# 14

# The Convolutional Functional Framework

In the convolutional functional framework, a third algebraic operation (after the addition and scalar multiplication, see Chapter 11, “The Algebraic and Order Functional Framework”), called the convolution, is introduced to spatially manipulate gray-tone images.

# 14.1. Paradigms

In the *convolutional functional framework*, a gray-tone image is considered as an integrable or a square-integrable gray-tone function. The specific algebraic operation that plays the pivotal role is the convolution which generalizes the basic idea of *sliding average.* This is an important special case of an integral transformation (see section 13.3).

# 14.2. Mathematical concepts and structures

## 14.2.1. *Mathematical disciplines*

The mathematical disciplines of reference are *Integral Calculus* [BOU 04a; Original ed., 1959-65-67] [BOU 04b; Original ed., 1963–69] (see Chapter 13 “The Integral Functional Framework”), *Functional Analysis* [KOL 99; Original ed., 1954 and 1957] [KAN 82] [KRE 89] and *Algebra* [LAN 04; 1st ed., 1971] [STR 05; 1st ed., 1976].

## 14.2.2. *Convolution of integrable gray-tone functions*

The *convolution* of two ^{1}(^{n}, ) gray-tone functions *f* and *g* is a third ^{1}(^{n}, ...