Rational Wavelet Frames
Overcomplete Discrete Wavelet Transforms with Rational Scaling Factors (pdf file, 312 KB)
I. Bayram and I. W. Selesnick.
Preprint, January 24, 2008
Abstract: This paper develops an overcomplete discrete wavelet transform (DWT) based on rational scaling factors. The proposed overcomplete rational DWT is implemented using self-inverting FIR filter banks, is approximately shift-invariant, and can provide a dense sampling of the time-frequency plane. A straightforward algorithm is described for the construction of minimal-length perfect reconstruction filters with a specified number of vanishing moments; whereas, in the non-redundant rational case, no such algorithm is available. The algorithm is based on matrix spectral factorization. The analysis/synthesis functions (discrete-time wavelets) can be very smooth and can be designed to closely approximate the derivatives of the Gaussian function.
Design of Orthonormal and Overcomplete Wavelet Transforms Based on Rational Sampling Factors (pdf file, 652 KB)
I. Bayram and I. W. Selesnick.
In Wavelet Applications in Industrial Processing V, Proceedings of
SPIE, volume 6763, September 11-12, 2007.
Abstract: Most wavelet transforms used in practice are based on integer sampling factors. Wavelet transforms based on rational sampling factors offer in principle the potential for time-scale signal representations having a finer frequency resolution. Previous work on rational wavelet transforms and filter banks includes filter design methods and frequency domain implementations. We present several specific examples of Daubechies-type filters for a discrete orthonormal rational wavelet transform (FIR filters having a maximum number of vanishing moments) obtained using Grobner bases. We also present the design of overcomplete rational wavelet transforms (tight frames) with FIR filters obtained using polynomial matrix spectral factorization.