An efficient multi-grid method for TV minimization problems

Zhenwei Zhang; Xue Li; Yuping Duan; Ke Yin; Xue-Cheng Tai

doi:10.3934/ipi.2021034

Article Contents

2021, Volume 15, Issue 5: 1199-1221. Doi: 10.3934/ipi.2021034

This issue Previous Article Boundary determination of electromagnetic and Lamé parameters with corrupted data Next Article Edge detection with mixed noise based on maximum a posteriori approach

An efficient multi-grid method for TV minimization problems

1.
Center for Applied Mathematics, Tianjin University, Tianjin, China
2.
Center for Mathematical Sciences, Huazhong University of Technology, Wuhan, China
3.
Department of Mathematics, Hong Kong Bapist University, Kowloon Tong, Hong Kong, China

^* Corresponding author: yuping.duan@tju.edu.cn
^* Corresponding author: yuping.duan@tju.edu.cn;
^* Corresponding author: yuping.duan@tju.edu.cn^* Corresponding author: kyin@hust.edu.cn
^* Corresponding author: kyin@hust.edu.cn

Received: August 2020

Revised: February 2021

Early access: April 2021

Published: October 2021

The work of the first three authors were supported by NSFC 11701418 and 12071345, Major Science and Technology Project of Tianjin 18ZXRHSY00160 and Recruitment Program of Global Young Expert. The work of Yin was supported by NSFC 11801200 and a startup grant from HUST. The work of the last author was supported by Hong Kong Baptist University startup grants RG(R)-RC/17-18/02-MATH and FRG2/17-18/033

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Abstract

We propose an efficient multi-grid domain decomposition method for solving the total variation (TV) minimization problems. Our multi-grid scheme is developed based on the piecewise constant function spanned subspace correction rather than the piecewise linear one in [17], which ensures the calculation of the TV term only occurs on the boundaries of the support sets. Besides, the domain decomposition method is implemented on each layer to enable parallel computation. Comprehensive comparison results are presented to demonstrate the improvement in CPU time and image quality of the proposed method on medium and large-scale image denoising and reconstruction problems.

Keywords:

Mathematics Subject Classification: Primary: 65K10, 68U10; Secondary: 68K10.

Citation:

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Figure 1. An illustration of the piecewise linear basic function and piecewise constant basic function $ \phi_j^i $ on $ \bar{\tau}_j^i $ of the layer $ j = 4 $. In (a), the $ \square $ defines the weight of $ 1 $ for the piecewise constant function and $ \lozenge $ defines the weight of $ 0 $ for the outer boundary $ \partial\bar\tau_j^i $. In (b), the weights of $ \circ $, $ \ast $, $ \triangledown $ and $ \triangle $ are $ 1/8 $, $ 3/8 $, $ 5/8 $, $ 7/8 $ respectively

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Figure 2. An illustration of $ \partial\bar{\tau}_j^{i,l} $, $ \partial\bar{\tau}_j^{i,t} $, $ \partial\tau_j^{i,b} $ and $ \partial\tau_j^{i,r} $ located on $ \bar{\tau}_j^i $ of the layer $ j = 4 $

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Figure 3. Illustration of the 4-color domain decomposition for a domain of size $ 8\times 8 $, where each color has one element on the coarse layer $ j = 3 $

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Figure 4. The test images with sizes and their identifiers

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Figure 5. Denoised results and the residual images obtained by different methods on image 'Baboon' ($ \#6 $) of the noise level $ \sigma = 20 $ and $ \alpha = 15 $, and image 'Building' ($ \#8 $) of noise level $ \sigma = 40 $ and $ \alpha = 35 $

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Figure 6. The decay of relative error and numerical energy for Baboon image ($ \#6 $) of noise level $ \sigma = 20 $ with $ \alpha = 15 $ and Building image ($ \#8 $) of noise level $ \sigma = 40 $ with $ \alpha = 35 $. From left to right on the first row is the relative error and numerical energy of Baboon image ($ \#6 $) while on the second row are relative error and numerical energy of Building image ($ \#8 $)

