Spatial-Frequency domain nonlocal total variation for image denoising

Haijuan Hu; Jacques Froment; Baoyan Wang; Xiequan Fan

doi:10.3934/ipi.2020059

Article Contents

2020, Volume 14, Issue 6: 1157-1184. Doi: 10.3934/ipi.2020059

This issue Previous Article A parallel operator splitting algorithm for solving constrained total-variation retinex Next Article

Spatial-Frequency domain nonlocal total variation for image denoising

1.
Northeastern University at Qinhuangdao, School of Mathematics and Statistics, Hebei, 066004, China
2.
Univ Bretagne-Sud, CNRS UMR 6205 LMBA, Campus de Tohannic, Vannes, F-56000, France
3.
Guangdong University of Petrochemical Technology, College of Computer Science, Guangdong, 525000, China
4.
Tianjin University, Center for Applied Mathematics, Tianjin, 300072, China

^* Corresponding author: Haijuan Hu
^* Corresponding author: Haijuan Hu

Received: November 2019

Revised: June 2020

Early access: October 2020

Published: December 2020

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Abstract

Following the pioneering works of Rudin, Osher and Fatemi on total variation (TV) and of Buades, Coll and Morel on non-local means (NL-means), the last decade has seen a large number of denoising methods mixing these two approaches, starting with the nonlocal total variation (NLTV) model. The present article proposes an analysis of the NLTV model for image denoising as well as a number of improvements, the most important of which being to apply the denoising both in the space domain and in the Fourier domain, in order to exploit the complementarity of the representation of image data. A local version obtained by a regionwise implementation followed by an aggregation process, called Local Spatial-Frequency NLTV (L-SFNLTV) model, is finally proposed as a new reference algorithm for image denoising among the family of approaches mixing TV and NL operators. The experiments show the great performance of L-SFNLTV in terms of image quality and of computational speed, comparing with other recently proposed NLTV-related methods.

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Mathematics Subject Classification: Primary: 35Q68, 65T50, 68U10, 62H12.

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Figure 1. An illustration of the set $ \mathcal{U}^2_i $ with $ D = 3 $

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Figure 2. PSNR values versus iterations for different images for NLTV

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Figure 3. PSNR values versus $ \lambda $ for different search window sizes $ D $ with $ \sigma = 20 $

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Figure 4. Denoised image by NLTV with $ D = 3, 11 $ with $ \sigma = 20 $

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Figure 5. Denoised image by NLTV with $ D = 11 $ for different values of $ \lambda $ with $ \sigma = 20 $

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Figure 6. Denoised image by NLTV for different values of $ d $ with $ \sigma = 20 $

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Figure 7. The top row is Lena image, with regions of size 16$ \times $16 highlighted; the following two rows are enlarged regions of the top row; the last two rows are the corresponding estimated MSE and true MSE for the corresponding regions

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Figure 8. Left: image denoised by choosing $ \lambda $ with smallest estimated MSE; right: choosing $ \lambda $ randomly. Top: region size 16$ \times $ 16; bottom region size 32$ \times $ 32

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Figure 9. Left: noisy image and denoised image by FNLTV. Middle: the corresponding Fourier transforms of left column, where the bottom one can also be considered as denoised Fourier transform of noisy image. Right: the Fourier transforms of noise and method noise

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Figure 10. Top row: images denoised by FNLTV with different $ \lambda_f $; the third row: images denoised by NLTV with different $ \lambda $; the second row and bottom row: the corresponding method noise images of the top row and the third row

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Figure 11. Top: Root mean square of method noise versus different $ \lambda $ (for NLTV) or $ \lambda_f $ (for FNLTV); Bottom: PSNR values versus different $ \lambda $ (for NLTV) or $ \lambda_f $ (for FNLTV)

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Figure 12. Left: image denoised by NLTV and FNLTV globally; Middle: image denoised by NLTV and FNLTV locally with no-overlapping regions of size 64$ \times $64; Right: image denoised by NLTV and FNLTV locally with overlapping regions of size 64$ \times $64 for moving step $ n_s = 50 $ (top) and $ n_s = 10 $ (third row). The second and bottom rows are the method noise images of the corresponding images of the top and third rows

