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# 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.

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 ...

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