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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:

\begin{displaymath}x=\{u_1,...,u_n\},\; u_i\in U\end{displaymath}

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Configurations space: $\Omega$

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

Poisson process :


\begin{displaymath}f(x)\propto \beta ^{n(x)}\end{displaymath}

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

Markov Object Process :


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


 
Markov Object Process simulation



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


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


$\rightarrow$ 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, Xt=x :

1.
simulate y with density q(.,.)
2.
compute:

\begin{displaymath}r=\frac{f(y)}{f(x)}\;\,{\bf\frac{q(y,x)}{q(x,y)}}\end{displaymath}

3.
with probability $\alpha=\min(1,r)$, set Xt+1=y, otherwise Xt+1=x


next up previous
Next: Road network extraction Up: Roads Extraction using a Previous: Roads Extraction using a
Radu Stoica
2000-04-17