Laplace-Beltrami eigenvalues and topological features of eigenfunctions for statistical shape analysis

  • Authors:

  • Martin Reuter;Franz-Erich Wolter;Martha Shenton;Marc Niethammer

  • Affiliations:

  • Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, United States and A.A. Martinos Center for Biomedical Imaging, Massachusetts General Hospital, Harvard Medic ...;Inst. für Mensch-Maschine-Kommunikation, Leibniz Universität Hannover, Germany;Brigham and Women's Hospital, Harvard Medical School, Boston, United States;Department of Computer Science, UNC-Chapel Hill, United States and Biomedical Research Imaging Center, School of Medicine, UNC - Chapel Hill, United States

  • Venue:
  • Computer-Aided Design
  • Year:
  • 2009

Quantified Score

Hi-index 0.00

Visualization

Abstract

This paper proposes the use of the surface-based Laplace-Beltrami and the volumetric Laplace eigenvalues and eigenfunctions as shape descriptors for the comparison and analysis of shapes. These spectral measures are isometry invariant and therefore allow for shape comparisons with minimal shape pre-processing. In particular, no registration, mapping, or remeshing is necessary. The discriminatory power of the 2D surface and 3D solid methods is demonstrated on a population of female caudate nuclei (a subcortical gray matter structure of the brain, involved in memory function, emotion processing, and learning) of normal control subjects and of subjects with schizotypal personality disorder. The behavior and properties of the Laplace-Beltrami eigenvalues and eigenfunctions are discussed extensively for both the Dirichlet and Neumann boundary condition showing advantages of the Neumann vs. the Dirichlet spectra in 3D. Furthermore, topological analyses employing the Morse-Smale complex (on the surfaces) and the Reeb graph (in the solids) are performed on selected eigenfunctions, yielding shape descriptors, that are capable of localizing geometric properties and detecting shape differences by indirectly registering topological features such as critical points, level sets and integral lines of the gradient field across subjects. The use of these topological features of the Laplace-Beltrami eigenfunctions in 2D and 3D for statistical shape analysis is novel.