Laplace-distributed increments, the Laplace prior, and edge-preserving regularization

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Abstract

For a given two-dimensional image, we define the horizontal and vertical increments at a pixel location to be the difference between the intensity values at that pixel and at the neighboring pixels to the right and above, respectively. For a typical image, it makes intuitive sense that the increments will usually be near zero, corresponding to areas of smooth variation in image intensity, but will often have large magnitude, corresponding to edges where sharp intensity changes occur. In this paper, we explore the use of the Laplace increment model, in which the increments are assumed to be independent and identically distributed Laplace random variables - a distribution with heavy tails allowing for large increment values - with zero mean. The prior constructed from the Laplace increment model is very similar to the total variation (TV) prior. We perform a theoretical analysis of its properties, which shows that the Laplace prior yields a regularization scheme with regularized solutions contained in the space of bounded variation, just as for the TV prior. Moreover, numerical experiments indicate that the Laplace prior yields reconstructions that are qualitatively very similar to those obtained using TV.

Original languageEnglish
Pages (from-to)271-285
Number of pages15
JournalJournal of Inverse and Ill-Posed Problems
Volume20
Issue number3
DOIs
StatePublished - Sep 2012

Keywords

  • Bayesian inference
  • Inverse problems
  • Markov random fields
  • Regularization
  • Total variation

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