A new hybrid $ l_p $-$ l_2 $ model for sparse solutions with applications to image processing

Xuerui Gao; Yanqin Bai; Shu-Cherng Fang; Jian Luo; Qian Li

doi:10.3934/jimo.2021211

Article Contents

2023, Volume 19, Issue 2: 890-915. Doi: 10.3934/jimo.2021211

This issue Previous Article A novel algorithm for approximating common solution of a system of monotone inclusion problems and common fixed point problem Next Article Distributionally robust chance constrained svm model with $\ell_2$-Wasserstein distance

A new hybrid $ l_p $-$ l_2 $ model for sparse solutions with applications to image processing

1.
Department of Mathematics, Shanghai University, Shanghai 200444, China
2.
Edward P. Fitts Department of Industrial and, Systems Engineering, North Carolina State University, Raleigh, NC 27695-7906, USA
3.
School of Management Science and, Engineering, Dongbei University of Finance and Economics, Dalian 116025, China
4.
School of Mathematics, Physics and, Statistics, Shanghai University of Engineering Science, Shanghai 201620, China

^* Corresponding author: Yanqin Bai (yqbai@shu.edu.cn)
^* Corresponding author: Yanqin Bai (yqbai@shu.edu.cn)

Received: August 2021

Revised: October 2021

Early access: December 2021

Published: February 2023

This work has been sponsored by the National Natural Science Foundations of China Grant 12171307, 11901382 and 71701035; The US Army Research Office Grant W911NF-15-1-0223; Chinese NNF Grant 71620107003 for Major International Joint Research Project

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Abstract

Finding sparse solutions to a linear system has many real-world applications. In this paper, we study a new hybrid of the $ l_p $ quasi-norm ($ 0 <p< 1 $) and $ l_2 $ norm to approximate the $ l_0 $ norm and propose a new model for sparse optimization. The optimality conditions of the proposed model are carefully analyzed for constructing a partial linear approximation fixed-point algorithm. A convergence proof of the algorithm is provided. Computational experiments on image recovery and deblurring problems clearly confirm the superiority of the proposed model over several state-of-the-art models in terms of the signal-to-noise ratio and computational time.

Keywords:

Sparse optimization,
hybrid of the l_p quasi-norm and l₂ norm,
optimality conditions,
image processing.

Mathematics Subject Classification: Primary: 90C26, 90C46, 90C90; Secondary: 65K05.

Citation:

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Figure 1. Comparison for $ Q(x) $ with $ p = 1/3 $ by taking different $ \alpha $

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Figure 2. Comparison for $ Q^*({{\mathit{\boldsymbol{x}}}}) $ with different $ p $

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Figure 3. Comparison between $ Q^*(x) $ and $ Q^*_t(x) $ with $ p = 1/3 $

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Figure 4. Contours of $ F(x) $ with different $ \lambda $ in two dimension

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Figure 5. 256 $ \times $ 256 Shepp-Logan head phantom and the images recovered by PLAFPA$ _p $

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Figure 6. Comparison results of recoverability

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Figure 7. The original images and the observed blurred noise images

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Figure 8. The values of SNR obtained by PLAFPA$ _p $ with different $ p $ at different numbers of iterations for image deblurring

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Figure 9. The values of SNR obtained by tested methods at different numbers of iterations for image deblurring

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Figure 10. The results of "Cameraman" and "Lena" image deblurring obtained by different tested methods

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Figure 11. The choice of $ \lambda $ for each tested method and the values of SNR obtained by tested methods at different numbers of iterations for the "Cameraman" image deblurring

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Figure 12. SNR value trend graph with different $ p $ for the method PLAFPA$ _p $

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Figure 13. $ \Phi(\eta) $ for $ \eta>0 $

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Table 1. Results of image recovery for the Shepp-Logan head phantom

Method		$ l/n_s $=0.51			$ l/n_s $=0.39			$ l/n_s $=0.25
Method		SNR	time	Iter	SNR	time	Iter	SNR	time	Iter
Soft		136.60	77.70	813	122.18	103.09	1000	89.72	94.73	1000
Half		173.20	10.20	94	169.09	15.87	139	161.68	33.04	306
PQA		160.46	18.36	183	155.84	27.23	285	52.80	42.77	438
H$ L_2 $-$ L_{p} $	$ p=1/5 $	173.80	10.06	94	169.43	13.63	135	162.50	28.59	298
	$ p=1/2 $	173.53	11.10	94	169.25	13.95	137	161.76	29.70	301
	$ p=4/5 $	173.92	9.66	97	169.64	13.91	143	162.10	31.96	322
PLAFPA$ _{p} $	$ p=1/5 $	175.08	6.19	76	170.14	8.63	113	163.09	21.10	256
	$ p=1/2 $	175.53	6.36	77	171.70	9.57	113	163.14	20.31	251
	$ p=4/5 $	175.35	6.30	80	171.58	9.86	121	163.99	21.98	279

