Anisotropic Smoothing of Posterior Probabilities

Abstract

In a large number of segmentation problems, the number of different objects or classes present in the image is known a priori. Examples are magnetic resonance images of the cortex and SAR data. A technique to introduce this prior knowledge into the segmentation process is presented and analyzed in this paper. The basic idea is to perform edge preserving anisotropic smoothing of posterior probabilities, computed via Bayes rule, followed by an independent pixelwise maximum aposterior probability (MAP) classification. In this paper, we describe the technique and develop the mathematical theory underlying it. We demonstrate that prior anisotropic smoothing of the posterior probabilities yields the MAP solution of a discrete Markov random field (MRF) with a non-interacting, analog discontinuity field. In contrast, isotropic smoothing of the posterior probabilities is equivalent to computing the MAP solution of a single, discrete MRF using continuous relaxation labeling. Combining a discontinuity field with a discrete MRF is important as it allows the disabling of clique potentials across discontinuities. Furthermore, explicit representation of the discontinuity field suggests new algorithms that incorporate properties like hysteresis and non-maximal suppression.

This work was partially supported by ONR Grant N00014-97-1-0509, ONR Young Investigator Program Award, and NSF-LIS.