Advances in Multimedia

Volume 2018, Article ID 6182953, 8 pages

https://doi.org/10.1155/2018/6182953

## Method of Camera Calibration Using Concentric Circles and Lines through Their Centres

Institute of Mathematics and Statistics, Yunnan University, Cuihu, Kunming 650091, China

Correspondence should be addressed to Yue Zhao; ten.haey@5866oahz

Received 10 December 2017; Accepted 13 March 2018; Published 22 April 2018

Academic Editor: Martin Reisslein

Copyright © 2018 Yue Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A novel method for camera calibration is proposed based on an analysis of lens distortion in camera imaging. In the method, a line through the centre of concentric circles is used as a template in which orthogonal directions can be determined from an angle of circumference that corresponds to a diameter. By using three lines through the centre of concentric circles, based on the invariance of the cross-ratio, an image at the centre of the concentric circles can be used to obtain the vanishing point. The intrinsic parameters of the camera can be computed based on the constraints of the orthogonal vanishing points and the imaged absolute conic. The lens distortion causes points in the template to have a position offset. In the proposed method, we optimize the positions of the distortion points such that they gradually approach those of the ideal points. The simulated and real-world experiments demonstrate that the proposed method is efficient and feasible.

#### 1. Introduction

Camera calibration is an important research topic in the field of pattern recognition because it is required for computer vision applications [1–3]. Meng and Hu [4] used a circle and several lines through the centre of the circle as a calibration template; however, a single circle contains little information. Wu et al. [5] proposed a method of camera calibration involving the affine invariance of parallel circles. If the intersection of two parallel circles is first computed to determine the circular points, then the intrinsic parameters can be determined. However, this method cannot be used to determine the centre of concentric circles and requires at least three images. And Bin [6] proposed a method to calculate the vanishing point by the theory of harmonic conjugate in projective geometry. The camera intrinsic parameters could be obtained by the relationship between the circular points and the image of absolute conic. In addition, lens distortion degrades the accuracy of the camera calibration [7]. Consequently, Ricolfe-Viala and Sánchez-Salmerón [8] proposed a nonlinear method that corrects the images based on the cross-ratio invariance, although this algorithm is more complex. To address the disadvantages in the above methods, we propose a method to compute the intrinsic parameters by employing a circle as a template, while the scale of the circle does not need to be known. Based on the property that an angle in a circular segment that corresponds to the diameter is 90°, if an image includes two pairs of orthogonal vanishing points, the intrinsic parameters can be calculated for the three images. This method reduces the complexity of camera calibration. We also propose a new method for correcting lens distortion, which corrects images using the least square method to fit a line that passes through the centre of concentric circles.

This paper is organized as follows. The underlying theory is introduced in Section 2. The camera calibration method is proposed in Section 3, and a method of determining the image of the circle centre is described using the concentric circles. The proposed method of correcting lens distortion is introduced in Section 4. In Section 5, the results of simulation experiments are presented to show whether the method described in Section 4 is valid. Then, an experiment that compares this method with other classic methods is conducted. Finally, Section 6 provides a summary of this paper.

#### 2. Preliminaries

Let denote the homogeneous coordinates of a 3D point and denote the homogeneous coordinates of the corresponding image point. The projection relationship between these points iswhere is a nonzero scale factor and is a 3 × 4 matrix that is defined as the projection matrix, which can be expressed aswhere is a 3D rotation, is a translation vector, and is the intrinsic parameters matrix [1].

#### 3. Use of the Orthogonal Vanishing Points to Solve K

##### 3.1. Computing the Image of the Centre of Concentric Circles

The circle intersects the line at two points , and circle intersects the line at two points , as shown in Figure 1. It can be shown that the centre of the circles is at the mid-point of lines . Let represent the points in the direction of infinity along line , and denote line as a calibration line.