A “theory” of optical flow

Abstract

We analyze curves in motion, be they edges, isobrightness curves, or silhouette contours. These curves are perspectively projected from opaque 3D scene surfaces in rigid or nonrigid motion. First, assume that we only know velocity components normal to the curves (so-called normal flow), except for a few (fewer than 4) velocity estimates at feature points. 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 catalog of ambiguous curves lies in telling us under what circumstances optical flow, motion, and structure parameters cannot be extracted from moving curves irrespective of the method used. We give a catalog of ambiguous curves specializing to a planar patch in motion. Ambiguous curves (excluding the case of a single line) correspond to either a one- or a three-parameter family of solution coefficients, so the dimensionality of the nonunique solution can be predicted. Second, we wish to find analytical descriptions of the optical flow arising from objects in rigid motion, or, deforming linearly. We 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 × 3 matrix P̂ 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 P̂, 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 P̂. The two subjects, ambiguous curves and field lines of the optical flow, are actually two sides of the Same coin.