3D shape comparison of cardiac geometries using a Laplace spectral-shape-matching approach

Abstract

The eigen-modes of the Laplace–Beltrami operator (LBO) applied to triangulated three-dimensional (3D) surface geometries have been shown to be effective parametric representations of overall shape and structural detail, which may constitute promising feature spaces for statistical shape comparison. The objective of this study is to explore a Laplace spectral-matching approach to compare pairs of similar 3D surface geometries using their respective Laplace spectral representations, obtained from the LBO or the graph Laplacian of the distributed points over each surface manifold. We demonstrate the efficacy of a greedy algorithm for appropriate selection of a set of Laplacian eigen-mode shape descriptors, while resolving their respective sign ambiguities, for optimal shape matching. We test our algorithm on three pairs of experimental shapes, as well as three sets of clinically relevant test geometries, i.e. surface models of two similar patient-specific models of the left atrial appendage of the atrial chamber of the heart, several pairs of patient-specific models of the left ventricular chamber of the heart functioning over the cardiac cycle, as well as a pair of similar coronary artery geometries. We also explore the relationship between accuracy of spectral shape matching and mesh refinement of compared geometries. Our results demonstrate good spectral matching efficacy for pairs of geometries, which differ only in terms of affine transformations relative to each other, but additionally illustrate the method's potential for biomedical applications such as shape-based patient information retrieval.