15.10 Haar wavelet basis

 Page 1 / 1
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 ${L}^{2}(\left[0 , T\right])$ 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 ${L}^{2}(\left[0 , T\right])$ and have many nice properties.

Basis comparisons

Fourier series - ${c}_{n}$ give frequency information. Basis functions last the entire interval.

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

In Fourier basis, the basis functions are harmonic multiples of $e^{i{\omega }_{0}t}$

In Haar wavelet basis , the basis functions are scaled and translated versions of a "mother wavelet" $\psi (t)$ .

Basis functions $\{{\psi }_{j,k}(t)\}$ are indexed by a scale j and a shift k.

Let $\forall , 0\le t< T\colon \phi (t)=1$ Then $\{\phi (t)2^{\left(\frac{j}{2}\right)}\psi (2^{j}t-k)\colon j\in ℤ\land (k=0,1,2,\dots ,{2}^{j}-1)\}$

$\psi (t)=\begin{cases}1 & \text{if 0\le t< \frac{T}{2}}\\ -1 & \text{if 0\le \frac{T}{2}< T}\end{cases}$

Let ${\psi }_{j,k}(t)=2^{\left(\frac{j}{2}\right)}\psi (2^{j}t-k)$

Larger $j$ → "skinnier" basis function, $j=\{0, 1, 2, \dots \}$ , $2^{j}$ shifts at each scale: $k=0,1,\dots ,{2}^{j}-1$

Check: each ${\psi }_{j,k}(t)$ has unit energy

$(\int {\psi }_{j,k}(t)^{2}\,d t=1)\implies ({\parallel {\psi }_{j,k}\left(t\right)\parallel }_{2}=1)$

Any two basis functions are orthogonal.

Also, $\{{\psi }_{j,k}, \phi \}$ span ${L}^{2}(\left[0 , T\right])$

Haar wavelet transform

Using what we know about Hilbert spaces : For any $f(t)\in {L}^{2}(\left[0 , T\right])$ , we can write

Synthesis

$f(t)=\sum_{j} \sum_{k} {w}_{j,k}{\psi }_{j,k}(t)+{c}_{0}\phi (t)$

Analysis

${w}_{j,k}=\int_{0}^{T} f(t){\psi }_{j,k}(t)\,d t$
${c}_{0}=\int_{0}^{T} f(t)\phi (t)\,d t$
the ${w}_{j,k}$ are real
The Haar transform is super useful especially in image compression

Haar wavelet demonstration

Questions & Answers

What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
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
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
Mueller Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Read also:

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Signals and systems. OpenStax CNX. Aug 14, 2014 Download for free at http://legacy.cnx.org/content/col10064/1.15
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Signals and systems' conversation and receive update notifications?

 By