Overview

The Discrete Wavelet Transform (DWT) is used in a variety of signal processing applications, such as video compression ^[28], Internet communications compression ^[29], object recognition ^[30], and numerical analysis. It can efficiently represent some signals, especially ones that have localized changes. Consider the example of representing a unit impulse function with the Fourier transform, which needs an infinite amount of terms because we are trying to represent a single quick change with a sum of sinusoids. However, the wavelet transform can represent this short-term signal with only a few terms.

This transform came about from different fields, including mathematics, physics, and image processing. Essentially, people in different areas were doing the same thing, but using different terminology. In the late 1980s, St phane Mallat unified the work into one topic ^[1].

This transform is discrete in time and scale. In other words, the DWT coefficients may have real (floating-point) values, but the time and scale values used to index these coefficients are integers. The wavelet transform is gaining popularity with the recent JPEG-2000 standard, which incorporates multiresolution analysis. This chapter explains how the wavelet transform works, and shows how the wavelet breaks a signal down into detail signals and an approximation.

We call an octave a level of resolution, where each octave can be envisioned as a pair of FIR filters, at least for the one-dimensional case. One filter of the analysis (wavelet transform) pair is a lowpass filter (LPF), while the other is a...