Tchebichef moment based restoration of Gaussian blurred images

Ahlad Kumar; Raveendran Paramesran; Chern-Loon Lim; Sarat C. Dass

doi:10.1364/AO.55.009006

Applied Optics
Vol. 55,
Issue 32,
pp. 9006-9016
(2016)
•https://doi.org/10.1364/AO.55.009006

Tchebichef moment based restoration of Gaussian blurred images

Ahlad Kumar, Raveendran Paramesran, Chern-Loon Lim, and Sarat C. Dass

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Ahlad Kumar, Raveendran Paramesran, Chern-Loon Lim, and Sarat C. Dass, "Tchebichef moment based restoration of Gaussian blurred images," Appl. Opt. 55, 9006-9016 (2016)

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Abstract

With the knowledge of how edges vary in the presence of a Gaussian blur, a method that uses low-order Tchebichef moments is proposed to estimate the blur parameters: sigma ( $σ$ ) and size ( $w$ ). The difference between the Tchebichef moments of the original and the reblurred images is used as feature vectors to train an extreme learning machine for estimating the blur parameters $(σ, w)$ . The effectiveness of the proposed method to estimate the blur parameters is examined using cross-database validation. The estimated blur parameters from the proposed method are used in the split Bregman-based image restoration algorithm. A comparative analysis of the proposed method with three existing methods using all the images from the LIVE database is carried out. The results show that the proposed method in most of the cases performs better than the three existing methods in terms of the visual quality evaluated using the structural similarity index.

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Images	Plain (%)	Edges (%)
Building2	3	97
paintedhouse	31	69
light house 2	51	49
parrots	85	15

		Edges		$Edges + Plains$				Edges		$Edges + Plain$
Images	$σ$	Estimated $σ$	Relative Error	Estimated $σ$	Relative Error	Images	$σ$	Estimated $σ$	Relative Error	Estimated $σ$	Relative Error
building2	0.5	0.51	0.02	1.51	2.02	lighthouse2	0.5	0.52	0.04	1.12	1.24
	1.0	0.99	0.01	2.14	1.14		1.0	1.05	0.05	2.54	0.54
	1.5	1.54	0.02	2.23	0.48		1.5	1.48	0.01	1.23	0.18
	2.0	2.07	0.03	2.87	0.43		2.0	2.08	0.04	3.22	0.61
	2.5	2.49	0.004	1.76	0.29		2.5	2.48	0.008	1.32	0.47
	3.0	3.11	0.03	3.95	0.31		3.0	3.12	0.04	4.13	0.37
	3.5	3.51	0.002	2.84	0.18		3.5	3.49	0.002	2.81	0.19
	4.0	4.11	0.02	4.52	0.13		4.0	4.15	0.03	3.21	0.19
paintedhouse	0.5	0.52	0.04	1.24	1.48	parrots	0.5	0.54	0.08	1.83	2.66
	1.0	1.04	0.04	0.78	0.22		1.0	1.07	0.07	0.32	0.68
	1.5	1.54	0.02	1.02	0.32		1.5	1.63	0.08	2.56	0.70
	2.0	1.89	0.05	1.32	0.34		2.0	2.19	0.09	1.67	0.17
	2.5	2.45	0.02	3.65	0.46		2.5	2.34	0.06	3.26	0.30
	3.0	3.16	0.05	2.15	0.28		3.0	3.26	0.08	2.24	0.25
	3.5	3.65	0.04	4.12	0.17		3.5	3.78	0.09	4.32	0.23
	4.0	4.16	0.04	3.23	0.19		4.0	3.67	0.08	3.27	0.182

		Edges	$Edges + Plains$			Edges	$Edges + Plain$
Images	( $w$ )	Estimated ( $w$ )	Estimated ( $w$ )	Images	( $w$ )	Estimated ( $w$ )	Estimated ( $w$ )
building2	$5 \times 5$	$7 \times 7$	$11 \times 11$	lighthouse2	$5 \times 5$	$5 \times 5$	$13 \times 13$
	$7 \times 7$	$5 \times 5$	$3 \times 3$		$7 \times 7$	$7 \times 7$	$11 \times 11$
	$9 \times 9$	$11 \times 11$	$13 \times 13$		$9 \times 9$	$11 \times 11$	$3 \times 3$
	$11 \times 11$	$11 \times 11$	$17 \times 17$		$11 \times 11$	$13 \times 13$	$17 \times 17$
	$13 \times 13$	$15 \times 15$	$7 \times 7$		$13 \times 13$	$11 \times 11$	$9 \times 9$
	$15 \times 15$	$15 \times 15$	$9 \times 9$		$15 \times 15$	$15 \times 15$	$7 \times 7$
	$17 \times 17$	$17 \times 17$	$11 \times 11$		$17 \times 17$	$15 \times 15$	$9 \times 9$
paintedhouse	$5 \times 5$	$5 \times 5$	$13 \times 13$	parrots	$5 \times 5$	$5 \times 5$	$7 \times 7$
	$7 \times 7$	$5 \times 5$	$3 \times 3$		$7 \times 7$	$7 \times 7$	$11 \times 11$
	$9 \times 9$	$7 \times 7$	$3 \times 3$		$9 \times 9$	$11 \times 11$	$15 \times 15$
	$11 \times 11$	$11 \times 11$	$17 \times 17$		$11 \times 11$	$11 \times 11$	$5 \times 5$
	$13 \times 13$	$15 \times 15$	$9 \times 9$		$13 \times 13$	$13 \times 13$	$7 \times 7$
	$15 \times 15$	$17 \times 17$	$9 \times 9$		$15 \times 15$	$13 \times 13$	$7 \times 7$
	$17 \times 17$	$17 \times 17$	$7 \times 7$		$17 \times 17$	$15 \times 15$	$9 \times 9$

