# 19

# The Variational Functional Framework

The variational functional framework deals with the use of “Calculus of Variations” on functionals defined on suitable classes of gray-tone functions.

# 19.1. Paradigms

In the *variational functional framework*, a gray-tone image is generally represented by a square-integrable gray-tone function. The basic idea is to consider that the resulting gray-tone image of a processing or an analysis is the solution of a *variational problem* or, in other words, that this gray-tone function minimizes a suitable functional operating in an appropriate gray-tone function space.

# 19.2. Mathematical structures

## 19.2.1. *Mathematical disciplines*

The mathematical discipline of reference is a branch of *Functional Analysis* [RUD 91; 1st ed., 1973], called *Calculus of Variations* [GEL 00, BRU 04], which deals with functionals (i.e. functions of functions in the present framework), as opposed to the calculus of functions dealing with functions. The interest is specifically about *extremal functions* that correspond to the extrema of a functional, or stationary functions, for which the rate of change of the functional is equal to, or at least close to, zero.

The other mathematical disciplines of reference are *Integral Calculus* [BOU 04a; Original ed., 1959-65-67] [BOU 04b; Original ed., 1963-69] (see Chapter 13) and *differential calculus* [KOL 99; Original ed., 1957 and 1961] [CAR 83; 1st ed., 1971] (see Chapter 15).

The *Theory of Generalized Functions* [SOB 36, SCH 51, VLA 02] ...