Comparison of Statistical Shape Models Built on Correspondence Probabilities and One-to-One Correspondences

In this paper, we present a method to compute a statistical shape model based on shapes which are represented by unstructured point sets with arbitrary point numbers. A fundamental problem when computing statistical shape models is the determination of correspondences between the observations of the associated data set. Often, homologies between points that represent the surfaces are assumed. When working merely with point clouds, this might lead to imprecise mean shape and variability results. To overcome this problem, we propose an approach where exact correspondences are replaced by evolving correspondence probabilities. These are the basis for a novel algorithm that computes a generative statistical shape model. We developed a unified Maximum A Posteriori (MAP) framework to compute the model parameters ('mean shape' and 'modes of variation') and the nuisance parameters which leads to an optimal adaption of the model to the set of observations. The registration of the model on the observations is solved using the Expectation Maximization - Iterative Closest Point algorithm which is based on probabilistic correspondences and proved to be robust and fast. The alternated optimization of the MAP explanation with respect to the observation and the generative model parameters leads to very efficient and closed-form solutions for nearly all parameters. A comparison with a statistical shape model which is built using the Iterative Closest Point (ICP) registration algorithm and a Principal Component Analysis (PCA) shows that our approach leads to better SSM quality measures.