Nonconvex regularization for blurred images with Cauchy noise

Xiao Ai; Guoxi Ni; Tieyong Zeng

doi:10.3934/ipi.2021065

Article Contents

2022, Volume 16, Issue 3: 625-646. Doi: 10.3934/ipi.2021065

This issue Previous Article An inverse problem for a fractional diffusion equation with fractional power type nonlinearities Next Article Automated filtering in the nonlinear Fourier domain of systematic artifacts in 2D electrical impedance tomography

Nonconvex regularization for blurred images with Cauchy noise

1.
School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China
2.
LCP, Institute of Applied Physics and Computational Mathematics, Beijing 100088, China
3.
Department of Mathematics, The Chinese University of Hong Kong, Hong Kong 999077

^* Corresponding author: Guoxi Ni (nijiusuo09@163.com)
^* Corresponding author: Guoxi Ni (nijiusuo09@163.com)

Received: December 31, 2020

Revised: July 31, 2021

Published: June 2022

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Abstract

In this paper, we propose a nonconvex regularization model for images damaged by Cauchy noise and blur. This model is based on the method of the total variational proposed by Federica, Dong and Zeng [SIAM J. Imaging Sci.(2015)], where a variational approach for restoring blurred images with Cauchy noise is used. Here we consider the nonconvex regularization, namely a weighted difference of $ l_1 $-norm and $ l_2 $-norm coupled with wavelet frame, the alternating direction method of multiplier is carried out for this minimization problem, we describe the details of the algorithm and prove its convergence. Numerical experiments are tested by adding different levels of noise and blur, results show that our method can denoise and deblur the image better.

Keywords:

Cauchy noise,
nonconvex,
alternating direction method of multiplier,
wavelet frame.

Mathematics Subject Classification: 58F15, 58F17.

Citation:

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Figure 1. Testing images

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Figure 2. Restored images from blurred and noised images by different methods. (a):Images with Cauchy noise ($ \xi = 0.02 $) and Gaussian blur; (b): Results by median filter (MD); (c): Results by convex TV method; (d): Results by our method

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Figure 3. Restored images from blurred and noised images by different methods. (a):Images with Cauchy noise ($ \xi = 0.04 $) and Gaussian blur; (b): Results by median filter (MD); (c): Results by convex TV method; (d): Results by our method

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Figure 4. Restored images from blurred and noised images by different methods. (a):Images destroyed by Cauchy noise($ \xi = 0.02 $) and motion blur; (b): Results by median filter (MD); (c): Results by convex TV method; (d): Results by our method

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Figure 5. Restored images from blurred and noised images by different methods. (a):Images destroyed by Cauchy noise($ \xi = 0.02 $) and average blur; (b): Results by median filter (MD); (c): Results by convex TV method; (d): Results by our method

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Table 1. The SNR, PSNR and SSIM values for corrupted images and recovered images given by different methods ($ \xi = 0.02 $, Gaussian blur)

Image	Value	Corrupted	MD	TV	Ours
Lena	SNR	5.76	12.43	14.31	14.93
	PSNR	18.47	24.39	27.64	28.21
	SSIM	0.1951	0.7884	0.8168	0.8389
Cameraman	SNR	6.16	11.72	13.69	14.57
	PSNR	18.19	24.39	26.08	26.84
	SSIM	0.1566	0.7483	0.8061	0.8083
Parrot	SNR	6.65	12.15	14.85	15.72
	PSNR	18.27	24.42	26.72	27.50
	SSIM	0.1909	0.7833	0.8239	0.8394
Boat	SNR	6.80	13.85	15.40	16.52
	PSNR	18.02	25.70	27.02	28.07
	SSIM	0.1981	0.7935	0.8286	0.8559
Kitten	SNR	6.15	10.13	11.86	13.01
	PSNR	17.98	22.67	24.03	25.06
	SSIM	0.2496	0.7195	0.7696	0.8161
House	SNR	5.03	13.70	16.11	17.05
	PSNR	18.38	28.48	30.52	31.44
	SSIM	0.1425	0.7908	0.8304	0.8494

