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The second statement of the theorem differs from the first in the following respect: when K < M < 2 K , there will necessarily exist K -sparse signals x that cannot be uniquely recovered from the M -dimensional measurement vector y = Φ x . However, these signals form a set of measure zero within the set of all K -sparse signals and can safely be avoided if Φ is randomly generated independently of x .

Unfortunately, as discussed in Nonlinear Approximation from Approximation , solving this 0 optimization problem is prohibitively complex. Yet another challenge is robustness; in the setting ofTheorem "Recovery via ℓ 0 optimization" , the recovery may be very poorly conditioned. In fact, both of these considerations (computational complexity and robustness) can be addressed, but atthe expense of slightly more measurements.

Recovery via convex optimization

The practical revelation that supports the new CS theory is that it is not necessary to solve the 0 -minimization problem to recover α . In fact, a much easier problem yields an equivalent solution (thanks again to the incoherency of thebases); we need only solve for the 1 -sparsest coefficients α that agree with the measurements y [link] , [link] , [link] , [link] , [link] , [link] , [link] , [link]

α ^ = arg min α 1 s.t. y = Φ Ψ α .
As discussed in Nonlinear Approximation from Approximation , this optimization problem, also known as Basis Pursuit [link] , is significantly more approachable and can be solved with traditionallinear programming techniques whose computational complexities are polynomial in N .

There is no free lunch, however; according to the theory, more than K + 1 measurements are required in order to recover sparse signals via Basis Pursuit. Instead, one typically requires M c K measurements, where c > 1 is an oversampling factor . As an example, we quote a result asymptotic in N . For simplicity, we assume that the sparsity scales linearly with N ; that is, K = S N , where we call S the sparsity rate .


[link] , [link] , [link] Set K = S N with 0 < S 1 . Then there exists an oversampling factor c ( S ) = O ( log ( 1 / S ) ) , c ( S ) > 1 , such that, for a K -sparse signal x in the basis Ψ , the following statements hold:

  1. The probability of recovering x via Basis Pursuit from ( c ( S ) + ϵ ) K random projections, ϵ > 0 , converges to one as N .
  2. The probability of recovering x via Basis Pursuit from ( c ( S ) - ