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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.

Questions & Answers

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algebra 2 Inequalities:If equation 2 = 0 it is an open set?
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At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
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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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