Lattice Image Processing: A Unification of Morphological and Fuzzy Algebraic Systems

  • Authors:

  • Petros Maragos

  • Affiliations:

  • National Technical University of Athens, School of Electrical & Computer Engineering, Athens, Greece 15773

  • Venue:
  • Journal of Mathematical Imaging and Vision
  • Year:
  • 2005

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Abstract

This paper explores some aspects of the algebraic theory of mathematical morphology from the viewpoints of minimax algebra and translation-invariant systems and extends them to a more general algebraic structure that includes generalized Minkowski operators and lattice fuzzy image operators. This algebraic structure is based on signal spaces that combine the sup-inf lattice structure with a scalar semi-ring arithmetic that possesses generalized `additions' and 驴-`multiplications'. A unified analysis is developed for: (i) representations of translation-invariant operators compatible with these generalized algebraic structures as nonlinear sup-驴 convolutions, and (ii) kernel representations of increasing translation-invariant operators as suprema of erosion-like nonlinear convolutions by kernel elements. The theoretical results of this paper develop foundations for unifying large classes of nonlinear translation-invariant image and signal processing systems of the max or min type. The envisioned applications lie in the broad intersection of mathematical morphology, minimax signal algebra and fuzzy logic.