A Theory on Optical Velocity Fields and Ambiguous Motion of Curves

Abstract

We analyze curves in motion. First, assume that we do not know the motion of points in the image plane, i.e. the optical flow. Then, define “ambiguous curves” as those that can give us nothing but ambiguous information about the underlying motion and - hence - the optical flow. The importance of a catalogue of am.bzguoas ciirues lies in telling us under what circumstances optical flow, motion and structure parameters cannot be extracted from moving curves (without point-to-point correspondence), irrespective of the method used. We give a catalogue of ambiguous curves , specializing to a planar patch in motion. Ambiguous curves (excluding the case of a single line) correspond either to a one- or a two-parameter family of solution coefficients, so the dimensionality of the non-unique solution can be predicted. Second, we wish to find analytical descriptions of the optical flow arising from objects in rigid motion, or, deforming linearly. M‘e stress the importance of using visual directions for optical flow descriptions ~ instead of image points. Modelling visual directions as elastic strings attached to a moving plane, we get a unique 3 x 3 matrix P describing the incremental transformation of visual directions. This description is more general than image point descriptions, and a convenient transform giving us concrete geometric interpretations when dealing with complicated dynamics. Using P, we can link critical-point analysis to the analysis of eigenvectors of P. The projected field lines for the optical flow, for a moving planar patch, can easily be derived from P. The two subjects, ambiguous curves and field lines of the optical flou are actually two sides of the same coin.