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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

anyone have book of Abdel Salam Hamdy Makhlouf book in pdf Fundamentals of Nanoparticles: Classifications, Synthesis
Naeem Reply
what happen with The nano material on The deep space.?
pedro Reply
It could change the whole space science.
puvananathan
the characteristics of nano materials can be studied by solving which equation?
sibaram Reply
plz answer fast
sibaram
synthesis of nano materials by chemical reaction taking place in aqueous solvents under high temperature and pressure is call?
sibaram
hydrothermal synthesis
ISHFAQ
how can chip be made from sand
Eke Reply
is this allso about nanoscale material
Almas
are nano particles real
Missy Reply
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
Lale Reply
no can't
Lohitha
where is the latest information on a no technology how can I find it
William
currently
William
where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
STM - Scanning Tunneling Microscope.
puvananathan
How we are making nano material?
LITNING Reply
Some times this process occur naturally. if not, nano engineers build nano materials. they have different physical and chemical properties.
puvananathan
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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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