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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

Complementary angles
Idrissa Reply
Commplementary angles
Idrissa Reply
Complementary angles
Idrissa
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Sherica
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Sherica
Complementary angles
Idrissa
yes
Sherica
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
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a perfect square v²+2v+_
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Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
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or infinite solutions?
Kim
y=10×
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ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
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No. 7x -4y is simplified from 4x + (3y + 3x) -7y
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is it 3×y ?
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J, combine like terms 7x-4y
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
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Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
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AMJAD
what is system testing
AMJAD
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Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
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Prasenjit
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Damian
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Damian
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Azam
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Prasenjit Reply
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.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Intro to digital signal processing. OpenStax CNX. Jan 22, 2004 Download for free at http://cnx.org/content/col10203/1.4
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