Introducing genetic modification concept to optimize rational function models (RFMs) for georeferencing of satellite imagery

Figures & data

Figure 1. GA coding for RFM, presence and absence of a parameter in the structure of RFM is demonstrated by values ‘1’ (active genes) and ‘0’ (inactive genes), respectively.

Figure 2. (a) Schematic histogram of the frequency response of genes in the qualified offspring and (b) unqualified offspring, (c) an example of a transgenic chromosome produced based on most qualified genes, active genes are marked with grey.

Figure 3. Proposed GM optimization algorithm.

Table 1. Specifications of two datasets used.

Dataset	Hamedan	Isfahan
Platform	IKONOS-2	SPOT-1
Sensor	Panchromatic	HRV
Product	Geo	Level 1A
Acquisition	October 2000	June 1987	August 1987
Off-nadir angle	20.4°	20.84° E	24.7° W
Sampling	1 m	10 m
Base to height ratio	–	0.974
Number of GCPs	74	35
Elevation relief	121.68 m	618.70 m

Figure 4. Distribution of GCPs: (a) SPOT stereo images, (b) IKONOS-Geo image.

Table 2. Comparison of the results obtained from standard GA and the proposed GM algorithm, Hamedan dataset.

		Mean ± Std in GCPs			Mean ± Std in ICPs			Optimum results
Algorithm	GCPs, DCPs, ICPs	RMSE	δr	δc	RMSE	δr	δc	RMSE	δr	δc	# of coeffs. P1,P2,P3,P4	df	Iter.
Standard GA	6, 10, 58	0.43 ± 0.01	0.54 ± 0.02	0.63 ± 0.01	0.81 ± 0.03	0.87 ± 0.01	0.90 ± 0.00	0.69	0.85	0.90	4,2,3,2	1	224
	8, 10, 56	0.82 ± 0.00	0.87 ± 0.02	0.98 ± 0.09	0.91 ± 0.00	1.23 ± 0.06	1.37 ± 0.11	0.83	1.14	1.28	4,4,4,2	2	107
	10, 10, 54	0.54 ± 0.03	0.73 ± 0.11	0.94 ± 0.11	0.81 ± 0.04	1.17 ± 0.13	1.44 ± 0.10	0.74	0.67	1.47	5,3,6,4	2	134
	12, 10, 52	0.46 ± 0.05	0.51 ± 0.01	1.03 ± 0.05	0.79 ± 0.04	0.85 ± 0.00	1.42 ± 0.06	0.69	0.66	1.03	5,5,6,3	5	116
	14, 10, 50	0.39 ± 0.00	0.41 ± 0.05	0.62 ± 0.02	0.58 ± 0.01	0.69 ± 0.07	0.95 ± 0.02	0.58	0.78	0.91	6,6,4,5	7	105
	16, 10, 48	0.28 ± 0.00	0.39 ± 0.01	0.68 ± 0.01	0.59 ± 0.03	0.67 ± 0.02	0.98 ± 0.03	0.58	0.58	0.93	7,5,6,6	8	98
Proposed MG	6, 10, 58	0.51 ± 0.02	0.58 ± 0.01	0.62 ± 0.02	0.80 ± 0.01	0.86 ± 0.00	0.90 ± 0.00	0.65	0.78	0.90	5,1,3,2	1	19
	8, 10, 56	0.65 ± 0.02	0.81 ± 0.02	0.90 ± 0.04	0.85 ± 0.00	1.14 ± 0.03	1.33 ± 0.08	0.74	1.03	0.95	3,4,3,3	3	6
	10, 10, 54	0.53 ± 0.00	0.90 ± 0.08	1.06 ± 0.12	0.82 ± 0.00	1.16 ± 0.17	1.43 ± 0.18	0.76	0.87	1.44	4,5,4,3	4	5
	12, 10, 52	0.43 ± 0.02	0.57 ± 0.06	0.95 ± 0.04	0.75 ± 0.01	0.82 ± 0.05	1.27 ± 0.07	0.56	0.65	0.88	4,7,3,1	9	7
	14, 10, 50	0.24 ± 0.01	0.42 ± 0.00	0.60 ± 0.00	0.59 ± 0.00	0.66 ± 0.03	0.95 ± 0.01	0.57	0.78	0.87	5,4,5,4	10	4
	16, 10, 48	0.24 ± 0.00	0.43 ± 0.01	0.59 ± 0.00	0.58 ± 0.00	0.65 ± 0.03	0.94 ± 0.02	0.57	0.46	0.90	6,4,7,4	11	3

Table 3. Comparison of the results obtained from standard GA and the proposed GM algorithm, Isfahan dataset (left and right images).