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Figure 7. Image deburring results obtained by our MMC and PD method on 'Boat' and 'Lena' with motion blur

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Figure 8. Both CT reconstruction results (Row one) and residual images (Row two) obtained by MMC and PD method for 'Forbil-gen' phantom with $ N_p = 36 $ and $ 72 $

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Figure 9. MRI reconstruction results (Row one) and residual images (Row two) obtained by our MMC and PD method on the brain image with radial sampling patterns and sampling lines $ N_s = 30 $ and $ 50 $

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Table 1. The results on selected images ($ \#1 $, $ \#3 $, $ \#5 $ and $ \#7 $) with different $ J $ for noise level $ \sigma = 20 $

	$ \alpha $	10			15			20
MML	$ J $	Iter	Energy	CPU(s)	Iter	Energy	CPU(s)	Iter	Energy	CPU(s)
#1	1	54	2.32E7	1.10	86	3.12E7	2.05	128	3.80E7	3.12
	2	23	2.31E7	1.21	50	3.05E7	2.54	80	3.62E7	3.49
	3	24	2.29E7	1.48	40	3.03E7	2.46	31	3.53E7	3.67
	4	22	2.29E7	1.53	32	3.03E7	3.35	29	3.52E7	3.21
	5	24	2.29E7	1.85	32	3.03E7	4.88	29	3.52E7	4.51
	6	22	2.29E7	2.08	23	3.03E7	5.73	29	3.52E7	5.71
#3	1	35	8.60E7	5.64	82	1.14E8	11.32	139	1.38E8	19.35
	2	18	8.57E7	5.87	36	1.11E8	10.32	65	1.29E8	16.43
	3	18	8.56E7	6.18	44	1.07E8	10.65	32	1.25E8	14.47
	4	18	8.56E7	6.79	31	1.07E8	9.31	32	1.24E8	11.28
	5	18	8.56E7	6.77	31	1.07E8	9.78	32	1.24E8	12.02
	6	18	8.56E7	7.25	31	1.07E8	10.17	32	1.24E8	12.56
#5	1	49	3.54E8	27.32	71	4.72E8	42.35	198	5.63E8	79.56
	2	31	3.54E8	31.47	53	4.60E8	50.61	70	5.34E8	71.35
	3	28	3.53E8	32.06	43	4.59E8	49.35	54	5.24E8	60.21
	4	21	3.53E8	33.37	36	4.59E8	48.35	52	5.22E8	52.13
	5	21	3.53E8	34.25	36	4.59E8	50.22	52	5.22E8	66.32
	6	21	3.53E8	38.24	36	4.59E8	51.78	52	5.22E8	75.21
#7	1	57	1.38E9	121.32	120	1.82E9	237.21	189	2.21E9	369.75
	2	31	1.38E9	117.24	61	1.76E9	225.76	102	2.05E9	366.14
	3	26	1.37E9	115.78	44	1.76E9	193.35	65	1.99E9	299.78
	4	26	1.37E9	122.73	39	1.75E9	192.49	56	1.94E9	280.32
	5	26	1.37E9	126.25	39	1.75E9	208.15	53	1.94E9	288.25
	6	26	1.37E9	139.35	39	1.75E9	229.29	53	1.94E9	282.12

