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

verfasst von: Martin Reuter, Franz Erich Wolter, Martha Shenton, Marc Niethammer
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.
Externe Organisation(en): Massachusetts Institute of Technology (MIT)
Harvard University
University of North Carolina
Typ: Artikel
Journal: CAD Computer Aided Design
Band: 41
Seiten: 739-755
Anzahl der Seiten: 17
ISSN: 0010-4485
Publikationsdatum: 10.2009
Publikationsstatus: Veröffentlicht
Peer-reviewed: Ja
ASJC Scopus Sachgebiete: Angewandte Informatik, Computergrafik und computergestütztes Design, Wirtschaftsingenieurwesen und Fertigungstechnik
Elektronische Version(en): https://europepmc.org/article/PMC/2753296 (Zugang: Offen)
https://doi.org/10.1016/j.cad.2009.02.007 (Zugang: Geschlossen)

BibTeX

@article{109c7cd00ecc41ddbcadf9f326c5cb13,
title = "Laplace-Beltrami eigenvalues and topological features of eigenfunctions for statistical shape analysis",
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.",
keywords = "Brain structure, Caudate nucleus, Eigenfunctions, Eigenvalues, Laplace-Beltrami spectra, Morse-Smale complex, Nodal domains, Reeb graph, Schizotypal personality disorder",
author = "Martin Reuter and Wolter, {Franz Erich} and Martha Shenton and Marc Niethammer",
year = "2009",
month = oct,
doi = "10.1016/j.cad.2009.02.007",
language = "English",
volume = "41",
pages = "739--755",
journal = "CAD Computer Aided Design",
issn = "0010-4485",
publisher = "Elsevier Ltd.",
number = "10",
}