Ambiguity in Structure from Motion: Sphere versus Plane

  • Authors:

  • Cornelia Fermüller;Yiannis Aloimonos

  • Affiliations:

  • Computer Vision Laboratory, Center for Automation Research, Institute for Advanced Computer Studies and Computer Science Department, University of Maryland, College Park, MD 20742-3275. E-mail: ...;Computer Vision Laboratory, Center for Automation Research, Institute for Advanced Computer Studies and Computer Science Department, University of Maryland, College Park, MD 20742-3275. E-mail: ...

  • Venue:
  • International Journal of Computer Vision
  • Year:
  • 1998

Quantified Score

Hi-index 0.00

Visualization

Abstract

If 3D rigid motion can be correctly estimated from image sequences,the structure of the scene can be correctly derived using the equationsfor image formation. However, an error in the estimation of 3D motion willresult in the computation of a distorted version of the scene structure. Ofcomputational interest are these regions in space where the distortions aresuch that the depths become negative, because in order for the scene to bevisible it has to lie in front of the image, and thus the correspondingdepth estimates have to be positive. The stability analysis for thestructure from motion problem presented in this paper investigates theoptimal relationship between the errors in the estimated translational and rotational parameters of a rigid motion that results in the estimation of aminimum number of negative depth values. The input used is the value of theflow along some direction, which is more general than optic flow orcorrespondence. For a planar retina it is shown that the optimalconfiguration is achieved when the projections of the translational androtational errors on the image plane are perpendicular. Furthermore, theprojection of the actual and the estimated translation lie on a linethrough the center. For a spherical retina, given a rotational error, theoptimal translation is the correct one; given a translational error, the optimal rotational negative deptherror depends both in direction and valueon the actual and estimated translation as well as the scene in view. The proofs, besides illuminating the confounding of translation and rotation instructure from motion, have an important application to ecological optics.The same analysis provides a computational explanation of why it is easierto estimate self-motion in the case of a spherical retina and why shape canbe estimated easily in the case of a planar retina, thus suggesting thatnature‘s design of compound eyes (or panoramic vision) for flying systemsand camera-type eyes for primates (and other systems that perform manipulation) is optimal.