18 The Multiscale Functional Framework

The extra dimension, compared to the previous functional frameworks, is the scale dimension that will allow us to consider a gray-tone image from a multiscale viewpoint.

18.1. Paradigms

In the multiscale functional framework, a gray-tone image is represented by an integrable or a square-integrable gray-tone function f, which will be analyzed at different spatial scales. Roughly speaking, the sinusoids employed in the frequential functional framework (see Chapter 17) will be replaced by damped oscillating functions called wavelets that will allow us to localize the spatial analysis of a gray-tone function, in contrast to the Fourier analysis.

18.2. Mathematical concepts and structures

18.2.1. Mathematical disciplines

The main mathematical discipline of reference is Functional Analysis [KOL 99; Original ed., 1954 and 1957] [KAN 82] [KRE 89], with an important place held by Integral Calculus or modern Integration Theory [BOU 04a; Original ed., 1959-65-67] [BOU 04b; Original ed., 1963–69]. The basic idea is to proceed with a change of representation domain through an ad hoc transition (see section 5.4).

The other useful mathematical disciplines are Algebra [LAN 04; 1st ed., 1971] [STR 05; 1st ed., 1976], Differential calculus [KOL 99; Original ed., 1957 and 1961] [CAR 83; 1st ed., 1971] and the theory of generalized functions [SOB 36, SCH 51].