Non-rigid shape-from-motion for isometric surfaces using infinitesimal planarity

Abstract

This paper proposes a general framework to solve Non-Rigid Shape-from-Motion (NRSfM) with the perspective camera under isometric deformations. Contrary to the usual low-rank linear shape basis, isometry allows us to recover complex shape deformations from a sparse set of images. Existing methods suffer from ambiguities and may be very expensive to solve. We bring four main contributions. First, we formulate isometric NRSfM as a system of first-order Partial Differential Equations (PDE) involving the shape's depth and normal field and an unknown template. Second, we show this system cannot be locally resolved. Third, we introduce the concept of infinitesimal planarity and show that it makes the system locally solvable for at least three views. Fourth, we derive an analytic solution which involves convex, linear least-squares optimization only, and outperforms existing works.