Gramians, generalized inverses, and the least squares approximation of optical flow

This paper deals with the recovery of optical flow, that is to say, with the identification of a vector field, defined on some subset of the image plane, which accounts for the infinitessimal time evolution of the image of a particular object. Our formulation is general in that it allows for the vector field to be expressed as a linear combination of an arbitrary (but chosen in advance) finite collection of vector fields and it allows the measurements to include (a) the velocity of feature points, (b) the velocity normal to an evolving contour and/or (c) the velocity tangent to an intensity gradient. The method is based on least squares and an explicit formula for the generalized inverse of a class of integral operators. It involves a gramian whose invertibility is necessary and sufficient for the identification of a unique best fitting vector field. Various important subcases have been studied earlier and reported in the computer vision literature, the emphasis here is on the systematic development of a general tool.