Abstract and Applied Analysis
Volume 2004 (2004), Issue 10, Pages 831-880
doi:10.1155/S1085337504401043
Geometric data fitting
Instituto de Matemáticas, Universidad Nacional Autónoma de México, A.P. 273, Admon. de Correos #3, Cuernavaca C.P. 62251, Morelos, Mexico
Received 21 April 2003
Copyright © 2004 José L. Martínez-Morales. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Given a dense set of points lying on or near an embedded
submanifold M0⊂ℝn of Euclidean
space, the manifold fitting problem is to find an
embedding F:M→ℝn that approximates
M0 in the sense of least squares. When the dataset is modeled
by a probability distribution, the fitting problem reduces to
that of finding an embedding that minimizes Ed[F], the
expected square of the distance from a point in ℝn
to F(M). It is shown that this approach to the fitting problem
is guaranteed to fail because the functional Ed has no local
minima. This problem is addressed by adding a small multiple k
of the harmonic energy functional to the expected square of the
distance. Techniques from the calculus of variations are then
used to study this modified functional.