Entropy-based particle correspondence for shape populations

Abstract

Purpose: Statistical shape analysis of anatomical structures plays an important role in many medical image analysis applications such as understanding the structural changes in anatomy in various stages of growth or disease. Establishing accurate correspondence across object populations is essential for such statistical shape analysis studies.

Methods: In this paper, we present an entropy-based correspondence framework for computing point-based correspondence among populations of surfaces in a groupwise manner. This robust framework is parameterization-free and computationally efficient. We review the core principles of this method as well as various extensions to deal effectively with surfaces of complex geometry and application-driven correspondence metrics.

Results: We apply our method to synthetic and biological datasets to illustrate the concepts proposed and compare the performance of our framework to existing techniques.

Conclusions: Through the numerous extensions and variations presented here, we create a very flexible framework that can effectively handle objects of various topologies, multi-object complexes, open surfaces, and objects of complex geometry such as high-curvature regions or extremely thin features.

Keywords: Correspondence; Entropy; Shape analysis.

Fig. 1

Initialization scheme based on recursive splitting

Fig. 2

Geometrical primitives such as the spheres and the plane define the open surface boundaries

Fig. 3

Spatial proximity can be a false indicator of correspondence. Point A is closer to C than to $B(\overline{AC}<\overline{AB})$. However, intuitively A should correspond to B rather than to C, as C is located on the opposite bank of the sulcus. A’s position is replicated on the right brain for ease of comparison

Fig. 4

“Box with a bump” experiment. The original formulation of the particle correspondence algorithm, shown in top row, captures the shape variation

Fig. 5

Particle correspondence between synthetically generated tori. Corresponding particles between the two shapes have matching colors

Fig. 6

“Coffee bean” experiment. Top first PCA mode; bottom, second PCA mode. Using normal penalty and geodesic distances (left) significantly improves the results in the high-curvature areas that are troublesome for the original formulation of the particle correspondence algorithm (right)

Fig. 7

Mean brain structure complexes with average pose as reconstructed from the Euclidean averages of the correspondence points. The length in the surface normal direction of each of the pointwise discriminant vector components for the autism data is given by the colormap. Yellow indicates a negative (inward) direction, and blue indicates a positive (outward) direction. Each structure is displayed in its mean orientation, position, and scale in the global coordinate frame

Fig. 8

Changes in head shape as a function of log(age in months). Corresponding particles are shown

Fig. 9

The sulcal depth (SD) and connectivity features on a select subset of subjects. Sulcal depth is defined as the length of the path traveled by each vertex during the inflation process. The connectivity features are computed via a probabilistic connectivity algorithm and projected to the cortical surface