Robust and Efficient Recovery of Rigid Motion from Subspace Constraints Solved using Recursive Identification of Nonlinear Implicit Systems

Abstract

The problem of estimating rigid motion from projections may be characterized using a nonlinear dynamical system, composed of the rigid motion transformation and the perspective map. The time derivative of the output of such a system, which is also called the "motion field", is bilinear in the motion parameters, and may be used to specify a subspace constraint on either the direction of translation or the inverse depth of the observed points. Estimating motion may then be formulated as an optimization task constrained on such a subspace. Heeger and Jepson [5], who first introduced this constraint, solve the optimization task using an extensive search over the possible directions of translation. We reformulate the optimization problem in a systems theoretic framework as the the identification of a dynamic system in exterior differential form with parameters on a differentiable manifold, and use techniques which pertain to nonlinear estimation and identification theory to perform the optimization task in a principled manner. The general technique for addressing such identification problems [14] has been used successfully in addressing other problems in computational vision [13, 12]. The application of the general method [14] results in a recursive and pseudo-optimal solution of the motion problem, which has robustness properties far superior to other existing techniques we have implemented. By releasing the constraint that the visible points lie in front of the observer, we may explain some psychophysical effects on the nonrigid percept of rigidly moving shapes. Experiments on real and synthetic image sequences show very promising results in terms of robustness, accuracy and computational efficiency.