			Mean ± Std in GCPs			Mean ± Std in ICPs			Optimum results
Image	Algorithm	GCPs, DCPs, ICPs	RMSE	δr	δc	RMSE	δr	δc	RMSE	δr	δc	# of coeffs. P1,P2,P3,P4	df	Iter.
Left	Standard GA	8, 7, 20	0.61 ± 0.00	0.75 ± 0.06	1.31 ± 0.03	0.97 ± 0.01	1.08 ± 0.11	1.88 ± 0.01	0.95	0.93	1.73	4,2,5,3	2	241
		10, 7, 18	0.65 ± 0.4	0.59 ± 0.19	1.30 ± 0.09	0.95 ± 0.10	0.78 ± 0.21	1.82 ± 0.08	0.92	0.85	1.56	3,4,4,5	4	200
		15, 7, 13	0.70 ± 0.03	1.02 ± 0.11	1.44 ± 0.17	0.97 ± 0.07	1.36 ± 0.12	1.89 ± 0.21	0.92	1.36	1.47	6,4,6,5	7	135
		20, 7, 8	0.63 ± 0.02	0.95 ± 0.20	1.07 ± 0.19	0.91 ± 0.02	1.12 ± 0.24	1.35 ± 0.17	0.89	1.10	1.26	9,8,8,7	8	113
	Proposed MG	8, 7, 20	0.71 ± 0.03	0.80 ± 0.02	1.41 ± 0.03	0.95 ± 0.00	0.96 ± 0.01	1.78 ± 0.01	0.94	0.86	1.67	3,2,4,3	4	47
		10, 7, 18	0.76 ± 0.00	0.68 ± 0.01	1.36 ± 0.05	0.94 ± 0.00	0.84 ± 0.00	1.79 ± 0.01	0.93	0.71	1.71	4,1,6,2	7	11
		15, 7, 13	0.75 ± 0.03	0.99 ± 0.04	1.26 ± 0.07	0.97 ± 0.00	1.16 ± 0.08	1.78 ± 0.09	0.92	1.41	1.36	5,3,5,6	11	12
		20, 7, 8	0.69 ± 0.01	1.01 ± 0.07	1.11 ± 0.02	0.91 ± 0.00	1.21 ± 0.06	1.35 ± 0.06	0.88	1.08	1.19	7,6,7,7	13	11
Right	Standard GA	8, 7, 20	1.09 ± 0.04	1.89 ± 0.03	1.95 ± 0.17	1.36 ± 0.05	2.26 ± 0.01	2.28 ± 0.21	1.31	2.09	2.71	4,3,5,2	2	280
		10, 7, 18	1.12 ± 0.9	1.47 ± 0.02	2.18 ± 0.35	1.21 ± 0.10	1.84 ± 0.01	2.51 ± 0.37	1.20	1.97	2.96	6,3,5,3	3	180
		15, 7, 13	0.99 ± 0.02	2.08 ± 0.15	2.23 ± 0.05	1.24 ± 0.06	2.40 ± 0.13	2.57 ± 0.07	1.24	2.04	2.15	7,5,7,4	7	120
		20, 7, 8	1.04 ± 0.02	1.47 ± 0.00	1.84 ± 0.05	1.23 ± 0.01	1.82 ± 0.02	2.18 ± 0.04	1.17	1.85	2.04	8,6,9,8	9	105
	Proposed MG	8, 7, 20	1.13 ± 0.03	2.00 ± 0.01	2.06 ± 0.10	1.33 ± 0.01	2.21 ± 0.02	2.30 ± 0.14	1.31	2.08	2.33	4,1,4,3	4	37
		10, 7, 18	0.99 ± 0.01	1.44 ± 0.01	2.09 ± 0.15	1.23 ± 0.00	1.82 ± 0.03	2.38 ± 0.14	1.22	1.92	2.33	4,3,4,3	6	17
		15, 7, 13	1.01 ± 0.05	1.60 ± 0.03	2.19 ± 0.05	1.24 ± 0.00	1.91 ± 0.07	2.21 ± 0.07	1.24	1.40	1.63	5,3,5,4	13	12
		20, 7, 8	1.00 ± 0.06	1.41 ± 0.02	1.73 ± 0.08	1.22 ± 0.00	1.78 ± 0.03	2.12 ± 0.04	1.16	1.73	1.68	6,7,10,5	12	9

Figure 5. Comparison between number of iterations of standard GA and proposed GM algorithm: (a) Hamedan dataset, (b) Isfahan dataset (left image), (c) Isfahan dataset (right image).

