Definitions and Notations 1/3

Marked point processes

Let $S$ be the space of interest, called the state space, typically a subset of $\mathbb{R}^{n}$. A configuration of objects in $S$ is an unordered list of objects. A point process $\mathit{X}\;$in $S$ is a random variable whose realizations are random configurations of points.

The most obvious example of point processes is the homogeneous Poisson process (cf Fig. (2)), which induces a complete spatial randomness on $S$, given the fact that the positions are uniformly and independently distributed.

Figure 2 : Realizations of an homogeneous discs Poisson process of mean 100 (click to enlarge).

To apply point processes to object extractions in images, the idea is to model the observed data $\mathcal{I}$(cf Fig. (1)) as a realization of a marked point process of simple geometric objects. The space of the positions $\mathcal{P}$ is given by the image size, while the space of the marks $\mathcal{K}$ is a compact set of $\mathbb{R}^{d}$.