# Mathematical modeling of real-world images

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## Abstract

The problem of finding appropriate mathematical objects to model images is considered. Using the notion of a*completed graph* of a bounded function, which is a closed and bounded point set in the three-dimensional Euclidean space*R* _{3}, and exploring the*Hausdorff distance* between these point sets, a metric space*IM* _{D} of functions is defined. The main purpose is to show that the functions*f∈IM* _{D}, defined on the square*D*=[0,1]^{2}, are appropriate mathematical models of real world images.

The properties of the metric space*IM* _{D} are studied and methods of approximation for the purpose of image compression are presented.

The metric space*IM* _{D} contains the so-called*pixel functions* which are produced through digitizing images. It is proved that every function*f∈IM* _{D} may be digitized and represented by a pixel function*p* _{n}, with*n* pixels, in such a way that the distance between*f* and*p* _{n} is no greater than 2*n* ^{−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 classification

Primary 41A46, 68U10 Secondary 41A99, 54D35### Key words and phrases

Hausdorff distance Hausdorff continuity Completed graph Pixel function Image compression Fractal transform operator## Preview

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