11
The Algebraic and Order Functional Framework
The algebraic and order functional framework is the first to be taken into account when addressing mathematical imaging problems.
11.1. Paradigms
Within the algebraic and order functional framework, a gray-tone image is considered as a gray-tone function that will be treated as a partially ordered vector. Therefore, gray-tone images will be combined through their gray-tone functions, which will be submitted to the natural partial order relation. Roughly speaking, it will be allowed to manipulate and modify gray-tone images, as well as to combine and compare them.
11.2. Mathematical structures and notions for IPA
The mathematical disciplines of reference are the theory of sets, for short Set Theory [RUB 67, DEV 93, BOU 04c], and Algebra [LAN 04; 1st ed., 1971] [STR 05; 1st ed., 1976].
11.2.1. Gray-tone functions
A gray-tone function, denoted f, is a mapping defined on the spatial domain 
n and valued in the tonal domain 
, namely:
[11.1] 
where x denotes an arbitrary pixel in n (see Chapter 9), and f(x) is a gray-tone value belonging to ...
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