsin, S. Osher and W. Liu, s-SMOOTH: Sparsity and smoothness enhanced EEG brain tomography, Frontiers Neurosci., 10 (2016). doi: 10.3389/fnins.2016.00543.  Google Scholar

[28]

Y. Li, J. Qin, S. Osher and W. Liu, Graph fractional-order total variation EEG source reconstruction, 38th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), Orlando, FL, 2016, 101–104. doi: 10.1109/EMBC.2016.7590650.  Google Scholar

[29]

X. Liu, Augmented Lagrangian method for total generalized variation based Poissonian image restoration, Comput. Math. Appl., 71 (2016), 1694-1705.  doi: 10.1016/j.camwa.2016.03.005.  Google Scholar

[30]

X. Liu and L. Huang, Total bounded variation-based Poissonian images recovery by split Bregman iteration, Math. Methods Appl. Sci., 35 (2012), 520-529.  doi: 10.1002/mma.1588.  Google Scholar

[31]

X. G. Lv, J. Lee and J. Liu, Deblurring Poisson noisy images by total variation with overlapping group sparsity, Appl. Math. Comput., 289 (2016), 132-148.  doi: 10.1016/j.amc.2016.03.029.  Google Scholar

[32]

M. Lysaker, A. Lundervold and X. C. Tai, Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time, IEEE Trans. Image Process., 12 (2003), 1579-1590.  doi: 10.1109/TIP.2003.819229.  Google Scholar

[33]

J. Ma and D. Gemechu, Anisotropic total fractional order variation model in seismic data denoising, World Academy of Science, Engineering and Technology, International J. Geological and Environmental Engineering, 12 (2018), 40-44.   Google Scholar

[34]

L. Ma, T. Zeng and G. Li, Hybrid variational model for texture image restoration, East Asian J. Appl. Math., 7 (2017), 629-642.  doi: 10.4208/eajam.090217.300617a.  Google Scholar

[35]

M. Makitalo and A. Foi, Optimal inversion of the Anscombe transformation in low-count Poisson image denoising, IEEE Trans. Image Process., 20 (2011), 99-109.  doi: 10.1109/TIP.2010.2056693.  Google Scholar

[36]

S. Osher, A. Solé and L. Vese, Image decomposition and restoration using total variation minimization and the $H^1$, Multiscale Model. Simul., 1 (2003), 349-370.  doi: 10.1137/S1540345902416247.  Google Scholar

[37]

Y. Pu, Fractional calculus approach to texture of digital image, Proc. 8th Int. Conf. Signal Process., Beijing, China, 2006, 1002–1006. doi: 10.1109/ICOSP.2006.345713.  Google Scholar

[38]

Y. Pu, Fractional differential analysis for texture of digital image, J. Alg. Comput. Tech., 1 (2007), 357-380.  doi: 10.1260/174830107782424075.  Google Scholar

[39]

Y. Pu, W. Wang, J. Zhou, Y. Wang and H. Jia, Fractional differential approach to detecting textural features of digital image and its fractional differential filter implementation, Sci. China Ser. F, 51 (2008), 1319-1339.  doi: 10.1007/s11432-008-0098-x.  Google Scholar

[40]

J. Qin and W. Guo, An efficient compressive sensing MR image reconstruction scheme, IEEE 10th International Symposium on Biomedical Imaging, San Francisco, CA, 2013,306–309. doi: 10.1109/ISBI.2013.6556473.  Google Scholar

[41]

J. Qin, F. Liu, S. Wang and J. Rosenberger, EEG source imaging based on spatial and temporal graph structures, Seventh International Conference on Image Processing Theory, Tools and Applications (IPTA), Montreal, Canada, 2017, 1–6. doi: 10.1109/IPTA.2017.8310089.  Google Scholar

[42]

J. Qin, T. Wu, Y. Li, W. Yin, S. Osher and W. Liu, Accelerated high-resolution EEG source imaging, 8th International IEEE/EMBS Conference on Neural Engineering (NER), Shanghai, China, 2017, 1–4. doi: 10.1109/NER.2017.8008277.  Google Scholar

[43]

J. Qin, X. Yi and S. Weiss, A novel fluorescence microscopy image deconvolution approach, IEEE 15th International Symposium on Biomedical Imaging, Washington DC, 2018,441–444. doi: 10.1109/ISBI.2018.8363611.  Google Scholar

[44]

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.  Google Scholar

[45]

