Improved AO optimization algorithm for distortion parameter estimation of catadioptric omnidirectional lens

算法	参数设置
GWO^[26]	a was linearly decreased from 2 to 0, rate=3
WOA^[27]	α decreased from 2 to 0, b=1
HHO^[28]	t=0
ALO^[29]	I ratio=10^ω, ω=[2,6]
AO^[18]	S=1.5, r₁ take a fixed index between 1 and 20, G₂ decreased from 2 to 0

Table 2. Benchmark function

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Table 2. Benchmark function

函数序号	函数名称	维度	范围	最优值
F1	Sphere	30	[−100,100]	0
F2	Schwefel 2.22	30	[−10,10]	0
F3	Schwefel 1.2	30	[−100,100]	0
F4	Schwefel 2.21	30	[−100,100]	0
F5	Ackley	10	[−32,32]	0
F6	Generalized Penalized	30	[−50,50]	0
F7	Shekel's Foxholes	2	[−65.536,65.536]	1
F8	Six-Hump Camel-Back	4	[−5,5]	−1.0316
F9	Goldstein-Price	2	[−2,2]	3
F10	Shekel's Family	4	[0,1]	−10.4028

Table 3. Optimization results of different algorithms for benchmark functions

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Table 3. Optimization results of different algorithms for benchmark functions

函数		GWO	WOA	HHO	ALO	AO	Improved AO
F1	AVG	1.40×10⁻¹²⁷	7.91×10⁻¹²¹	4.76×10⁻¹¹⁷	2.67×10⁻¹⁰	3.71×10⁻¹⁷⁴	3.95×10⁻³²³
	STD	6.08×10⁻¹²⁷	2.45×10⁻¹²⁰	2.21×10⁻¹¹⁶	1.24×10⁻¹⁰	0.00	0.00
	Best	5.36×10⁻¹³⁵	1.12×10⁻¹²⁷	2.56×10⁻¹²⁹	9.62×10⁻¹¹	9.38×10⁻¹⁸²	0.00
F2	AVG	2.31×10⁻⁷¹	4.86×10⁻⁶⁶	6.93×10⁻⁶⁵	1.94×10⁻⁵	7.45×10⁻⁸⁸	5.37×10⁻¹⁶⁶
	STD	6.58×10⁻⁷¹	2.58×10⁻⁶⁵	2.29×10⁻⁶⁴	4.16×10⁻⁵	2.29×10⁻⁸⁷	0.00
	Best	3.10×10⁻⁷⁴	6.20×10⁻⁷³	1.69×10⁻⁷⁵	5.99×10⁻⁶	4.82×10⁻⁹⁴	1.17×10⁻¹⁷⁴
F3	AVG	2.76×10⁻⁶⁷	1.17×10⁻²	2.11×10⁻¹⁰⁶	2.53×10⁻⁹	1.13×10⁻¹⁷⁰	0.00
	STD	6.84×10⁻⁶⁷	2.78×10⁻²	1.16×10⁻¹⁰⁵	1.27×10⁻⁹	0.00	0.00
	Best	1.91×10⁻⁷⁴	3.49×10⁻⁶	1.89×10⁻¹²³	3.96×10⁻¹⁰	5.12×10⁻¹⁸¹	0.00
F4	AVG	6.10×10⁻⁴⁴	2.27×10⁻⁹	2.90×10⁻⁵⁸	1.54×10⁻⁵	6.46×10⁻⁸⁸	8.25×10⁻¹⁶⁶
	STD	1.49×10⁻⁴³	7.34×10⁻⁹	1.11×10⁻⁵⁷	2.98×10⁻⁶	2.33×10⁻⁸⁷	0.00
	Best	1.01×10⁻⁴⁷	3.01×10⁻²⁵	1.68×10⁻⁶⁴	1.06×10⁻⁵	7.89×10⁻⁹⁵	2.23×10⁻¹⁸²
F5	AVG	4.00×10⁻¹⁵	2.46×10⁻¹⁵	4.44×10⁻¹⁶	5.49×10⁻²	4.44×10⁻¹⁶	4.44×10⁻¹⁶
	STD	0.00	2.02×10⁻¹⁵	0.00	3.01×10⁻¹	0.00	0.00
	Best	4.00×10⁻¹⁵	4.44×10⁻¹⁶	4.44×10⁻¹⁶	3.63×10⁻⁶	4.44×10⁻¹⁶	4.44×10⁻¹⁶
F6	AVG	3.00	3.39×10⁻²	4.18×10⁻⁷	2.16×10²	1.44×10⁻⁷	2.37×10⁻⁴
	STD	4.72×10⁻¹	3.06×10⁻²	6.39×10⁻⁷	1.72×10¹	2.16×10⁻⁷	4.32×10⁻⁴
	Best	2.08	9.24×10⁻³	1.19×10⁻¹¹	1.83×10²	4.47×10⁻¹¹	1.18×10⁻⁸
F7	AVG	9.98×10⁻¹	9.98×10⁻¹	9.98×10⁻¹	9.98×10⁻¹	9.98×10⁻¹	9.98×10⁻¹
	STD	1.13×10⁻¹¹	8.48×10⁻¹⁵	1.04×10⁻¹⁵	2.31×10⁻¹⁶	8.48×10⁻¹¹	3.92×10⁻⁹
	Best	9.98×10⁻¹	9.98×10⁻¹	9.98×10⁻¹	9.98×10⁻¹	9.98×10⁻¹	9.98×10⁻¹
F8	AVG	−1.03	−1.03	−1.03	−1.03	−1.03	−1.03
	STD	1.01×10⁻⁹	7.23×10⁻¹⁵	4.25×10⁻¹⁶	8.87×10⁻¹⁵	5.38×10⁻⁶	2.44×10⁻⁴
	Best	−1.03	−1.03	−1.03	−1.03	−1.03	−1.03
F9	AVG	2.99	3.00	2.98	2.98	3.00	3.00
	STD	1.71×10⁻⁷	1.00×10⁻⁹	3.01×10⁻¹⁴	4.99×10⁻¹⁴	3.24×10⁻⁴	6.16×10⁻¹⁵
	Best	2.99	3.00	2.98	2.98	3.00	3.00
F10	AVG	−1.02×10¹	−1.04×10¹	−6.86	−9.35	−1.04×10¹	−1.04×10¹
	STD	9.70×10⁻¹	8.75×10⁻⁷	2.55	2.15	5.73×10⁻⁵	3.22×10⁻²
	Best	−1.04×10¹	−1.04×10¹	−1.04×10¹	−1.04×10¹	−1.04×10¹	−1.04×10¹