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Figure 13. PSNR values for different images with different versions of FNLTV as in Figure 12 and the corresponding versions of NLTV

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Figure 14. Denoised images by ROF model, NL-means, NLTV model and SFNLTV model for Barbara

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Figure 15. Denoised images by ROF model, NL-means, NLTV model and SFNLTV model for Lena

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Figure 16. Denoised Lena images by L-SFNLTV, NLSTV [15], RNLTV [17], BNLTV [18] and SFNLTV in the case $ \sigma = 10 $

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Figure 17. Denoised Lena images by L-SFNLTV, NLSTV [15], RNLTV [17], BNLTV [18] and SFNLTV in the case $ \sigma = 20 $

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Figure 18. Denoised Lena images by L-SFNLTV, NLSTV [15], RNLTV [17], BNLTV [18] and SFNLTV in the case $ \sigma = 50 $

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Figure 19. Denoised images Peppers and House by L-FNLTV and L-SFNLTV in the case $ \sigma = 20 $

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Table 1. PSNR values by choosing $ \lambda $ randomly (the first and third lines) and choosing $ \lambda $ according to the estimated MSE (the second and the fourth lines). The first two lines are with region size 16$ \times $ 16; the last two lines are with region size 32$ \times $ 32

Lena	Barbara	Peppers	Boats	Bridge	House	Cameraman
28.65	26.50	28.14	27.36	25.08	28.92	27.49
30.59	28.02	29.55	29.02	26.54	30.73	29.10
28.96	26.50	28.41	27.53	25.08	28.52	27.80
30.89	28.22	29.82	29.19	26.65	30.98	29.27

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Table 2. PSNR values for different images with NLTV, NL-means, ROF, and SFNLTV in the case $ \sigma = 20 $

Image	Lena	Barbara	Peppers	Boats	Bridge	House	Cameraman
NLTV	31.56	28.48	30.16	29.51	26.66	31.68	29.41
NL-means	31.61	$ \bf{29.68} $	30.28	29.47	26.41	31.78	29.27
ROF	31.00	26.70	29.65	29.19	26.43	31.09	28.77
SFNLTV	$ \bf{31.77} $	29.19	$ \bf{30.29} $	$ \bf{29.89} $	$ \bf{ 26.92} $	$ \bf{ 32.14} $	$ \bf{29.64} $

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Table 3. Choice of parameters of NLTV, SF(SFNLTVL) and L-SF(L-SFNLTV)

	NLTV/SF/L-SF			L-SF	NLTV	$ \lambda=2+0.6\sigma $
$ \sigma $	$ d $	$ D $	$ \sigma_r $	$ \lambda_f $	SF	$ D_f=5 $	$ d_f=9 $
10	9	3	$ \sigma $	6		$ \sigma_{rf}=0.8\sigma $
20	9			14		$ \lambda=0.55\sigma $	$ \lambda_f=1.6+0.02\sigma $
30	11			25	L-SF	$ D_f=3 $	$ d_f=5 $
50	15			49	L-SF	$ \sigma_{rf}=\sigma $	$ \lambda=4 $

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Table 4. Comparisons of PSNR values for $ \sigma = 10 $ and $ \sigma = 20 $

