# 0.7 Generalizations of the basic multiresolution wavelet system  (Page 16/28)

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There are at least two descriptions of the problem. We may want a single expansion system to handle several different classes of signals, each ofwhich are well-represented by a particular basis system or we may have a single class of signals, but the elements of that class are linearcombinations of members of the well-represented classes. In either case, there are several criteria that have been identified as important [link] , [link] :

• Sparsity: The expansion should have most of the important information in the smallest number of coefficients so that the others aresmall enough to be neglected or set equal to zero. This is important for compression and denoising.
• Separation: If the measurement consists of a linear combination of signals with different characteristics, the expansion coefficientsshould clearly separate those signals. If a single signal has several features of interest, the expansion should clearly separate thosefeatures. This is important for filtering and detection.
• Superresolution: The resolution of signals or characteristics of a signal should be much better than with a traditional basis system. Thisis likewise important for linear filtering, detection, and estimation.
• Stability: The expansions in terms of our new overcomplete systems should not be significantly changed by perturbations or noise.This is important in implementation and data measurement.
• Speed: The numerical calculation of the expansion coefficients in the new overcomplete system should be of order $O\left(N\right)$ or $O\left(Nlog\left(N\right)\right)$ .

These criteria are often in conflict with each other, and various compromises will be made in the algorithms and problem formulations for anacceptable balance.

## Overcomplete representations

This section uses the material in  Chapter: Bases, Orthogonal Bases, Biorthogonal Bases, Frames, Right Frames, and unconditional Bases on bases and frames. One goal is to represent a signal using a “dictionary" ofexpansion functions that could include the Fourier basis, wavelet basis, Gabor basis, etc. We formulate a finite dimensional version of thisproblem as

$y\left(n\right)=\sum _{k}{\alpha }_{k}\phantom{\rule{0.166667em}{0ex}}{x}_{k}\left(n\right)\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}n,k\in \mathbf{Z}$

for $n=0,1,2,\cdots ,N-1$ and $k=0,1,2,\cdots ,K-1$ . This can be written in matrix form as

$\mathbf{y}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\mathbf{X}\phantom{\rule{0.166667em}{0ex}}\alpha$

where $\mathbf{y}$ is a $N×1$ vector with elements being the signal values $y\left(n\right)$ , the matrix $\mathbf{X}$ is $N×K$ the columns of which are made up of all the functions in the dictionaryand $\alpha$ is a $K×1$ vector of the expansion coefficients ${\alpha }_{k}$ . The matrix operator has the basis signals ${\mathbf{x}}_{\mathbf{k}}$ as its columns so that the matrix multiplication [link] is simply the signal expansion [link] .

For a given signal representation problem, one has two decisions: what dictionary to use (i.e., choice of the $\mathbf{X}$ ) and how to represent the signal in terms of this dictionary (i.e., choice of $\alpha$ ). Since the dictionary is overcomplete, there are several possible choices of $\alpha$ and typically one uses prior knowledge or one or more of the desired properties we saw earlier to calculatethe $\alpha$ .

## A matrix example

Consider a simple two-dimensional system with orthogonal basis vectors

${\mathbf{x}}_{\mathbf{1}}=\left[\begin{array}{c}\mathbf{1}\\ \mathbf{0}\end{array}\right]\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\text{and}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{\mathbf{x}}_{\mathbf{2}}=\left[\begin{array}{c}\mathbf{0}\\ \mathbf{1}\end{array}\right]$

which gives the matrix operator with ${\mathbf{x}}_{\mathbf{1}}$ and ${\mathbf{x}}_{\mathbf{2}}$ as columns

$\mathbf{X}=\phantom{\rule{0.166667em}{0ex}}\left[\begin{array}{cc}\hfill \mathbf{1}& \hfill \mathbf{0}\\ \hfill \mathbf{0}& \hfill \mathbf{1}\end{array}\right].$

how do they get the third part x = (32)5/4
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
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Uday
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
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algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
is it 3×y ?
J, combine like terms 7x-4y
how do you translate this in Algebraic Expressions
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
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
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
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
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.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
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