Table 4. Wilcoxon rank sum test result

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Table 4. Wilcoxon rank sum test result

对比算法	单峰函数	多峰函数	固定维数多峰函数
Improved AO vs. GWO	6.39×10⁻⁴	2.86×10⁻²	1.19×10⁻²
Improved AO vs. WOA	2.48×10⁻⁴	6.43×10⁻²	7.80×10⁻³
Improved AO vs. HHO	1.55×10⁻³	6.26×10⁻²	3.97×10⁻³
Improved AO vs. ALO	1.00×10⁻⁵	2.86×10⁻²	7.80×10⁻³
Improved AO vs. AO	9.58×10⁻³	7.62×10⁻²	5.69×10⁻²

Table 5. Sensitivity analysis of the improved AO for the number of population members (N)

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Table 5. Sensitivity analysis of the improved AO for the number of population members (N)

函数	种群数量值
函数	100	200	300	400
F1	0.00	0.00	0.00	0.00
F2	5.13×10⁻¹⁶⁷	1.16×10⁻¹⁶⁷	7.23×10⁻¹⁷¹	3.09×10⁻¹⁷⁶
F3	0.00	0.00	0.00	0.00
F4	3.87×10⁻¹⁸¹	3.87×10⁻¹⁸⁷	5.83×10⁻¹⁹⁵	0.00
F5	4.44×10⁻¹⁶	4.44×10⁻¹⁶	3.45×10⁻¹⁸	1.12×10⁻²¹
F6	1.49×10⁻⁵	1.35×10⁻⁶	2.58×10⁻⁶	2.33×10⁻⁷
F7	9.98×10⁻¹	9.98×10⁻¹	9.98×10⁻¹	9.98×10⁻¹
F8	−1.03	−1.03	−1.03	−1.03
F9	3.02	3.02	3.01	3.01
F10	−1.04×10⁻¹	−1.04×10⁻¹	−1.04×10⁻¹	−1.04×10⁻¹

Table 6. Sensitivity analysis of the improved AO for the number of iterations (T)

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Table 6. Sensitivity analysis of the improved AO for the number of iterations (T)

