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

  • Shen Cai (蔡 棽)
  • Chen-hao Wang (王宸昊)
  • Yan Yan (阎 炎)
  • Yun-cai Liu (刘允才)


In this paper, we introduce a novel class of coplanar conics, 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.

Key words

camera calibration homography con-axes and con-eccentricity ellipse concentric circle polynomial optimization problem (POP) 

CLC number

TP 391 


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Copyright information

© Shanghai Jiaotong University and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Shen Cai (蔡 棽)
    • 1
  • Chen-hao Wang (王宸昊)
    • 1
  • Yan Yan (阎 炎)
    • 1
    • 2
  • Yun-cai Liu (刘允才)
    • 1
  1. 1.Institute of Image Processing and Pattern RecognitionShanghai Jiaotong UniversityShanghaiChina
  2. 2.Information and Communication Technology DepartmentUniversity of TrentoTrentoItaly

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