Analysis of the pencil of conics with double complex contact and its application to camera calibration

In this paper,we introduce a novel class of coplanar conies,the pencil of which can doubly contact to calibrate camera and estimate pose.We first analyze the properties of con-axes and con-eccentricity ellipses, which consist of a natural extending pattern of concentric circles.Then the general case that two ellipses have two repeated complex intersection points is presented.This degenerate configuration results in a one-parameter family of homographies which map the planar pattern to its image.Although it is unable to compute the complete homography,an indirect 3-degree polynomial or 5-degree polynomial constraint on intrinsic parameters from one image can also be used for camera calibration and pose estimation under the minimal conditions.Furthermore, this nonlinear problem can be treated as a polynomial optimization problem(POP) and the global optimization solution can be also obtained by using SparsePOP(a sparse semidefinite programming relaxation of POPs). Finally,the experiments with simulated data and real images are shown to verify the correctness and robustness of the proposed technique.