Definability and stability of multiscale decompositions for manifold-valued data

TL;DR

This paper investigates multiscale representations for manifold-valued data, focusing on stable constructions based on interpolating upscaling operators, and discusses their definability and stability properties.

Contribution

It introduces a stability analysis for definable multiscale decompositions using interpolating upscaling operators for manifold-valued data.

Findings

Stability results for definable multiscale decompositions

Focus on interpolating and midpoint-interpolating upscaling operators

Insights into limitations of manifold-analogues of biorthogonal wavelets

Abstract

We discuss multiscale representations of discrete manifold-valued data. As it turns out that we cannot expect general manifold-analogues of biorthogonal wavelets to possess perfect reconstruction, we focus our attention on those constructions which are based on upscaling operators which are either interpolating or midpoint-interpolating. For definable multiscale decompositions we obtain a stability result.