MMC	$ J $	Iter	Energy	CPU(s)	Iter	Energy	CPU(s)	Iter	Energy	CPU(s)
#1	1	33	2.31E7	0.85	72	3.12E7	1.72	112	3.79E7	2.77
	2	21	2.31E7	0.69	41	3.05E7	1.22	37	3.63E7	1.21
	3	23	2.31E7	0.83	33	3.03E7	1.15	46	3.56E7	1.54
	4	21	2.31E7	0.88	33	3.03E7	1.21	34	3.54E7	1.20
	5	21	2.31E7	0.83	29	3.03E7	1.31	33	3.54E7	1.28
	6	20	2.31E7	0.82	23	3.03E7	1.95	33	3.54E7	1.55
#3	1	39	8.59E7	4.96	80	1.14E8	9.65	134	1.39E8	15.84
	2	28	8.58E7	4.07	36	1.11E8	5.13	54	1.31E8	7.54
	3	27	8.58E7	4.25	42	1.07E8	6.41	46	1.26E8	6.99
	4	27	8.58E7	4.27	37	1.07E8	5.97	34	1.24E8	5.75
	5	27	8.58E7	4.73	33	1.07E8	5.68	38	1.24E8	6.56
	6	26	8.58E7	4.56	30	1.07E8	5.93	32	1.24E8	6.73
#5	1	51	3.54E8	23.32	99	4.71E8	44.23	161	5.64E8	74.36
	2	33	3.54E8	20.67	49	4.60E8	29.95	66	5.37E8	42.74
	3	31	3.53E8	23.07	46	4.59E8	31.26	47	5.26E8	34.33
	4	30	3.53E8	22.07	40	4.59E8	28.98	48	5.23E8	34.28
	5	28	3.53E8	25.88	42	4.59E8	31.20	43	5.23E8	37.25
	6	29	3.53E8	30.15	39	4.59E8	36.59	44	5.23E8	42.67
#7	1	57	1.38E9	121.96	120	1.82E9	237.88	189	2.21E9	369.02
	2	36	1.38E9	100.06	56	1.76E9	124.64	84	2.05E9	183.47
	3	34	1.38E9	86.18	44	1.75E9	110.30	62	1.99E9	150.35
	4	34	1.36E9	81.35	43	1.74E9	105.42	43	1.95E9	114.78
	5	37	1.36E9	90.06	43	1.74E9	109.73	39	1.95E9	117.75
	6	36	1.36E9	98.81	43	1.74E9	115.06	40	1.95E9	123.42

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Table 2. The denoising results by MMC and MML methods for Rudin-Osher-Fatemi model with $ \alpha = 10,\ 15,\ 20 $ for noise level $ \sigma = 20 $.

