Mathematical modeling of real-world images
type: Item
instance of: scholarly article
title: Mathematical modeling of real-world images
zbMATH Open document ID: 0853.41012
DOI: 10.1007/BF02432854
published in: Constructive Approximation
publication date: 1996-09-12T00:00:00Z
review text: 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).
Mathematics Subject Classification ID: 41A46
Mathematics Subject Classification ID: 68U10
Mathematics Subject Classification ID: 41A99
Mathematics Subject Classification ID: 54D35
zbMATH DE Number: 895636
zbMATH Keywords: Hausdorff distance
zbMATH Keywords: fractal geometry
author: Blagovest Sendov
reviewed by: Alessandra Iozzi
MaRDI profile type: Publication
cites work: Q4040874
cites work: Q5611294
cites work: Q3731958
cites work: A limit theorem for Markov chains in compact metric spaces with applications to products of random matrices
cites work: Über die beste Annäherung von Funktionen einer gegebenen Funktionenklasse
cites work: Q3231506
cites work: Q4209136
cites work: Q3999282
cites work: Q4324443
description: scientific article; zbMATH DE number 895636