Shape modeling and analysis with entropy-based particle systems

Abstract

This paper presents a new method for constructing compact statistical point-based models of ensembles of similar shapes that does not rely on any specific surface parameterization. The method requires very little preprocessing or parameter tuning, and is applicable to a wider range of problems than existing methods, including nonmanifold surfaces and objects of arbitrary topology. The proposed method is to construct a point-based sampling of the shape ensemble that simultaneously maximizes both the geometric accuracy and the statistical simplicity of the model. Surface point samples, which also define the shape-to-shape correspondences, are modeled as sets of dynamic particles that are constrained to lie on a set of implicit surfaces. Sample positions are optimized by gradient descent on an energy function that balances the negative entropy of the distribution on each shape with the positive entropy of the ensemble of shapes. We also extend the method with a curvature-adaptive sampling strategy in order to better approximate the geometry of the objects. This paper presents the formulation; several synthetic examples in two and three dimensions; and an application to the statistical shape analysis of the caudate and hippocampus brain structures from two clinical studies.

Fig. 1

A system of 100 particles on a sphere, produced by particle splitting.

Fig. 2

The box-bump experiment.

Fig. 3

The mean and ±3 std. deviations of the top 3 modes of the hand models.

Fig. 4

Surface meshes of the mean and two modes of variation at ±3 standard deviations of the right hippocampus model.

Fig. 5

P-value maps for the hippocampus and caudate shape analyses, shown on the mean shape. Red indicates significant group differences (p <= .05)