Poisson image denoising based on fractional-order total variation

Mujibur Rahman Chowdhury; Jun Zhang; Jing Qin; Yifei Lou

doi:10.3934/ipi.2019064

Article Contents

2020, Volume 14, Issue 1: 77-96. Doi: 10.3934/ipi.2019064

This issue Previous Article A partial data inverse problem for the convection-diffusion equation Next Article Electrocommunication for weakly electric fish

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
^* Corresponding author: Yifei Lou

Received: February 28, 2019

Revised: July 31, 2019

Published: November 2019

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Abstract

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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Mathematics Subject Classification: Primary: 65F22, 65Y04; Secondary: 52A41, 49N45.

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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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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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