A Closed-Form Uncertainty Propagation in Non-Rigid Structure From Motion

Abstract

Semi-Definite Programming (SDP) with low-rank prior has been widely applied in Non-Rigid Structure from Motion (NRSfM). A low-rank constraint avoids the inherent ambiguity of the basis number selection in conventional base-shape or base-trajectory methods. Despite SDP-based NRSfM’s efficiency, it remains unclear how to propagate the noisy tracked feature points’ uncertainty to the 3D recovered shape in SDP-based NRSfM formulation. This paper presents a closed-form statistical inference for the element-wise uncertainty propagation of the estimated deforming 3D shape points in the exact low-rank SDP-based NRSfM. Then, we extend the exact low-rank uncertainty propagation to the approximate low-rank scenario with an optimal numerical rank selection method. The proposed method provides an independent module to the SDP-based method and only requires the statistical information of the input 2D trackings. Extensive experiments show that the major uncertainty in the recovered 3D points follows normal distribution, the proposed method quantifies the uncertainty accurately, and it has desirable effects on the routinely SDP low-rank based NRSfM solver.