Recursive 3-D Visual Motion Estimation Using Subspace Constraints

Abstract

The 3-D motion of a camera within a static environment produces a sequence of time-varying images that can be used for reconstructing the relative motion between the scene and the viewer. The problem of reconstructing rigid motion from a sequence of perspective images may be characterized as the estimation of the state of a nonlinear dynamical system, which is defined by the rigidity constraint and the perspective measurement map. The time-derivative of the measured output of such a system, which is called the “2-D motion field” and is approximated by the “optical flow”, is bilinear in the motion parameters, and may be used to specify a subspace constraint on the direction of heading independent of rotation and depth, and a pseudo-measurement for the rotational velocity as a function of the estimated heading. The subspace constraint may be viewed as an implicit dynamical model with parameters on a differentiable manifold, and the visual motion estimation problem may be cast in a system-theoretic framework as the identification of such an implicit model. We use techniques which pertain to nonlinear estimation and identification theory to recursively estimate 3-D rigid motion from a sequence of images independent of the structure of the scene. Such independence from scene-structure allows us to deal with a variable number of visible feature-points and occlusions in a principled way. The further decoupling of the direction of heading from the rotational velocity generates a filter with a state that belongs to a two-dimensional and highly constrained state-space. As a result, the filter exhibits robustness properties which are highlighted in a series of experiments on real and noisy synthetic image sequences. While the position of feature-points is not part of the state of the model, the innovation process of the filter describes how each feature is compatible with a rigid motion interpretation, which allows us to test for outliers and makes the filter robust with respect to errors in the feature tracking/optical flow, reflections, T-junctions. Once motion has been estimated, the 3-D structure of the scene follows easily. By releasing the constraint that the visible points lie in front of the viewer, one may explain some psychophysical effects on the nonrigid percept of rigidly moving objects.