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Journal of Mathematical Imaging and Vision

, Volume 60, Issue 1, pp 128–144 | Cite as

An Extended Perona–Malik Model Based on Probabilistic Models

  • L. M. MeschederEmail author
  • D. A. Lorenz
Article
  • 467 Downloads

Abstract

The Perona–Malik model has been very successful at restoring images from noisy input. In this paper, we reinterpret the Perona–Malik model in the language of Gaussian scale mixtures and derive some extensions of the model. Specifically, we show that the expectation–maximization (EM) algorithm applied to Gaussian scale mixtures leads to the lagged-diffusivity algorithm for computing stationary points of the Perona–Malik diffusion equations. Moreover, we show how mean field approximations to these Gaussian scale mixtures lead to a modification of the lagged-diffusivity algorithm that better captures the uncertainties in the restoration. Since this modification can be hard to compute in practice, we propose relaxations to the mean field objective to make the algorithm computationally feasible. Our numerical experiments show that this modified lagged-diffusivity algorithm often performs better at restoring textured areas and fuzzy edges than the unmodified algorithm. As a second application of the Gaussian scale mixture framework, we show how an efficient sampling procedure can be obtained for the probabilistic model, making the computation of the conditional mean and other expectations algorithmically feasible. Again, the resulting algorithm has a strong resemblance to the lagged-diffusivity algorithm. Finally, we show that a probabilistic version of the Mumford–Shah segmentation model can be obtained in the same framework with a discrete edge prior.

Keywords

Perona–Malik denoising Probabilistic models Mean field approximation Gaussian scale mixtures 

Notes

Acknowledgements

We would like to thank Sebastian Nowozin from Microsoft Research for some helpful literature hints.

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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.Autonomous Vision GroupMPI TübingenTübingenGermany
  2. 2.Institute for Analysis and AlgebraTU BraunschweigBraunschweigGermany

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