Interpreting Sphere Images Using the Double-Contact Theorem

Abstract

An occluding contour of a sphere is projected to a conic in the perspective image, and such a conic is called a sphere image. Recently, it has been discovered that each sphere image is tangent to the image of the absolute conic at two double-contact image points. The double-contact theorem describes the properties of three conics which all have double contact with another conic. This theorem provides a linear algorithm to find the another conic if these three conics are given. In this paper, the double-contact theorem is employed to interpret the properties among three sphere images and the image of the absolute conic. The image of the absolute conic can be determined from three sphere images using the double-contact theorem. Therefore, a linear calibration method using three sphere images is obtained. Only three sphere images are required, and all five intrinsic parameters are recovered linearly without making assumptions, such as, zero-skew or unitary aspect ratio. Extensive experiments on simulated and real data were performed and shown that our calibration method is an order of magnitude faster than previous optimized methods and a little faster than former linear methods while maintaining comparable accuracy.