Mathematical Foundations of Image Processing and Analysis, Volume 1

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 ...