Symmetry in Perspective

Abstract

When a symmetric object in 3D is projected to an image, the symmetry properties are lost except for specific relative viewpoints. For sufficiently complex objects however, it is still possible to infer symmetry in 3D from a single image of an uncalibrated camera. In this paper we give a general computational procedure for computing 3D structure and finding image symmetry constraints from single view projections of symmetric 3D objects. For uncalibrated cameras these constraints take the form of polynomials in brackets (determinants) and by using projectively invariant shape constraints relating 3D and image structure, the symmetry constraints can be derived easily by considering the effects of the corresponding symmetry transformation group in 3D. The constraints can also be given a useful geometric interpretation in terms of projective incidence using the Grassmann-Cayley algebra formalism. We demonstrate that in specific situations geometric intuition and the use of GC-algebra is a very simple and effective tool for finding image symmetry constraints.