Journal of Mathematical Imaging and Vision

, Volume 24, Issue 3, pp 341–348

Some Results on Minimal Euclidean Reconstruction from Four Points

  • Long Quan
  • Bill Triggs
  • Bernard Mourrain


Methods for reconstruction and camera estimation from miminal data are often used to boot-strap robust (RANSAC and LMS) and optimal (bundle adjustment) structure and motion estimates. Minimal methods are known for projective reconstruction from two or more uncalibrated images, and for “5 point” relative orientation and Euclidean reconstruction from two calibrated parameters, but we know of no efficient minimal method for three or more calibrated cameras except the uniqueness proof by Holt and Netravali. We reformulate the problem of Euclidean reconstruction from minimal data of four points in three or more calibrated images, and develop a random rational simulation method to show some new results on this problem. In addition to an alternative proof of the uniqueness of the solutions in general cases, we further show that unknown coplanar configurations are not singular, but the true solution is a double root. The solution from a known coplanar configuration is also generally unique. Some especially symmetric point-camera configurations lead to multiple solutions, but only symmetry of points or the cameras gives a unique solution.


3D reconstruction relative orientation structure from motion polynomial methods algebraic geometry 


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Copyright information

© Springer Science + Business Media, Inc. 2006

Authors and Affiliations

  • Long Quan
    • 1
  • Bill Triggs
    • 2
  • Bernard Mourrain
    • 3
  1. 1.Department of Computer ScienceHKUSTHong Kong
  2. 2.INRIA Rhône-AlpesFrance
  3. 3.INRIA Sophia AntipolisFrance

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