J. Salmon, Z. Harmany, C. A. Deledalle and R. Willett, Poisson noise reduction with non-local PCA, J. Math. Imaging Vision, 48 (2014), 279-294.  doi: 10.1007/s10851-013-0435-6.  Google Scholar

[46]

A. Sawatzky, C. Brune, T. Kosters, F. Wubbeling and M. Burger, EM-TV methods for inverse problems with Poisson noise, in Level Set and PDE Based Reconstruction Methods in Imaging, Lecture Notes in Math., 2090, Springer, Cham, 2013, 71–142. doi: 10.1007/978-3-319-01712-9_2.  Google Scholar

[47]

A. Sawatzky, C. Brune, J. Muller and M. Burger, Total variation processing of images with Poisson statistics, in Computer Analysis of Images and Patterns, Lecture Notes in Computer Science, 5702, Springer, Berlin, Heidelberg, 2009,533–540. doi: 10.1007/978-3-642-03767-2_65.  Google Scholar

[48]

S. Setzer, G. Steidl and T. Teuber, Deblurring Poissonian images by split Bregman techniques, J. Vis. Commun. Image, 21 (2010), 193-199.  doi: 10.1016/j.jvcir.2009.10.006.  Google Scholar

[49]

J. Shen, S. H. Kang and T. F. Chan, Euler's elastica and curvature-based inpainting, SIAM J. Appl. Math., 63 (2002), 564-592.  doi: 10.1137/S0036139901390088.  Google Scholar

[50]

X. C. Tai, J. Hahn and G. J. Chung, A fast algorithm for Euler's elastica model using augmented Lagrangian method, SIAM J. Imaging Sci., 4 (2011), 313-344.  doi: 10.1137/100803730.  Google Scholar

[51]

A. N. Tikhonov, A. Goncharsky, V. V. Stepanov and A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems, Mathematics and its Applications, 328, Kluwer Academic Publishers Group, Dordrecht, 1995. doi: 10.1007/978-94-015-8480-7.  Google Scholar

[52]

A. Ullah, W. Chen and M. A. Khan, Fracto-integer order total variation based multiplicative noise removal model, International Conference on Information Science and Technology (ICIST), Changsha, China, 2015,160–165. doi: 10.1109/ICIST.2015.7288960.  Google Scholar

[53]

A. Ullah, W. Chen and M. A. Khan, A new variational approach for restoring images with multiplicative noise, Comput. Math. Appl., 71 (2016), 2034-2050.  doi: 10.1016/j.camwa.2016.03.024.  Google Scholar

[54]

Y. Vardi, L. A. Shepp and L. Kaufman, A statistical model for positron emission tomography, J. Amer. Statist. Assoc., 80 (1985), 8-37.  doi: 10.1080/01621459.1985.10477119.  Google Scholar

[55]

X. D. Wang, X. C. Feng, W. Wang and W. J. Zhang, Iterative reweighted total generalized variation based Poisson noise removal model, Appl. Math. Comput., 223 (2013), 264-277.  doi: 10.1016/j.amc.2013.07.090.  Google Scholar

[56]

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.  Google Scholar

[57]

Z. Wang, A. C. Bovik, H. R. Sheikh and E. P. Simoncelli, Image quality assessment: From error visibility to structural similarity, IEEE Trans. Image Process., 13 (2004), 600-612.  doi: 10.1109/TIP.2003.819861.  Google Scholar

[58]

Y. Wen, R. Chan and T. Zeng, Primal-dual algorithms for total variation based image restoration under Poisson noise, Sci. China Math., 59 (2016), 141-160.  doi: 10.1007/s11425-015-5079-0.  Google Scholar

[59]

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), 300-339.  doi: 10.1137/090767558.  Google Scholar

[60]

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 Sci., 8 (2015), 2487-2518.  doi: 10.1137/14097121X.  Google Scholar

[61]

J. Zhang, R. Chen, C. Deng and S. Wang, Fast linearized augmented Lagrangian method for Euler's elastica model, Numer. Math. Theory Methods Appl., 10 (2017), 98-115.  doi: 10.4208/nmtma.2017.m1611.  Google Scholar

[62]

J. Zhang, M. Ma, Z. Wu and C. Deng, High-order total bounded variation model and its fast algorithm for Poissonian image restoration, Math. Probl. Eng., 2019, 11pp. doi: 10.1155/2019/2502731.  Google Scholar

[63]

J. Zhang, Z. Wei and L. Xiao, Adaptive fractional-order multi-scale method for image denoising, J. Math. Imaging Vision, 43 (2012), 39-49.  doi: 10.1007/s10851-011-0285-z.  Google Scholar

