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The Perona–Malik Method as an Edge Pruning Algorithm

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Abstract

The Perona–Malik equation (PM), in the continuum limit, is interpreted as the gradient flow for a functional, corresponding to the reconstruction of an image with edges with non-zero thickness. This result is based on an image model (u,Γ) where Γ is an edge set, and u is a slowly-varying function. PM simplifies the image by reducing the jump across each component of Γ, resulting in an automatic edge pruning procedure. The initial-value problem thus defined is well-posed, but practically stable only for small times: it leads to a semi-group with exponential growth. The rigorous analysis gives a mathematical basis for empirical observations, including edge localization and the need to use a small number of iterations. The variational formulation enables an easy comparison with earlier methods.

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  1. Laboratoire de Mathématiques (UMR 6056), CNRS & Université de Reims Champagne-Ardenne, Moulin de la Housse, B.P. 1039, F-51687, Reims Cedex 2, France

    Satyanad Kichenassamy

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Kichenassamy, S. The Perona–Malik Method as an Edge Pruning Algorithm. J Math Imaging Vis 30, 209–219 (2008). https://doi.org/10.1007/s10851-007-0029-2

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