Multiresolution Gauss Markov Random Field Models

Abstract

This paper presents multiresolution models for Gauss Markov random fields (GMRF) with applications to texture segmentation. Coarser resolution sample fields are obtained by either subsampling or local averaging the sample field at the fine resolution. Al though Markovianity is lost under such resolution transformation, coarse resolution non-Markov random fields can be effectively approximated by Markov fields. We present two techniques to estimate the GMRF parameters at coarser resolutions from the fine resolution parameters, one by minimizing the Kullback-Leibler distance and another based on local conditional distribution invariance. We show the validity of the estimators by comparing the power spectral densities of the Markov approximation and the exac t non-Markov measures. We also allude to the fact that different measures (different GMRF parameters) on the fine resolution can result in the same probability measure after subsampling and show the results for the first and second order cases.

We apply this multiresolution model to texture segmentation. Different texture regions in an image are modeled by GMRFs and the associated parameters are assumed to be known. Parameters at lower resolutions are estimated from the fine resolution paramete rs. The coarsest resolution data is first segmented and the segmentation results are propagated upwards to the finer resolution. We use iterated conditional mode (ICM) minimization at all resolutions. A confidence measure is attached to the segmentation r esult at each pixel and passed on to the higher resolution. At each resolution, ICM is restricted only to pixels with low confidence measure. Our experiments with synthetic, Brodatz texture and real satellite images show that the multiresolution technique results in a better segmentation and requires lesser computation than the single resolution algorithm. (Also cross-referenced as UMIACS-TR-94-136)