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:
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show all references

References:
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[13]

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[14]

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[15]

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[17]

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[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]

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[21]

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[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

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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]

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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
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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)
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Figure 3.  PSNR versus fractional orders for the synthetic image (top) and Barbara image (bottom) with peak values at 55 (left) and 255 (right)
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Figure 4.  Poisson denoising results of Train image with peak at 55
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Figure 5.  Poisson denoising results of Barbara image with peak at 255
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Figure 6.  Poisson denoising results of Mandril image with peak at 55
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Figure 7.  Poisson denoising results of Penguin image with peak at 155 and Cameraman image with peak at 255
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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
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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
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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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