Global Medical Shape Analysis Using the Volumetric Laplace Spectrum

Abstract

This paper proposes to use the volumetric Laplace spectrum as a global shape descriptor for medical shape analysis. The approach allows for shape comparisons using minimal shape preprocessing. In particular, no registration, mapping, or remeshing is necessary. All computations can be performed directly on the voxel representations of the shapes. The discriminatory power of the method is tested on a population of female caudate shapes (brain structure) of normal control subjects and of subjects with schizotypal personality disorder. The behavior and properties of the volumetric Laplace spectrum are discussed extensively for both the Dirichlet and Neumann boundary condition showing advantages of the Neumann spectra. Both, the computations of spectra on 3D voxel data for shape matching as well as the use of the Neumann spectrum for shape analysis are completely new.

Hierarchical Shape Segmentation and Registration via Topological Features of Laplace-Beltrami Eigenfunctions

“…The Reeb graph can also be helpful to improve this approach for surfaces of higher genus by incorporating the pairs of essential homology classes to include handles (extended persistence). The presented method for shape segmentation can also be extended to 3D solids using the 3D Laplacian on tetrahedra or voxel representations as described in [23,26].…”

“…The higher order approach used here is already described in [53,23,36] for triangle meshes, NURBS patches, other parametrized surfaces and (parametrized) 3D tetrahedra meshes and in [26] for voxels. Here we will explain the computation for the specific case of piecewise flat triangulations and give closed formulas in the Appendix.…”

“…Note that [22] employs level sets of eigenfunctions to implictly construct correspondence of brain structures for statistical shape analysis. Other work employing spectral entities of the Laplace Beltrami operator includes retrieval [23,24], medical shape analysis [25][26][27]22], filtering/smoothing [28][29][30] of which specifically [28] mentions the use of zero level sets of specific eigenfunctions for mesh segmentation. This idea is later analyzed in more detail in [16], where also a comparison of different common discrete Laplace Beltrami operators is given.…”

Discrete Laplace–Beltrami operators for shape analysis and segmentation

“…This was done first for triangle surface meshes with a linear finite element method (FEM) in [20], for parametrized surfaces, triangle and tetrahedra meshes using higher order FEM in [48,49] and for voxel data in [47]. The discrete setting (6) with linear finite elements is equivalent to the generalized eigenvalue problem…”

Posture-invariant statistical shape analysis using Laplace operator

“…The Laplace operator has previously been used for mesh smoothing and regularization [12], for surface editing [13], and for 3D mesh fitting [14]. Furthermore, the spectrum of the Laplace operator has been used for mesh processing in a similar way to the Fourier transforms of images [15], and for shape matching and dissimilarity computation of volumetric data [16] and of triangle meshes [17]. Baran et al [18] used differential coordinates over local connected patches of a mesh to define a shape representation that is used to transfer mesh deformations from one character to another one while preserving the semantic meaning.…”