	$ \alpha=10 $					$ \alpha=15 $				$ \alpha=20 $
		MMC	MML	Dual	P-D	MMC	MML	Dual	P-D	MMC	MML	Dual	P-D
PSNR	#1	28.77	28.77	28.71	28.70	29.26	29.27	29.23	29.27	29.14	29.14	29.13	29.17
	#2	28.43	28.42	28.40	28.41	28.64	28.64	28.60	28.62	27.61	27.59	27.55	27.58
	#3	29.73	29.73	29.70	29.73	30.79	30.77	30.72	30.75	30.60	30.63	30.58	30.60
	#4	29.84	29.86	29.83	29.83	31.14	31.13	31.09	31.11	30.28	30.32	30.28	30.30
	#5	29.18	29.16	29.15	29.15	29.64	29.64	29.63	29.63	29.35	29.34	29.30	29.33
	#6	27.81	27.82	27.80	27.79	27.95	27.95	27.89	27.94	26.58	26.58	26.55	26.55
	#7	30.70	30.65	30.60	30.68	32.45	32.44	32.43	32.40	32.33	32.33	32.26	32.38
	#8	30.43	30.40	30.39	30.41	32.28	32.25	32.23	32.27	31.27	31.26	31.20	31.25
SSIM	#1	0.7786	0.7771	0.7735	0.7757	0.8511	0.8521	0.8515	0.8523	0.8492	0.8499	0.8491	0.8501
	#2	0.7473	0.7451	0.7376	0.7391	0.8303	0.8279	0.8299	0.8255	0.8243	0.8216	0.8255	0.8299
	#3	0.7342	0.7277	0.7225	0.7245	0.8256	0.8221	0.8242	0.8253	0.8195	0.8207	0.8217	0.8228
	#4	0.7544	0.7431	0.7419	0.7433	0.8548	0.8562	0.854	0.8537	0.8518	0.8548	0.8520	0.8521
	#5	0.7287	0.7298	0.7267	0.7294	0.7649	0.7631	0.7622	0.7641	0.7381	0.7394	0.7388	0.7395
	#6	0.7953	0.7909	0.7943	0.7952	0.8031	0.8131	0.8124	0.8122	0.7883	0.7894	0.7788	0.7793
	#7	0.7447	0.7412	0.7368	0.7342	0.8865	0.8877	0.8788	0.8791	0.8725	0.8717	0.8736	0.8744
	#8	0.7386	0.7252	0.7237	0.7241	0.9203	0.9107	0.9154	0.9165	0.9154	0.9199	0.9198	0.9192
CPU(s)	#1	0.88	1.53	2.41	1.40	1.21	3.35	4.49	2.15	1.20	3.61	7.61	2.71
	#2	1.00	1.64	1.56	1.72	1.17	2.01	2.81	1.97	1.18	3.11	9.01	2.21
	#3	4.27	6.94	8.26	5.62	5.97	9.83	14.42	8.60	5.81	11.28	20.67	13.94
	#4	4.22	6.81	8.56	6.13	5.13	9.13	15.01	8.11	5.22	12.10	21.35	11.65
	#5	22.68	33.19	30.12	19.73	28.68	48.35	54.08	29.93	34.28	52.13	63.59	41.57
	#6	18.33	23.19	31.01	20.21	26.78	50.14	51.35	28.86	28.48	56.13	66.01	39.73
	#7	88.80	122.73	184.93	113.37	106.43	192.49	463.12	161.99	115.83	280.32	560.32	188.35
	#8	116.35	133.84	190.21	121.61	123.28	183.17	454.35	159.73	134.78	244.35	555.38	198.46
Energy	#1	2.31E7	2.29E7	2.31E7	2.31E7	3.03E7	3.01E7	3.03E7	3.03E7	3.52E7	3.52E7	3.53E7	3.53E7
	#2	2.44E7	2.43E7	2.43E7	2.43E7	3.23E7	3.21E7	3.23E7	3.23E7	3.70E7	3.71E7	3.71E7	3.70E7
	#3	1.28E8	1.29E8	1.29E8	1.28E8	2.77E8	2.78E8	2.78E8	2.78E8	4.66E8	4.67E8	4.67E8	4.67E8
	#4	8.39E7	8.37E7	8.39E7	8.39E7	1.08E8	1.09E8	1.09E8	1.09E8	1.22E8	1.23E8	1.22E8	1.23E8
	#5	3.53E8	3.51E8	3.53E8	3.53E8	4.59E8	4.57E8	4.57E8	4.57E8	5.23E8	5.21E8	5.22E8	5.22E8
	#6	3.71E8	3.72E8	3.72E8	3.72E8	4.71E8	4.72E8	4.71E8	4.71E8	5.97E8	5.97E8	5.98E8	5.98E8
	#7	1.37E9	1.36E9	1.37E9	1.36E9	1.75E9	1.75E9	1.75E9	1.75E9	1.95E9	1.94E9	1.95E9	1.95E9
	#8	1.40E9	1.42E9	1.40E9	1.41E9	1.81E9	1.80E9	1.81E9	1.80E9	1.97E9	1.98E9	1.98E9	1.97E9

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Table 3. The denoising results for Rudin-Osher-Fatemi model with different noise levels $ \sigma = 20,\ 30,\ 40 $.

