Statistical Applications with Deformable M-Reps

Abstract

There are many uses of the means of representing objects by discrete m-reps and of estimating probability distributions on them by extensions of linear statistical techniques to nonlinear manifolds describing the associated nonlinear transformations that were detailed in Chapter 8. Two important ones are described in this chapter: segmentation by posterior optimization and determining the significant shape distinctions that can be found in two different probability distributions on an m-rep with the same topology but from two different classes. Both uses require facing issues of probabilities on geometry at multiple levels of spatial scale. The segmentation problem requires the estimation of the probability of image intensity distributions given the object description; we describe a way of doing that by an extension of principal component analysis to regional intensity summaries produced using the object-relative coordinates provided by m-reps. Applications of both segmentation and determination of shape distinctions to anatomic objects in medical images are described. Also described is a variant on the segmentation program used in estimating the probability density on an m-rep; this program fits an m-rep to a binary image in a way that is intended to achieve correspondence of medial atoms across the training population.