Linear camera calibration from a single view of two concentric semicircles for augmented reality applications

Abstract - A linear method that calibrates a camera from a single view of two concentric semicircles of known radii is presented. Using the estimated centers of the projected semicircles and the four corner points on the projected semicircles, the focal length and the pose of the camera are accurately estimated in real-time. Our method is applied to augmented reality applications and its validity is verified.

Keywords: Linear camera calibration, concentric semicircles, augmented reality

In the field of augmented reality, Inexpensive and effective techniques for calibrating a camera(or estimating the pose of a camera) are needed. In this paper, we present a novel method that easily calibrates a camera using only a single view of two concentric semicircles of known radii. When we consider the ellipse as the perspective projection of a circle, the coefficients of the ellipse equation are function of intrinsic and extrinsic parameters as well as the projected circle center. Therefore the camera intrinsic and extrinsic parameters can be estimated if the coefficients of the ellipse and the coordinates of the projected circle center are known.

The equation of a 3D (semi-)circle in the image plane is

The intrinsic and extrinsic parameters are associated with the camera parameters and the projected center of a 3D semicircle as follows.

where N[.] designates normalization to a unit vector and u_c, v_c represent the image coordinates of the projected semicircle center and t_x, t_y, and t_z represent the elements of the translation vector t. Therefore, all the camera parameters except two rotation vectors r₁ and r₃ are computed.

Four corner points on semicircles to determine one of r₁ and r₃ are used. By the properties of rotation matrix, we have

where a, b, and c represent the variables to be estimated. From the relationship between 3D points (M) and their corresponding 2D points (m), we have

It may be expressed in terms of the vector cross product as

It can be rewritten as

This means that each point correspondence (m↔M) gives a set of two independent equations in a, b, and c. We therefore have eight equations from four corner points on concentric semicircles and their projected points. a, b, and c can be computed by least squares techniques. The 3D coordinates of four points are given from the radii of semicircles and 2D coordinates are estimated from location of the points having four largest curvatures on semicircles.

Figure 1. Calibration process. (a) Calibration image, (b) recovered ellipses using ellipse fitting algorithm, (c) the line through the centers of two ellipses (black line) and the center of concentric semicircles (yellow circle), (d) application of the calibration results to augmented reality applications.

Publication

• • 1. Hanhoon Park and Jong-Il Park, "Linear Camera Calibration from a Single View of Two Concentric Semicircles for Augmented Realtiy Applications," Proceedings of SPIE-IS&T Electronic Imaging 2005, vol. 5665, pp. 353-361, San Jose, California, USA, Jan. 2005 [paper link]