On Smooth Whitney Extensions of almost isometries with small distortion, Interpolation and Alignment in $\Bbb R^D$-Part 1

TL;DR

This paper investigates conditions under which a small distortion map defined on a finite subset of Euclidean space can be extended smoothly to the entire space, with applications to data interpolation and alignment.

Contribution

It provides new criteria and methods for extending small distortion maps from finite sets to smooth global maps in Euclidean spaces.

Findings

Established conditions for smooth extension of small distortions

Developed approximation techniques for rigid and non-rigid motions

Applied results to data interpolation and alignment problems

Abstract

In this paper, we study the following problem: Let $D\geq 2$ and let $E\subset \mathbb R^D$ be finite satisfying certain conditions. Suppose that we are given a map $\phi:E\to \mathbb R^D$ with $\phi$ a small distortion on $E$. How can one decide whether $\phi$ extends to a smooth small distortion $\Phi:\mathbb R^D\to \mathbb R^D$ which agrees with $\phi$ on $E$. We also ask how to decide if in addition $\Phi$ can be approximated well by certain rigid and non-rigid motions from $\mathbb R^D\to \mathbb R^D$. Since $E$ is a finite set, this question is basic to interpolation and alignment of data in $\mathbb R^D$. The work in this paper appears in the research memoir [10].