# 0.3 Low-dimensional signal models  (Page 2/3)

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Indeed, these results suggest that the largest Fourier or wavelet coefficients of smooth signals tend to concentrate at the coarsest scales (lowest-frequencies). In Linear Approximation from Approximation , we see that linear approximations formed from just the lowest frequencyelements of the Fourier or wavelet dictionaries (i.e., the truncation of the Fourier or wavelet representation to only the lowest frequency terms) provide very accurate approximations to smooth signals. Put differently, smooth signals live near the subspace spanned by just the lowest frequency Fourier or wavelet basis functions.

## Sparse (nonlinear) models

Sparse signal models can be viewed as a generalization of linear models. The notion of sparsity comes from the fact that, by theproper choice of dictionary $\Psi$ , many real-world signals $x=\Psi \alpha$ have coefficient vectors $\alpha$ containing few large entries, but across different signals the locations (indicesin $\alpha$ ) of the large entries may change. We say a signal is strictly sparse (or “ $K$ -sparse”) if all but $K$ entries of $\alpha$ are zero.

Sparsity is a nonlinear model. In particular, let ${\Sigma }_{K}$ denote the set of all $K$ -sparse signals for a given dictionary. It is easy to see that the set ${\Sigma }_{K}$ is not closed under addition. (In fact, ${\Sigma }_{K}+{\Sigma }_{K}={\Sigma }_{2K}$ .) From a geometric perspective, the set of all $K$ -sparse signals from the dictionary $\Psi$ forms not a hyperplane but rather a union of $K$ -dimensional hyperplanes, each spanned by $K$ vectors of $\Psi$ (see [link] (b)). For a dictionary $\Psi$ with $Z$ entries, there are $\left(\genfrac{}{}{0pt}{}{Z}{K}\right)$ such hyperplanes. (The geometry of sparse signal collections has alsobeen described in terms of orthosymmetric sets; see  [link] .)

Signals that are not strictly sparse but rather have a few “large” and many “small” coefficients are known as compressible signals. The notion of compressibility can be made more precise by considering the rate at which the sorted magnitudes of the coefficients $\alpha$ decay, and this decay rate can in turn be related to the ${\ell }_{p}$ norm of the coefficient vector $\alpha$ . Letting $\stackrel{˜}{\alpha }$ denote a rearrangement of the vector $\alpha$ with the coefficients ordered in terms of decreasing magnitude, then the reordered coefficientssatisfy  [link]

${\stackrel{˜}{\alpha }}_{k}\le {\parallel \alpha \parallel }_{{\ell }_{p}}{k}^{-1/p}.$
As we discuss in Nonlinear Approximation from Approximation , these decay rates play an important role in nonlinear approximation , where adaptive, $K$ -sparse representations from the dictionary are used to approximate a signal.

We recall from [link] that for a smooth signal $f$ , the largest Fourier and wavelet coefficients tend to cluster at coarse scales (low frequencies). Suppose, however, thatthe function $f$ is piecewise smooth; i.e., it is ${\mathcal{C}}^{H}$ at every point $t\in \mathbb{R}$ except for one point ${t}_{0}$ , at which it is discontinuous. Naturally, this phenomenon will be reflected in the transform coefficients.In the Fourier domain, this discontinuity will have a global effect, as the overall smoothness of the function $f$ has been reduced dramatically from $H$ to 0. Wavelet coefficients, however, depend only on local signalproperties, and so the wavelet basis functions whose supports do not include ${t}_{0}$ will be unaffected by the discontinuity. Coefficients surrounding the singularity will decay only as ${2}^{-j/2}$ , but there are relatively few such coefficients. Indeed, at each scale there are only $O\left(1\right)$ wavelets that include ${t}_{0}$ in their supports, but these locations are highly signal-dependent. (For modeling purposes, these significantcoefficients will persist through scale down the parent-child tree structure.) After reordering by magnitude, the waveletcoefficients of piecewise smooth signals will have the same general decay rate as those of smooth signals. In Nonlinear Approximation from Approximation , we see that the quality of nonlinear approximations offered by wavelets for smooth 1-D signals is nothampered by the addition of a finite number of discontinuities.

#### Questions & Answers

what is the stm
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
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
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?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
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?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
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
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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