Bayesian Estima tion of Mo tion Vector F ields

Abstract-This paper presents a new approach to the esti-mation of 2-D motion vector fields from time-varying images. The approach is stochastic both in its formulation and in the solution method. The formulation involves the specification of a deterministic structural model along ivith stochastic observation and motion field models. Two motion models are proposed: a globally smooth model based on vector Markov random fields and a piecewise smooth model derived from coupled vector-binary Markov random fields. Two estimation criteria are studied. In the max imum a posteriori probability (MAP) estimation, the a posteriori probability of motion given data is maximized, whereas in the min imum expected cost (MEC) estimation, the expectation of a certain cost function is minimized. The MAP estimation is performed via simulated annealing, whereas the MEC algorithm performs iteration-wise averaging. Both algorithms generate sam-ple fields by means of stochustic relaxation implemented via the Gibbs sampler. Two versions are developed: one for a discrete state space and the other for a continuous state space. The MAP estimation is incorporated into a hierarchical environment to deal efficiently with large displacements. Numerous experi-mental results of application of these algorithms to natural and computer-generated images with natural and synthetic motion are shown. Index Terms- Bayesian estimation, Markov random fields, motion estimation, motion modeling, optical flow, simulated an-nealing, stochastic relaxation, 2-D motion. I