A fast explicit diffusion algorithm of fractional order anisotropic diffusion for image denoising

Zhiguang Zhang; Qiang Liu; Tianling Gao

doi:10.3934/ipi.2021018

Article Contents

2021, Volume 15, Issue 6: 1451-1469. Doi: 10.3934/ipi.2021018 open access

This issue Previous Article An adaptive total variational despeckling model based on gray level indicator frame Next Article Preface

A fast explicit diffusion algorithm of fractional order anisotropic diffusion for image denoising

1.
College of Mathematics and Statistics, Chongqing Three Gorges University, Chongqing, 404100, China
2.
College of Mathematics and Computational Science, Shenzhen University, Shenzhen, 518060, China

^* Corresponding author: Tianling Gao
^* Corresponding author: Tianling Gao

Received: October 31, 2019

Revised: September 30, 2020

Early access: February 2021

Published: December 2021

The first author is supported by Young Foundation of Three Gorges University(19QN09). The last two authors are supported by NSF of Guandong grant 2018A030310454, 2020A1515010554.

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Abstract

In this paper, we mainly show a novel fast fractional order anisotropic diffusion algorithm for noise removal based on the recent numerical scheme called the Fast Explicit Diffusion. To balance the efficiency and accuracy of the algorithm, the truncated matrix method is used to deal with the iterative matrix in the model and its error is also estimated. In particular, we obtain the stability condition of the iteration by the spectrum analysis method. Through implementing the fast explicit format iteration algorithm with periodic change of time step size, the efficiency of the algorithm is greatly improved. At last, we show some numerical results on denoising tasks. Many experimental results confirm that the algorithm can more quickly achieve satisfactory denoising results.

Keywords:

Mathematics Subject Classification: Primary: 68U10; Secondary: 35R11.

Citation:

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Figure 1. The truncation error of the truncated polynomial $ {{\rho _N}\left( t \right)} $ at $ z = 1 $ with truncated lengths $ t $ from 10 to 50 and fractional-orders $ \alpha $ from 1.1 to 1.9

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Figure 2. First row to last row: comparing the convergence rate of three algorithms for the camera image, the Lena image, the butterfly image and the house image with additive Gauss noise levels of 10, 20 and 40

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Figure 3. First column to last column: Origin images, noisy images with $ \sigma {\rm{ = }}10 $, and the restored images by the FOFED algorithm, FOFED$ _f $ algorithm and FOFFT algorithm in [6], respectively

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Figure 4. First column to last column: Origin images, noisy images with $ \sigma {\rm{ = }}20 $, and the restored images by the FOFED algorithm, FOFED$ _f $ algorithm and FOFFT algorithm in [6], respectively

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Figure 5. First column to last column: Origin images, noisy images with $ \sigma {\rm{ = }}40 $, and the restored images by the FOFED algorithm, FOFED$ _f $ algorithm and FOFFT algorithm in [6], respectively

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Table 1. The truncation error of the truncated polynomial $ {{\rho _N}\left( t \right)} $ at $ z = 1 $ with truncated lengths $ t $ from 10 to 50 (rows) and fractional-orders $ \alpha $ from 1.1 to 1.9 (columns)

t	$ \alpha = 1.1 $	$ \alpha = 1.2 $	$ \alpha = 1.3 $	$ \alpha = 1.4 $	$ \alpha = 1.5 $	$ \alpha = 1.6 $	$ \alpha = 1.7 $	$ \alpha = 1.8 $	$ \alpha = 1.9 $
10	0.0084	0.0125	0.0136	0.0128	0.0109	0.0085	0.0060	0.0036	0.0016
20	0.0037	0.0051	0.0051	0.0044	0.0035	0.0025	0.0016	0.0009	0.0004
30	0.0023	0.0030	0.0029	0.0024	0.0018	0.0013	0.0008	0.0004	0.0002
40	0.0017	0.0021	0.0020	0.0016	0.0012	0.0008	0.0005	0.0002	0.00009
50	0.0013	0.0016	0.0015	0.0012	0.0008	0.0005	0.0003	0.00016	0.00005

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Table 2. PSNR, MAE, and SSIM results for the camera image, the Lena image, the butterfly image and the house image with additive Gauss noise levels of 10, 20 and 40. The PSNR, MAE and SSIM values are similar

	PSNR			MAE			SSIM
$ \sigma $	10	20	40	10	20	40	10	20	40
Cameraman image ($ 1024 \times 1024 $)
FOFED	39.4084	35.4073	33.7938	2.0434	3.2504	5.1143	0.5978	0.4621	0.3333
FOFED$ _{f} $	39.6005	35.6151	33.9866	1.9789	3.1111	4.8185	0.6030	0.4671	0.3374
FOFFT	39.4881	35.5515	33.8737	1.9958	3.1240	4.8238	0.6021	0.4665	0.3370
Lena image ($ 512 \times 512 $)
FOFED	34.2268	30.9968	28.5389	3.7184	5.2972	7.3706	0.6056	0.4904	0.3856
FOFED$ _{f} $	34.2882	31.0802	28.6008	3.6588	5.0912	7.1460	0.6082	0.4957	0.3908
FOFFT	34.2816	31.0847	28.5937	3.6564	5.0856	7.1361	0.6095	0.4962	0.3910
Butterfly image ($ 256 \times 256 $)
FOFED	32.8641	28.5347	26.6302	4.2297	6.8386	10.5351	0.7961	0.6960	0.5777
FOFED$ _{f} $	32.8944	28.5595	26.6618	4.1910	6.7488	10.3155	0.8126	0.7198	0.5960
FOFFT	32.8459	28.5339	26.6119	4.2067	6.7536	10.2976	0.8134	0.7209	0.5973
House image ($ 128 \times 128 $)
FOFED	31.9328	27.8935	27.0384	4.7282	7.3484	11.1483	0.5607	0.4463	0.3439
FOFED$ _{f} $	31.9543	27.9203	27.0600	4.6964	7.2673	10.6816	0.5610	0.4470	0.3446
FOFFT	31.9849	27.9562	27.0898	4.6672	7.2156	10.6107	0.5624	0.4476	0.3453

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Table 3. Number of Iteration and CPU time for the camera image, the Lena image, the butterfly image and the house image with additive Gauss noise levels of 10, 20 and 40. The best CPU times are in black

	Number of Iteration			CPU time(s)
$ \sigma $	10	20	40	10	20	40
Cameraman image ($ 1024 \times 1024 $)
FOFED	41	71	116	$ 12.0397 $	$ 15.5452 $	$ 25.8294 $
FOFED$ _{f} $	100	230	553	32.8190	77.5531	187.0590
FOFFT	114	258	612	98.8958	227.8830	518.8290
Lena image ($ 512 \times 512 $)
FOFED	30	56	129	$ 1.9445 $	$ 3.5040 $	$ 8.1629 $
FOFED$ _{f} $	58	157	420	4.6834	12.4698	32.5470
FOFFT	69	181	473	14.8598	39.0244	101.7675
Butterfly image ($ 256 \times 256 $)
FOFED	25	55	88	$ 0.3193 $	$ 0.6546 $	$ 1.0031 $
FOFED$ _{f} $	47	112	289	0.6994	1.5716	4.0302
FOFFT	56	134	336	2.8438	6.7368	16.9259
House image ($ 128 \times 128 $)
FOFED	40	54	27.0384	$ 0.3604 $	$ 0.4095 $	$ 0.7060 $
FOFED$ _{f} $	41	104	27.0600	0.3820	0.8661	2.2998
FOFFT	50	125	27.0898	0.7193	1.6759	3.9392

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