Singularities in the Recovery of Rigid Body Motions from Optical Flow Data

Abstract

This paper addresses the problem of recovering the six-degree-of-freedom motions of a rigid body from a sequence of images. We address the mathematics of tracking several points of the rigid object, each feature point provides optical flow data in the image plane. First, we show that three feature points are not enough to avoid singularities in the process of recovering the relative motion of the rigid body with respect to the camera: we claim that, for any set of three points on a rigid object, there always exists certain positions and orientations of the object with respect to the camera, for which non-zero motions of the object cannot be detected on the image. We also show that it is su cient to track four or more feature points, provided that the selected four points belong to the same circular cylinder and satisfy a geometric condition detailed in the paper. Once the geometric condition is satis ed, for all positions and poses of the object with respect to the camera, any motion of the bject can be mathematically recovered from the optical flow data of the projections of the feature points on the image plane. We formally prove our claims by decomposing the well-established 2x 6 optical flow matrix associated to every feature point, into the product of a 2x 3 matrix by a 3x 6 matrix.