	NLTV	NLSTV	RNLTV	BNLTV	SFNLTV	L-SFNLTV
$ \sigma=10 $
Lena	34.74	34.61	34.17	34.57	35.05	$ \bf{35.58} $
Barbara	32.79	31.29	32.79	33.77	33.93	$ \bf{34.46} $
Peppers	33.80	34.06	33.11	32.31	33.82	$ \bf{34.28} $
Boats	32.80	33.15	32.68	32.83	33.42	$ \bf{33.57} $
Bridge	30.56	30.62	29.14	29.96	30.86	$ \bf{30.97} $
House	34.94	34.52	34.43	34.97	35.49	$ \bf{35.62} $
Cameraman	33.25	33.30	31.97	32.52	33.45	$ \bf{33.65 } $
Monarch	32.98	33.00	31.49	32.41	33.51	$ \bf{33.73 } $
Couple	32.73	33.07	32.78	32.96	33.21	$ \bf{33.57 } $
Fingerprint	30.82	31.17	29.44	30.09	32.05	$ \bf{32.34 } $
Hill	32.66	32.41	32.15	32.89	33.11	$ \bf{33.29 } $
Man	33.18	33.00	32.19	32.96	33.40	$ \bf{33.75} $
$ \sigma=20 $
Lena	31.56	31.18	30.40	31.71	$ {31.77} $	$ \bf{32.54} $
Barbara	28.48	27.23	29.19	30.40	29.19	$ \bf{30.75} $
Peppers	30.16	30.16	29.64	28.38	$ {30.29} $	$ \bf{30.55} $
Boats	29.51	29.80	29.50	29.55	$ {29.89} $	$ \bf{30.42} $
Bridge	26.66	27.03	26.63	26.06	$ { 26.92} $	$ \bf{27.06} $
House	31.68	30.93	30.21	32.17	$ {32.14} $	$ \bf{32.54} $
Cameraman	29.41	29.41	28.54	28.85	$ \bf{29.64} $	$ {29.63} $
Monarch	29.30	28.56	27.82	28.78	$ \bf{29.66} $	29.61
Couple	29.02	29.50	29.36	29.58	29.36	$ \bf{30.19} $
Fingerprint	26.71	26.83	26.94	26.22	27.50	$ \bf{28.55 } $
Hill	29.58	29.13	28.90	29.92	29.84	$ \bf{30.38 } $
Man	29.77	29.42	28.93	29.66	29.88	$ \bf{30.31 } $

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Table 5. Comparisons of PSNR values for $ \sigma = 30 $ and $ \sigma = 50 $

	NLTV	NLSTV	RNLTV	BNLTV	SFNLTV	L-SFNLTV
$ \sigma=30 $
Lena	29.67	29.86	27.89	29.98	29.82	$ \bf{30.64} $
Barbara	26.16	24.84	26.74	28.59	26.55	$ \bf{28.63} $
Peppers	27.96	$ \bf{28.51} $	27.26	26.58	28.13	$ {28.44} $
Boats	27.73	28.14	27.18	27.94	27.93	$ \bf{28.54} $
Bridge	24.86	25.04	24.93	24.69	25.01	$ \bf{25.23} $
House	29.69	29.89	27.78	30.36	29.97	$ \bf{30.65} $
Cameraman	27.48	$ \bf{27.76} $	26.55	27.21	27.58	27.65
Monarch	27.09	26.89	25.95	27.00	$ \bf{27.29 } $	$ \bf{27.29} $
Couple	27.11	27.62	27.01	27.88	27.31	$ \bf{28.26} $
Fingerprint	24.37	25.05	25.14	24.84	24.97	$ \bf{26.48 } $
Hill	28.06	27.79	26.68	28.44	28.22	$ \bf{28.76 } $
Man	28.03	27.93	26.74	28.10	28.08	$ \bf{28.47 } $
$ \sigma=50 $
Lena	27.51	27.67	24.40	27.92	27.61	$ \bf{28.28} $
Barbara	24.00	23.17	23.30	$ \bf{26.21} $	24.11	$ {26.00} $
Peppers	25.31	$ {26.00} $	23.91	24.39	25.48	$ \bf{ 26.03} $
Boats	25.62	25.96	23.94	25.92	25.69	$ \bf{26.28} $
Bridge	23.09	23.12	22.50	23.03	23.18	$ \bf{23.41} $
House	27.23	27.57	24.12	28.10	27.40	$ \bf{28.22} $
Cameraman	24.87	$ \bf{25.42} $	23.41	25.08	24.83	25.20
Monarch	24.39	24.33	22.97	$ \bf{24.69} $	24.47	24.64
Couple	25.12	25.36	23.72	25.75	25.21	$ \bf{26.00} $
Fingerprint	21.71	22.36	22.48	22.96	22.16	$ \bf{23.96 } $
Hill	26.35	25.87	23.50	26.60	26.46	$ \bf{26.86} $
Man	26.11	25.89	23.59	26.18	26.13	$ \bf{26.41} $

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Table 6. Running time in second with grayscale images of size $ 256\times 256 $, where NLSTV is run under Linux system, and other algorithms are run under Windows system on another computer with a slightly faster processor

NLSTV	RNLTV	BNLTV	SFNLTV	L-SFNLTV
22	3344	11.7	2.4	8.3

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