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.