Double distortion correction method in a catadioptric vision system with a conic mirror

Fuqiang Zhou; Xin Chen; Haishu Tan

doi:10.1364/JOSAA.34.001651

Journal of the Optical Society of America A
Vol. 34,
Issue 9,
pp. 1651-1658
(2017)
•https://doi.org/10.1364/JOSAA.34.001651

Double distortion correction method in a catadioptric vision system with a conic mirror

Fuqiang Zhou, Xin Chen, and Haishu Tan

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Fuqiang Zhou, Xin Chen, and Haishu Tan, "Double distortion correction method in a catadioptric vision system with a conic mirror," J. Opt. Soc. Am. A 34, 1651-1658 (2017)

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Abstract

Because of the low calibration precision caused by reflector distortion in catadioptric vision systems with conic mirrors, we proposed a double distortion correction model based on the panoramic image rectification. The lens distortion and the reflector distortion are analyzed independently in the mathematic model of the double distortion correction. Moreover, the modified model in this paper added the trigonometric functions into the polynomial model to solve the combined distortion. Additionally, extensive experiments, both simulative and real data, have been carried out and show that the proposed model achieves comparable measurement accuracy in contrast with the traditional distortion method.

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Camera Parameters	Zhang’s Method	Scaramuzza’s Method	Proposed Method
$(f_{x}, f_{y}) / pixel$	(215.93,216.43)	(225.95,219.72)	(220.46, 216.96)
$(u_{0}, v_{0}) / pixel$	(632.26,530.98)	(605.55,546.49)	(639.49,535.50)
	Lens distortion	Lens distortion	Lens distortion	Reflector distortion
$(k_{1}, k_{2}, k_{3})$	$[- 0.1298, 0.0131, - 0.0051]$	$[- 0.1987, 0.0122, - 0.0032]$	$[- 0.1347, 0.0125, - 0.0063]$	$[- 0.228, 0.089, - 0.0358]$
$(p_{1}, p_{2})$	$[- 3.33 \times 10^{- 4}, - 1.16 \times 10^{- 5}]$	$[- 2.57 \times 10^{- 4}, - 2.31 \times 10^{- 5}]$	$[- 2.96 \times 10^{- 4}, - 1.89 \times 10^{- 5}$ ]	$[- 1.52 \times 10^{- 5}, - 2.19 \times 10^{- 5}]$
Reprojection error (pixel)	0.513	0.477	0.402

Number of Image Point	Zhang’s Method/(Pixel)	Scaramuzza Method/(Pixel)	Our Method/(Pixel)	Error in $Δ x / (Pixel)$	Error in $Δ y / (Pixel)$	Error in $Δ x / (Pixel)$	Error in $Δ y / (Pixel)$
1	(760.13,340.96)	(758.57,340.45)	(758.01,340.33)	1.56	0.51	2.12	0.63
2	(772.99,355.86)	(773.26,355.24)	(773.74,355.30)	0.27	0.62	0.75	0.56
3	(788.14,373.40)	(789.62,372.69)	(789.19,372.04)	1.48	0.71	1.05	0.36
4	(804.81,391.67)	(803.81,390.81)	(803.53,390.52)	1	0.86	1.28	1.15
5	(815.63,411.77)	(816.58,410.53)	(816.49,410.71)	0.95	1.24	0.86	1.06
6	(826.71,430.77)	(827.47,431.55)	(827.59,431.31)	0.76	0.78	0.88	0.54
7	(836.06,452.73)	(837.69,452.98)	(837.01,452.44)	1.63	0.25	0.95	0.29
8	(844.50,472.85)	(845.38,473.92)	(844.86,473.16)	0.88	1.07	0.36	0.31
9	(743.24,352.03)	(742.55,352.68)	(741.95,352.16)	0.69	0.65	1.29	0.13
10	(755.86,368.18)	(757.93,367.31)	(757.36,367.17)	2.07	0.87	1.5	1.01
11	(773.86,383.66)	(772.50,384.79)	(772.01,384.11)	1.36	1.13	1.85	0.45
12	(786.37,402.91)	(786.52,402.37)	(786.14,402.24)	0.15	0.54	0.23	0.67
	Distortion correction efforts (pixels)			1.07	0.77	1.09	0.59

Camera Parameters	Zhang’s Method	Scaramuzza’s Method	Proposed Method
$(f_{x}, f_{y}) / pixel$	(215.93,216.43)	(225.95,219.72)	(220.46, 216.96)
$(u_{0}, v_{0}) / pixel$	(632.26,530.98)	(605.55,546.49)	(639.49,535.50)
	Lens distortion	Lens distortion	Lens distortion	Reflector distortion
$(k_{1}, k_{2}, k_{3})$	$[- 0.1298, 0.0131, - 0.0051]$	$[- 0.1987, 0.0122, - 0.0032]$	$[- 0.1347, 0.0125, - 0.0063]$	$[- 0.228, 0.089, - 0.0358]$
$(p_{1}, p_{2})$	$[- 3.33 \times 10^{- 4}, - 1.16 \times 10^{- 5}]$	$[- 2.57 \times 10^{- 4}, - 2.31 \times 10^{- 5}]$	$[- 2.96 \times 10^{- 4}, - 1.89 \times 10^{- 5}$ ]	$[- 1.52 \times 10^{- 5}, - 2.19 \times 10^{- 5}]$
Reprojection error (pixel)	0.513	0.477	0.402

Number of Image Point	Zhang’s Method/(Pixel)	Scaramuzza Method/(Pixel)	Our Method/(Pixel)	Error in $Δ x / (Pixel)$	Error in $Δ y / (Pixel)$	Error in $Δ x / (Pixel)$	Error in $Δ y / (Pixel)$
1	(760.13,340.96)	(758.57,340.45)	(758.01,340.33)	1.56	0.51	2.12	0.63
2	(772.99,355.86)	(773.26,355.24)	(773.74,355.30)	0.27	0.62	0.75	0.56
3	(788.14,373.40)	(789.62,372.69)	(789.19,372.04)	1.48	0.71	1.05	0.36
4	(804.81,391.67)	(803.81,390.81)	(803.53,390.52)	1	0.86	1.28	1.15
5	(815.63,411.77)	(816.58,410.53)	(816.49,410.71)	0.95	1.24	0.86	1.06
6	(826.71,430.77)	(827.47,431.55)	(827.59,431.31)	0.76	0.78	0.88	0.54
7	(836.06,452.73)	(837.69,452.98)	(837.01,452.44)	1.63	0.25	0.95	0.29
8	(844.50,472.85)	(845.38,473.92)	(844.86,473.16)	0.88	1.07	0.36	0.31
9	(743.24,352.03)	(742.55,352.68)	(741.95,352.16)	0.69	0.65	1.29	0.13
10	(755.86,368.18)	(757.93,367.31)	(757.36,367.17)	2.07	0.87	1.5	1.01
11	(773.86,383.66)	(772.50,384.79)	(772.01,384.11)	1.36	1.13	1.85	0.45
12	(786.37,402.91)	(786.52,402.37)	(786.14,402.24)	0.15	0.54	0.23	0.67
	Distortion correction efforts (pixels)			1.07	0.77	1.09	0.59

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Journal of the Optical Society of America A