#### Title

Image analysis with partially ordered Markov models

#### RIS ID

72971

#### Abstract

Statistical approaches to image analysis, such as image restoration, segmentation, object classification, and reconstruction often require specification of a distributional model for the variability of the pixel intensities around the true image and a prior distributional model for the true image itself. Spatial dependence (i.e., nearby values tend to be more - or less - alike than those far apart) is often modeled by assuming a Markov random field (MRF) for the prior model and sometimes for the pixel-intensity model. When dealing with MRFs, there is typically an unwieldy normalizing constant that can cause inference to be either inefficient or computationally intensive. In this article, we propose a class of models that are a subset of the class of MRFs but whose members have probability distributions that can be written in closed form. This class, called the partially ordered Markov models (POMMs), contains as a special case the Markov mesh models (MMMs) and is seen to be an important subclass of graphical models used in the analysis of (Bayesian) networks. POMMs are used in experiments for both the forward problem of texture synthesis and the inverse problem of parameter estimation. Various images of textures are generated using POMMs and are seen not to exhibit any obvious directional patterns. Also, parameter estimates from maximum likelihood estimators are found using a real texture image, and the estimates are then used to generate a texture that is similar to the real data. | Statistical approaches to image analysis, such as image restoration, segmentation, object classification, and reconstruction often require specification of a distributional model for the variability of the pixel intensities around the true image and a prior distributional model for the true image itself. Spatial dependence (i.e., nearby values tend to be more - or less - alike than those far apart) is often modeled by assuming a Markov random field (MRF) for the prior model and sometimes for the pixel-intensity model. When dealing with MRFs, there is typically an unwieldy normalizing constant that can cause inference to be either inefficient or computationally intensive. In this article, we propose a class of models that are a subset of the class of MRFs but whose members have probability distributions that can be written in closed form. This class, called the partially ordered Markov models (POMMs), contains as a special case the Markov mesh models (MMMs) and is seen to be an important subclass of graphical models used in the analysis of (Bayesian) networks. POMMs are used in experiments for both the forward problem of texture synthesis and the inverse problem of parameter estimation. Various images of textures are generated using POMMs and are seen not to exhibit any obvious directional patterns. Also, parameter estimates from maximum likelihood estimators are found using a real texture image, and the estimates are then used to generate a texture that is similar to the real data.

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## Publication Details

Cressie, N. A. & Davidson, J. L. (1998). Image analysis with partially ordered Markov models. Computational Statistics and Data Analysis, 29 (1), 1-26.