Mathematical Foundations of Image Processing and Analysis, Volume 1

16

The Generalized Functional Framework

In the generalized functional framework, the classical notion of derivation is enlarged in order to give a meaning to the derivation of gray-tone functions with possible discontinuities. The concept of generalized gray-tone function will be widely used to formulate generalized solutions of partial differential equations (PDEs).

16.1. Paradigms

In the generalized functional framework, a gray-tone image f is no longer considered as a gray-tone function, but as a linear functional on the space of +∞-times continuously differentiable gray-tone functions with compact supports in ⁿ, which will be named a generalized gray-tone function. Although a generalized gray-tone function may not look like any ordinary gray-tone function, this notion offers a great flexibility, and allows us to model a large number of encountered real physical situations [CHA05b; p. 93]. However, generalized gray-tone functions are so general that further regularity conditions are needed.

16.2. Mathematical concepts and structures

16.2.1. Mathematical disciplines

The mathematical disciplines of reference are Differential Calculus [KOL 99; Original ed., 1957 and 1961] [CAR 83; 1st ed., 1971], Integral Calculus [BOU 04a; Original ed., 1959-65-67] [BOU 04b; Original ed., 1963-69] and Functional Analysis [RUD 91; 1st ed., 1973] (see Chapters 15 and 13).

The Theory of Generalized ...