# 20

# The Probabilistic Functional Framework

The extra dimension compared to the previous functional frameworks is the probabilistic dimension, which will allow us to consider a gray-tone image as the realization of a random process.

# 20.1. Paradigms

In the *probabilistic functional framework*, an image is represented by a gray-tone function considered as a *random function*, whose spatial behavior is studied in terms of occurrences, local dependencies or/and local arrangements. The statistical properties of the images are mainly expressed by: (1) the distribution of the gray- tones, (2) the correlation between the gray-tones of spatially close pixels, and (3) the frequency of occurrence of certain spatial structures. Statistical measurements provide data (numbers and functions) on which probabilistic models can be built.

# 20.2. Mathematical concepts and structures

## 20.2.1. *Mathematical disciplines*

The mathematical discipline of reference is the *Calculus of Probability*, or more modernly called the *Probability Theory* [KOL 00; Original ed., 1933] [BIL 12; 1st ed., 1979], which focuses on the study of random realizations of a probabilistic quantity; here a random gray-tone, or a random gray-tone function.

## 20.2.2. *Random gray-tones*

A gray-tone random variable, or simply a *random gray-tone* or even a -valued random variable, denoted as , is a random variable defined on a set Ω of events, denoted ...