<< Chapter < Page Chapter >> Page >

Mallat proposes a scheme for computing an approximation of the continuous wavelet transform [link] that turns out to be equivalent to the method described above. This has been realized and proved byShensa [link] . Moreover, Shensa shows that Mallat's algorithm exhibits the same structure as the so-called algorithm à trous.Interestingly, Mallat's intention in [link] was not in particular to overcome the shift variance of the DWT but to get an approximation ofthe continuous wavelet transform.

In the following, we shall refer to the algorithm for computing the SIDWT as the Beylkin algorithm However, it should be noted that Mallat published his algorithm earlier. since this is the one we have implemented. Alternative algorithms for computing a shift-invariantwavelet transform [link] are based on the scheme presented in [link] . They explicitly or implicitly try to find an optimal, signal-dependent shift of the input signal. Thus, the transform becomesshift-invariant and orthogonal but signal dependent and, therefore, nonlinear. We mention that the generalization of the Beylkin algorithm tothe multidimensional case, to an M -band multiresolution analysis, and to wavelet packets is straightforward.

Combining the shensa-beylkin-mallat-à trous algorithms and wavelet denoising

It was Coifman who suggested that the application of Donoho's method to several shifts of the observation combined with averagingyields a considerable improvement. A similar remark can be found in [link] , p. 53. This statement first lead us to the following algorithm: 1) apply Donoho's method not onlyto “some” but to all circular shifts of the input signal 2) average the adjusted output signals. As has beenshown in the previous section, the computation of all possible shifts can be effectively done using Beylkin's algorithm. Thus,instead of using the algorithm just described, one simply applies thresholding to the SIDWT of the observation and computes theinverse transform.

Before going into details, we want to briefly discuss the differences between using the traditional orthogonal and the shift-invariant wavelettransform. Obviously, by using more than N wavelet coefficients, we introduce redundancy. Several authors stated that redundant wavelettransforms, or frames, add to the numerical robustness [link] in case of adding white noise in the transform domain; e.g., by quantization. Thisis, however, different from the scenario we are interested in, since 1) we have correlated noise due to the redundancy, and 2) we try to remove noisein the transform domain rather than considering the effect of adding some noise [link] , [link] .

Performance analysis

The analysis of the ideal risk for the SIDWT is similar to that by Guo [link] . Define the sets A and B according to

A = { i | | X i | ϵ } B = { i | | X i | < ϵ }

and an ideal diagonal projection estimator, or oracle,

X ˜ = Y i = X i + N i i A 0 i B .

The pointwise estimation error is then

X ˜ i - X i = N i i A - X i i B .

In the following, a vector or matrix indexed by A (or B ) indicates that only those rows are kept that have indices out of A (or B ). All others are set to zero. With these definitions and [link] , the ideal risk for the SIDWT can be derived

Questions & Answers

Do somebody tell me a best nano engineering book for beginners?
s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
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.
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
many many of nanotubes
what is the k.e before it land
what is the function of carbon nanotubes?
I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
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
what is system testing
what is the application of nanotechnology?
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
anybody can imagine what will be happen after 100 years from now in nano tech world
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
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
silver nanoparticles could handle the job?
not now but maybe in future only AgNP maybe any other nanomaterials
I'm interested in Nanotube
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
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.
Ali 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
Got questions? Join the online conversation and get instant answers!
QuizOver.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?