Finite Element Based Sequential Bayesian Non-Rigid Structure from Motion

Abstract

Navier’s equations modelling linear elastic solid deformations are embedded within an Extended Kalman Filter (EKF) to compute a sequential Bayesian estimate for the Non-Rigid Structure from Motion problem. The algorithm processes every single frame of a sequence gathered with a full perspective camera. No prior data association is assumed because matches are computed within the EKF prediction-match-update cycle. Scene is coded as a Finite Element Method (FEM) � elastic thin-plate solid, where the discretization nodes are the sparse set of scene points salient in the image. It is assumed a set of Gaussian forces acting on solid nodes a to r cause scene deformation. The EKF combines xv in a feedback loop an approximate FEM model and the frame rate measurements from the camera, resulting in anxefficient method r to embed Navier’s equations without resorting to expensive non-linear FEM models. Classical FEM modelling has implied an interactive identification of boundary points to constrain the a scene rigid motion, in this work this dissatisfying prior knowledge is no longer needed. The scene and camera a = rigid a nr motion are combined in a unique pose vector and the xr = estimation xv +ar is coded relative to the camera. Additionally, the deforming effect of the Gaussian forces on the thin-plate is computed by means of the Moore-Penrose pseudoinverse of the FEM stiffness matrix. The proposed algorithm is validated with three real sequences gathered with hand-held camera observing isometric and non-isometric deformations. It is also shown the consistency of the EKF estimation with respect to ground truth computed from stereo. xvxv xvxv W W W W a a