An adaptive total variational despeckling model based on gray level indicator frame

Yu Zhang; Songsong Li; Zhichang Guo; Boying Wu

doi:10.3934/ipi.2020068

Article Contents

2021, Volume 15, Issue 6: 1421-1450. Doi: 10.3934/ipi.2020068 open access

This issue Previous Article Image fusion network for dual-modal restoration Next Article A fast explicit diffusion algorithm of fractional order anisotropic diffusion for image denoising

An adaptive total variational despeckling model based on gray level indicator frame

1.
School of Mathematics, Harbin Institute of Technology, Harbin 150001, China
2.
School of Management, Harbin Institute of Technology, Harbin 150001, China

^* Corresponding author: Zhichang Guo
^* Corresponding author: Zhichang Guo

Received: September 30, 2019

Revised: June 30, 2020

Early access: November 2020

Published: December 2021

This work is partially supported by the National Natural Science Foundation of China (11971131, U1637208, 61873071, 51476047, 11871133, 71773024), the Natural Sciences Foundation of Heilongjiang Province of China (LC2018001, G2018006) and the Heilongjiang Postdoctoral Scientific Research Developmental Fund (LBH-Q18064)

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Abstract

For the characteristics of the degraded images with multiplicative noise, the gray level indicators for constructing adaptive total variation are proposed. Based on the new regularization term, we propose the new convex adaptive variational model. Then, considering the existence, uniqueness and comparison principle of the minimizer of the functional. The finite difference method with rescaling technique and the primal-dual method with adaptive step size are used to solve the minimization problem. The paper ends with a report on numerical tests for the denoising of images subject to multiplicative noise, the comparison with other methods is provided as well.

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Mathematics Subject Classification: Primary:68U10, 68Q25;Secondary:32A70.

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Figure 1. The relation between the influence of the noise and the gray levels. (a)1D original ladder-shaped signal; (b)1D speckle noise with mean of 1 and $ L = 4 $; (c) the comparison of original signal and speckle-corrupted signal; (d) The gray area shows values between the first and third quartiles, the dots represent the expectation and the blue line a single noisy realization. Expectation and quartiles are computed from 10000 noisy realizations

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Figure 2. Pretreatment effect map. The black curve is the original signal, and the green curve is the observed signal which has polluted by the Gamma noise with a standard deviation of $ 1/\sqrt{6} $ ($ L $ = 6). The red curve is result of preprocessing

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Figure 3. Adjustment for $ \alpha $. In the first row, the red curves are the observed signal which have polluted by the Gamma noise with a standard deviation of 1/2 (L = 4). In the second line, the blue and green curves represent the corresponding $ \alpha $ and $ \alpha^{'} $ respectively

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Figure 4. Comparison of AA, DZ and our model for multiplicative noise removal. The red curve is the denoising relization of the observed signals by each model. The random Gamma noise with a standard deviation of 1/$ \sqrt{6} $ ($ L $ = 6)

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Figure 5. Comparison of AA, DZ and our model for multiplicative noise removal. The random Gamma noise with a mean of 1 and a standard deviation of 1/2 (L = 4)

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Figure 6. Pie image (214 $ \times $ 216, $ L $ = 1.)

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Figure 7. Pie image (214 $ \times $ 216, $ L $=4.)

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Figure 8. Pie image (214 $ \times $ 216, $ L $ = 10.)

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Figure 9. Cameraman image (256 $ \times $ 256, $ L $ = 1)

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Figure 10. Cameraman image (256 $ \times $ 256, $ L $=4)

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Figure 11. Cameraman image (256 $ \times $ 256, $ L $ = 10)

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Figure 12. Parrot image (512 $ \times $ 512, $ L $ = 1.)

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Figure 13. Parrot image (512 $ \times $ 512, $ L $=4.)

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Figure 14. Parrot image (512 $ \times $ 512, $ L $ = 10.)

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Figure 15. Synthetic image (300 $ \times $ 300, $ L $ = 1.)