Database	Original Images	Total Images	Training	Testing
LIVE	29	6612	4600	2012
CSIQ	30	6840	4788	2052
CIDIQ	23	5244	3670	1574
VCL	23	5244	3670	1574
Caltech	34	7752	5425	2327
Berkeley	32	7296	5107	2189

	Testing
Training	VCL	Berkeley	LIVE	Caltech	CSIQ	CIDIQ
VCL	—	0.8461	0.9388	0.8242	0.9223	0.8596
Berkeley	0.8628	—	0.8718	0.7901	0.8769	0.8555
LIVE	0.8649	0.8295	—	0.8102	0.8482	0.8026
Caltech	0.8358	0.8336	0.8020	—	0.8474	0.7781
CSIQ	0.8055	0.8440	0.8998	0.7963	—	0.8152
CIDIQ	0.8297	0.7983	0.7812	0.8444	0.8964	—

			SSIM
Images	Actual $(σ, w)$	Estimated $(σ, w)$	Proposed	[34]	[26]	[22]
building2	(0.5,7)	(0.51,5)	0.9742	0.9645	0.9567	0.9357
	(1.0,9)	(0.99,11)	0.9512	0.9354	0.9234	0.9134
	(1.5,11)	(1.54,11)	0.9132	0.8956	0.8867	0.8678
	(2.0,11)	(2.07,11)	0.8876	0.8582	0.8478	0.8356
	(2.5,13)	(2.49,15)	0.8612	0.8476	0.8354	0.8112
	(3.0,13)	(3.11,15)	0.8329	0.8251	0.8121	0.8092
	(3.5,15)	(3.51,15)	0.8113	0.7989	0.7923	0.7843
	(4.0,17)	(4.11,17)	0.8067	0.8145	0.7767	0.7512
paintedhouse	(0.5,7)	(0.52,5)	0.9736	0.9631	0.9434	0.9315
	(1.0,9)	(1.04,7)	0.9331	0.9236	0.9125	0.8926
	(1.5,11)	(1.54,11)	0.9223	0.9175	0.9067	0.8824
	(2.0,11)	(1.89,11)	0.9165	0.9031	0.8882	0.8752
	(2.5,13)	(2.45,15)	0.8678	0.8569	0.8361	0.8318
	(3.0,13)	(3.16,17)	0.8354	0.8236	0.8153	0.7925
	(3.5,15)	(3.65,17)	0.8143	0.8371	0.8049	0.7851
	(4.0,17)	(4.16,17)	0.7832	0.8178	0.7843	0.7721
lighthouse2	(0.5,7)	(0.52,7)	0.9832	0.9643	0.9587	0.9432
	(1.0,9)	(1.05,11)	0.9456	0.9303	0.9278	0.9272
	(1.5,11)	(1.48,11)	0.9165	0.9015	0.8996	0.8732
	(2.0,11)	(2.08,11)	0.8856	0.8784	0.8578	0.8341
	(2.5,13)	(2.48,13)	0.8621	0.8697	0.8486	0.8271
	(3.0,13)	(3.12,15)	0.8556	0.8612	0.8265	0.8072
	(3.5,15)	(3.49,15)	0.8376	0.8359	0.7974	0.7745
	(4.0,17)	(4.15,17)	0.8032	0.8118	0.7786	0.7632
parrots	(0.5,7)	(0.54,7)	0.9692	0.9594	0.9487	0.9332
	(1.0,9)	(1.07,11)	0.9465	0.9364	0.9212	0.9173
	(1.5,11)	(1.63,11)	0.9367	0.9252	0.8996	0.8732
	(2.0,11)	(2.19,11)	0.9132	0.9013	0.8778	0.8541
	(2.5,13)	(2.34,13)	0.8732	0.9108	0.8486	0.8271
	(3.0,13)	(3.26,13)	0.8456	0.8381	0.8265	0.8072
	(3.5,15)	(3.78,13)	0.8212	0.8280	0.7974	0.7745
	(4.0,17)	(3.67,15)	0.7897	0.8167	0.7786	0.7632

	Testing
Training	VCL	Berkeley	LIVE	Caltech	CSIQ	CIDIQ
VCL	—	0.8843	0.9241	0.8462	0.8967	0.8241
Berkeley	0.8423	—	0.8921	0.8321	0.8462	0.8872
LIVE	0.8376	0.8963	—	0.8367	0.8856	0.8346
Caltech	0.8571	0.8642	0.8528	—	0.8698	0.8134
CSIQ	0.8278	0.8571	0.9237	0.8341	—	0.7823
CIDIQ	0.7926	0.8472	0.8931	0.8647	0.8856	—

	Testing
Training	VCL	Berkeley	LIVE	Caltech	CSIQ	CIDIQ
VCL	—	0.8843	0.9241	0.8462	0.8967	0.8241
Berkeley	0.8423	—	0.8921	0.8321	0.8462	0.8872
LIVE	0.8376	0.8963	—	0.8367	0.8856	0.8346
Caltech	0.8571	0.8642	0.8528	—	0.8698	0.8134
CSIQ	0.8278	0.8571	0.9237	0.8341	—	0.7823
CIDIQ	0.7926	0.8472	0.8931	0.8647	0.8856	—

Abstract

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Applied Optics