Alternating method based on framelet l₀-norm and TV regularization for image restoration

Figures & data

Figure 1. Relatively errors for the DAL method (dash line) and the MDAL method (solid line).

Figure 2. Original images for ‘Barbara’,‘Kitten’,‘Bridge’ and ‘Mandrill’, respectively.

Figure 3. Blur kernels: GB(9,2), AB(9,9), MB(20,0), respectively.

Table 1. SNR, PSNR(dB), SSIM values, iteration numbers, and CPU time of different methods for the different images corrupted by Gaussian blur and Gaussian noise.

Image	Blur/ BSNR	Method	SNR/PSNR	SSIM	Ites	Time
Barbara	$\mathrm{GB}(3,2)/4010.28$	Fl0pro	19.93/33.04	0.9497	12	0.78
		Fl0	18.23/31.33	0.9423	15	1.14
		Fast-TV	18.68/31.79	0.9434	12	0.09
	$\mathrm{GB}(5,2)/408.42$	Fl0pro	15.91/29.02	0.8889	11	0.73
		Fl0	15.26/28.36	0.8755	15	1.08
		Fast-TV	15.11/28.22	0.8852	14	0.09
	$\mathrm{GB}(7,2)/408.29$	Fl0pro	11.82/24.92	0.7846	10	0.61
		FL0	11.52/24.62	0.7647	21	1.35
		Fast-TV	10.90/24.01	0.7728	17	0.12
	$\mathrm{GB}(9,2)/308.10$	Fl0pro	9.34/22.44	0.6844	13	0.80
		FL0	9.10/22.20	0.6822	20	1.28
		Fast-TV	9.23/22.34	0.6512	19	0.13
Kitten	$\mathrm{GB}(3,2)/4013.42$	Fl0pro	21.62/33.55	0.9523	9	0.87
		FL0	20.38/32.30	0.9503	8	0.74
		Fast-TV	21.09/33.02	0.9444	11	0.12
	$\mathrm{GB}(5,2)/4010.17$	Fl0pro	17.10/29.03	0.9046	8	0.77
		FL0	16.46/28.39	0.8977	11	1.01
		Fast-TV	16.91/28.84	0.9023	13	0.14
	$\mathrm{GB}(7,2)/409.12$	Fl0pro	14.14/26.07	0.8348	13	1.22
		FL0	13.70/25.63	0.8237	14	1.23
		Fast-TV	13.81/25.74	0.8358	17	0.18
	$\mathrm{GB}(9,2)/308.75$	Fl0pro	11.78/23.70	0.7378	12	1.11
		FL0	10.77/22.70	0.7395	24	2.13
		Fast-TV	11.66/23.59	0.7135	20	0.21
Bridge	$\mathrm{GB}(3,2)/4010.53$	Fl0pro	18.71/32.54	0.9532	9	0.84
		FL0	18.38/32.21	0.9491	8	0.72
		Fast-TV	18.54/32.37	0.9559	11	0.12
	$\mathrm{GB}(5,2)/408.45$	Fl0pro	14.42/28.25	0.8803	11	1.04
		FL0	14.25/28.08	0.8650	11	0.97
		Fast-TV	13.92/27.75	0.8658	18	0.19
	$\mathrm{GB}(7,2)/407.67$	Fl0pro	11.84/25.67	0.7827	17	1.59
		FL0	11.65/25.49	0.7738	15	1.36
		Fast-TV	11.25/25.08	0.7476	26	0.27
	$\mathrm{GB}(9,2)/307.37$	Fl0pro	9.73/23.56	0.6710	13	1.16
		FL0	9.25/23.08	0.6671	24	2.08
		Fast-TV	9.71/23.54	0.6567	28	0.29
Mandrill	$\mathrm{GB}(3,2)/407.90$	Fl0pro	16.88/32.48	0.9454	8	3.15
		FL0	16.59/32.19	0.9434	6	2.25
		Fast-TV	16.44/32.04	0.9483	12	0.58
	$\mathrm{GB}(5,2)/406.15$	Fl0pro	12.49/28.09	0.8731	9	3.53
		FL0	12.25/27.85	0.8593	9	3.37
		Fast-TV	11.64/27.24	0.8632	13	0.62
	$\mathrm{GB}(7,2)/405.69$	Fl0pro	9.53/25.13	0.7630	10	3.87
		FL0	9.48/25.08	0.7622	11	4.13
		Fast-TV	8.61/24.22	0.7299	15	0.72
	$\mathrm{GB}(9,2)/305.54$	Fl0pro	7.15/22.75	0.6275	6	2.32
		FL0	6.83/22.43	0.6261	16	5.92
		Fast-TV	7.04/22.64	0.6260	24	1.14

Figure 4. (a1), (b1), (c1), (d1) blurred images degraded by GB(5)/40, GB(7)/40, GB(7)/40, GB(9)/30, respectively; (a2), (b2), (c2), (d2) corresponding restored images by Fast-TV; (a3), (b3), (c3), (d3) corresponding restored images by Fl0; (a4), (b4), (c4), (d4) corresponding restored images by Fl0pro.

Table 2. SNR, PSNR(dB), SSIM values, iteration numbers, and CPU time(second) of different methods for the different images corrupted by average blur and Gaussian noise.

