Simultaneous Bayesian inference of motion velocity fields and probabilistic models in successive video-frames described by spatio-temporal MRFs

We numerically investigate a mean-field Bayesian approach with the assistance of the Markov chain Monte Carlo method to estimate motion velocity fields and probabilistic models simultaneously in consecutive digital images described by spatio-temporal Markov random fields. Preliminary to construction of our procedure, we find that mean-field variables in the iteration diverge due to improper normalization factor of regularization terms appearing in the posterior. To avoid this difficulty, we rescale the regularization term by introducing a scaling factor and optimizing it by means of minimization of the mean-square error. We confirm that the optimal scaling factor stabilizes the mean-field iterative process of the motion velocity estimation. We next attempt to estimate the optimal values of hyper-parameters including the regularization term, which define our probabilistic model macroscopically, by using the Boltzmann-machine type learning algorithm based on gradient descent of marginal likelihood (type-II likelihood) with respect to the hyper-parameters. In our framework, one can estimate both the probabilistic model (hyper-parameters) and motion velocity fields simultaneously. We find that our motion estimation is much better than the result obtained by Zhang and Hanouer (1995) in which the hyper-parameters are set to some ad-hoc values without any theoretical justification.