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The CS theory tells us that when certain conditions hold, namely that the functions { φ m } cannot sparsely represent the elements of the basis { ψ n } (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 { α ( n ) } (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 R N , y R M , and M < N ), it is indeed possible to recover sparse signals from CS measurements. Given the measurements y = Φ x , there exist an infinite number of candidate signals in the shifted nullspace N ( Φ ) + 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 Ψ , then recovery of x from y can be formulated as the 0 minimization

α ^ = arg min α 0 s.t. y = Φ Ψ α .
Given some technical conditions on Φ and Ψ (see Theorem [link] below), then with high probability this optimization problem returns the proper K -sparse solution α , from which the true x may be constructed. (Thanks to the incoherence between the two bases, if the originalsignal is sparse in the α coefficients, then no other set of sparse signal coefficients α ' 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 Φ Ψ , 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 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] .)


Let Ψ be an orthonormal basis for R N , and let 1 K < N . Then the following statements hold:

  1. Let Φ be an M × N measurement matrix with i.i.d. Gaussian entries with M 2 K . Then with probability one the following statement holds: all signals x = Ψ α having expansion coefficients α R N that satisfy α 0 = K can be recovered uniquely from the M -dimensional measurement vector y = Φ x via the 0 optimization [link] .
  2. Let x = Ψ α such that α 0 = K . Let Φ be an M × N measurement matrix with i.i.d. Gaussian entries (notably, independent of x ) with M K + 1 . Then with probability one the following statement holds: x can be recovered uniquely from the M -dimensional measurement vector y = Φ x via the 0 optimization [link] .
  3. Let Φ be an M × N measurement matrix, where M K . Then, aside from pathological cases (specified in the proof), no signal x = Ψ α with α 0 = K can be uniquely recovered from the M -dimensional measurement vector y = Φ x .

Questions & Answers

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
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
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 ?
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
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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