Mathematical modeling of real-world images
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.
AMS classificationPrimary 41A46, 68U10 Secondary 41A99, 54D35
Key words and phrasesHausdorff distance Hausdorff continuity Completed graph Pixel function Image compression Fractal transform operator
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