# 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