TR2010-081

Analytical Forward Projection for Axial Non-Central Dioptric & Catadioptric Cameras


    •  Agrawal, A.; Taguchi, Y.; Ramalingam, S., "Analytical Forward Projection for Axial Non-Central Dioptric & Catadioptric Cameras", European Conference on Computer Vision (ECCV), September 2010.
      BibTeX Download PDF
      • @inproceedings{Agrawal2010sep,
      • author = {Agrawal, A. and Taguchi, Y. and Ramalingam, S.},
      • title = {Analytical Forward Projection for Axial Non-Central Dioptric & Catadioptric Cameras},
      • booktitle = {European Conference on Computer Vision (ECCV)},
      • year = 2010,
      • month = sep,
      • url = {http://www.merl.com/publications/TR2010-081}
      • }
  • Research Area:

    Computer Vision


We present a technique for modeling non-central catadioptric cameras consisting of a perspective camera and a rotationally symmetric conic reflector. While previous approaches use a central approximation and/or iterative methods for forward projection, we present an analytical solution. This allows computation of the optical path from a given 3D point to the given viewpoint by solving a 6th degree forward projection equation for general conic mirrors. For a spherical mirror, the forward projection reduces to a 4th degree equation, resulting in a closed form solution. We also derive the forward projection equation for imaging through a refractive sphere (non-central dioptric camera) and show that it is a 10th degree equation. While central catadioptric cameras lead to conic epipolar curves, we show the existence of a quartic epipolar curve for catadioptric systems using a spherical mirror. The analytical forward projection leads to accurate and fast 3D reconstruction via bundle adjustment. Simulations and real results on single image sparse 3D reconstruction are presented. We demonstrate ~ 100 times speed up using the analytical solution over iterative forward projection for 3D reconstruction using spherical mirrors.