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Figure 16. Synthetic image (300 $ \times $ 300, $ L $=4.)

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Figure 17. Synthetic image (300 $ \times $ 300, $ L $ = 10.)

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Figure 18. Piglet image ($ 341 \times 199 $, $ L $ = 1.)

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Figure 19. Piglet image ($ 341 \times 199 $, $ L $=4.)

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Figure 20. Piglet image ($ 341 \times 199 $, $ L $ = 10.)

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Figure 21. Plots of Energy and PSNR of $ \tilde{u}^{k} $. The first column is about Pie(L=10) and the second column is about Cameraman(L=4)

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Table 1. Parameters used in the comparison study

	Parameters
Algorithm	$ L=1 $	$ L=4 $	$ L=10 $
The Pie image (214 $ \times $ 216)
AA ($ \lambda $)	5e-4	0.2	22
DZ ($ \alpha,\lambda $)	0.3, 0.2	0.5, 0.04	5, 0.03
$ E_{1} $-FDM ($ \lambda,\beta,p $)	3e-7, 3e-8, 1.3	2e-3, 1e-3, 1.3	2e-3, 1e-3, 1.3
$ E_{2} $-FDM ($ \lambda,\beta,p,a $)	1e-7, 1e-8, 1.39, 0.74	2e-3, 1e-3, 1.39, 0.74	2e-3, 3e-3, 1.39, 0.74
$ E_{1} $-PDM ($ \lambda,\beta,p $)	4e-3, 6e-3, 1.3	4e-3, 5e-3, 1.3	3e-3, 6e-3, 1.3
$ E_{2} $-PDM ($ \lambda,\beta,p,a $)	4e-2, 6e-3, 1.39, 0.74	4e-3, 5e-3, 1.38, 0.74	4e-3, 4e-3, 1.34, 0.74
The Cameraman image($ 256\times256 $)
AA ($ \lambda $)	0.18	40	75
DZ ($ \alpha,\lambda $)	0.21, 0.25	1.6, 0.11	2.5, 0.03
$ E_{1} $-FDM ($ \lambda,\beta,p $)	1e-7, 5e-7, 1.3	2e-3, 2e-3, 1.3	3e-2, 2e-2, 1.3
$ E_{2} $-FDM ($ \lambda,\beta,p,a $)	3e-7, 2e-7, 1.3, 0.8	3e-3, 2e-3, 1.33, 0.76	3e-2, 2e-2, 1.4, 0.78
