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A partial data inverse problem for the convection-diffusion equation
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 |
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.
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show all references
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A. Chambolle and T. Pock,
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[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.
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doi: 10.1137/15M1029357. |
[10] |
D. Chen, Y. Q. Chen and D. Xue,
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[15] |
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[18] |
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A new detail-preserving regularization scheme, SIAM J. Imaging Sci., 7 (2014), 1309-1334.
doi: 10.1137/120904263. |
[20] |
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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.
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[23] |
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A variational approach to reconstructing images corrupted by Poisson noise, J. Math. Imaging Vision, 27 (2007), 257-263.
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[25] |
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Adaptive fractional-order total variation image restoration with split Bregman iteration, ISA Transactions, 82 (2017), 210-222.
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[26] |
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[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. Google Scholar |
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L. Ma, T. Zeng and G. Li,
Hybrid variational model for texture image restoration, East Asian J. Appl. Math., 7 (2017), 629-642.
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[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. |
[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. |
[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. |
[38] |
Y. Pu,
Fractional differential analysis for texture of digital image, J. Alg. Comput. Tech., 1 (2007), 357-380.
doi: 10.1260/174830107782424075. |
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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 |
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(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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