The common self-polar triangle of concentric circles and its application to camera calibration

Abstract

In projective geometry, the common self-polar triangle has often been used to discuss the position relationship of two planar conics. However, there are few researches on the properties of the common self-polar triangle, especially when the two planar conics are special conics. In this paper, we explore the properties of the common self-polar triangle, when the two conics happen to be concentric circles. We show there exist infinite many common self-polar triangles of two concentric circles, and provide a method to locate the vertices of these triangles. By investigating all these triangles, we find that they encode two important properties. The first one is all triangles share one common vertex, and the opposite side of the common vertex lies on the same line, which are the circle center and the line at the infinity of the support plane. The second is all triangles are right triangles. Based on these two properties, the imaged circle center and the varnishing line of support plane can be recovered simultaneously, and many conjugate pairs on vanishing line can be obtained. These allow to induce good constraints on the image of absolute conic. We evaluate two calibration algorithms, whereby accurate results are achieved. The main contribution of this paper is that we initiate a new perspective to look into circle-based camera calibration problem. We believe that other calibration methods using different circle patterns can benefit from this perspective, especially for the patterns which involve more than two circles.