# 0.7 Compressed sensing  (Page 2/5)

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The CS theory tells us that when certain conditions hold, namely that the functions $\left\{{\phi }_{m}\right\}$ cannot sparsely represent the elements of the basis $\left\{{\psi }_{n}\right\}$ (a condition known as incoherence of the two dictionaries [link] , [link] , [link] , [link] ) and the number of measurements $M$ is large enough, then it is indeed possible to recover the set of large $\left\{\alpha \left(n\right)\right\}$ (and thus the signal $x$ ) from a similarly sized set of measurements $y$ . This incoherence property holds for many pairs of bases, including forexample, delta spikes and the sine waves of a Fourier basis, or the Fourier basis and wavelets. Significantly, this incoherencealso holds with high probability between an arbitrary fixed basis and a randomly generated one.

## Methods for signal recovery

Although the problem of recovering $x$ from $y$ is ill-posed in general (because $x\in {\mathbb{R}}^{N}$ , $y\in {\mathbb{R}}^{M}$ , and $M ), it is indeed possible to recover sparse signals from CS measurements. Given the measurements $y=\Phi x$ , there exist an infinite number of candidate signals in the shifted nullspace $\mathcal{N}\left(\Phi \right)+x$ that could generate the same measurements $y$ (see Linear Models from Low-Dimensional Signal Models ). Recovery of the correct signal $x$ can be accomplished by seeking a sparse solution among these candidates.

## Recovery via combinatorial optimization

Supposing that $x$ is exactly $K$ -sparse in the dictionary $\Psi$ , then recovery of $x$ from $y$ can be formulated as the ${\ell }_{0}$ minimization

$\stackrel{^}{\alpha }=argmin{\parallel \alpha \parallel }_{0}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\text{s.t.}\phantom{\rule{4.pt}{0ex}}y=\Phi \Psi \alpha .$
Given some technical conditions on $\Phi$ and $\Psi$ (see Theorem [link] below), then with high probability this optimization problem returns the proper $K$ -sparse solution $\alpha$ , from which the true $x$ may be constructed. (Thanks to the incoherence between the two bases, if the originalsignal is sparse in the $\alpha$ coefficients, then no other set of sparse signal coefficients ${\alpha }^{\text{'}}$ can yield the same projections $y$ .) We note that the recovery program [link] can be interpreted as finding a $K$ -term approximation to $y$ from the columns of the dictionary $\Phi \Psi$ , which we call the holographic basis because of the complex pattern in which it encodes the sparse signal coefficients [link] .

In principle, remarkably few incoherent measurements are required to recover a $K$ -sparse signal via ${\ell }_{0}$ minimization. Clearly, more than $K$ measurements must be taken to avoid ambiguity; the following theorem (which is proved in [link] ) establishes that $K+1$ random measurements will suffice. (Similar results were established by Venkataramani and Bresler  [link] .)

Theorem

Let $\Psi$ be an orthonormal basis for ${\mathbb{R}}^{N}$ , and let $1\le K . Then the following statements hold:

1. Let $\Phi$ be an $M×N$ measurement matrix with i.i.d. Gaussian entries with $M\ge 2K$ . Then with probability one the following statement holds: all signals $x=\Psi \alpha$ having expansion coefficients $\alpha \in {\mathbb{R}}^{N}$ that satisfy ${\parallel \alpha \parallel }_{0}=K$ can be recovered uniquely from the $M$ -dimensional measurement vector $y=\Phi x$ via the ${\ell }_{0}$ optimization [link] .
2. Let $x=\Psi \alpha$ such that ${\parallel \alpha \parallel }_{0}=K$ . Let $\Phi$ be an $M×N$ measurement matrix with i.i.d. Gaussian entries (notably, independent of $x$ ) with $M\ge K+1$ . Then with probability one the following statement holds: $x$ can be recovered uniquely from the $M$ -dimensional measurement vector $y=\Phi x$ via the ${\ell }_{0}$ optimization [link] .
3. Let $\Phi$ be an $M×N$ measurement matrix, where $M\le K$ . Then, aside from pathological cases (specified in the proof), no signal $x=\Psi \alpha$ with ${\parallel \alpha \parallel }_{0}=K$ can be uniquely recovered from the $M$ -dimensional measurement vector $y=\Phi x$ .

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
Do somebody tell me a best nano engineering book for beginners?
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
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
so some one know about replacing silicon atom with phosphorous in semiconductors device?
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 ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
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
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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