函数	最大迭代次数
函数	200	400	600	800
F1	1.68×10⁻²²⁵	0.00	0.00	0.00
F2	5.67×10⁻¹⁰⁸	3.64×10⁻¹⁴⁵	1.52×10⁻¹⁶⁷	6.26×10⁻¹⁷⁰
F3	2.28×10⁻²¹⁷	0.00	0.00	0.00
F4	5.91×10⁻¹⁰⁹	5.61×10⁻¹⁵⁹	5.85×10⁻¹⁶⁵	2.99×10⁻¹⁶⁸
F5	4.44×10⁻¹⁶	4.44×10⁻¹⁶	3.25×10⁻¹⁷	1.93×10⁻¹⁷
F6	3.93×10⁻⁵	1.79×10⁻⁵	1.45×10⁻⁶	9.95×10⁻⁷
F7	9.98×10⁻¹	9.98×10⁻¹	9.98×10⁻¹	9.98×10⁻¹
F8	−1.03	−1.03	−1.03	−1.03
F9	3.01	3.01	3.01	3.01
F10	−1.04×10⁻¹	−1.04×10⁻¹	−1.04×10⁻¹	−1.04×10⁻¹

Table 7. Radial distortion parameter estimation results

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Table 7. Radial distortion parameter estimation results

序列	Ref [8]	Ref [11]	Ref [12]	Ours
(a)	−1.03×10⁻⁵	−1.01×10⁻⁵	−1.01×10⁻⁵	−1.01×10⁻⁵
(b)	−1.02×10⁻⁶	−1.02×10⁻⁶	−1.02×10⁻⁶	−1.02×10⁻⁶
(c)	−1.05×10⁻⁷	−0.96×10⁻⁷	−0.97×10⁻⁷	−1.02×10⁻⁷
(d)	−1.07×10⁻⁸	−1.04×10⁻⁸	−1.04×10⁻⁸	−1.02×10⁻⁸
(e)	−1.02×10⁻⁶	−1.03×10⁻⁶	−1.01×10⁻⁶	−1.01×10⁻⁶
(f)	−1.02×10⁻⁵	−1.01×10⁻⁵	−1.01×10⁻⁵	−1.01×10⁻⁵

Table 8. Estimation error of the distortion parameter
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Table 8. Estimation error of the distortion parameter
方法 D/(pixel) R(%)
(a) (b) (c) (d) (e) (f)
Ref [8] 1.2412 3.2 2.8 5.1 7.0 2.5 2.3
Ref [11] 0.8430 1.5 2.8 4.6 4.3 3.0 1.5
Ref [12] 0.6762 1.2 2.6 3.2 4.3 1.3 1.2
Ours 0.4618 1.2 2.4 2.2 2.2 1.5 1.2

Table 9. Omnidirectional image distortion parameter estimation results

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Table 9. Omnidirectional image distortion parameter estimation results

序列	图像大小	畸变参数	畸变中心
10 (a)	$2\;048 \times 1\;536$	2.2613×10−6	(1091.72,695.14)
10 (b)	$2\;048 \times 1\;536$	2.1983×10−6	(1095.62,692.57)
10 (c)	$2\;048 \times 1\;536$	2.3564×10−6	(1096.34,695.83)
10 (d)	$2\;048 \times 1\;536$	2.2083×10−6	(1094.61,696.53)
10 (e)	$4\;352 \times 3\;264$	3.2517×10−6	（517.86,382.15）
10 (f)	$4\;352 \times 3\;264$	2.8476×10−6	（516.43,380.93）

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Yue ZHANG, Ning ZHANG, Xi-ping XU. Improved AO optimization algorithm for distortion parameter estimation of catadioptric omnidirectional lens[J]. Chinese Optics, 2025, 18(1): 89

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Paper Information

Category:

Received: Jun. 27, 2024

Accepted: Sep. 4, 2024

Published Online: Mar. 14, 2025

The Author Email:

DOI:10.37188/CO.2024-0118

Topics

Table 1. Parameters setting

Table 1. Parameters setting

Table 2. Benchmark function

Table 2. Benchmark function

Table 3. Optimization results of different algorithms for benchmark functions

Table 3. Optimization results of different algorithms for benchmark functions

Table 4. Wilcoxon rank sum test result

Table 4. Wilcoxon rank sum test result

Table 5. Sensitivity analysis of the improved AO for the number of population members (N)

Table 5. Sensitivity analysis of the improved AO for the number of population members (N)

Table 6. Sensitivity analysis of the improved AO for the number of iterations (T)

Table 6. Sensitivity analysis of the improved AO for the number of iterations (T)

Table 7. Radial distortion parameter estimation results

Table 7. Radial distortion parameter estimation results

Table 8. Estimation error of the distortion parameter

Table 8. Estimation error of the distortion parameter

Table 9. Omnidirectional image distortion parameter estimation results

Table 9. Omnidirectional image distortion parameter estimation results