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# The Tonal Domain

Before processing and analyzing gray-tone images, it should be necessary to mathematically structure the tonal domain, that is to say, roughly speaking, the space of gray tones, or for short the gray-tone space. The purpose of this chapter is to present the first mathematical frameworks associated with the tonal domain.

The gray tones are considered as algebraic and ordered values, i.e. the addition of two gray tones, the multiplication of gray tone by a real (or complex) number and the comparison of two gray tones (with the classic order relation on or , or even ) are defined within the tonal domain.

# 10.2. Mathematical concepts and structures

## 10.2.1. Mathematical disciplines

The mathematical discipline of reference is Algebra [LAN 04; 1st ed., 1966] [STR 05; 1st ed., 1976] that deals with the operations and relations, rules, properties and associated concepts. The vector structure is the most classical to perform combinations of vectors with the two basic operations of addition and scalar multiplication.

## 10.2.2. Gray tones

The range of gray tones, ...

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