Image	Blur/ BSNR	Method	SNR/PSNR	SSIM	Ites	Time
Barbara	$\mathrm{AB}(5,5)/407.84$	Fl0pro	16.12/29.22	0.8895	11	0.67
		Fl0	15.15/28.26	0.8756	13	0.83
		Fast-TV	15.34/28.44	0.8811	14	0.10
	$\mathrm{AB}(7,7)/407.67$	Fl0pro	13.65/26.75	0.8385	9	0.55
		Fl0	13.01/26.11	0.8105	18	1.13
		Fast-TV	12.81/25.92	0.8301	15	0.11
	$\mathrm{AB}(9,9)/307.11$	Fl0pro	10.43/23.54	0.7158	12	0.97
		FL0	10.25/23.36	0.6937	21	1.38
		Fast-TV	10.18/23.29	0.7020	20	0.14
Kitten	$\mathrm{AB}(5,5)/409.31$	Fl0pro	17.37/29.30	0.9024	10	0.94
		FL0	16.16/28.09	0.8842	10	0.94
		Fast-TV	16.98/28.91	0.8975	12	0.13
	$\mathrm{AB}(7,7)/407.73$	Fl0pro	15.12/27.05	0.8559	13	1.21
		FL0	14.62/26.54	0.8456	11	1.03
		Fast-TV	14.88/26.81	0.8597	14	0.15
	$\mathrm{AB}(9,9)/306.88$	Fl0pro	11.58/23.50	0.7186	11	1.06
		FL0	11.34/23.26	0.7015	17	1.49
		Fast-TV	11.33/23.26	0.7123	20	0.22
Bridge	$\mathrm{AB}(5,5)/407.74$	Fl0pro	15.17/29.01	0.8945	11	1.01
		FL0	14.83/28.66	0.8783	11	0.98
		Fast-TV	15.10/28.93	0.8944	14	0.15
	$\mathrm{AB}(7,7)/406.44$	Fl0pro	13.48/27.31	0.8447	13	1.25
		FL0	13.25/27.08	0.7998	15	1.31
		Fast-TV	13.14/26.97	0.8351	15	0.16
	$\mathrm{AB}(9,9)/305.60$	Fl0pro	10.39/24.22	0.6990	16	1.50
		FL0	10.28/24.11	0.6522	20	1.77
		Fast-TV	10.20/24.03	0.6851	24	0.25
Mandrill	$\mathrm{AB}(5,5)/405.57$	Fl0pro	13.36/28.96	0.8856	9	3.39
		FL0	13.09/28.69	0.8821	7	2.57
		Fast-TV	12.50/28.10	0.8842	13	0.63
	$\mathrm{AB}(7,7)/404.88$	Fl0pro	11.44/27.04	0.8345	9	3.33
		FL0	11.19/26.79	0.8164	8	2.96
		Fast-TV	10.35/25.95	0.8147	14	0.68
	$\mathrm{AB}(9,9)/304.51$	Fl0pro	8.08/23.68	0.6631	11	4.05
		FL0	7.96/23.56	0.6370	13	4.73
		Fast-TV	8.02/23.63	0.6590	16	0.78

Figure 5. (a1), (b1), (c1), (d1) blurred images degraded by AB(5)/40, AB(7)/40, AB(7)/40, AB(9)/30, respectively; (a2), (b2), (c2), (d2) corresponding restored images by Fast-TV; (a3), (b3), (c3), (d3) corresponding restored images by Fl0; (a4), (b4), (c4), (d4) corresponding restored images by Fl0pro.

Table 3. SNR, PSNR(dB), SSIM values, iteration numbers, and CPU time(second) of different methods for the different images corrupted by motion blur and Gaussian noise.

Image	Blur/ BSNR	Method	SNR/PSNR	SSIM	Ites	Time
Barbara	$\mathrm{MB}(9,0)/407.54$	Fl0pro	17.92/31.02	0.9050	12	0.78
		Fl0	17.18/30.29	0.8920	16	1.08
		Fast-TV	16.71/29.82	0.9043	21	0.15
	$\mathrm{MB}(20,0)/305.77$	Fl0pro	9.59/22.69	0.6427	16	0.98
		Fl0	9.19/22.30	0.6361	23	1.53
		Fast-TV	9.09/22.20	0.6383	38	0.27
Kitten	$\mathrm{MB}(9,0)/409.13$	Fl0pro	20.13/32.06	0.9372	13	1.21
		FL0	19.24/31.17	0.9215	14	1.24
		Fast-TV	19.83/31.76	0.9298	12	0.13
	$\mathrm{MB}(20,0)/306.52$	Fl0pro	13.03/24.96	0.7742	13	1.21
		FL0	12.58/24.51	0.7781	25	2.18
		Fast-TV	12.42/24.34	0.7661	28	0.29
Bridge	$\mathrm{MB}(9,0)/408.19$	Fl0pro	18.25/32.08	0.9415	13	1.25
		FL0	17.94/31.77	0.9322	15	1.33
		Fast-TV	17.76/31.59	0.9387	21	0.23
	$\mathrm{MB}(20,0)/305.93$	Fl0pro	11.08/24.91	0.7422	21	1.90
		FL0	10.92/24.75	0.7357	26	2.27
		Fast-TV	11.03/24.86	0.7408	23	0.24
Mandrill	$\mathrm{MB}(9,0)/406.49$	Fl0pro	17.35/32.95	0.9407	7	2.81
		FL0	17.14/32.74	0.9405	8	2.97
		Fast-TV	16.28/31.88	0.9406	19	0.92
	$\mathrm{MB}(20,0)/304.64$	Fl0pro	10.04/25.64	0.7655	11	4.15
		FL0	9.83/25.43	0.7629	13	4.79
		Fast-TV	9.24/24.84	0.7419	19	0.91

Figure 6. (a1), (b1), (c1), (d1) blurred images degraded by MB(9)/40, MB(9)/40, MB(20)/30, MB(20)/30, respectively; (a2), (b2), (c2), (d2) corresponding restored images by Fast-TV; (a3), (b3), (c3), (d3) corresponding restored images by Fl0; (a4), (b4), (c4), (d4)corresponding restored images by Fl0pro.