$ E_{1} $-PDM ($ \lambda,\beta,p $)	6e-3, 8e-3, 1.3	8e-3, 6e-3, 1.3	8e-3, 3e-3, 1.3
$ E_{2} $-PDM ($ \lambda,\beta,p,a $)	1e-2, 6e-3, 1.3, 0.79	8e-3, 4e-3, 1.3, 0.75	6e-3, 6e-3, 1.3, 0.75
The Parrot image($ 512\times512 $)
AA ($ \lambda $)	0.6	1.5	8
DZ ($ \alpha,\lambda $)	0.11, 0.2	1, 0.2	15, 0.3
$ E_{1} $-FDM ($ \lambda,\beta,p $)	2e-6, 2e-6, 1.3	1e-4, 1e-4, 1.3	3e-3, 3e-3, 1.3
$ E_{2} $-FDM ($ \lambda,\beta,p,a $)	1e-6, 1e-6, 1.36, 0.74	1e-4, 1e-4, 1.35, 0.74	1e-3, 1e-3, 1.32, 0.74
$ E_{1} $-PDM ($ \lambda,\beta,p $)	6e-3, 6e-3, 1.3	6e-3, 6e-3, 1.3	6e-3, 6e-3, 1.3
$ E_{2} $-PDM ($ \lambda,\beta,p,a $)	6e-3, 6e-3, 1.36, 0.74	6e-3, 6e-3, 1.34, 0.74	6e-3, 6e-3, 1.32, 0.74
The Synthetic image($ 300\times300 $)
AA ($ \lambda $)	7e-4	4e-3	1e-3
DZ ($ \alpha,\lambda $)	0.117, 0.125	0.155, 0.1	0.11, 0.2
$ E_{1} $-FDM ($ \lambda,\beta,p $)	1e-7, 1e-7, 1.3	2e-4, 2e-4, 1.3	1e-3, 1e-3, 1.3
$ E_{2} $-FDM ($ \lambda,\beta,p,a $)	1e-7, 1e-7, 1.35, 0.78	1e-4, 1e-4, 1.36, 0.74	1e-3, 1e-3, 1.35, 0.74
$ E_{1} $-PDM ($ \lambda,\beta,p $)	4e-3, 5e-3, 1.3	4e-3, 4e-3, 1.3	4e-3, 4e-3, 1.3
$ E_{2} $-PDM ($ \lambda,\beta,p,a $)	4e-3, 5e-3, 1.4, 0.75	4e-3, 5e-3, 1.4, 0.74	4e-3, 4e-3, 1.32, 0.74
The Piglet image($ 341\times199 $)
AA ($ \lambda $)	5e-3	80	190
DZ ($ \alpha,\lambda $)	3e-3, 0.16	2e-3, 3e-2	0.1, 7e-3
$ E_{1} $-FDM ($ \lambda,\beta,p $)	1e-7, 9e-8, 1.3	1e-4, 1e-4, 1.3	1e-3, 1e-3, 1.3
$ E_{2} $-FDM ($ \lambda,\beta,p,a $)	2e-7, 9e-8, 1.38, 0.74	1e-4, 1e-4, 1.37, 0.74	1e-3, 1e-3, 1.39, 0.74
$ E_{1} $-PDM ($ \lambda,\beta,p $)	4e-3, 4e-3, 1.3	4e-3, 7e-3, 1.3	6e-3, 8e-3, 1.3
$ E_{2} $-PDM ($ \lambda,\beta,p,a $)	4e-3, 7e-3, 1.35, 0.74	4e-3, 7e-3, 1.35, 0.74	6e-3, 8e-3, 1.32, 0.74

