A Swarm Optimization Algorithm for Multimodal Functions and Its Application in Multicircle Detection

Table 6

The averaged execution time, detection rate, and the averaged multiple error for the GA-based algorithm, the BFOA method, and the proposed CAB algorithm, considering six test images (shown by Figures 6 and 7).

Image	Averaged execution time ± standard deviation (s)			Success rate (DR) (%)			Averaged ME ± standard deviation
	GA	BFOA	CAB	GA	BFOA	CAB	GA	BFOA	CAB
Synthetic images
(a)	2.23 ± (0.41)	1.71 ± (0.51)	0.21 ± (0.22)	88	99	100	0.41 ± (0.044)	0.33 ± (0.052)	0.22 ± (0.033)
(b)	3.15 ± (0.39)	2.80 ± (0.65)	0.36 ± (0.24)	79	92	99	0.51 ± (0.038)	0.37 ± (0.032)	0.26 ± (0.041)
(c)	4.21 ± (0.11)	3.18 ± (0.36)	0.20 ± (0.19)	74	88	100	0.48 ± (0.029)	0.41 ± (0.051)	0.15 ± (0.036)
Natural images
(a)	5.11 ± (0.43)	3.45 ± (0.52)	1.10 ± (0.24)	90	96	100	0.45 ± (0.051)	0.41 ± (0.029)	0.25 ± (0.037)
(b)	6.33 ± (0.34)	4.11 ± (0.14)	1.61 ± (0.17)	83	89	100	0.81 ± (0.042)	0.77 ± (0.051)	0.37 ± (0.055)
(c)	7.62 ± (0.97)	5.36 ± (0.17)	1.95 ± (0.41)	84	92	99	0.92 ± (0.075)	0.88 ± (0.081)	0.41 ± (0.066)