Two-point calibration method for a zoom camera with an approximate focal-invariant radial distortion model

Pei An; Pei An; Jie Ma; Jie Ma; Tao Ma; Bin Fang; Bin Fang; Kun Yu; Kun Yu; Xiaomao Liu; Jun Zhang; Jun Zhang

doi:10.1364/JOSAA.414504

Journal of the Optical Society of America A
Vol. 38,
Issue 4,
pp. 504-514
(2021)
•https://doi.org/10.1364/JOSAA.414504

Two-point calibration method for a zoom camera with an approximate focal-invariant radial distortion model

Pei An, Jie Ma, Tao Ma, Bin Fang, Kun Yu, Xiaomao Liu, and Jun Zhang

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Pei An, Jie Ma, Tao Ma, Bin Fang, Kun Yu, Xiaomao Liu, and Jun Zhang, "Two-point calibration method for a zoom camera with an approximate focal-invariant radial distortion model," J. Opt. Soc. Am. A 38, 504-514 (2021)

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Abstract

A zoom camera can change its focal length and track moving objects with an adjustable resolution. To extract precise geometric information for the tracked objects, a zoom camera requires an accurate calibration method. High-precision camera calibration methods, however, usually require a number of control points that are not guaranteed in some practical situations. Most zoom cameras suffer radial distortion. Athough a traditional method can recover an undistorted image with known intrinsic parameters, it fails to work for a zoom camera with an unknown focal length. Motivated by these problems, we propose a two-point calibration method (TPCM). In this scheme, we first propose an approximate focal-invariant radial distortion (AFRD) model. With the AFRD model, an RGB image can be undistorted with an unknown focal length. After that, the TPCM method is presented to estimate the focal length and rotation matrix with only two control points for one image. Synthetic experiments demonstrate that the AFRD model is efficient. In the real data experiment, the mean reprojection error of the TPCM method is less than one pixel, which is smaller than current state-of-the-art methods, and we believe meets the demand for high-precision calibration.

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Calibration Conditions	Chen [3]	TPCM
Known camera position	✓	✓
Using two control points	✓	✓
Working without base orientation	✗	✓
Using general intrinsic parameters	✗	✓
Considering radial distortion	✗	✓
Estimating rotation matrix	2-DoF	3-DoF

Requirements	Model [11]	Model [12]	AFRD
Establishment	PICS	NICS	PICS
Focal length	✗		✗
Pixel physical size	✓	✗	✗
Offline calibration	✓	✓	✓

Camera Parameter	Value
$k_{1, A F R D}$ /( ${p i x e l}^{- 2}$ )	$- 1.2 \times 10^{- 7}$
$k_{2, A F R D}$ /( ${p i x e l}^{- 4}$ )	$6.2 \times 10^{- 15}$
$f_{u}$ /(pixel)	1000.0
$u_{0}$ /(pixel)	512.0
$v_{0}$ /(pixel)	512.0

	X/(m)	Y/(m)	Z/(m)	u/(pixel)	v/(pixel)
$O$	0.20	–0.80	0.10	—	—
$P (1)$	4.52	−4.21	9.19	841.0	326.2
$P (2)$	−2.69	−1.74	12.58	216.9	814.2

Parameter	12 mm	13 mm	14 mm	15 mm
$f_{u} / p i x e l s$	2868.736	3102.209	3362.857	3811.638
$u_{0} / p i x e l s$	2599.246	2590.946	2639.012	2605.208
$v_{0} / p i x e l s$	1747.834	1760.528	1716.216	1803.532
$a$	0.926	0.925	0.927	0.926
$s$	2.733 $E$ -4	2.729 $E$ -4	2.733 $E$ -4	2.731 $E$ -4
$k_{1}$	−0.372	−0.378	−0.379	−0.409
$k_{2}$	0.158	0.171	0.174	0.201

	$Z_1$	$Z_2$	$Z_3$	$Z_4$	AFRD
12 mm	0.1576	0.5314	0.9032	1.1725	0.3436
13 mm	0.6632	0.1710	0.5178	0.8505	0.2995
14 mm	1.2449	0.5873	0.2157	0.4434	0.2674
15 mm	1.8330	1.1169	0.6148	0.3919	0.4204
Mean	0.9759	0.6016	0.5628	0.7145	0.3327
St. Dev	0.7248	0.3899	0.2834	0.3677	0.0662

	X/(m)	Y/(m)	Z/(m)	u/(pixel)	v/(pixel)
$O$	1.243	0.818	0.891	—	—
$P [1]$	0.663	0.074	3.341	262.7	155.6
$P [2]$	1.624	0.028	3.491	930.8	147.5
$P [3]$	0.844	1.232	2.958	311.6	745.4
$P [4]$	2.068	1.122	3.100	905.1	674.1

Method	$f_{u}$ /pixel	$f_{v}$ /pixel	$f_{s}$ /pixel	$P a n$ /degree	$T i l t$ /degree	Reprojection Error/pixel
Ground Truth	1027.52	1028.34	1.850	27.23	−0.81	0.0
TPCM	1027.74	1028.56	1.849	27.22	−0.82	0.84211
Chen	1057.30	1057.30	0	27.27	−0.93	11.39341
Chen++	1028.07	1028.07	0	27.20	−0.83	3.23349

Point id pair	$(1, 2)$	$(1, 3)$	$(1, 4)$	$(2, 3)$	$(2, 4)$	$(3, 4)$	$M e a n$	$S t d . d e v i a t i o n$
$f_{u}$ /(pixel)	1027.92	1028.12	1027.21	1027.96	1027.23	1027.98	1027.74	0.405
$f_{v}$ /(pixel)	1028.74	1028.94	1028.03	1028.78	1028.05	1028.80	1028.56	0.406
$f_{s}$ /(pixel)	1.852	1.851	1.849	1.850	1.852	1.845	1.849	$2.64 \times 10^{- 3}$
$P a n$ /(degree)	27.28	27.24	27.25	27.21	27.19	27.20	27.22	$3.43 \times 10^{- 2}$
$T i l t$ /(degree)	−0.83	−0.80	−0.84	−0.81	−0.79	−0.82	−0.81	$1.87 \times 10^{- 2}$
Reprojection error/(pixel)	1.65326	2.17843	1.25962	1.61651	1.22542	1.76112	1.61573	0.352

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Journal of the Optical Society of America A