Mathematical Foundations of Image Processing and Analysis, Volume 1

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, ...