[64]

J. Zhang, Z. Wei and L. Xiao, A fast adaptive reweighted residual-feedback iterative algorithm for fractional-order total variation regularized multiplicative noise removal of partly-textured images, Signal Process., 98 (2014), 381-395.  doi: 10.1016/j.sigpro.2013.12.009.  Google Scholar

[65]

J. Zhang, Z. Wei and L. Xiao, Bi-component decomposition based hybrid regularization method for partly-textured CS-MR image reconstruction, Signal Process., 128 (2016), 274-290.  doi: 10.1016/j.sigpro.2016.04.012.  Google Scholar

[66]

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.  Google Scholar

[67]

W. Zhou and O. Li, Poisson noise removal scheme based on fourth-order PDE by alternating minimization algorithm, Abstr. Appl. Anal., 2012 (2012), 14pp. doi: 10.1155/2012/965281.  Google Scholar

[68]

W. Zhou and Q. Li, Adaptive total variation regularization based scheme for Poisson noise removal, Math. Methods Appl. Sci., 36 (2013), 290-299.  doi: 10.1002/mma.2587.  Google Scholar

[69]

W. Zhu and T. Chan, Image denoising using mean curvature of image surface, SIAM J. Imaging Sci., 5 (2012), 1-32.  doi: 10.1137/110822268.  Google Scholar

show all references

References:
[1]

J. Bai and X. C. Feng, Fractional-order anisotropic diffusion for image denoising, IEEE Trans. Image Process., 16 (2007), 2492-2502.  doi: 10.1109/TIP.2007.904971.  Google Scholar

[2]

S. Bonettini and V. Ruggiero, An alternating extragradient method for total variation-based image restoration from Poisson data, Inverse Problems, 27 (2011), 26pp. doi: 10.1088/0266-5611/27/9/095001.  Google Scholar

[3]

S. Boyd, N. Parikh, E. Chu, B. Peleato and J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers, Found. Trends Mach. Learn., 3 (2011), 1-122.  doi: 10.1561/2200000016.  Google Scholar

[4]

K. Bredies, K. Kunisch and T. Pock, Total generalized variation, SIAM J. Imaging Sci., 3 (2010), 492-526.  doi: 10.1137/090769521.  Google Scholar

[5]

A. Chambolle, An algorithm for total variation minimization and applications, J. Math. Imaging Vision, 20 (2004), 89-97.  doi: 10.1023/B:JMIV.0000011325.36760.1e.  Google Scholar

[6]

A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, J. Math. Imaging Vision, 40 (2011), 120-145.  doi: 10.1007/s10851-010-0251-1.  Google Scholar

[7]

R. Chan, A. Lanza, S. Morigi and F. Sgallari, An adaptive strategy for the restoration of textured images using fractional order regularization, Numer. Math. Theory Methods Appl., 6 (2013), 276-296.  doi: 10.4208/nmtma.2013.mssvm15.  Google Scholar

[8]

T. Chan, A. Marquina and P. Mulet, High-order total variation-based image restoration, SIAM J. Sci. Comput., 22 (2000), 503-516.  doi: 10.1137/S1064827598344169.  Google Scholar

[9]

H. Chang, Y. Lou, M. K. Ng and T. Zeng, Phase retrieval from incomplete magnitude information via total variation regularization, SIAM J. Sci. Comput., 38 (2016), A3672–A3695. doi: 10.1137/15M1029357.  Google Scholar

[10]

D. Chen, Y. Q. Chen and D. Xue, Fractional-order total variation image denoising based on proximity algorithm, Appl. Math. Comput., 257 (2015), 537-545.  doi: 10.1016/j.amc.2015.01.012.  Google Scholar

[11]

H. Chen, J. Song and X. C. Tai, A dual algorithm for minimization of the LLT model, Adv. Comput. Math., 31 (2009), 115-130.  doi: 10.1007/s10444-008-9097-0.  Google Scholar

[12]

K. Dabov, A. Foi, V. Katkovnik and K. Egiazarian, Image denoising by sparse 3D transform-domain collaborative filtering, IEEE Trans. Image Process., 16 (2007), 2080-2095.  doi: 10.1109/TIP.2007.901238.  Google Scholar

[13]

D. di Serafino, G. Landi and M. Viola, ACQUIRE: An inexact iteratively reweighted norm approach for TV-based Poisson image restoration, Appl. Math. Comput., 364 (2020), 23pp. doi: 10.1016/j.amc.2019.124678.  Google Scholar

[14]