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Table 2. The SNR, PSNR and SSIM values for corrupted images and recovered images given by different methods ($ \xi = 0.04 $, Gaussian blur)

Image	Value	Corrupted	MD	TV	Ours
Lena	SNR	3.8	11.24	12.94	13.35
	PSNR	15.87	23.53	26.31	26.65
	SSIM	0.0928	0.6944	0.7675	0.7928
Cameraman	SNR	4.16	10.88	12.39	13.09
	PSNR	15.67	23.53	24.94	25.44
	SSIM	0.0761	0.6241	0.7532	0.7602
Parrot	SNR	4.58	11.19	13.46	14.12
	PSNR	15.75	23.55	25.40	25.91
	SSIM	0.0964	0.6752	0.7763	0.7847
Boat	SNR	4.65	12.59	14.13	14.81
	PSNR	15.33	24.50	25.83	26.41
	SSIM	0.0932	0.6936	0.7843	0.8074
Kitten	SNR	4.24	9.52	10.48	11.37
	PSNR	15.46	22.02	22.76	23.48
	SSIM	0.1235	0.6611	0.6967	0.7413
House	SNR	3.19	11.83	14.61	15.29
	PSNR	15.53	26.49	29.16	29.70
	SSIM	0.0636	0.6628	0.8159	0.8216

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Table 3. The SNR, PSNR and SSIM values for corrupted images and recovered images given by different methods($ \xi = 0.02 $, motion blur)

Image	Value	Corrupted	MD	TV	Ours
Lena	SNR	4.15	7.78	11.66	12.40
	PSNR	17.37	22.05	25.16	25.73
	SSIM	0.1294	0.6209	0.7328	0.7969
Cameraman	SNR	4.98	8.60	11.60	12.42
	PSNR	17.19	21.53	24.13	24.60
	SSIM	0.1149	0.6478	0.7777	0.7797
Parrot	SNR	5.73	9.61	12.16	13.28
	PSNR	17.46	22.00	24.15	24.73
	SSIM	0.1574	0.7011	0.7782	0.7848
Boat	SNR	5.93	10.80	13.02	13.68
	PSNR	17.23	22.77	24.77	25.27
	SSIM	0.1458	0.6731	0.7652	0.7675
Kitten	SNR	4.93	7.63	10.15	10.76
	PSNR	16.92	20.37	22.44	22.87
	SSIM	0.1802	0.5619	0.6901	0.7216
House	SNR	4.54	10.72	14.85	15.65
	PSNR	18.01	25.60	29.34	30.03
	SSIM	0.1272	0.7211	0.8192	0.8264

| Show Table

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Table 4. The SNR, PSNR and SSIM values for corrupted images and recovered images given by different methods($ \xi = 0.02 $, average blur)

Image	Value	Corrupted	MD	TV	Ours
Lena	SNR	5.87	12.88	14.56	15.35
	PSNR	18.55	26.47	27.87	28.64
	SSIM	0.1991	0.8031	0.8203	0.8534
Cameraman	SNR	6.20	12.04	14.17	15.16
	PSNR	18.15	24.65	26.54	27.40
	SSIM	0.1571	0.7623	0.8077	0.8309
Parrot	SNR	6.78	12.64	15.35	16.11
	PSNR	18.34	24.83	27.20	27.88
	SSIM	0.1938	0.7974	0.8228	0.8542
Boat	SNR	6.86	14.24	15.79	16.90
	PSNR	18.00	26.04	27.38	28.45
	SSIM	0.2014	0.8075	0.8345	0.8701
Kitten	SNR	6.31	10.62	12.46	13.39
	PSNR	18.06	23.08	24.56	25.38
	SSIM	0.2598	0.7467	0.7956	0.8327
House	SNR	5.09	14.08	16.54	17.41
	PSNR	18.39	28.70	30.98	31.81
	SSIM	0.1436	0.8013	0.8448	0.8581

| Show Table

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