<< Chapter < Page Chapter >> Page >

Parameterization of the scaling coefficients

The case where φ ( t ) and h ( n ) have compact support is very important. It aids in the time localization properties of the DWT andoften reduces the computational requirements of calculating the DWT. If h ( n ) has compact support, then the filters described in  Chapter: Filter Banks and the Discrete Wavelet Transform are simple FIR filters. We have stated that N , the length of the sequence h ( n ) , must be even and h ( n ) must satisfy the linear constraint of [link] and the N 2 bilinear constraints of [link] . This leaves N 2 - 1 degrees of freedom in choosing h ( n ) that will still guarantee the existence of φ ( t ) and a set of essentially orthogonal basis functions generated from φ ( t ) .

Length-2 scaling coefficient vector

For a length-2 h ( n ) , there are no degrees of freedom left after satisfying the required conditions in [link] and [link] . These requirements are

h ( 0 ) + h ( 1 ) = 2

and

h 2 ( 0 ) + h 2 ( 1 ) = 1

which are uniquely satisfied by

h D 2 = h ( 0 ) , h ( 1 ) = 1 2 , 1 2 .

These are the Haar scaling functions coefficients which are also the length-2 Daubechies coefficients [link] used as an example in Chapter: A multiresolution formulation of Wavelet Systems and discussed later in this book.

Length-4 scaling coefficient vector

For the length-4 coefficient sequence, there is one degree of freedom or one parameter that gives all the coefficients that satisfy the requiredconditions:

h ( 0 ) + h ( 1 ) + h ( 2 ) + h ( 3 ) = 2 ,
h 2 ( 0 ) + h 2 ( 1 ) + h 2 ( 2 ) + h 2 ( 3 ) = 1

and

h ( 0 ) h ( 2 ) + h ( 1 ) h ( 3 ) = 0

Letting the parameter be the angle α , the coefficients become

h ( 0 ) = ( 1 - cos ( α ) + sin ( α ) ) / ( 2 2 ) h ( 1 ) = ( 1 + cos ( α ) + sin ( α ) ) / ( 2 2 ) h ( 2 ) = ( 1 + cos ( α ) - sin ( α ) ) / ( 2 2 ) h ( 3 ) = ( 1 - cos ( α ) - sin ( α ) ) / ( 2 2 ) .

These equations also give the length-2 Haar coefficients [link] for α = 0 , π / 2 , 3 π / 2 and a degenerate condition for α = π . We get the Daubechies coefficients (discussed later in this book) for α = π / 3 . These Daubechies-4 coefficients have a particularly clean form,

h D 4 = 1 + 3 4 2 , 3 + 3 4 2 , 3 - 3 4 2 , 1 - 3 4 2

Length-6 scaling coefficient vector

For a length-6 coefficient sequence h ( n ) , the two parameters are defined as α and β and the resulting coefficients are

h ( 0 ) = [ ( 1 + cos ( α ) + sin ( α ) ) ( 1 - cos ( β ) - sin ( β ) ) + 2 sin ( β ) cos ( α ) ] / ( 4 2 ) h ( 1 ) = [ ( 1 - cos ( α ) + sin ( α ) ) ( 1 + cos ( β ) - sin ( β ) ) - 2 sin ( β ) cos ( α ) ] / ( 4 2 ) h ( 2 ) = [ 1 + cos ( α - β ) + sin ( α - β ) ] / ( 2 2 ) h ( 3 ) = [ 1 + cos ( α - β ) - sin ( α - β ) ] / ( 2 2 ) h ( 4 ) = 1 / 2 - h ( 0 ) - h ( 2 ) h ( 5 ) = 1 / 2 - h ( 1 ) - h ( 3 )

Here the Haar coefficients are generated for any α = β and the length-4 coefficients [link] result if β = 0 with α being the free parameter. The length-4 Daubechies coefficients are calculatedfor α = π / 3 and β = 0 . The length-6 Daubechies coefficients result from α = 1 . 35980373244182 and β = - 0 . 78210638474440 .

The inverse of these formulas which will give α and β from the allowed h ( n ) are

α = arctan 2 ( h ( 0 ) 2 + h ( 1 ) 2 ) - 1 + ( h ( 2 ) + h ( 3 ) ) / 2 2 ( h ( 1 ) h ( 2 ) - h ( 0 ) h ( 3 ) ) + 2 ( h ( 0 ) - h ( 1 ) )
β = α - arctan h ( 2 ) - h ( 3 ) h ( 2 ) + h ( 3 ) - 1 / 2

As α and β range over - π to π all possible h ( n ) are generated. This allows informative experimentation to better see whatthese compactly supported wavelets look like. This parameterization is implemented in the Matlab programs in Appendix C and in the Aware, Inc. software, UltraWave [link] .

Since the scaling functions and wavelets are used with integer translations, the location of their support is not important, only thesize of the support. Some authors shift h ( n ) , h 1 ( n ) , φ ( t ) , and ψ ( t ) to be approximately centered around the origin. This is achieved by having the initial nonzero scaling coefficient start at n = - N 2 + 1 rather than zero. We prefer to have the origin at n = t = 0 .

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
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
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
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Wavelets and wavelet transforms. OpenStax CNX. Aug 06, 2015 Download for free at https://legacy.cnx.org/content/col11454/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Wavelets and wavelet transforms' conversation and receive update notifications?

Ask