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The Haar wavelet transform
Kristian Sandberg
Dept. of Applied Mathematics
University of Colorado at Boulder




A fractal along with a two level Haar (standard) decomposition. Image source: Carlson's Fractal Gallery.




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1 Goal
The goals with this lab are:

1.
Understand the properties of the Haar basis.
2.
Understand the averaging and differencing technique and how it relates to a wavelet decomposition.
3.
Use the Haar transform to compress and de-noise one dimensional signals and images.

2 Introduction
This lab will illustrate some fundamental concepts about wavelets and multiresolution theory. Most examples and definitions will be made with respect to the Haar scaling function and the Haar wavelet. However, the concepts covered in this lab are true for more ``advanced'' wavelets such as the Daubechies wavelets.

3 The Haar basis


Definition 1 (The Haar scaling function) Let be defined by


(1)




Define as



(2)




Define the vector space Vj as



(3)




where denotes the linear span.
The index j refers to dilation and i refers to translation.


Warning! This definition may be different in different texts! Sometimes one defines the scaling function for negative integers as well and uses a different sign convention. Whenever you use a text on wavelets, make sure you know what sign convention is being used.

The constant is chosen such that . If one considers the scaling function on other intervals...