From Adaptive Averaging to Accelerated Nonlinear Diffusion Filtering

  • Stephan Didas
  • Joachim Weickert
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4174)


Weighted averaging filters and nonlinear partial differential equations (PDEs) are two popular concepts for discontinuity-preserving denoising. In this paper we investigate novel relations between these filter classes: We deduce new PDEs as the scaling limit of the spatial step size of discrete weighted averaging methods. In the one-dimensional setting, a simple weighted averaging of both neighbouring pixels leads to a modified Perona-Malik-type PDE with an additional acceleration factor that provides sharper edges. A similar approach in the two-dimensional setting yields PDEs that lack rotation invariance. This explains a typical shortcoming of many averaging filters in 2-D. We propose a modification leading to a novel, anisotropic PDE that is invariant under rotations. By means of the example of the bilateral filter, we show that involving a larger number of neighbouring pixels improves rotational invariance in a natural way and leads to the same PDE formulation. Numerical examples are presented that illustrate the usefulness of these processes.


Rotational Invariance Image Denoising Scaling Limit Adaptive Smoothing Adaptive Average 
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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Stephan Didas
    • 1
  • Joachim Weickert
    • 1
  1. 1.Mathematical Image Analysis Group, Faculty of Mathematics and Computer ScienceSaarland UniversitySaarbrückenGermany

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