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Recovery via greedy pursuit

At the expense of slightly more measurements, iterative greedy algorithms such as Orthogonal Matching Pursuit(OMP) [link] , Matching Pursuit (MP) [link] , and Tree Matching Pursuit (TMP) [link] , [link] have also been proposed to recover the signal x from the measurements y (see Nonlinear Approximation from Approximation ). In CS applications, OMP requires c 2 ln ( N ) [link] to succeed with high probability. OMP is also guaranteed to converge within M iterations. We note that Tropp and Gilbert require the OMP algorithm to succeed in the first K iterations [link] ; however, in our simulations, we allow the algorithm to run up to the maximum of M possible iterations. The choice of an appropriate practical stopping criterion (likely somewhere between K and M iterations) is a subject of current research in the CS community.

Impact and applications

CS appears to be promising for a number of applications in signal acquisition and compression. Instead of sampling a K -sparse signal N times, only c K incoherent measurements suffice, where K can be orders of magnitude less than N . Therefore, a sensor can transmit far fewer measurements to a receiver, which can reconstruct the signal and then process itin any manner. Moreover, the c K measurements need not be manipulated in any way before being transmitted, except possiblyfor some quantization. Finally, independent and identically distributed (i.i.d.) Gaussian or Bernoulli/Rademacher (random ± 1 ) vectors provide a useful universal basis that is incoherent with all others. Hence, when using a random basis, CSis universal in the sense that the sensor can apply the same measurement mechanism no matter what basis the signal is sparse in(and thus the coding algorithm is independent of the sparsity-inducing basis) [link] , [link] , [link] .

These features of CS make it particularly intriguing for applications in remote sensing environments that might involvelow-cost battery operated wireless sensors, which have limited computational and communication capabilities. Indeed, in many suchenvironments one may be interested in sensing a collection of signals using a network of low-cost signals.

Other possible application areas of CS include imaging  [link] , medical imaging  [link] , [link] , and RF environments (where high-bandwidth signals may containlow-dimensional structures such as radar chirps)  [link] . As research continues into practical methods for signal recovery (see [link] ), additional work has focused on developing physical devices foracquiring random projections. Our group has developed, for example, a prototype digital CS camera based on a digitalmicromirror design  [link] . Additional work suggests that standard components such as filters (with randomized impulseresponses) could be useful in CS hardware devices  [link] .

The geometry of compressed sensing

It is important to note that the core theory of CS draws from a number of deep geometric arguments. For example, when viewedtogether, the CS encoding/decoding process can be interpreted as a linear projection Φ : R N R M followed by a nonlinear mapping Δ : R M R N . In a very general sense, one may naturally ask for a given classof signals F R N (such as the set of K -sparse signals or the set of signals with coefficients α p 1 ), what encoder/decoder pair Φ , Δ will ensure the best reconstruction (minimax distortion) of all signals in F . This best-case performance is proportional to what is known as the Gluskin n -width  [link] , [link] of F (in our setting n = M ), which in turn has a geometric interpretation. Roughly speaking, the Gluskin n -width seeks the ( N - n ) -dimensional slice through F that yields signals of greatest energy. This n -width bounds the best-case performance of CS on classes of compressible signals, and one of the hallmarks of CS is that,given a sufficient number of measurements this optimal performance is achieved (to within a constant)  [link] , [link] .

Questions & Answers

what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
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
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Akash Reply
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s. Reply
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
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.
Do you know which machine is used to that process?
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
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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
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I'm interested in nanotube
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
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Sravani Reply
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preparation of nanomaterial
Victor Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:  OpenStax, Concise signal models. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10635/1.4
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