The Discrete Periodic Wavelet Transform

In the papers below the discrete wavelet transform (DWT) is extended to functions on the discrete circle to create a fast and complete discrete periodic wavelet transform (DPWT). It is proven that the same filter coefficients used with the DWT may be used with the DPWT to create an orthonormal basis of discrete periodic wavelets. Unlike the DWT, the DPWT is perfectly invertible when applied to sequences of finite length.

The illustration shows examples of elements of a periodic wavelet basis for length 128 sequences. This basis set is based on Daubechies filters of length 4. The sequences have been normalized to have peak magnitudes of 1. The outer dotted circle corresponds to a sample value of 1 and the inner to a sample value of -1. The intermediate solid circle corresponds to 0.

Papers

Neil H. Getz, "A Fast Discrete Periodic Wavelet Transform", Memorandum UCB/ERL M92-138, Electronic Research Laboratory Berkeley, California 22 December 1992.

Neil H. Getz, "A Perfectly Invertible, Fast, and Complete Wavelet Transform for Finite Length Sequences: The Discrete Periodic Wavelet Transform", Mathematical Imaging: Wavelet Applications in Signal and Image Processing Proceedings of the SPIE - The International Society for Optical Engineering vol.2034:332-48 July 1993 San Diego, July 1993.

This paper is a slightly abbreviated version of the Memorandum UCB/ERL M92-138 above.

Tools

Neil H. Getz, "Discrete Periodic Wavelet Transform Toolbox", University of California at Berkeley December 1992.

Abstract

This is a collection of Matlab routines designed to enable easy experimentation with the DPWT and its inverse. Some knowledge of the background contained in either of the above two publications is suggested.

Matlab files