	$ \alpha=15,\ \sigma=20 $					$ \alpha=25,\ \sigma=30 $				$ \alpha=35,\ \sigma=40 $
		MMC	MML	Dual	P-D	MMC	MML	Dual	P-D	MMC	MML	Dual	P-D
PSNR	#1	29.26	29.27	29.23	29.27	26.73	26.73	26.73	26.73	25.04	25.04	25.04	25.04
	#2	28.64	28.64	28.60	28.62	26.49	26.44	26.40	26.42	25.01	24.95	24.96	24.96
	#3	30.79	30.77	30.72	30.75	28.78	28.78	28.78	28.78	27.25	27.21	27.21	27.21
	#4	31.14	31.13	31.09	31.11	29.13	29.11	29.11	29.12	27.21	27.21	27.21	27.21
	#5	29.64	29.64	29.63	29.63	28.73	27.75	27.75	27.74	26.40	26.38	26.39	26.40
	#6	27.95	27.95	27.89	27.94	25.44	25.45	25.42	24.13	23.93	23.94	23.92	23.94
	#7	32.45	32.44	32.43	32.40	29.92	29.91	29.90	29.91	28.20	28.20	28.20	28.20
	#8	32.28	32.25	32.23	32.27	29.95	29.93	29.85	29.87	27.52	27.52	27.51	27.51
SSIM	#1	0.8511	0.8521	0.8515	0.8523	0.8051	0.8048	0.8046	0.8084	0.7561	0.7563	0.7601	0.7623
	#2	0.8303	0.8279	0.8299	0.8255	0.7782	0.7755	0.7749	0.7811	0.7499	0.7476	0.7377	0.7492
	#3	0.8256	0.8221	0.8242	0.8253	0.7842	0.7827	0.7841	0.7851	0.7555	0.7519	0.7549	0.7537
	#4	0.8548	0.8562	0.8542	0.8537	0.8046	0.8011	0.8044	0.8012	0.7717	0.7782	0.7739	0.7717
	#5	0.7649	0.7631	0.7622	0.7641	0.7162	0.7144	0.7088	0.7101	0.6772	0.6766	0.6688	0.6768
	#6	0.8031	0.8131	0.8124	0.8122	0.7311	0.7353	0.7344	0.7349	0.6601	0.6721	0.6687	0.6699
	#7	0.8865	0.8877	0.8788	0.8791	0.8478	0.8516	0.8496	0.8502	0.8224	0.8236	0.8235	0.8247
	#8	0.9203	0.9107	0.9154	0.9165	0.9002	0.8997	0.8981	0.8994	0.8811	0.8868	0.8714	0.8726
CPU(s)	#1	1.21	3.35	4.49	2.15	1.62	3.88	4.35	3.99	2.55	5.95	5.69	5.81
	#2	1.17	2.01	2.81	1.97	1.79	3.61	7.13	4.23	3.05	11.78	7.13	8.68
	#3	5.97	9.83	14.42	8.60	6.93	16.06	17.14	11.86	9.88	37.98	17.19	15.06
	#4	5.13	9.13	15.01	8.11	7.68	15.76	19.06	8.99	9.85	20.74	18.99	15.04
	#5	28.68	48.35	54.08	29.93	35.66	76.67	64.43	49.30	48.53	190.87	78.30	56.35
	#6	26.78	50.14	51.35	28.86	38.63	79.12	78.84	58.84	50.25	180.62	64.30	55.44
	#7	106.43	192.49	463.12	161.99	161.37	334.45	452.12	188.14	186.92	491.83	443.35	269.35
	#8	123.28	183.17	454.35	159.73	164.28	303.28	443.18	191.18	208.15	482.78	482.06	275.05
Energy	#1	3.03E7	3.01E7	3.03E7	3.03E7	6.56E7	6.57E7	6.57E7	6.56E7	1.18E8	1.17E8	1.18E8	1.17E8
	#2	3.23E7	3.21E7	3.23E7	3.23E7	6.84E7	6.85E7	6.85E7	6.84E7	1.15E8	1.16E8	1.16E8	1.15E8
	#3	2.77E8	2.78E8	2.78E8	2.78E8	2.87E8	2.87E8	2.86E8	2.86E8	4.35E8	4.36E8	4.36E8	4.35E8
	#4	1.08E8	1.09E8	1.09E8	1.09E8	2.42E8	2.43E8	2.43E8	2.43E8	4.30E8	4.29E8	4.31E8	4.31E8
	#5	4.59E8	4.57E8	4.57E8	4.57E8	1.01E9	1.02E9	1.02E9	1.01E9	1.77E9	1.78E9	1.78E9	1.77E9
	#6	4.72E8	4.71E8	4.71E8	4.71E8	1.08E9	1.09E9	1.09E9	1.08E9	1.87E9	1.88E9	1.88E9	1.87E9
	#7	1.75E9	1.75E9	1.75E9	1.75E9	3.94E9	3.93E9	3.93E9	3.93E9	6.95E9	6.94E9	6.95E9	6.94E9
	#8	1.81E9	1.80E9	1.81E9	1.80E9	3.98E9	3.99E9	3.99E9	3.99E9	7.31E9	7.32E9	7.31E9	7.31E9