F. Dong and Y. Chen, A fractional-order derivative based variational framework for image denoising, Inverse Probl. Imaging, 10 (2016), 27-50.  doi: 10.3934/ipi.2016.10.27.  Google Scholar

[15]

Y. Duan, Y. Wang and J. Hahn, A fast augmented Lagrangian method for Euler's elastica models, Numer. Math. Theory Methods Appl., 6 (2013), 47-71.  doi: 10.4208/nmtma.2013.mssvm03.  Google Scholar

[16]

J. Eckstein and D. Bertsekas, On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators, Math. Programming, 55 (1992), 293-318.  doi: 10.1007/BF01581204.  Google Scholar

[17]

M. Figueiredo and J. Bioucas-Dias, Restoration of Poissonian images using alternating direction optimization, IEEE Trans. Image Process., 19 (2010), 3133-3145.  doi: 10.1109/TIP.2010.2053941.  Google Scholar

[18]

M. Figueiredo and J. M. Bioucas-Dias, Deconvolution of Poissonian images using variable splitting and augmented Lagrangian optimization, preprint, arXiv: math/0904.4868. doi: 10.1109/TIP.2010.2045029.  Google Scholar

[19]

W. Guo, J. Qin and W. Yin, A new detail-preserving regularization scheme, SIAM J. Imaging Sci., 7 (2014), 1309-1334.  doi: 10.1137/120904263.  Google Scholar

[20]

L. Jiang, J. Huang, X. G. Lv and J. Liu, Alternating direction method for the high-order total variation-based Poisson noise removal problem, Numer. Algorithms, 69 (2015), 495-516.  doi: 10.1007/s11075-014-9908-y.  Google Scholar

[21]

S. H. Kayyar and P. Jidesh, Non-local total variation regularization approach for image restoration under a Poisson degradation, J. Modern Optics, 65 (2018), 2231-2242.  doi: 10.1080/09500340.2018.1506058.  Google Scholar

[22]

G. Landi and E. L. Piccolomini, NPTool: A MATLAB software for nonnegative image restoration with Newton projection methods, Numer. Algorithms, 62 (2013), 487-504.  doi: 10.1007/s11075-012-9602-x.  Google Scholar

[23]

H. Lantéri and C. Theys, Restoration of astrophysical images: The case of Poisson data with additive Gaussian noise, EURASIP J. Adv. Signal Process., 2005 (2005), 2500-2513.  doi: 10.1155/ASP.2005.2500.  Google Scholar

[24]

T. Le, R. Chartrand and T. J. Asaki, A variational approach to reconstructing images corrupted by Poisson noise, J. Math. Imaging Vision, 27 (2007), 257-263.  doi: 10.1007/s10851-007-0652-y.  Google Scholar

[25]

D. Li, X. Tian, Q. Jin and K. Hirasawa, Adaptive fractional-order total variation image restoration with split Bregman iteration, ISA Transactions, 82 (2017), 210-222.  doi: 10.1016/j.isatra.2017.08.014.  Google Scholar

[26]

H. Li, J. Wang and H. Dou, Second-order TGV model for Poisson noise image restoration, SpringerPlus, 5 (2016). doi: 10.1186/s40064-016-2929-3.  Google Scholar

[27]

Y. Li, J. Qin, Y. Hsin, S. Osher and W. Liu, s-SMOOTH: Sparsity and smoothness enhanced EEG brain tomography, Frontiers Neurosci., 10 (2016). doi: 10.3389/fnins.2016.00543.  Google Scholar

[28]

Y. Li, J. Qin, S. Osher and W. Liu, Graph fractional-order total variation EEG source reconstruction, 38th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC), Orlando, FL, 2016, 101–104. doi: 10.1109/EMBC.2016.7590650.  Google Scholar

[29]

X. Liu, Augmented Lagrangian method for total generalized variation based Poissonian image restoration, Comput. Math. Appl., 71 (2016), 1694-1705.  doi: 10.1016/j.camwa.2016.03.005.  Google Scholar

[30]

X. Liu and L. Huang, Total bounded variation-based Poissonian images recovery by split Bregman iteration, Math. Methods Appl. Sci., 35 (2012), 520-529.  doi: 10.1002/mma.1588.  Google Scholar

[31]

X. G. Lv, J. Lee and J. Liu, Deblurring Poisson noisy images by total variation with overlapping group sparsity, Appl. Math. Comput., 289 (2016), 132-148.  doi: 10.1016/j.amc.2016.03.029.  Google Scholar

[32]

