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Can one hear shape?
Author(s) -
Reuter Martin
Publication year - 2007
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.200700694
Subject(s) - isometry (riemannian geometry) , invariant (physics) , eigenvalues and eigenvectors , computer science , representation (politics) , parametrization (atmospheric modeling) , object (grammar) , artificial intelligence , mathematics , computer vision , pure mathematics , physics , quantum mechanics , politics , political science , law , mathematical physics , radiative transfer
Abstract The question “Can one hear the shape of a drum” has been asked in several contexts before (e.g., by Bers and Kac). It is a pictorial way of asking if the eigenvalues of the Laplacian on a given domain completely characterize its shape, in other words, if the spectrum is a complete shape descriptor (which it is not in general). Since the spectrum contains geometrical information and since it is an isometry invariant and therefore independent of the object's representation, parametrization, spatial position, and optionally of its size, it is well suited to be used as a fingerprint (Shape‐DNA) in contemporary computer graphics applications like database retrieval, quality assessment, and shape matching in fields like CAD, engineering or medicine. We will explain why isometry invariance is so important and point out future directions of research. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)