ISSN:
1432-0940
Keywords:
Primary 41A46, 68U10
;
Secondary 41A99, 54D35
;
Hausdorff distance
;
Hausdorff continuity
;
Completed graph
;
Pixel function
;
Image compression
;
Fractal transform operator
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract The problem of finding appropriate mathematical objects to model images is considered. Using the notion of acompleted graph of a bounded function, which is a closed and bounded point set in the three-dimensional Euclidean spaceR 3, and exploring theHausdorff distance between these point sets, a metric spaceIM D of functions is defined. The main purpose is to show that the functionsf∈IM D, defined on the squareD=[0,1]2, are appropriate mathematical models of real world images. The properties of the metric spaceIM D are studied and methods of approximation for the purpose of image compression are presented. The metric spaceIM D contains the so-calledpixel functions which are produced through digitizing images. It is proved that every functionf∈IM D may be digitized and represented by a pixel functionp n, withn pixels, in such a way that the distance betweenf andp n is no greater than 2n −1/2. It is advocated that the Hausdorff distance is the most natural one to measure the difference between two pixel representations of a given image. This gives a natural mathematical measure of the quality of the compression produced through different methods.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF02432854
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