Introduction

 

Probabilistic methods in image processing

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Bayesian framework, Markov Random Fields

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Random variables: pixels

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Local interaction

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Correlated noise ?

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No geometric constraints

 

Markov Object Processes

Notations :

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Image space: T

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Object space: U (object parameters)

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Objet support: R(u)

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Configuration:

$$x=\{u_1,...,u_n\},\; u_i\in U$$

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Configurations space: $\Omega$

$\Omega$ is a measurable space, with measure $\mu$, corresponding to a uniform Poisson process.

Poisson process :

$$f(x)\propto \beta ^{n(x)}$$

n(x) = number of objects of the configuration x.

Markov Object Process :

$$f(x) \propto \beta ^{n(x)} \prod_{x_i \in x}{g(x_i)} \prod_{i \sim j}{h(x_i,x_j)} ...$$

 

Markov Object Process simulation

Simulation using a Markov Chain : find a Markov chain X_t such that

$$\lim_{t \rightarrow +\infty} P(X_t\;\vert\;X_0=x) = f \qquad \forall x\in \Omega$$

$\rightarrow$ MCMC methods

 

RJMCMC algorithm

Description:

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general scheme

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every transition can be defined: at each step, a transition from the current state x to a new state y is proposed

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the transition is accepted with a probability depending on an acceptance ratio which depends on the law f

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Challenge: define ``good'' transitions

We consider a proposal density q(.,.) which can be easily simulated.

At step t, X_t=x :

1.

simulate y with density q(.,.)

2.

compute:

$$r=\frac{f(y)}{f(x)}\;\,{\bf\frac{q(y,x)}{q(x,y)}}$$

3.

with probability $\alpha=\min(1,r)$, set X_t+1=y, otherwise X_t+1=x