Linear multiscale transforms based on even-reversible subdivision operators

TL;DR

This paper introduces a new class of linear multiscale transforms based on even-reversible subdivision operators, extending multiscale analysis to non-interpolatory cases with proven invertibility and stability properties.

Contribution

It defines and analyzes even-reversible subdivision operators for multiscale transforms, including explicit formulas for spline-based cases and proofs of invertibility using Wiener's lemma.

Findings

Even-reversible subdivision operators include spline and pseudo-spline masks.

Explicit formulas for inverse operators are derived for quadratic and cubic splines.

Multiscale transforms exhibit stability and controlled detail decay.

Abstract

Multiscale transforms for real-valued data, based on interpolatory subdivision operators have been studied in recent year. They are easy to define, and can be extended to other types of data, for example to manifold-valued data. In this paper we define linear multiscale transforms, based on certain linear, non-interpolatory subdivision operators, termed "even-reversible". For such operators, we prove, using Wiener's lemma, the existence of an inverse to the linear operator defined by the even part of the subdivision mask, and termed it "even-inverse". We show that the non-interpolatory subdivision operators, with spline or with pseudo-spline masks, are even-reversible, and derive explicitly, for the quadratic and cubic spline subdivision operators, the symbols of the corresponding even-inverse operators. We also analyze properties of the multiscale transforms based on even-reversible subdivision operators, in particular, their stability and the rate of decay of the details.