June 2014
Intermediate to advanced
464 pages
11h 15m
English
book
Mathematical Foundations of Image Processing and Analysis, Volume 1
by Jean-Charles Jean-Charles Pinoli Pinoli
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12
The Morphological Functional Framework
In the morphological functional framework gray-tone images are studied by means of their local maxima and minima.
12.1. Paradigms
Within the morphological functional framework, a gray-tone image is considered as a simple gray-tone function, for which the local behavior is studied by using its local maxima and minima, and their combinations.
12.2. Mathematical concepts and structures
12.2.1. Mathematical disciplines
The mathematical disciplines of reference are Set Theory [RUB 67, DEV 93, BOU 04c], Order Theory [LUX 71], [DAV 02; 1st ed., 1990] [SCH 03], Algebra [LAN 04; 1st ed., 1971] [STR 05; 1st ed., 1976] and Topology [KEL 75, JÄN 84].
12.2.2. Local maxima and minima of a gray-tone function
A pixel x in the domain of definition S in 
n of a gray-tone function f is a (strict) local maximum (respectively, (a strict) local minimum) of f if there exists a non-empty neighborhood U (x) in S with the property that:
[12.1a] 
[12.1b] 
12.2.3. Semi-continuity of extended gray-tone functions
Semi-continuity is a property of (extended) gray-tone functions (see section 10.2.3.3) , which is weaker than continuity. An extended gray-tone function f
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