Orthonormal shift-invariant wavelet packet decomposition and representation

Abstract

In this work, a shifted wavelet packet (SWP) library, containing all the time shifted wavelet packet bases, is defined. A corresponding shift-invariant wavelet packet decomposition (SIWPD) search algorithm for a ‘best basis’ is introduced. The search algorithm is representable by a binary tree, in which a node symbolizes an appropriate subspace of the original signal. We prove that the resultant ‘best basis’ is orthonormal and the associated expansion, characterized by the lowest information cost, is shift-invariant. The shift invariance stems from an additional degree of freedom, generated at the decomposition stage and incorporated into the search algorithm. The added dimension is a relative shift between a given parent node and its respective children nodes. We prove that for any subspace it suffices to consider one of two alternative decompositions, made feasible by the SWP library. These decompositions correspond to a zero shift and a 2 −ℓ relative shift where ℓ denotes the resolution level. The optimal relative shifts, which minimize the information cost, are estimated using finite depth subtrees. By adjusting their depth, the quadratic computational complexity associated with SIWPD may be controlled at the expense of the attained information cost down to O ( Nlog 2 N).