In defense of orthonormality constraints for nonrigid structure from motion

In factorization approaches to nonrigid structure from motion, the 3D shape of a deforming object is usually modeled as a linear combination of a small number of basis shapes. The original approach to simultaneously estimate the shape basis and nonrigid structure exploited ortho-normality constraints for metric rectification. Recently, it has been asserted that structure recovery through ortho-normality constraints alone is inherently ambiguous and cannot result in a unique solution. This assertion has been accepted as conventional wisdom and is the justification of many remedial heuristics in literature. Our key contribution is to prove that ortho-normality constraints are in fact sufficient to recover the 3D structure from image observations alone. We characterize the true nature of the ambiguity in using ortho-normality constraints for the shape basis and show that it has no impact on structure reconstruction. We conclude from our experimentation that the primary challenge in using shape basis for nonrigid structure from motion is the difficulty in the optimization problem rather than the ambiguity in ortho-normality constraints.