M. Lysaker, A. Lundervold and X. C. Tai, Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time, IEEE Trans. Image Process., 12 (2003), 1579-1590.  doi: 10.1109/TIP.2003.819229.  Google Scholar

[33]

J. Ma and D. Gemechu, Anisotropic total fractional order variation model in seismic data denoising, World Academy of Science, Engineering and Technology, International J. Geological and Environmental Engineering, 12 (2018), 40-44.   Google Scholar

[34]

L. Ma, T. Zeng and G. Li, Hybrid variational model for texture image restoration, East Asian J. Appl. Math., 7 (2017), 629-642.  doi: 10.4208/eajam.090217.300617a.  Google Scholar

[35]

M. Makitalo and A. Foi, Optimal inversion of the Anscombe transformation in low-count Poisson image denoising, IEEE Trans. Image Process., 20 (2011), 99-109.  doi: 10.1109/TIP.2010.2056693.  Google Scholar

[36]

S. Osher, A. Solé and L. Vese, Image decomposition and restoration using total variation minimization and the $H^1$, Multiscale Model. Simul., 1 (2003), 349-370.  doi: 10.1137/S1540345902416247.  Google Scholar

[37]

Y. Pu, Fractional calculus approach to texture of digital image, Proc. 8th Int. Conf. Signal Process., Beijing, China, 2006, 1002–1006. doi: 10.1109/ICOSP.2006.345713.  Google Scholar

[38]

Y. Pu, Fractional differential analysis for texture of digital image, J. Alg. Comput. Tech., 1 (2007), 357-380.  doi: 10.1260/174830107782424075.  Google Scholar

[39]

Y. Pu, W. Wang, J. Zhou, Y. Wang and H. Jia, Fractional differential approach to detecting textural features of digital image and its fractional differential filter implementation, Sci. China Ser. F, 51 (2008), 1319-1339.  doi: 10.1007/s11432-008-0098-x.  Google Scholar

[40]

J. Qin and W. Guo, An efficient compressive sensing MR image reconstruction scheme, IEEE 10th International Symposium on Biomedical Imaging, San Francisco, CA, 2013,306–309. doi: 10.1109/ISBI.2013.6556473.  Google Scholar

[41]

J. Qin, F. Liu, S. Wang and J. Rosenberger, EEG source imaging based on spatial and temporal graph structures, Seventh International Conference on Image Processing Theory, Tools and Applications (IPTA), Montreal, Canada, 2017, 1–6. doi: 10.1109/IPTA.2017.8310089.  Google Scholar

[42]

J. Qin, T. Wu, Y. Li, W. Yin, S. Osher and W. Liu, Accelerated high-resolution EEG source imaging, 8th International IEEE/EMBS Conference on Neural Engineering (NER), Shanghai, China, 2017, 1–4. doi: 10.1109/NER.2017.8008277.  Google Scholar

[43]

J. Qin, X. Yi and S. Weiss, A novel fluorescence microscopy image deconvolution approach, IEEE 15th International Symposium on Biomedical Imaging, Washington DC, 2018,441–444. doi: 10.1109/ISBI.2018.8363611.  Google Scholar

[44]

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.  Google Scholar

[45]

J. Salmon, Z. Harmany, C. A. Deledalle and R. Willett, Poisson noise reduction with non-local PCA, J. Math. Imaging Vision, 48 (2014), 279-294.  doi: 10.1007/s10851-013-0435-6.  Google Scholar

[46]

A. Sawatzky, C. Brune, T. Kosters, F. Wubbeling and M. Burger, EM-TV methods for inverse problems with Poisson noise, in Level Set and PDE Based Reconstruction Methods in Imaging, Lecture Notes in Math., 2090, Springer, Cham, 2013, 71–142. doi: 10.1007/978-3-319-01712-9_2.  Google Scholar

[47]

A. Sawatzky, C. Brune, J. Muller and M. Burger, Total variation processing of images with Poisson statistics, in Computer Analysis of Images and Patterns, Lecture Notes in Computer Science, 5702, Springer, Berlin, Heidelberg, 2009,533–540. doi: 10.1007/978-3-642-03767-2_65.  Google Scholar

[48]

S. Setzer, G. Steidl and T. Teuber, Deblurring Poissonian images by split Bregman techniques, J. Vis. Commun. Image, 21 (2010), 193-199.  doi: 10.1016/j.jvcir.2009.10.006.  Google Scholar

[49]

