Time Warping and Interpolation Operators for Piecewise Smooth Maps

TL;DR

This paper develops analytical models and algorithms for time warping and interpolation of signals using piecewise smooth maps, extending previous frequency warping frameworks to non-energy-preserving cases.

Contribution

It introduces a unified approach for time warping and interpolation with efficient algorithms, generalizing existing models to non-energy-preserving transformations.

Findings

Provides analytical models for time warping with piecewise smooth maps.

Develops fast algorithms for inverse warping and interpolation.

Extends theoretical framework from frequency to time domain warping.

Abstract

A warping operator consists of an invertible axis deformation applied either in the signal domain or in the corresponding Fourier domain. Additionally, a warping transformation is usually required to preserve the signal energy, thus preserving orthogonality and being invertible by its adjoint. Initially, the design of such operators has been motivated by the idea of suitably generalizing the properties of orthogonal time-frequency decompositions such as wavelets and filter banks, hence the energy preservation property was essential. Recently, warping operators have been employed for frequency dispersion compensation in the Fourier domain or the identification of waveforms similarity in the time domain. For such applications, the energy preservation requirement can be given up, thus making warping a special case of interpolation. In this context, the purpose of this work is to provide analytical models and efficient computational algorithms for time warping with respect to piecewise smooth warping maps by transposing and extending a theoretical framework which has been previously introduced for frequency warping. Moreover, the same approach is generalized to the case of warping without energy preservation, thus obtaining a fast interpolation operator with analytically defined and fast inverse operator.