Mathematical modeling of real-world images
In ``Mathematical modeling of real-world images'' the author considers the problem of approximating images using appropriate mathematical objects. If \(D = [0,1]^2\), the author defines the metric space \(IM_D\) of completed graphs of bounded functions (as closed bounded subsets of \(\mathbb{R}^3)\) with the Hausdorff distance. Properties of \(IM_D\) are studied with an eye toward two particular directions. On the one hand it is shown that every function \(f \in IM_D\) can be approximated arbitrarily well by a pixel function. On the other hand, methods of approximation with the purpose of image compression are considered. Here the goal is to maximize the data computed, while maintaining the quality of the reproduction of the image. Drawing from fractal geometry, the authors asserts that the Hausdorff metric is the natural tool to measure continuity in this context and lays the foundations for a treatment of the mathematical modeling of images (which by their nature are Hausdorff continuous but not continuous in the classical sense).
scientific article; zbMATH DE number 895636
scientific article