How to avoid ambiguity is a challenging problem for conic-based homography estimation. In this paper, we address the problem of homography estimation from two separate ellipses. We find that any two ellipses have a unique common self-polar triangle, which can provide three line correspondences. Furthermore, by investigating the location features of the common self-polar triangle, we show that one vertex of the triangle lies outside of both ellipses, while the other two vertices lies inside the ellipses separately. Accordingly, one more line correspondence can be obtained from the intersections of the conics and the common self-polar triangle. Therefore, four line correspondences can be obtained based on the common self-polar triangle, which can provide enough constraints for the homography estimation. The main contributions in this paper include: (1) A new discovery on the location features of the common self-polar triangle of separate ellipses. (2) A novel approach for homography estimation. Simulate experiments and real experiments are conducted to demonstrate the feasibility and accuracy of our approach.

This paper investigates the properties of the common self-polar triangle of separate coplanar circles and applies them to camera calibration. We find that any two separate circles have a unique common self-polar triangle. In particular, we show that one vertex of the common self-polar triangle lies on the line at infinity. Given three separate circles, the line at infinity can be recovered using the vertices of the common self-polar triangles. Accordingly, the vanishing line of the support plane can be obtained in their images. This allows recovering the imaged circular points, which provides good constraints on the image of the absolute conic. Compared to previous calibration methods using separate circles, our approach can avoid solving quartic equation, which often causes numerical instability. In the application, we test one calibration algorithm and accurate results are achieved.

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.