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Keywords:

  • spectral analysis;
  • Laplace-Beltrami operator;
  • medial axis;
  • shape descriptor;
  • shape matching;
  • I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Curve, surface, solid, and object representations

Abstract

Shape analysis finds many important applications in shape understanding, matching and retrieval. Among the various shape analysis methods, spectral shape analysis aims to study the spectrum of the Laplace–Beltrami operator of some well-designed shape-dependent equations and obtain a spectral shape descriptor that can in turn be used for shape analysis purposes. The success of such approaches depends greatly on the discriminating power of a shape descriptor. On the other hand, the medial axis of a shape is widely known for its complete shape representation. It is sensitive to small perturbation of the boundary of a shape which often poses difficulty in its effective use for shape analysis. In this paper, we propose a new spectral shape descriptor, called the medial axis spectrum for 2D shapes, which directly applies spectral analysis to the medial axes of the shapes. We extend the Laplace–Beltrami operator onto the medial axis, and take the solution to an extended Laplacian eigenvalue problem defined on the axis as the medial axis spectrum. The medial axis spectrum is robust in the presence of shape boundary noise, and is invariant under rigid transformations, uniform scaling and isometry of the medial axis. We demonstrate these benefits of such a medial axis spectrum representation through extensive experiments. The medial axis spectrum is further used for 2D shape retrieval, and its superiority over previous work is shown by comparison.