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Table 2. Results of image deblurring by PLAFPA$ _{8/9} $ at 500 iterations

	t	10	20	30	40	50	60	70	80	90	100
"Cameraman"	SNR	12.10	12.80	13.05	13.15	13.18	13.19	13.17	13.15	13.12	13.09
	time	10.47	12.53	14.09	15.46	15.92	17.11	17.68	18.31	19.26	20.62
"Lena"	t	10	20	30	40	50	60	70	80	90	100
	SNR	13.96	14.69	15.03	15.17	15.21	15.19	15.15	15.09	15.03	14.97
	time	49.38	54.88	60.33	64.82	68.99	72.83	76.09	78.31	80.59	83.25

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Table 3. "Cameraman" image deblurring at different numbers of iterations

	Iter	100	200	300	400	500	600	700	800	900	1000
Soft	SNR	11.84	12.30	12.46	12.51	12.52	12.52	12.50	12.49	12.47	12.45
	time	1.68	2.75	3.98	4.95	6.12	7.29	8.43	9.51	10.57	11.84
Half	Iter	100	200	300	400	500	600	700	800	900	1000
	SNR	11.37	11.55	11.55	11.51	11.46	11.41	11.36	11.31	11.27	11.23
	time	3.67	5.24	7.68	10.30	13.50	15.88	18.47	20.42	23.21	25.10
PQA	Iter	100	200	300	400	500	600	700	800	900	1000
	SNR	11.98	12.55	12.74	12.77	12.72	12.64	12.54	12.42	12.30	12.18
	time	1.92	2.67	3.91	5.13	6.20	7.46	9.76	9.96	11.27	13.58
H$ L_2 $-$ L_{p} $ with $ p=8/9 $	Iter	100	200	300	400	500	600	700	800	900	1000
	SNR	11.99	12.60	12.81	12.88	12.87	12.82	12.76	12.68	12.59	12.50
	time	6.25	11.89	16.77	21.83	26.35	29.46	34.17	38.54	42.34	46.18
PLAFPA$ _{p} $ with $ p=8/9 $	Iter	100	200	300	400	500	600	700	800	900	1000
	SNR	11.87	12.56	12.88	13.05	13.15	13.20	13.22	13.28	13.22	13.22
	time	5.30	8.71	11.65	14.87	15.92	19.57	22.27	24.01	28.26	29.52

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Table 4. "Lena" image deblurring at different numbers of iteration

	Iter	100	200	300	400	500	600	700	800	900	1000
Soft	SNR	14.86	14.81	14.67	14.53	14.43	14.35	14.29	14.23	14.18	14.13
	time	7.68	15.21	21.94	27.80	37.62	42.08	48.75	56.46	63.33	69.26
Half	Iter	100	200	300	400	500	600	700	800	900	1000
	SNR	14.02	13.85	13.70	13.55	13.43	13.34	13.26	13.19	13.13	13.07
	time	12.50	24.50	35.50	47.05	59.63	71.80	83.47	94.40	107.80	118.27
PQA	Iter	100	200	300	400	500	600	700	800	900	1000
	SNR	15.13	15.15	14.82	14.41	14.00	13.62	13.27	12.94	12.65	12.36
	time	7.63	14.91	22.60	29.37	33.68	43.74	52.27	57.88	65.31	73.68
H$ L_2 $-$ L_{p} $ with $ p=8/9 $	Iter	100	200	300	400	500	600	700	800	900	1000
	SNR	15.18	15.28	15.02	14.69	14.35	14.02	13.72	13.44	13.18	12.94
	time	27.11	47.28	66.64	85.65	95.97	123.68	143.52	159.69	176.60	192.09
PLAFPA$ _{p} $ with $ p=8/9 $	Iter	100	200	300	400	500	600	700	800	900	1000
	SNR	15.13	15.45	15.43	15.33	15.21	15.09	14.96	14.84	14.73	14.62
	time	24.76	38.04	50.43	62.75	68.99	85.64	95.82	105.12	116.96	120.00

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