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# 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.

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] ...

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