J. Shen, S. H. Kang and T. F. Chan, Euler's elastica and curvature-based inpainting, SIAM J. Appl. Math., 63 (2002), 564-592.  doi: 10.1137/S0036139901390088.  Google Scholar

[50]

X. C. Tai, J. Hahn and G. J. Chung, A fast algorithm for Euler's elastica model using augmented Lagrangian method, SIAM J. Imaging Sci., 4 (2011), 313-344.  doi: 10.1137/100803730.  Google Scholar

[51]

A. N. Tikhonov, A. Goncharsky, V. V. Stepanov and A. G. Yagola, Numerical Methods for the Solution of Ill-Posed Problems, Mathematics and its Applications, 328, Kluwer Academic Publishers Group, Dordrecht, 1995. doi: 10.1007/978-94-015-8480-7.  Google Scholar

[52]

A. Ullah, W. Chen and M. A. Khan, Fracto-integer order total variation based multiplicative noise removal model, International Conference on Information Science and Technology (ICIST), Changsha, China, 2015,160–165. doi: 10.1109/ICIST.2015.7288960.  Google Scholar

[53]

A. Ullah, W. Chen and M. A. Khan, A new variational approach for restoring images with multiplicative noise, Comput. Math. Appl., 71 (2016), 2034-2050.  doi: 10.1016/j.camwa.2016.03.024.  Google Scholar

[54]

Y. Vardi, L. A. Shepp and L. Kaufman, A statistical model for positron emission tomography, J. Amer. Statist. Assoc., 80 (1985), 8-37.  doi: 10.1080/01621459.1985.10477119.  Google Scholar

[55]

X. D. Wang, X. C. Feng, W. Wang and W. J. Zhang, Iterative reweighted total generalized variation based Poisson noise removal model, Appl. Math. Comput., 223 (2013), 264-277.  doi: 10.1016/j.amc.2013.07.090.  Google Scholar

[56]

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.  Google Scholar

[57]

Z. Wang, A. C. Bovik, H. R. Sheikh and E. P. Simoncelli, Image quality assessment: From error visibility to structural similarity, IEEE Trans. Image Process., 13 (2004), 600-612.  doi: 10.1109/TIP.2003.819861.  Google Scholar

[58]

Y. Wen, R. Chan and T. Zeng, Primal-dual algorithms for total variation based image restoration under Poisson noise, Sci. China Math., 59 (2016), 141-160.  doi: 10.1007/s11425-015-5079-0.  Google Scholar

[59]

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), 300-339.  doi: 10.1137/090767558.  Google Scholar

[60]

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 Sci., 8 (2015), 2487-2518.  doi: 10.1137/14097121X.  Google Scholar

[61]

J. Zhang, R. Chen, C. Deng and S. Wang, Fast linearized augmented Lagrangian method for Euler's elastica model, Numer. Math. Theory Methods Appl., 10 (2017), 98-115.  doi: 10.4208/nmtma.2017.m1611.  Google Scholar

[62]

J. Zhang, M. Ma, Z. Wu and C. Deng, High-order total bounded variation model and its fast algorithm for Poissonian image restoration, Math. Probl. Eng., 2019, 11pp. doi: 10.1155/2019/2502731.  Google Scholar

[63]

J. Zhang, Z. Wei and L. Xiao, Adaptive fractional-order multi-scale method for image denoising, J. Math. Imaging Vision, 43 (2012), 39-49.  doi: 10.1007/s10851-011-0285-z.  Google Scholar

[64]

J. Zhang, Z. Wei and L. Xiao, A fast adaptive reweighted residual-feedback iterative algorithm for fractional-order total variation regularized multiplicative noise removal of partly-textured images, Signal Process., 98 (2014), 381-395.  doi: 10.1016/j.sigpro.2013.12.009.  Google Scholar

[65]

J. Zhang, Z. Wei and L. Xiao, Bi-component decomposition based hybrid regularization method for partly-textured CS-MR image reconstruction, Signal Process., 128 (2016), 274-290.  doi: 10.1016/j.sigpro.2016.04.012.  Google Scholar

[66]

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.  Google Scholar

[67]

W. Zhou and O. Li, Poisson noise removal scheme based on fourth-order PDE by alternating minimization algorithm, Abstr. Appl. Anal., 2012 (2012), 14pp. doi: 10.1155/2012/965281.  Google Scholar

[68]

W. Zhou and Q. Li, Adaptive total variation regularization based scheme for Poisson noise removal, Math. Methods Appl. Sci., 36 (2013), 290-299.  doi: 10.1002/mma.2587.  Google Scholar

[69]

