Overcomplete Discrete Wavelet Transforms With Rational Dilation Factors

• Published in: IEEE transactions on signal processing 2009-01, Vol.57 (1), p.131-145
• Main Authors: Bayram, I., Selesnick, I.W.
• Format: Article
• Language: English
• Subjects: Algorithms; Applied sciences; Approximation; Derivatives; Detection, estimation, filtering, equalization, prediction; Dilation; Discrete Wavelet Transform; Discrete wavelet transforms; Exact sciences and technology; Factorization; Filter bank; Finite impulse response filter; frame; Gaussian; Information, signal and communications theory; matrix spectral factorization; Miscellaneous; Multiresolution analysis; rational dilation factor; Sampling methods; Sampling, quantization; Signal and communications theory; Signal processing; Signal processing algorithms; Signal resolution; Signal synthesis; Signal, noise; Spectra; Telecommunications and information theory; Time frequency analysis; Wavelet analysis; wavelet transforms
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2008.2007097

This paper develops an overcomplete discrete wavelet transform (DWT) based on rational dilation factors for discrete-time signals. 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 nonredundant 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.