Geometric data fitting
type: Item
instance of: scholarly article
title: Geometric data fitting
zbMATH Open document ID: 1078.41026
DOI: 10.1155/S1085337504401043
published in: Abstract and Applied Analysis
publication date: 2005-04-29T00:00:00Z
full work available at URL: https://eudml.org/doc/51936
review text: The author solves the following problem: \(M\) be a smooth, compact manifold without boundary; \(M_0\) be the image of a smooth embedding \(F_0:M\to \mathbb R^n\). Let \(Y=\{y_1,-,y_n\}\subset \mathbb R^n\) be a collection of points contained in a smooth tubular neighborhood \(\Omega\) of \(M_0\). The manifold fitting problem is to find an embedding \(F:M\to \mathbb R^n\) such that \(F(M)\) is a good approximation to \(M_0\) in the sense of least squares. It is shown that this approach to the fitting problem is guaranteed to fail because the functional of expected square of the distance from a point in \(\mathbb R^n\) to \(F(M)\) has no local minima. This problem is addressed by adding a small multiple \(k\) of the energy functional to the expected square of the distance. The calculus of variations is used for the study of the problem.
reviewed by: Josef Kofroň
Mathematics Subject Classification ID: 41A63
zbMATH DE Number: 2163606
zbMATH Keywords: manifold data fitting
zbMATH Keywords: Euclidean space
author: J. L. Martínez Morales
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 2163606