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This module gives an overview of wavelets and their usefulness as a basis in image processing. In particular we look at the properties of the Haar wavelet basis.

Introduction

Fourier series is a useful orthonormal representation on L 2 0 T especiallly for inputs into LTI systems. However, it is ill suited for some applications, i.e. image processing (recall Gibb's phenomena ).

Wavelets , discovered in the last 15 years, are another kind of basis for L 2 0 T and have many nice properties.

Basis comparisons

Fourier series - c n give frequency information. Basis functions last the entire interval.

Fourier basis functions

Wavelets - basis functions give frequency info but are local in time.

Wavelet basis functions

In Fourier basis, the basis functions are harmonic multiples of ω 0 t

basis 1 T ω 0 n t

In Haar wavelet basis , the basis functions are scaled and translated versions of a "mother wavelet" ψ t .

Basis functions ψ j , k t are indexed by a scale j and a shift k.

Let 0 t T φ t 1 Then φ t 2 j 2 ψ 2 j t k j k 0 , 1 , 2 , , 2 j - 1 φ t 2 j 2 ψ 2 j t k

ψ t 1 0 t T 2 -1 0 T 2 T

Let ψ j , k t 2 j 2 ψ 2 j t k

Larger j → "skinnier" basis function, j 0 1 2 , 2 j shifts at each scale: k 0 , 1 , , 2 j - 1

Check: each ψ j , k t has unit energy

t ψ j , k t 2 1 ψ j , k ( t ) 2 1

Any two basis functions are orthogonal.

Same scale
Different scale
Integral of product = 0

Also, ψ j , k φ span L 2 0 T

Haar wavelet transform

Using what we know about Hilbert spaces : For any f t L 2 0 T , we can write

Synthesis

f t j j k k w j , k ψ j , k t c 0 φ t

Analysis

w j , k t 0 T f t ψ j , k t
c 0 t 0 T f t φ t
the w j , k are real
The Haar transform is super useful especially in image compression

Haar wavelet demonstration

HaarDemo
Interact (when online) with a Mathematica CDF demonstrating the Haar Wavelet as an Orthonormal Basis.

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Source:  OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
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