Decomposition of Separable Concave Structuring Functions

  • Authors:

  • E. A. Engbers;R. Van Den Boomgaard;A. W. M. Smeulders

  • Affiliations:

  • Intelligent Sensory Information Systems, University of Amsterdam, Kruislaan 403, 1098 SJ Amsterdam, The Netherlands;Intelligent Sensory Information Systems, University of Amsterdam, Kruislaan 403, 1098 SJ Amsterdam, The Netherlands;Intelligent Sensory Information Systems, University of Amsterdam, Kruislaan 403, 1098 SJ Amsterdam, The Netherlands

  • Venue:
  • Journal of Mathematical Imaging and Vision
  • Year:
  • 2001

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Abstract

This paper presents a decomposition scheme for a large class of greyscale concave structuring functions from mathematical morphology. In contrast with many existing decomposition schemes, our method is valid in the continuous domain. Conditions are given under which this continuous method can be properly discretized. The class of functions that can be decomposed with our method contains all concave 2D-functions that are separable in two 1D-functions. This class contains the class of quadratic functions, that are of major importance in, for instance, distance transforms and morphological scale space. In the continuous domain, the size of the structuring elements resulting from the decomposition, can be chosen arbitrarily small. Conditions under which the discrete version of the decomposition can be guaranteed are given. For functions from the mentioned class, that can be separated along the standard image axes, a discrete decomposition in elements of n × n pixels is always possible, with n ≥ 3. The parabola fall in the atter category.