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@ARTICLE{Reuter:280030,
      author       = {Reuter, Martin and Wolter, Franz-Erich and Peinecke,
                      Niklas},
      title        = {{L}aplace–{B}eltrami spectra as ‘{S}hape-{DNA}’ of
                      surfaces and solids},
      journal      = {Computer aided design},
      volume       = {38},
      number       = {4},
      issn         = {0010-4485},
      address      = {Amsterdam [u.a.]},
      publisher    = {Elsevier Science},
      reportid     = {DZNE-2025-00874},
      pages        = {342 - 366},
      year         = {2006},
      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.},
      ddc          = {600},
      pnm          = {899 - ohne Topic (POF4-899)},
      pid          = {G:(DE-HGF)POF4-899},
      typ          = {PUB:(DE-HGF)16},
      doi          = {10.1016/j.cad.2005.10.011},
      url          = {https://pub.dzne.de/record/280030},
}