We introduce a new image texture segmentation algorithm, HMTseg, based on wavelets and the hidden Markov tree (HMT) model. The HMT is a tree-structured probabilistic graph that captures the statistical properties of the coefficients of the wavelet transform. Since the HMT is particularly well suited to images containing singularities (edges and ridges), it provides a good classifier for distinguishing between textures. Utilizing the inherent tree structure of the wavelet HMT and its fast training and likelihood computation algorithms, we perform texture classification at a range of different scales. We then fuse these multiscale classifications using a Bayesian probabilistic graph to obtain reliable final segmentations. Since HMTseg works on the wavelet transform of the image, it can directly segment wavelet-compressed images without the need for decompression into the space domain. We demonstrate the performance of HMTseg with synthetic, aerial photo, and document image segmentations.

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Multiscale image segmentation using wavelet-domain hidden Markov models

Reviews a significant component of the rich field of statistical multiresolution (MR) modeling and processing. These MR methods have found application and permeated the literature of a widely scattered set of disciplines, and one of our principal objectives is to present a single, coherent picture of this framework. A second goal is to describe how this topic fits into the even larger field of MR methods and concepts-in particular, making ties to topics such as wavelets and multigrid methods. A third goal is to provide several alternate viewpoints for this body of work, as the methods and concepts we describe intersect with a number of other fields. The principle focus of our presentation is the class of MR Markov processes defined on pyramidally organized trees. The attractiveness of these models stems from both the very efficient algorithms they admit and their expressive power and broad applicability. We show how a variety of methods and models relate to this framework including models for self-similar and 1/f processes. We also illustrate how these methods have been used in practice.

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Multiresolution Markov models for signal and image processing

We present a tree-based reparameterization (TRP) framework that provides a new conceptual view of a large class of algorithms for computing approximate marginals in graphs with cycles. This class includes the belief propagation (BP) or sum-product algorithm as well as variations and extensions of BP. Algorithms in this class can be formulated as a sequence of reparameterization updates, each of which entails refactorizing a portion of the distribution corresponding to an acyclic subgraph (i.e., a tree, or more generally, a hypertree). The ultimate goal is to obtain an alternative but equivalent factorization using functions that represent (exact or approximate) marginal distributions on cliques of the graph. Our framework highlights an important property of the sum-product algorithm and the larger class of reparameterization algorithms: the original distribution on the graph with cycles is not changed. The perspective of tree-based updates gives rise to a simple and intuitive characterization of the fixed points in terms of tree consistency. We develop interpretations of these results in terms of information geometry. The invariance of the distribution, in conjunction with the fixed-point characterization, enables us to derive an exact expression for the difference between the true marginals on an arbitrary graph with cycles, and the approximations provided by belief propagation. More broadly, our analysis applies to any algorithm that minimizes the Bethe free energy. We also develop bounds on the approximation error, which illuminate the conditions that govern their accuracy. Finally, we show how the reparameterization perspective extends naturally to generalizations of BP (e.g., Kikuchi (1951) approximations and variants) via the notion of hypertree reparameterization.

Tree-based reparameterization framework for analysis of sum-product and related algorithms

This paper presents multiresolution models for Gauss-Markov random fields (GMRFs) with applications to texture segmentation. Coarser resolution sample fields are obtained by subsampling the sample field at fine resolution. Although the Markov property 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 also allude to the fact that different GMRF parameters at the fine resolution can result in the same probability measure after subsampling and present the results for 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 parameters. The coarsest resolution data is first segmented and the segmentation results are propagated upward to the finer resolution. We use the iterated conditional mode (ICM) minimization at all resolutions. 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.

Multiresolution Gauss-Markov random field models for texture segmentation

The basic frameworks and techniques of the Bayesian approach to image restoration are reviewed from the statistical-mechanical point of view. First, a few basic notions in digital image processing are explained to convince the reader that statistical mechanics has a close formal similarity to this problem. Second, the basic formulation of the statistical estimation from the observed degraded image by using the Bayes formula is demonstrated. The relationship between Bayesian statistics and statistical mechanics is also listed. Particularly, it is explained that some correlation inequalities on the Nishimori line of the random spin model also play an important role in Bayesian image restoration. Third, the framework of Bayesian image restoration for binary images by means of the Ising model is reviewed. Some practical algorithms for binary image restoration are given by employing the mean-field and the Bethe approximations. Finally, Bayesian image restoration for a grey-level image using the Gaussian model is reviewed, and the Gaussian model is extended to a more practical probabilistic model by introducing the line state to treat the effects of edges. The line state is also extended to quantized values.

https://iopscience.iop.org/article/10.1088/0305-4470/35/37/201/pdf

Statistical-mechanical approach to image processing

Methods for approximately computing the marginal probability mass functions and means of a Markov random field (MRF) by approximating the lattice by a tree are described. Applied to the a posteriori MRF these methods solve Bayesian spatial pattern classification and image restoration problems. The methods are described, several theoretical results concerning fixed-point problems are proven, and four numerical examples are presented, including comparison with optimal estimators and the iterated conditional mode estimator and including two agricultural optical remote sensing problems. >

C.-H. Wu

Papers

Tree approximations to Markov random fields