Optimal Metric Projections for Deformable and Articulated Structure-from-Motion

Abstract

This paper describes novel algorithms for recovering the 3D shape and motion of deformable and articulated objects purely from uncalibrated 2D image measurements using a factorisation approach. Most approaches to deformable and articulated structure from motion require to upgrade an initial affine solution to Euclidean space by imposing metric constraints on the motion matrix. While in the case of rigid structure the metric upgrade step is simple since the constraints can be formulated as linear, deformability in the shape introduces non-linearities. In this paper we propose an alternating bilinear approach to solve for non-rigid 3D shape and motion, associated with a globally optimal projection step of the motion matrices onto the manifold of metric constraints. Our novel optimal projection step combines into a single optimisation the computation of the orthographic projection matrix and the configuration weights that give the closest motion matrix that satisfies the correct block structure with the additional constraint that the projection matrix is guaranteed to have orthonormal rows (i.e. its transpose lies on the Stiefel manifold). This constraint turns out to be non-convex. The key contribution of this work is to introduce an efficient convex relaxation for the non-convex projection step. Efficient in the sense that, for both the cases of deformable and articulated motion, the proposed relaxations turned out to be exact (i.e. tight) in all our numerical experiments. The convex relaxations are semi-definite (SDP) or second-order cone (SOCP) programs which can be readily tackled by popular solvers. An important advantage of these new algorithms is their ability to handle missing data which becomes crucial when dealing with real video sequences with self-occlusions. We show successful results of our algorithms on synthetic and real sequences of both deformable and articulated data. We also show comparative results with state of the art algorithms which reveal that our new methods outperform existing ones.