Showing papers by "Niklas Peinecke published in 2006"
TL;DR: This paper introduces a method to extract 'Shape-DNA', a numerical fingerprint or signature, of any 2d or 3d manifold by taking the eigenvalues (i.e. the spectrum) of its Laplace-Beltrami operator and succeeds in computing eigen values for smoothly bounded objects without discretization errors caused by approximation of the boundary.
Abstract: This paper introduces a method to extract 'Shape-DNA', a numerical fingerprint or signature, of any 2d or 3d manifold (surface or solid) by taking the eigenvalues (i.e. the spectrum) of its Laplace-Beltrami operator. Employing the Laplace-Beltrami spectra (not the spectra of the mesh Laplacian) as fingerprints of surfaces and solids is a novel approach. Since the spectrum is an isometry invariant, it is independent of the object's representation including parametrization and spatial position. Additionally, the eigenvalues can be normalized so that uniform scaling factors for the geometric objects can be obtained easily. Therefore, checking if two objects are isometric needs no prior alignment (registration/localization) of the objects but only a comparison of their spectra. In this paper, we describe the computation of the spectra and their comparison for objects represented by NURBS or other parametrized surfaces (possibly glued to each other), polygonal meshes as well as solid polyhedra. Exploiting the isometry invariance of the Laplace-Beltrami operator we succeed in computing eigenvalues for smoothly bounded objects without discretization errors caused by approximation of the boundary. Furthermore, we present two non-isometric but isospectral solids that cannot be distinguished by the spectra of their bodies and present evidence that the spectra of their boundary shells can tell them apart. Moreover, we show the rapid convergence of the heat trace series and demonstrate that it is computationally feasible to extract geometrical data such as the volume, the boundary length and even the Euler characteristic from the numerically calculated eigenvalues. This fact not only confirms the accuracy of our computed eigenvalues, but also underlines the geometrical importance of the spectrum. With the help of this Shape-DNA, it is possible to support copyright protection, database retrieval and quality assessment of digital data representing surfaces and solids. A patent application based on ideas presented in this paper is pending.
789 citations
Patent•
18 May 2006
TL;DR: A method for characterization of objects has the steps of: describing an object with an elliptical self-adjoint eigenvalue problem in order to form an isometrically invariant model, determining eigenvalues of the eigen value problem, and characterizing the object by the Eigenvalues as discussed by the authors.
Abstract: A method for characterization of objects has the steps of:
a) describing an object with an elliptical self-adjoint eigenvalue problem in order to form an isometrically invariant model; b) determining eigenvalues of the eigenvalue problem; and c) characterizing the object by the eigenvalues.
3 citations
Patent•
18 May 2006
TL;DR: In this paper, a Verfahren zur Charakterisierung von Objekten hat die Schritte: a) Beschreiben eines Objektes with einem elliptischen, selbstadjungierten Eigenwertproblem zur Bildung eines isometrieinvarianten Modells; b) Bestimmen von Eigen werten des EigenWertproblems; and c) CharAKterisieren des Objekte durch die Eigenwerte.
Abstract: Ein Verfahren zur Charakterisierung von Objekten hat die Schritte: a) Beschreiben eines Objektes mit einem elliptischen, selbstadjungierten Eigenwertproblem zur Bildung eines isometrieinvarianten Modells; b) Bestimmen von Eigenwerten des Eigenwertproblems; und c) Charakterisieren des Objektes durch die Eigenwerte.
1 citations