816 Y.Chen et al. non-planar scene structures.In this paper.we make an attempt to calibrate the camera from a single image of planar scene.In our approach,coplanar circles are adopted as basic calibration patters and vanishing points function in a hrand new way.Conic based planar rectification(1 1]and conic based pose estimation[4]are two previous approaches most related to our work.The work of [4]estimates the focal length and the camera pose from two coplanar circles in the image.However,this method implicitly assumes that the principal point is known beforehand and cannot treat non-unit aspect ratio.The work of [1I]makes accurate Euclidean measures from coplanar circles in a calibration-free manner.Nevertheless,the analysis is limited on the target plane and cannot be extended to applications in need of camera parameters. In our work,we propose a full calibration scheme which statistically estimates the focal length,the principal point,the aspect ratio as well as the extrinsic parameters.In particular,we show that circle is a powerful conic in that a single view of two coplanar cireles is capable of providing adequate information to do metric calibration.A coarse pipeline of our algorithm is as follows:First,calibration-free planar rectification re- ported in [11]is performed and extended to recover the vanishing line,the centers of the circles and many orthogonal vanishing point pairs.Second.under different guesses of the principal point the distribution of the focal length are computed from all orthogonal vanishing point pairs.Third,based on the previously computed focal length distribution a statistical optimization routine is designed to estimate the focal length,the principal point and the aspect ratio simultaneously.Fourth,the conic based pose estimation[4,6] are employed to compute the extrinsic parameters.Finally,the calibration result is vali- dated by comparison with the ground truth for synthetic scenes or by augmented reality tests for real scenes. The major advantage of our work lics in that the algorithm uses only a single image of simple planar scene to achieve a full solution of camera calibration.Therefore,the scene requirement is low in comparison with previous methods.This approach is very practical and works well for many scenes which previous methods fail to treat. The rest of the paper is organized as follows.Section 2 briefly reviews and extends the previous work of coplanar cireles based scene analysis.Section 3 elaborates a fea- sible scheme which statistically estimates the focal length,the principal point and the aspect ratio simultaneously.Some discussions are also provided in this section.Section 4 presents the experimental results on both synthetic and real scenes.Finally.conclud- ing remarks are given in Section 5. 2 Preliminaries We will present a calibration algorithm step by step under the practical assumption of zero skew.Our algorithm solves the camera projection matrix P=K[]where K is the zero-skew calibration matrix containing 4 intrinsic parameters as defined in equation (1)and the metric matrix fully encodes the 6 extrinsic parameters. (1) 001 816 Y. Chen et al. non-planar scene structures. In this paper, we make an attempt to calibrate the camera from a single image of planar scene. In our approach, coplanar circles are adopted as basic calibration patterns and vanishing points function in a brand new way. Conic based planar rectification[11] and conic based pose estimation[4] are two previous approaches most related to our work. The work of [4] estimates the focal length and the camera pose from two coplanar circles in the image. However, this method implicitly assumes that the principal point is known beforehand and cannot treat non-unit aspect ratio. The work of [11] makes accurate Euclidean measures from coplanar circles in a calibration-free manner. Nevertheless, the analysis is limited on the target plane and cannot be extended to applications in need of camera parameters. In our work, we propose a full calibration scheme which statistically estimates the focal length, the principal point, the aspect ratio as well as the extrinsic parameters. In particular, we show that circle is a powerful conic in that a single view of two coplanar circles is capable of providing adequate information to do metric calibration. A coarse pipeline of our algorithm is as follows: First, calibration-free planar rectification re￾ported in [11] is performed and extended to recover the vanishing line, the centers of the circles and many orthogonal vanishing point pairs. Second, under different guesses of the principal point the distribution of the focal length are computed from all orthogonal vanishing point pairs. Third, based on the previously computed focal length distribution a statistical optimization routine is designed to estimate the focal length, the principal point and the aspect ratio simultaneously. Fourth, the conic based pose estimation[4,6] are employed to compute the extrinsic parameters. Finally, the calibration result is vali￾dated by comparison with the ground truth for synthetic scenes or by augmented reality tests for real scenes. The major advantage of our work lies in that the algorithm uses only a single image of simple planar scene to achieve a full solution of camera calibration. Therefore, the scene requirement is low in comparison with previous methods. This approach is very practical and works well for many scenes which previous methods fail to treat. The rest of the paper is organized as follows. Section 2 briefly reviews and extends the previous work of coplanar circles based scene analysis. Section 3 elaborates a fea￾sible scheme which statistically estimates the focal length, the principal point and the aspect ratio simultaneously. Some discussions are also provided in this section. Section 4 presents the experimental results on both synthetic and real scenes. Finally, conclud￾ing remarks are given in Section 5. 2 Preliminaries We will present a calibration algorithm step by step under the practical assumption of zero skew. Our algorithm solves the camera projection matrix P = K[R|t] where K is the zero-skew calibration matrix containing 4 intrinsic parameters as defined in equation (1) and the metric matrix [R|t] fully encodes the 6 extrinsic parameters. K = ⎛ ⎝ αf 0 u 0 f v 0 01 ⎞ ⎠ (1)