Article Contents

2023, Volume 17, Issue 4: 870-889. Doi: 10.3934/ipi.2023007

This issue Previous Article Learning spectral windowing parameters for regularization using unbiased predictive risk and generalized cross validation techniques for multiple data sets Next Article On inverse problems for uncoupled space-time fractional operators involving time-dependent coefficients

Image denoising based on a new anisotropic mean curvature model

Yuan Wang^1, and
Zhi-Feng Pang^2, ,

1.
School of Mathematical Sciences, Zhejiang University, Hangzhou, China
2.
School of Mathematics and Statistics, Henan University, Kaifeng, China

^*Corresponding author: Zhi-Feng Pang
^*Corresponding author: Zhi-Feng Pang

Received: June 2022

Revised: February 2023

Early access: March 2023

Published: August 2023

Abstract / Introduction Full Text(HTML) Figure(10) / Table(3) Related Papers Cited by

Abstract

A number of variational models for image denoising have been proposed in the last few years in order to advance the denoising performance. To improve the denoising quality, it is very significant to describe the local structure of image in the proposed models. To this end, this paper proposes a novel denoising model which combines the gradient operator $ \nabla $ with the adaptive weighted matrix $ W $ in the mean curvature regularized term such that the proposed model can describe the local features in image efficiently. Since the proposed model is a high-order nonlinear and nonconvex optimization problem, we need to use the operator splitting method to tansform it into a multi-variable optimization problem and then the alternating direction method of multipliers (ADMM) can be applied to solve it. Numerical experiments demonstrate that the proposed model yields good performance compared with other well-known gradient-based models.

Keywords:

Image denoising,
anisotropic mean curvature,
augmented Lagrangian method,
weighted matrix,
alternating direction method of multipliers.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

Citation:

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Figure 1. 1st: affine images with different angels $ 0, \pi/8, \pi/4, $$ 3\pi/8, \pi/2 $. 2nd: the ratio of $ \log(W_x/W_y) $

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Figure 2. Original images

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Figure 3. Denoising results are shown by different models. Images from left to right are noisy images, denoising images of TV, HOTV, TGV, ATV, MC and the proposed model (AMC). 1st row: the noising level of variance $ \sigma = 0.05 $; 2nd row: the noising level of variance $ \sigma = 0.1 $; 3rd row: the noising level of $ \sigma = 0.2 $

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Figure 4. The comparisons of the 180th row plots generated from the restored image and the original image. 1st row: restored image with variance $ \sigma = 0.05 $; 2nd row: restored image with variance $ \sigma = 0.1 $; 3rd row: restored image with variance $ \sigma = 0.2 $. Images from left to right are restored results of TV, HOTV, TGV, ATV, MC and AMC

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Figure 5. Comparisons among six models in terms of the SNR and the SSIM for the brain image

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Figure 6. Comparisons of six methods on the brain MRI with different levels of noise

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Figure 7. Visual examples of restored images with the noise variance as $ \sigma = 0.2 $. The noisy images are displayed in the first column. The second through seventh columns show the restored results of TV, HOTV, TGV, ATV, MC and AMC model

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Figure 8. Denoising results of six models. Images from left to right are original images, restored results of TV, HOTV, TGV, ATV, MC and the proposed model (AMC). The first and third rows are the restored images in order to efficiently show the restored results. The second and fourth rows are the contour plots. White Gaussian noise with variance as $ \sigma = 0.05 $

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Figure 9. Visual examples of restored images with the noise variance as $ \sigma = 0.1 $. The first, fourth and seventh rows show the denoising results by using the different models. The second, fifth and eighth rows show the contours. The third, sixth and ninth rows show the difference images between restored images and the original clean images

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Figure 10. The plots of relative residuals, relative error in $ u^k $ and energy for the brain image with variance as $ \sigma = 0.2 $

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Table 1. SNR and SSIM of the Texmos image with different levels of white Gaussian noise

		$ 0.05 $		$ 0.1 $		$ 0.2 $
		SNR	SSIM	SNR	SSIM	SNR	SSIM
Texmos $ (512\times512) $	TV	34.9389	0.9900	30.7660	0.9864	25.8750	0.9740
	HOTV	31.8945	0.9859	26.3419	0.9657	21.4325	0.9079
	TGV	30.1237	0.9524	27.6509	0.9469	20.2697	0.7842
	ATV	36.5850	0.9959	31.5363	0.9912	26.7421	0.9859
	MC	36.7772	0.9928	32.8587	0.9897	28.1378	0.9859
	AMC	39.5339	0.9974	33.7164	0.9920	28.6771	0.9865

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Table 2. SNR and SSIM of the natural images with different levels of white Gaussian noise

