Automatic Circle Detection on Images Based on an Evolutionary Algorithm That Reduces the Number of Function Evaluations

Table 2

The averaged execution-time, detection rate, the averaged multiple error for the GA-based algorithm, the BFOA method, and the proposed APRE algorithm, considering six test images are shown by Figures 8 and 9.

	Performance indexes	Synthetic images			Natural images
		(a)	(b)	(c)	(a)	(b)	(c)
GA	Averaged execution time	2.23	3.15	4.21	5.11	6.33	7.62
	Averaged number of function evaluations	14,000	14,000	14,000	14,000	14,000	14,000
	Success rate (DR) (%)	88	79	74	90	83	84
	Averaged ME	0.41	0.51	0.48	0.45	0.81	0.92
BFOA	Averaged execution time	1.71	2.80	3.18	3.45	4.11	5.36
	Averaged number of function evaluations	17,500	17,500	17,500	17,500	17,500	17,500
	Success rate (DR) (%)	99	92	88	96	89	92
	Averaged ME	0.33	0.37	0.41	0.41	0.77	0.37
APRE	Averaged execution time	0.21	0.36	0.20	1.10	1.61	1.95
	Averaged number of function evaluations	2,321	2,756	3,191	4,251	3,768	3,834
	Success rate (DR) (%)	100	100	100	100	100	100
	Averaged ME	0.22	0.26	0.15	0.25	0.37	0.41