W. Zhu and T. Chan, Image denoising using mean curvature of image surface, SIAM J. Imaging Sci., 5 (2012), 1-32.  doi: 10.1137/110822268.  Google Scholar

Figure 1.  A synthetic example. Top: the synthetic ground-truth image and two noisy images with peak values 55 and 255, respectively. Bottom: FOTV Poisson denoising results (via Algorithm 3) with the respective fractional order $ 1 $, $ 1.8 $, and $ 2.4 $ for the case with peak value 255
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  Algorithmic comparison in terms of energy (left) and PSNR (right) by denoising the noisy synthetic image with peak value of 255 (top) and the Barbara image with peak value of 55 (bottom)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  PSNR versus fractional orders for the synthetic image (top) and Barbara image (bottom) with peak values at 55 (left) and 255 (right)
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 4.  Poisson denoising results of Train image with peak at 55
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 5.  Poisson denoising results of Barbara image with peak at 255
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 6.  Poisson denoising results of Mandril image with peak at 55
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 7.  Poisson denoising results of Penguin image with peak at 155 and Cameraman image with peak at 255
Figure Options

Download full-size image

Download as PowerPoint slide

Table 1.  Poisson denoising comparison. Each entry contains PSNR and SSIM values. The best results are highlighted in bold
Test Image Peak Noisy NPTool NLPCA BM3D Proposed
Cameraman 55 20.63 27.81/0.87 27.36/0.85 28.80/0.89 28.15/0.87
155 25.18 30.65/0.91 29.52/0.87 29.06/0.90 30.90/0.92
255 27.38 32.06/0.93 30.58/0.91 29.45/0.90 32.31/0.94
Penguin 55 19.53 30.86/0.91 30.49/0.89 31.50/0.92 31.46/0.92
155 24.01 33.34 /0.94 32.14/0.91 32.37/0.93 33.86/0.95
255 26.74 34.27/0.95 33.21/0.93 32.69/0.93 34.73/0.96
Train 55 20.94 28.04/0.88 27.20/0.81 28.01/0.89 28.25/0.88
155 25.47 30.99 /0.92 29.64/0.88 28.31/0.89 31.06/0.92
255 27.85 32.38/0.94 30.76/0.88 29.16/0.90 32.43/0.94
Mandril 55 19.75 22.57/0.76 22.00 /0.71 20.39/0.56 22.80/0.76
155 24.26 25.73/0.86 25.06/0.84 20.58/0.58 26.00/0.87
255 26.46 27.67/0.90 26.46/0.87 20.95/0.60 27.76/0.91
Barbara 55 20.57 25.48/0.81 27.97/0.86 29.44/0.91 25.87/0.81
155 25.14 28.52/0.87 30.29/0.90 30.77/0.93 29.14/0.89
255 27.31 30.14/0.91 31.01/0.92 31.47/0.94 30.78/0.92
Table Options

Download as excel

Test Image Peak Noisy NPTool NLPCA BM3D Proposed
Cameraman 55 20.63 27.81/0.87 27.36/0.85 28.80/0.89 28.15/0.87
155 25.18 30.65/0.91 29.52/0.87 29.06/0.90 30.90/0.92
255 27.38 32.06/0.93 30.58/0.91 29.45/0.90 32.31/0.94
Penguin 55 19.53 30.86/0.91 30.49/0.89 31.50/0.92 31.46/0.92
155 24.01 33.34 /0.94 32.14/0.91 32.37/0.93 33.86/0.95
255 26.74 34.27/0.95 33.21/0.93 32.69/0.93 34.73/0.96
Train 55 20.94 28.04/0.88 27.20/0.81 28.01/0.89 28.25/0.88
155 25.47 30.99 /0.92 29.64/0.88 28.31/0.89 31.06/0.92
255 27.85 32.38/0.94 30.76/0.88 29.16/0.90 32.43/0.94
Mandril 55 19.75 22.57/0.76 22.00 /0.71 20.39/0.56 22.80/0.76
155 24.26 25.73/0.86 25.06/0.84 20.58/0.58 26.00/0.87
255 26.46 27.67/0.90 26.46/0.87 20.95/0.60 27.76/0.91
Barbara 55 20.57 25.48/0.81 27.97/0.86 29.44/0.91 25.87/0.81
155 25.14 28.52/0.87 30.29/0.90 30.77/0.93 29.14/0.89
255 27.31 30.14/0.91 31.01/0.92 31.47/0.94 30.78/0.92
Table 2.  Computation time (in sec)
Test image(size) Peak NPTool NLPCA BM3D Proposed
Barbara (512 × 512) 55 5.68 72.73 2.26 8.68
Train (512 × 357) 155 5.49 91.70 1.63 6.36
Mandril (256 × 256) 255 1.99 3.77 0.71 1.32
Table Options

