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A semidiscrete nonlinear scale-space theory and its relation to the Perona—Malik paradox

  • Joachim Weickert
  • Brahim Benhamouda
Part of the Advances in Computing Science book series (ACS)

Abstract

Although much effort has been spent in the recent decade to establish a theoretical foundation of certain partial differential equations (PDEs) as scale-spaces, it is almost never taken into account that, in practice, images are sampled on a fixed pixel grid1. For nonlinear PDE-based filters, usually straightforward finite difference discretizations are applied in the hope that they reflect the nice properties of the continuous equations. Since scale-spaces cannot perform better than their numerical discretizations, however, it would be desirable to have a genuinely discrete nonlinear framework which reflects the discrete nature of digital images. In this paper we discuss a semidiscrete scale-space framework for nonlinear diffusion filtering. It keeps the scale-space idea of having a continuous time parameter, while taking into account the spatial discretization on a fixed pixel grid. It leads to nonlinear systems of coupled ordinary differential equations. Conditions are established under which one can prove existence of a stable unique solution which preserves the average grey level. An interpretation as a smoothing scale-space transformation is introduced which is based on an extremum principle and the existence of a large class of Lyapunov functionals comprising for instance p-norms, even central moments and the entropy. They guarantee that the process is not only simplifying and information-reducing, but also converges to a constant image as the scale parameter t tends to infinity.

Keywords

Anisotropic Diffusion Nonlinear Diffusion Extremum Principle Lyapunov Functional Couple Ordinary Differential Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer-Verlag/Wien 1997

Authors and Affiliations

  • Joachim Weickert
  • Brahim Benhamouda

There are no affiliations available

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