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.
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Communicated by Albert Cohen
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Sendov, B. Mathematical modeling of real-world images. Constr. Approx 12, 31–65 (1996). https://doi.org/10.1007/BF02432854
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DOI: https://doi.org/10.1007/BF02432854