# 15

# The Differential Functional Framework

The differential functional framework focuses on the study of the global or local variations of gray-tone images.

# 15.1. Paradigms

In the **differential functional framework**, a gray-tone image *f* is considered to be a *k*-times (*k* ∈ _{0}) differentiable gray-tone function, whose local spatial ‘patterns’ in terms of local variations will be studied using its derivatives of order 1 (i.e. *f*^{(1)}), … order 2 (i.e. *f*^{(2)}), … order *k* (i.e. *f*^{(k)}) and so on. The more differentiable this gray-tone function, the more it will be called regular (or roughly speaking, smooth).

# 15.2. Mathematical concepts and structures

## 15.2.1. *Mathematical disciplines*

The mathematical discipline of reference is *Differential Calculus* [KOL 99; Original ed., 1957 and 1961] [CAR 83; 1st ed., 1971], which focuses on the study of the variations of a quantity (here, a gray-tone function).

## 15.2.2. *Differentiable gray-tone functions and partial derivatives*

A gray-tone function *f* defined on a non-empty open set *S* in ^{n} is said to be *Fréchet k-times differentiable* (respectively, Fréchet *k*-times continuously differentiable) at a pixel *x* in an open set *S* in ^{n} [FRÉ 11] if all its partial derivatives ...