Mathematical Foundations of Image Processing and Analysis, Volume 1

13

The Integral Functional Framework

The integral functional framework focuses on the study of the global or local cumulative spatial behaviors of gray-tone images.

13.1. Paradigms

In the integral functional framework, a gray-tone image f is considered as an integrable gray-tone function (or/and whose pth power is integrable), whose local (respectively, regional or global) cumulative spatial behavior is studied by using its local (respectively, regional or global) integrals. An integral operator is an operator that acts through an integral, and an integral equation is an equation that involves integrals of one or several gray-tone functions.

13.2. Mathematical structures

13.2.1. Mathematical disciplines

The mathematical discipline of reference is Integral Calculus, or modern Integration Theory [BOU 04a; Original ed., 1959-65-67] [BOU 04b; Original ed., 1963–69], that focuses on the study of accumulated behaviors of a quantity (here, a gray-tone function).

The other main mathematical discipline of reference is Functional Analysis [KOL 99; Original ed., 1954 and 1957] [KAN 82, KRE 89] that deals with the study of functions in concrete or abstract forms.

13.2.2. Lebesgue–Bochner gray-tone function spaces

The considered gray-tone function spaces are the Lebesgue–Bochner gray-tone function spaces [LEB 04, BOC 33], denoted by that is to say the space of Lebesgue–Bochner measurable gray-tone ...