Table 4. Results obtained from conventional RFMs regarding the maximum number of estimable parameters using available GCPs.

		GCPs (in pixels)			ICPs (in pixels)
Image	GCPs, ICPs	RMSE	δr	δc	RMSE	δr	δc	# of coeffs. P1,P2,P3,P4	df
IKONOS	6, 68	3.46	0.58	6.09	9.25	−3.77	9.77	3,2,3,2	2
	8, 66	0.21	−0.02	−0.38	1.84	−1.39	−1.63	4,3,4,3	2
	10, 64	0.18	0.43	0.11	1.34	−1.07	−1.61	5,4,5,4	2
	12, 42	0.29	0.45	−0.37	0.97	−1.01	1.20	6,5,6,5	2
	14, 60	0.16	0.06	0.31	1.34	−1.97	1.86	7,6,7,6	2
	16, 58	0.11	−0.21	−0.13	1.12	1.09	2.39	8,7,8,7	2
SPOT-L	8, 27	0.62	0.95	−0.77	3.74	−3.73	−1.93	4,3,4,3	2
	10, 25	0.47	−0.55	−0.51	2.74	−2.48	2.66	5,4,5,4	2
	15, 20	0.32	0.40	−0.54	2.18	−2.40	−3.19	7,6,7,6	4
	20, 15	0.06	0.16	0.01	6.25	−1.71	19.13	10,9,10,9	2
SPOT-R	8, 27	0.61	0.08	−1.11	4.35	4.08	2.71	4,3,4,3	2
	10, 25	0.60	−0.83	0.73	3.07	3.04	−3.88	5,4,5,4	2
	15, 20	0.40	−0.21	−0.88	2.24	2.83	−3.32	7,6,7,6	4
	20, 15	0.33	1.42	0.17	7.5	−14.46	21.73	10,9,10,9	2

Figure 6. Comparison between accuracies of standard GA and the proposed GM algorithm with traditional RFMs: (a) Hamedan dataset, (b) Isfahan dataset (left and right images).

Table 5. Results obtained from optimal conventional RFMs, surveyed in a try and error approach.

		GCPs (in pixels)			ICPs (in pixels)
Image	GCPs, ICPs	RMSE	δr	δc	RMSE	δr	δc	# of coeffs. P1,P2,P3,P4	df
IKONOS	6, 68	3.46	0.58	6.09	9.25	−3.77	9.77	3,2,3,2	2
	8, 66	0.21	−0.02	−0.38	1.84	−1.39	−1.63	4,3,4,3	2
	10, 64	0.39	0.45	−0.61	1.05	−1.03	−1.04	4,3,4,3	6
	12, 42	0.37	0.57	0.46	0.97	−1.01	−1.20	5,4,5,4	6
	14, 60	0.34	0.73	−0.36	0.81	0.78	1.08	6,5,6,5	6
	16, 58	0.40	−0.48	0.53	0.76	0.77	−0.97	5,4,5,4	14
SPOT-L	8, 27	0.62	0.95	−0.77	3.78	3.73	1.93	4,3,4,3	2
	10, 25	0.66	0.98	0.63	1.92	1.78	1.63	4,3,4,3	6
	15, 20	0.66	0.86	−0.81	1.24	1.39	−1.90	5,4,5,4	12
	20, 15	0.58	0.52	0.94	0.95	1.24	−1.43	6,5,6,5	18
SPOT-R	8, 27	0.61	0.08	−1.11	4.35	−4.08	−2.71	4,3,4,3	2
	10, 25	0.83	−1.22	0.85	2.63	2.71	2.45	4,3,4,3	6
	15, 20	0.66	−0.69	0.97	1.74	−2.60	−2.25	5,4,5,4	12
	20, 15	0.57	0.75	1.10	1.31	2.39	2.29	7,3,7,6	14