A Phase Space Approach to Minimax Entropy Learning and the Minutemax Approximations

There has been much recent work on measuring image statistics and on learning probability distributions on images. We observe that the mapping from images to statistics is many-to-one and show it can be quantified by a phase space factor. This phase space approach throws light on the Minimax Entropy technique for learning Gibbs distributions on images with potentials derived from image statistics and elucidates the ambiguities that are inherent to determining the potentials. In addition, it shows that if the phase factor can be approximated by an analytic distribution then this approximation yields a swift "Minutemax" algorithm that vastly reduces the computation time for Minimax entropy learning. An illustration of this concept, using a Gaussian to approximate the phase factor, gives a good approximation to the results of Zhu and Mumford (1997) in just seconds of CPU time. The phase space approach also gives insight into the multi-scale potentials found by Zhu and Mumford (1997) and suggests that the forms of the potentials are influenced greatly by phase space considerations. Finally, we prove that probability distributions learned in feature space alone are equivalent to Minimax Entropy learning with a multinomial approximation of the phase factor. 1 Introduction Bayesian probability theory gives a powerful framework for visual perception (Knill and Richards 1996). This approach, however, requires specifying prior probabilities and likelihood functions. Learning these probabilities is difficult because it requires estimating distributions on random variables of very high dimensions (for example, images with 200 x 200 pixels, or shape curves of length 400 pixels).