Wavelet Transforms with a Rational Dilation Factor
Ilker Bayram, Electrical and Computer Engineering Department Polytechnic Institute of NYU Brooklyn, New York (now with BIG)

Seminar • 22 June 2009 • BM 1.111

Abstract
The dyadic wavelet transform is an effective tool for processing piecewise smooth signals; however, its poor frequency resolution (its low Q-factor) limits its effectiveness for processing oscillatory signals like speech, music, EEG, and vibration measurements, etc. In this talk, I will describe a more flexible family of discrete-time wavelet transforms (i.e. iterated filter banks) for which the frequency resolution can be varied. The new wavelet transform can attain higher Q-factors (desirable for processing oscillatory signals) or the same low Q-factor of the dyadic wavelet transform. The new wavelet transform is modestly overcomplete and based on rational dilations. Like the dyadic wavelet transform, it is an easily invertible 'constant-Q' discrete transform implemented using iterated filter banks and can likewise be associated with a wavelet frame for L2(R). I will also briefly talk (as time permits) about the problems I have worked on in the past, involving (i) a 'packet' extension of the dual-tree complex wavelet transform, (ii) stability of (the frame bounds of) iterated filter banks and (iii) analysis prior (relevant for wavelet, TV, etc.) regularized inverse problems.