Geometric data fitting

José L Martínez-Morales

doi:10.1155/s1085337504401043

José L Martínez-Morales

Open Access

https://doi.org/10.1155/s1085337504401043

Copy DOI

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.

Highlights

In this paper we are concerned with the following problem
The manifold fitting problem is to find an embedding F : M → Rn such that F(M) is a good approximation to M0 in the sense of least squares. This situation occurs in a variety of contexts such as medical imaging [2, 14, 16, 17, 18], geography [1], computer graphics and vision [5, 6, 9, 10, 15, 19, 22], and mechanical engineering [12, 13, 20]
We show that the distance functional does not have local minima within the class of Ω-regular embeddings

Summary

Introduction

In this paper we are concerned with the following problem. Let M0 be the image of a smooth embedding F0 : M → Rn, where M is a smooth, compact manifold without boundary, and let Y = {y1, . . . , yq} ⊂ Rn be a collection of points that we assume to be contained in a smooth tubular neighborhood Ω of M0. Where the projection index λF : Rn → M is the map that assigns to each point x ∈ Rn a point in M such that F(λF(x)) realizes the distance from x to F(M) This reduces the fitting problem to the problem of finding local minima of this functional. This generalizes the results of Duchamp and Stuetzle on critical curves [4].

The geometry of regular embeddings

Extremal properties of Ed

Extremal properties of the energy functional

Conclusions