Time Warping and Interpolation Operators for Piecewise Smooth Maps

A warping operator consists of an invertible axisdeformation applied either in the signal domain or in thecorresponding Fourier domain. Additionally, a warping trans-formation is usually required to preserve the signal energy, thuspreserving orthogonality and being invertible by its adjoint.Initially, the design of such operators has been motivated bythe idea of suitably generalizing the properties of orthogonaltime-frequency decompositions such as wavelets and filter banks,hence the energy preservation property was essential. Recently,warping operators have been employed for frequency dispersioncompensation in the Fourier domain or the identification ofwaveforms similarity in the time domain. For such applica-tions, the energy preservation requirement can be given up,thus making warping a special case of interpolation. In thiscontext, the purpose of this work is to provide analytical modelsand efficient computational algorithms for time warping withrespect to piecewise smooth warping maps by transposing andextending a theoretical framework which has been previouslyintroduced for frequency warping. Moreover, the same approachis generalized to the case of warping without energy preservation,thus obtaining a fast interpolation operator with analyticallydefined and fast inverse operator.