Stochastic Geometry for Image Analysis

Chapter 4

Simulation and Optimization 1

In this chapter, we tackle the subject of marked point process simulation. This chapter is important because the relevance of a model, in image analysis, strongly depends on how effectively it can be simulated. Moreover, for most image analysis problems, the most probable configuration is being sought. Thus, we have an optimization problem. To solve such a problem, we use a simulated annealing method, which consists of simulating the density , where T is a parameter known as the temperature, which decreases with time. Point process models are a special case of optimization in the sense that the number of variables to be estimated is itself random. During simulation, it is therefore necessary to be able to realize dimensional jumps, which means moving from one configuration to another while changing the dimension of the configuration space. Put more simply, it must be possible to add or to remove objects to or from the current configuration.

In this chapter, we show how traditional sampling approaches can be generalized to apply to the case of marked point processes. We distinguish discrete approaches, which are founded on the theory of Markov chains, and continuous approaches, which are founded on stochastic differential equations. We finish by studying the different approaches to decreasing the temperature in the simulated annealing framework. ...