		$ 0.05 $		$ 0.1 $		$ 0.2 $
		SNR	SSIM	SNR	SSIM	SNR	SSIM
Cameraman $ (256\times256) $	TV	24.4261	0.9403	19.8933	0.8741	15.4665	0.7645
	HOTV	24.7059	0.9389	20.0565	0.8692	15.8390	0.7175
	TGV	24.8285	0.9407	20.2318	0.8660	16.0555	0.7590
	ATV	24.8216	0.9405	19.5423	0.8163	16.1089	0.7329
	MC	24.2884	0.9438	19.7386	0.8844	15.7260	0.8098
	AMC	24.8751	0.9452	20.2513	0.8882	16.8309	0.8301
Peppers $ (256\times256) $	TV	23.4557	0.9509	19.0540	0.9031	15.3310	0.8472
	HOTV	23.6027	0.9482	19.4940	0.9123	15.4403	0.8566
	TGV	23.6712	0.9487	19.7737	0.9127	15.4945	0.8449
	ATV	23.5929	0.9508	19.3028	0.8929	15.7272	0.8423
	MC	23.4629	0.9523	19.4053	0.9137	15.9017	0.8659
	AMC	23.8056	0.9535	20.1047	0.9204	16.4739	0.8687
Barbara $ (256\times256) $	TV	21.0820	0.9428	17.6311	0.8804	14.2412	0.8008
	HOTV	22.1298	0.9388	17.7057	0.8820	14.6666	0.8107
	TGV	21.7975	0.9455	17.7752	0.8663	14.7037	0.7913
	ATV	22.1881	0.9431	17.7769	0.8856	14.6581	0.8062
	MC	21.7009	0.9425	17.7023	0.8846	14.4138	0.8155
	AMC	22.2248	0.9458	17.7859	0.8878	14.7221	0.8178
House $ (256\times256) $	TV	22.7621	0.9302	19.1033	0.8786	16.1751	0.8317
	HOTV	22.8041	0.9263	19.3590	0.8724	15.9549	0.8131
	TGV	22.7787	0.9268	19.4610	0.8772	15.2764	0.7463
	ATV	22.9113	0.9311	19.2093	0.8832	16.4647	0.8369
	MC	22.3283	0.9200	19.4971	0.8845	16.4433	0.8408
	AMC	23.0546	0.9313	19.7522	0.8857	16.9050	0.8483
Lena $ (256\times256) $	TV	21.9420	0.9385	17.8192	0.8757	14.2216	0.7932
	HOTV	21.9930	0.9384	17.8427	0.8759	14.1715	0.8014
	TGV	21.7423	0.9405	17.9651	0.8751	14.0864	0.7777
	ATV	21.9480	0.9386	17.8277	0.8760	14.2312	0.8132
	MC	21.9025	0.9408	17.8036	0.8914	14.1209	0.8108
	AMC	22.1449	0.9413	17.9969	0.8924	14.3413	0.8196
Parrot $ (256\times256) $	TV	25.1037	0.9472	20.7407	0.9100	17.3382	0.8376
	HOTV	23.6234	0.9489	19.5562	0.9107	15.5233	0.8419
	TGV	25.0737	0.9415	21.3407	0.9063	17.0111	0.7845
	ATV	25.1060	0.9472	20.8167	0.8796	17.5184	0.8586
	MC	25.1114	0.9472	21.0438	0.9125	17.5632	0.8706
	AMC	25.2144	0.9512	21.3952	0.9151	17.9631	0.8707
Clown $ (256\times256) $	TV	25.4481	0.9541	21.0520	0.8967	17.2097	0.8068
	HOTV	25.3700	0.9460	21.0189	0.9019	17.2749	0.8228
	TGV	25.1128	0.9572	21.0398	0.8873	17.4512	0.8091
	ATV	25.4768	0.9501	21.0730	0.8969	17.4509	0.8104
	MC	25.2299	0.9557	21.0721	0.9066	16.9260	0.8210
	AMC	25.6417	0.9578	21.3543	0.9085	17.4636	0.8322
Butterfly $ (256\times256) $	TV	24.8939	0.9692	20.3859	0.9285	16.0945	0.8755
	HOTV	24.8206	0.9663	20.1207	0.9394	15.5635	0.8768
	TGV	24.7823	0.9656	20.2884	0.9414	15.7252	0.8538
	ATV	24.8984	0.9692	20.3992	0.9288	16.3346	0.8908
	MC	24.5480	0.9713	20.5937	0.9457	16.5272	0.9099
	AMC	25.2768	0.9729	20.9997	0.9495	16.9803	0.9126

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Table 3. The computation time of different models for the cameraman image with variance as $ \sigma = 0.05, 0.1, 0.2 $

		$ 0.05 $	$ 0.1 $	$ 0.2 $
Time(s)	TV	0.761	0.736	0.781
	HOTV	9.910	9.326	7.181
	TGV	18.405	19.147	18.587
	ATV	1.041	1.378	2.195
	MC	41.277	43.530	44.150
	AMC	42.806	43.994	42.667

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