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 ...
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