Download as excel

Test image(size) Peak NPTool NLPCA BM3D Proposed
Barbara (512 × 512) 55 5.68 72.73 2.26 8.68
Train (512 × 357) 155 5.49 91.70 1.63 6.36
Mandril (256 × 256) 255 1.99 3.77 0.71 1.32
[1]

Xianchao Xiu, Ying Yang, Wanquan Liu, Lingchen Kong, Meijuan Shang. An improved total variation regularized RPCA for moving object detection with dynamic background. Journal of Industrial & Management Optimization, 2020, 16 (4) : 1685-1698. doi: 10.3934/jimo.2019024
[2]

María J. Garrido-Atienza, Bohdan Maslowski, Jana  Šnupárková. Semilinear stochastic equations with bilinear fractional noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3075-3094. doi: 10.3934/dcdsb.2016088
[3]

Haili Qiao, Aijie Cheng. A fast high order method for time fractional diffusion equation with non-smooth data. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021073
[4]

Sandrine Anthoine, Jean-François Aujol, Yannick Boursier, Clothilde Mélot. Some proximal methods for Poisson intensity CBCT and PET. Inverse Problems & Imaging, 2012, 6 (4) : 565-598. doi: 10.3934/ipi.2012.6.565
[5]

Guangying Lv, Jinlong Wei, Guang-an Zou. Noise and stability in reaction-diffusion equations. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021005
[6]

Wensheng Yin, Jinde Cao, Guoqiang Zheng. Further results on stabilization of stochastic differential equations with delayed feedback control under $ G $-expectation framework. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021072
[7]

Tomasz Kosmala, Markus Riedle. Variational solutions of stochastic partial differential equations with cylindrical Lévy noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2879-2898. doi: 10.3934/dcdsb.2020209
[8]

Caifang Wang, Tie Zhou. The order of convergence for Landweber Scheme with $\alpha,\beta$-rule. Inverse Problems & Imaging, 2012, 6 (1) : 133-146. doi: 10.3934/ipi.2012.6.133
[9]

Alexandre B. Simas, Fábio J. Valentim. $W$-Sobolev spaces: Higher order and regularity. Communications on Pure & Applied Analysis, 2015, 14 (2) : 597-607. doi: 10.3934/cpaa.2015.14.597
[10]

Anton Schiela, Julian Ortiz. Second order directional shape derivatives of integrals on submanifolds. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021017
[11]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1631-1648. doi: 10.3934/dcdss.2020447
[12]

Kuan-Hsiang Wang. An eigenvalue problem for nonlinear Schrödinger-Poisson system with steep potential well. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021030
[13]

Jinyi Sun, Zunwei Fu, Yue Yin, Minghua Yang. Global existence and Gevrey regularity to the Navier-Stokes-Nernst-Planck-Poisson system in critical Besov-Morrey spaces. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3409-3425. doi: 10.3934/dcdsb.2020237
[14]

Qian Liu. The lower bounds on the second-order nonlinearity of three classes of Boolean functions. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020136
[15]

Pavel I. Naumkin, Isahi Sánchez-Suárez. Asymptotics for the higher-order derivative nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021028
[16]

Fumihiko Nakamura. Asymptotic behavior of non-expanding piecewise linear maps in the presence of random noise. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2457-2473. doi: 10.3934/dcdsb.2018055
[17]

Xiaohu Wang, Dingshi Li, Jun Shen. Wong-Zakai approximations and attractors for stochastic wave equations driven by additive noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2829-2855. doi: 10.3934/dcdsb.2020207
[18]

Lingyu Li, Zhang Chen. Asymptotic behavior of non-autonomous random Ginzburg-Landau equation driven by colored noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3303-3333. doi: 10.3934/dcdsb.2020233
[19]

Khosro Sayevand, Valeyollah Moradi. A robust computational framework for analyzing fractional dynamical systems. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021022
[20]

A. Aghajani, S. F. Mottaghi. Regularity of extremal solutions of semilinaer fourth-order elliptic problems with general nonlinearities. Communications on Pure & Applied Analysis, 2018, 17 (3) : 887-898. doi: 10.3934/cpaa.2018044

2019 Impact Factor: 1.373

Tools

Metrics

  • PDF downloads (426)
  • HTML views (411)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences