# American Institute of Mathematical Sciences

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

## Poisson image denoising based on fractional-order total variation

 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.

Citation: Mujibur Rahman Chowdhury, Jun Zhang, Jing Qin, Yifei Lou. Poisson image denoising based on fractional-order total variation. Inverse Problems and Imaging, 2020, 14 (1) : 77-96. doi: 10.3934/ipi.2019064
##### References:
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##### References:
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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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [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. [35] M. Makitalo and A. Foi, Optimal inversion of the Anscombe transformation in low-count Poisson image denoising, IEEE Trans. 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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
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)
PSNR versus fractional orders for the synthetic image (top) and Barbara image (bottom) with peak values at 55 (left) and 255 (right)
Poisson denoising results of Train image with peak at 55
Poisson denoising results of Barbara image with peak at 255
Poisson denoising results of Mandril image with peak at 55
Poisson denoising results of Penguin image with peak at 155 and Cameraman image with peak at 255
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
 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
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
 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
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