# MARKOV RANDOM FIELDS FOR VISION AND IMAGE PROCESSING PDF

Request PDF on ResearchGate | On Jan 1, , Andrew Blake and others published Markov random fields for vision and image processing. PDF | On Jan 1, , Oliver Moolan-Feroze and others published Review: Markov Random Fields for Vision and Image Processing, in Your Face: The New . PDF | INTRODUCTION Markov random field models have become useful in several areas of image processing. The success of Markov random fields (MRFs) .

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Markov Random Fields for Vision and Image Processing edited by Andrew Blake, Pushmeet Kohli, and Carsten Rother. The MIT Press. Cambridge. The Handbook of Brain Theory and Neural Networks, M. Arbib, editor, , MIT Press, Markov random eld models in image processing Anand. Ebook Pdf Markov Random Fields For Vision And Image Processing contains important information and a detailed explanation about Ebook Pdf Markov.

At each step these algorithms explore a space of possible changes also called a move space that can be made to the current solution xc , and choose a change move that leads to a new solution xn having the lowest energy. This move is referred to as the optimal move. The algorithm is said to converge when no solution with a lower energy can be found.

ICM works in a coordinate descent fashion. This greedy characteristic makes it prone to getting stuck in local minima. Intuitively, an energy minimization algorithm that can jump out of local minima will do well in problems with a large number of local optima. Simulated annealing SA is one such class of algorithms. Developed as a general optimization strategy by Kirkpatrick et al.

This probability is controlled by a parameter T that is called the temperature. A higher value of T implies a high probability of accepting changes to the solution that lead to a higher energy.

The algorithm starts with a high value of T and reduces it to 0 as it proceeds. Details are given in chapter 5. The max-product algorithm can be used to minimize general energy functions approximately. Since an energy function is the negative log of posterior probability, it is necessary only to replace the product operation with a sum operation, and the max operation with a min. For the pairwise energy function 4.

Those results have inspired a lot of work on the theoretical properties of the way in which BP operates in a general probabilistic model []. More recently, a number of researchers have shown the relationship between message-passing algorithms and linear programming relaxation-based methods for MAP inference [, ].

This relationship is the subject of chapter 6. The problem of minimizing a function of discrete variables is in general NP-hard, and has been well studied in the discrete optimization and operations research communities. An introduction to these methods is given in this section. Although minimizing a function of discrete variables is NP-hard in general, there are families of energy functions for which it can be done in polynomial time.

Submodular functions are one such well-known family [, ]. For instance, the Ising energy described earlier 1. This is quite remarkable, given how rare it is that realistic scale information problems in the general area of machine intelligence admit exact solutions. Even real-time operation is possible, in which several million pixels in an image or video are processed each second. This condition implies that the energy of a pairwise MRF 1.

Hence the entire Ising potential function is submodular. Although recent work has been partly successful in reducing the complexity of algorithms for general submodular function minimization, they are still quite expensive computationally and cannot practically be used for large problems. We now calculate the clique energies involving the site x by expanding the conditional probability density ij and collecting the terms.

There are cliques of order one and two.

They are x2 x x ij 2? This is of utmost G X x importance since it is the joint probability distribution and not the conditional that contains the complete image representation. Before relating the conditional and joint distributions, we introduce the concept of a Gibbs distribution which will turn out to be crucial in specifying the relationship.

X Energy functions have been widely used in spin glass models of statistical physics. G Homogeneity: The recipe for obtaining the joint density function is as follows: Then, all clique energies were summed taking care to count each clique only once yielding the energy function. Our presentation has been quite terse E x and further details on cliques and the transition from the conditional to the joint probability distribution can be found in Besag, ; Geman and Geman, ; Kinderman and Snell, These models can be used in a variety of image processing and pattern recognition tasks.

A Bayesian setup consists of two ingredients the prior and the degradation model. X In edge preserving image restoration for example, includes the set of image intensities X and a further set of binary-valued edge labels. In texture segmentation, includes theX image intensities and a set of texture labels at each location. Usually, we are faced with noisy and incomplete observations.

Denote the set of observations by Y and let the degradation model also be a Gibbs-Markov distribution: D 8 y In general, the partition function Z x is a function of the image attributes x. E x; y is D D the energy function corresponding to the degradation model. This type of degradation model routinely occurs in image restoration and in tomographic reconstruction. Rangarajan and Chellappa: ED x; y?

## Markov Random Fields for Vision and Image Processing

In the case of the MAP estimate, the entire Bayesian estimation engine reduces to minimizing just this pos- terior energy function E x since the partition function of the posterior is independent of x. However, when the MMSE estimate is desired, the expected value of X in the posterior distribution needs to be computed.

This computation is usually intractable since it involves computing the partition function of the posterior distribution. For example, in edge preserving image restoration Geman and Geman, the process X includes both continuous image intensities and binary-valued edge variables.

Deterministic Anneal- ing DA is a general method that has emerged recently. Note that the partition function is now a function of the inverse temperature. The terminology is inherited from statistical physics.

## Markov random field

In this manner, the posterior energy function is increasingly, closely approximated by a sequence of smooth, continuous energy functions. The main reason for doing this is based on the following statistical mechanics identity: Also, the expected value of the posterior energy goes to the minimum value of the energy. The key idea in deterministic annealing is to minimize the free energy F instead of E x while reducing the temperature to zero.

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However, the free energy involves the logarithm of the partition function which is intractable! While details are beyond the scope of this presentation the reader is referred to Geiger and Girosi, ; Lee et al.

Let the energy function contain only binary-valued variables and take the following form: The free energy consists of two terms. The third term in 15 is an ap- i j proximation to the entropy.

In this manner, a deterministic relaxation network is obtained. There are ques- tions regarding the choice of annealing schedules and the quality of the minima obtained, etc.

Parameter estimation So far, we have concentrated on estimating X given the noisy observations Y. This is the general form of the Gauss-Markov model. The model is a generalization of our earlier model 6 since it has the same clique form, albeit with a more general neighborhood structure.

## Markov Random Fields for Vision and Image Processing

The parameters can be estimated by maximizing the joint probability of X w. In most cases, this computation is intractable in its pure form and approximations have to be devised.

The pseudo-likelihood takes advantage of the local conditional probability structure of MRFs.

The parameters are now estimated by maximizing the product of the conditional distributions at each site s w. The availabilty of a suitable training set is critical to both likelihood and pseudo-likelihood parameter estimation.

When a training set is not available, parameter es- timation and cost minimization proceed in lockstep. Our exposition has been brief and we have ignored important issues like validation, choice of the order of MRF models and size of training sets.

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MRF models being parametric, introduce a certain kind of bias into the image representation. This seems to be the right kind of bias in terms of reducing variance for tasks like image restoration, tomographic reconstruction and texture segmentation.

However, if the order of the chosen model is incorrect, high bias could result. Also, there are interesting similarities between Gauss-Markov models and thin-plate splines Wahba, Finally, there are interesting relationships between MRFs and recurrent neural networks at both computational and algorithmic levels Rangarajan et al.This computation is usually intractable since it involves computing the partition function of the posterior distribution.

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Related Papers. Visual reconstruction is therefore often cast in terms of continuous variables [, 54], 1. First, the joint distribution of the MRF is obtained.

In this manner, the posterior energy function is increasingly, closely approximated by a sequence of smooth, continuous energy functions.

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