Two-Dimensional Parallel Thinning Algorithms Based on Critical Kernels

  • Authors:

  • G. Bertrand;M. Couprie

  • Affiliations:

  • Institut Gaspard-Monge, Laboratoire A2SI, Groupe ESIEE, Noisy-le-Grand cedex, France 93162;Institut Gaspard-Monge, Laboratoire A2SI, Groupe ESIEE, Noisy-le-Grand cedex, France 93162

  • Venue:
  • Journal of Mathematical Imaging and Vision
  • Year:
  • 2008

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Abstract

Critical kernels constitute a general framework in the category of abstract complexes for the study of parallel thinning in any dimension. The most fundamental result in this framework is that, if a subset Y of X contains the critical kernel of X, then Y is guaranteed to have "the same topology as X". Here, we focus on 2D structures in spaces of two and three dimensions. We introduce the notion of crucial pixel, which permits to link this work with the framework of digital topology. We prove simple local characterizations, which allow us to express thinning algorithms by way of sets of masks. We propose several new parallel algorithms, which are both fast and simple to implement, that yield symmetrical or non-symmetrical skeletons of 2D objects in 2D or 3D grids. We prove some properties of these skeletons, related to topology preservation, to minimality, and to the inclusion of the topological axis. The latter may be seen as a generalization of the medial axis. We also show how to use critical kernels in order to provide simple proofs of the topological soundness of existing thinning schemes. Finally, we clarify the link between critical kernels, minimal non-simple sets, and P-simple points.