Stochastic geometry

In mathematics, stochastic geometry is the study of random spatial patterns. At the heart of the subject lies the study of random point patterns. This leads to the theory of spatial point processes, hence notions of Palm conditioning, which extend to the more abstract setting of random measures.

Models

There are various models for point processes, typically based on but going beyond the classic homogeneous Poisson point process (the basic model for complete spatial randomness) to find expressive models which allow effective statistical methods.

The point pattern theory provides a major building block for generation of random object processes, allowing construction of elaborate random spatial patterns. The simplest version, the Boolean model, places a random compact object at each point of a Poisson point process. More complex versions allow interactions based in various ways on the geometry of objects. Different directions of application include: the production of models for random images either as set-union of objects, or as patterns of overlapping objects; also the generation of geometrically inspired models for the underlying point process

(for example, the point pattern distribution may be biased by an exponential factor involving the area of the union of the objects; this is related to the Widom–Rowlinson model of statistical mechanics).

Random object

What is meant by a random object? A complete answer to this question requires the theory of random closed sets, which makes contact with advanced concepts from measure theory.