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Book Sections Year : 2014

## Markov Models and MCMC Algorithms in Image Processing

Xavier Descombes

#### Abstract

Probabilistic approaches have been brought to image analysis starting with the famous paper by Besag [1], inspired by the Gibbs field theory developped in statistical physics. Later on, this approach mostly developped in the Bayesian framework has become very popular in the image processing community from the texture modeling proposed by Cross and Jain [2] and the restoration scheme proposed by Geman and Geman [3] to the recent developments based on marked point processes for multiple objects detection [4]. Adapted to solve ill posed inverse problems, the stochastic approach in image analysis includes, but is not restricted to image restoration, denoising, deblurring, classification, segmentation, feature extraction, surface reconstruction, stereo matching. The main idea is to model independently the data, including the sensors property and the noise, and some prior knowledge we may have on the solution. These two models, respectively named the likelihood and the prior, are combined to define the posterior distribution, by applying the Bayes rule. One key point is to consider Markovian models which can be rewritten as a Gibbs model, using the Hammersley- Clifford theorem. The posterior is then defined by an energy function written as a sum of local functions, so called the potentials. These functions can be interpreted as local constraints that are very intuitive. For example, in the case of an image restoration problem, the term issued from the likelihood measures the similarity between the solution and the data whereas the prior energy measures the similarity between neighbourgh pixels. The solution consists then of the configuration minimizing the energy, interpreted as a cost function, that is a compromise between an uniform image and the data. In a probabilistic setting, this solution maximizes the posterior and is referred as the Maximum A Posteriori (MAP). Once the model defined, the second issue consists of the solution computation. This optimization problem has three main characteristics. The different variables, for example associated to each pixel, are mutually dependent because of the interaction introduced in the potentials function. Therefore, we cannot perform the optimization independently on each variable. The posterior embeds a normalizing function, also named the partition function, which is neither anatically nor numerically computable. Therefore, the posterior cannot be directly simulated, it requires iterative methods such as Markov Chain Monte Carlo (MCMC) simulations. Finally, the energy function is rarely convex. Classical approaches based on gradient descent cannot be applied, or at most only provide an approximation of the solution. In some specific cases, some combinatorial algorithms, based on graph theory, can provide the exact solution. In this chapter, we derive the two main frameworks based on stochastic modeling in image analysis. The most traditional concerns Markov Random Fields. We briefly give the main definitions and derive associated algorithms to perform the optimization. Several examples illustrate the wide range of MRF modeling. In a second step, we generalize the random field approach to the point process approach that allows geometry to be taken into account. Here again, after a brief description of the main definitions and optimization techniques, we exemplify on several applications.

#### Domains

Image Processing [eess.IV]

### Dates and versions

hal-00904989 , version 1 (15-11-2013)

### Identifiers

• HAL Id : hal-00904989 , version 1

### Cite

Xavier Descombes. Markov Models and MCMC Algorithms in Image Processing. Rama Chelappa. Academic Press Library in Signal Processing, 4, Elsevier Ltd, 2014, Image, Video Processing and Analysis, Hardware, Audio, Acoustic and Speech 978-0-12-396501-1. ⟨hal-00904989⟩

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