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Mathematical Foundations of Image Processing and Analysis, Volume 1 by Jean-Charles Jean-Charles Pinoli Pinoli

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17

The Frequential Functional Framework

In the frequential functional framework, a gray-tone image is no longer considered as constituted of pixels, but composed of different mixed periodic spatial components.

17.1. Paradigms

In the frequency functional framework, a gray-tone image is considered as an integrable function or square-integrable gray-tone function, apprehended in terms of periodic spatial components (i.e. spatial frequencies). It is studied by expansion on a basis of periodic elementary gray-tone functions, like sinusoids (i.e. sinewaves), in order to determine their relative contributions to the information content of the gray-tone image.

17.2. Mathematical concepts and structures

17.2.1. Mathematical disciplines

The mathematical discipline of reference is Functional Analysis [KOL 99; Original ed., 1954 and 1957] [KAN 82] [KRE 89; 1st ed., 1978], with an important place held by Integral Calculus [BOU 04a; Original ed., 1959-65-67] [BOU 04b; Original ed., 1963–69]. The basic idea is to proceed with a change of representation domain through an ad hoc transition by passing from the spatial domain to the frequency domain (see section 5.4). Gray-tone functions will thus be studied in terms of spatial frequencies instead of spatial locations.

17.2.2. Fourier transformation of integrable gray-tone functions

The most classical approach is based on the Fourier transformation [FOU 95; Original ed., 1822], denoted , which is an operator that makes the transition from the spatial ...

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