Laplace-Beltrami spectra as ‘Shape-DNA’ of surfaces using the closest point method

The need to compare two separate manifolds arises in a wide range of applications. In this thesis, ‘Shape-DNA’, i.e. the eigenvalues of the Laplace-Beltrami operator, are used to create a numerical signature representing an individual object. The corresponding spectrum is isometry invariant, which means it is independent of manifold representations such as parameterization or spatial positioning. Therefore, checking if two objects are isometry invariants does not require any alignment (registration/localization) of the objects but only comparing their spectra. We determine the Shape-DNA using the closest point method on the manifold. In 3D we illustrate the process for triangulated mesh and point cloud surfaces. Convergence studies demonstrate that the convergence rates correspond to those of the underlying finite difference methods. A 2D multidimensional scaling plot illustrates how identical objects are mapped to the same spot and similar objects form groups based on the Shape-DNA’s.