Distributed Representation of Image Velocity

We describe a new form of representation of image velocities, which does not rely on vector elds. For each local spatio-temporal region of the input image, we desire a function over the space of velocities describing the presence of a given velocity in that region. This function may be interpreted as a probability distribution over velocity, although it is not necessary to do so. A primary advantage of this representation is that it is capable of representing more than one velocity at a given image location. A multi-modal distribution indicates the presence of multiple motions. Such situations occur frequently in natural scenes near occlusion boundaries, and in situations of transparency. We develop an example of this type of representation through a series of modi cations of current di erential approaches to motion estimation. We de ne an angular version of the standard gradient constraint equation, and then extend this to represent multiple motions. The derivation is rst done for one-dimensional signals and then extended to two dimensions. We implement an e cient version of this distributed representation, in which the entire distribution may be interpolated from a sparse set of samples. We then demonstrate its use on simple synthetic examples containing occlusion boundaries and transparent surfaces.