Generation of a statistical shape model with probabilistic point correspondences and the expectation maximization- iterative closest point algorithm

Abstract

Objective:

Identification of point correspondences between shapes is required for statistical analysis of organ shapes differences. Since manual identification of landmarks is not a feasible option in 3D, several methods were developed to automatically find one-to-one correspondences on shape surfaces. For unstructured point sets, however, one-to-one correspondences do not exist but correspondence probabilities can be determined.

Materials and methods:

A method was developed 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 points of the shape observations of the training data set. In the absence of landmarks, exact correspondences can only be determined between continuous surfaces, not between unstructured point sets. To overcome this problem, we introduce correspondence probabilities instead of exact correspondences. The correspondence probabilities are found by aligning the observation shapes with the affine expectation maximization-iterative closest points (EM-ICP) registration algorithm. In a second step, the correspondence probabilities are used as input to compute a mean shape (represented once again by an unstructured point set). Both steps are unified in a single optimization criterion which depe nds on the two parameters ‘registration transformation’ and ‘mean shape’. In a last step, a variability model which best represents the variability in the training data set is computed. Experiments on synthetic data sets and in vivo brain structure data sets (MRI) are then designed to evaluate the performance of our algorithm.

Results:

The new method was applied to brain MRI data sets, and the estimated point correspondences were compared to a statistical shape model built on exact correspondences. Based on established measures of “generalization ability” and “specificity”, the estimates were very satisfactory.

Conclusion:

The novel algorithm for building a generative statistical shape model (gSSM) does not need one-to-one point correspondences but relies solely on point correspondence probabilities for the computation of mean shape and eigenmodes. It is well-suited for shape analysis on unstructured point sets.