TY - JOUR

T1 - Laplace-distributed increments, the Laplace prior, and edge-preserving regularization

AU - Bardsley, Johnathan M.

PY - 2012/9

Y1 - 2012/9

N2 - 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.

AB - 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.

KW - Bayesian inference

KW - Inverse problems

KW - Markov random fields

KW - Regularization

KW - Total variation

UR - http://www.scopus.com/inward/record.url?scp=84869397666&partnerID=8YFLogxK

U2 - 10.1515/jip-2012-0017

DO - 10.1515/jip-2012-0017

M3 - Article

AN - SCOPUS:84869397666

SN - 0928-0219

VL - 20

SP - 271

EP - 285

JO - Journal of Inverse and Ill-Posed Problems

JF - Journal of Inverse and Ill-Posed Problems

IS - 3

ER -