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Table 2. Comparison of PSNR, MAE, SSIM and iterations

	PSNR			MAE			SSIM			#iter
$ L $	$ 1 $	$ 4 $	$ 10 $	$ 1 $	$ 4 $	$ 10 $	$ 1 $	$ 4 $	$ 10 $	$ 1 $	$ 4 $	$ 10 $
The Pie image($ 214\times216 $)
AA	$ 13.12 $	$ 18.43 $	$ 22.20 $	$ 42.51 $	$ 22.66 $	$ 13.40 $	0.57	0.77	0.88	1978	743	464
DZ	$ 14.25 $	$ 19.94 $	$ 23.45 $	$ 32.77 $	$ 16.26 $	$ 10.14 $	0.63	0.86	0.92	637	1162	1027
$ E_{1} $-FDM	$ 17.26 $	$ 21.35 $	$ 24.00 $	$ 23.71 $	$ 14.33 $	10.15	0.67	0.84	0.90	481	275	273
$ E_{2} $-FDM	$ 17.45 $	$ 21.30 $	24.08	$ 23.50 $	$ 14.44 $	10.10	0.69	0.84	0.90	362	256	211
$ E_{1} $-PDM	$ 17.22 $	$ 21.30 $	$ 23.86 $	$ 24.00 $	$ 14.29 $	$ 10.15 $	0.71	0.84	0.90	1973	609	312
$ E_{2} $-PDM	$ 17.46 $	$ 21.21 $	$ 23.87 $	$ 23.19 $	$ 14.53 $	$ 10.28 $	0.71	0.83	0.90	1475	485	239
The cameraman image($ 256\times256 $)
AA	19.06	22.96	25.09	16.79	10.09	7.94	0.56	0.69	0.74	1800	672	351
DZ	19.34	22.81	25.22	16.64	10.45	7.94	0.53	0.70	0.75	387	291	338
$ E_{1} $-FDM	21.56	24.44	26.37	12.95	9.19	7.29	0.63	0.71	0.77	442	361	289
$ E_{2} $-FDM	21.56	24.50	26.42	12.98	9.18	7.27	0.63	0.71	0.76	395	315	238
$ E_{1} $-PDM	21.70	24.28	26.21	12.60	9.01	7.29	0.65	0.73	0.78	1469	455	243
$ E_{2} $-PDM	21.76	24.30	26.23	11.96	9.00	7.35	0.66	0.73	0.77	1167	382	203
The Parrot image($ 512\times512 $)
AA	20.52	24.46	26.89	14.73	10.15	7.86	0.60	0.68	0.72	1833	666	359
DZ	20.59	25.19	27.52	14.22	9.38	7.29	0.57	0.71	0.76	533	187	69
$ E_{1} $-FDM	23.84	26.73	28.57	10.88	7.95	6.49	0.66	0.74	0.78	445	391	342
$ E_{2} $-FDM	23.84	26.75	28.61	10.96	8.00	6.48	0.65	0.73	0.78	372	309	227
$ E_{1} $-PDM	23.88	26.73	28.55	10.93	7.94	6.50	0.65	0.73	0.78	1907	629	323
$ E_{2} $-PDM	23.81	26.71	28.60	11.96	9.00	6.48	0.65	0.73	0.78	1572	523	240
The Synthetic image($ 300\times300 $)
AA	$ 22.31 $	$ 27.44 $	$ 30.43 $	$ 11.90 $	$ 6.32 $	$ 4.35 $	0.81	0.88	0.92	2664	1010	643
DZ	$ 22.43 $	$ 27.90 $	$ 30.54 $	$ 10.30 $	$ 5.42 $	$ 4.04 $	0.77	0.94	0.96	1195	570	291
$ E_{1} $-FDM	$ 24.71 $	$ 29.08 $	$ 31.68 $	$ 9.02 $	$ 4.88 $	3.30	0.84	0.93	0.95	507	442	371
$ E_{2} $-FDM	$ 24.77 $	$ 29.07 $	31.71	$ 8.61 $	$ 4.94 $	3.41	0.83	0.92	0.95	457	337	282
$ E_{1} $-PDM	$ 25.44 $	$ 28.89 $	$ 31.46 $	$ 7.30 $	$ 4.29 $	$ 3.24 $	0.89	0.93	0.96	2506	787	425
$ E_{2} $-PDM	$ 25.36 $	$ 28.93 $	$ 31.42 $	$ 7.43 $	$ 4.43 $	$ 3.24 $	0.89	0.92	0.96	1759	645	338
The Piglet image($ 341 \times 199 $)
AA	$ 21.70 $	$ 26.96 $	$ 30.08 $	$ 12.27 $	$ 6.88 $	$ 4.72 $	0.83	0.89	0.91	2363	1068	660
DZ	$ 21.74 $	$ 27.17 $	$ 30.18 $	$ 11.94 $	$ 6.67 $	$ 4.66 $	0.73	0.89	0.91	739	1540	4440
$ E_{1} $-FDM	$ 24.92 $	$ 28.45 $	$ 30.89 $	$ 9.65 $	$ 6.08 $	4.53	0.82	0.89	0.92	499	390	345
$ E_{2} $-FDM	$ 24.82 $	$ 28.51 $	30.75	$ 9.44 $	$ 5.88 $	4.58	0.78	0.88	0.91	432	321	297
$ E_{1} $-PDM	$ 24.96 $	$ 28.33 $	$ 30.68 $	$ 8.99 $	$ 5.95 $	$ 4.58 $	0.84	0.88	0.91	1904	666	335
$ E_{2} $-PDM	$ 24.89 $	$ 28.31 $	$ 30.68 $	$ 9.12 $	$ 5.96 $	$ 4.46 $	0.80	0.88	0.92	1463	530	280

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