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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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s. Reply
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
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Cied
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Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
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Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
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Yasmin
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Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
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what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
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
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
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Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
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.
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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, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
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