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Table 4. PSNR, SSIM values and CPUs for deblurring with different kernel

	Image	Gaussian Blur				Motion Blur
	Image	CPU	PSNR	SSIM	Energy	CPU	PSNR	SSIM	Energy
MMC	Lena	8.63	34.85	0.9207	1.4840	14.34	35.41	0.9401	1.5673
PD	Lena	16.22	34.82	0.9234	1.4713	19.41	35.36	0.9382	1.5610
MMC	Boat	16.69	33.83	0.9204	1.9612	18.98	33.43	0.9178	1.9549
PD	Boat	21.22	33.86	0.9222	1.9623	29.41	33.26	0.9166	1.9568

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Table 5. PSNR values and CPUs for CT reconstruction with different projection numbers of MMC and PD method

	Image	$ N_p $	18				36				72
	Image	Size	CPU(s)	PSNR	SSIM	Energy	CPU(s)	PSNR	SSIM	Energy	CPU(s)	PSNR	SSIM	Energy
MMC	Shepp	256	15.64	39.28	0.9908	0.0508	20.01	50.19	0.9995	0.0292	33.21	58.30	0.9998	0.0292
PD	Shepp	256	27.59	39.16	0.9901	0.0508	59.33	50.37	0.9995	0.0292	112.04	58.21	0.9998	0.0292
MMC	Fobild	256	21.33	27.96	0.9508	0.0595	24.15	37.66	0.9838	0.0421	40.21	45.06	0.9918	0.0423
PD	Fobild	256	25.02	27.92	0.9500	0.0600	71.21	37.17	0.9828	0.0421	140.10	45.16	0.9917	0.0425
MMC	Shepp	512	96.62	35.79	0.9881	0.0991	103.54	43.22	0.9911	0.0587	129.91	50.60	0.9995	0.0585
PD	Shepp	512	176.86	35.72	0.9879	0.0991	204.57	43.21	0.9911	0.0585	370.14	50.51	0.9995	0.0585
MMC	Fobild	512	130.01	31.29	0.9726	0.1501	153.52	37.90	0.9899	0.1557	179.58	48.88	0.9925	0.0861
PD	Fobild	512	187.02	31.23	0.9726	0.1501	248.01	37.86	0.9896	0.1557	457.51	48.83	0.9925	0.0861

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Table 6. PSNR values and CPUs for MRI reconstruction with different numbers of sampling lines for the MMC and PD method

	$ N_s $	30				50
	Pattern	CPU	PSNR	SSIM	Energy	CPU	PSNR	SSIM	Energy
MMC	Radial	4.78	32.15	0.9711	1.8840	10.19	38.41	0.9847	1.1673
PD	Radial	13.22	32.08	0.9701	1.8840	19.41	38.36	0.9837	1.1675
MMC	Random	5.84	27.03	0.9500	2.0915	11.98	35.73	0.9812	2.0070
PD	Random	15.34	27.01	0.9504	2.0913	25.65	35.72	0.9812	2.0070

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