Stochastic Geometry for Image Analysis

Chapter 5

Parametric Inference for Marked Point Processes in Image Analysis 1

5.1. Introduction

We observe a digital image d and we would like to detect, recognize, and characterize the objects that it contains.

This problem is very simple when the image contains only a single point. In reality, we come across images which instead contain a single object. In that situation, such an object, yi = (ki, mi), is described by its position in the image, ki, and its mark, mi. The mark contains information that relates to the geometric shape, the color, or even the texture of the object. Much more realistic images are obtained by considering configurations that contain a finite number of such objects. The most complex shapes can be obtained by starting from simple objects. In this spirit, an agricultural landscape, as seen by a satellite, can be approximated by a random configuration of polygons that form a mosaic. A road network can be seen as a set of small segments that are connected and aligned. A set of particles in physics can be modeled as hard spheres, which cannot intersect.

One of the most suitable mathematical frameworks for handling these sets of objects is the theory of marked point processes. An image, or a configuration of objects as previously described, corresponds to a realization, y = {y1y2,…,yn}, of a random process which is called a marked point process.

This process is characterized by its probability density Pd(y|θ), where θ is the vector of parameters. These parameters ...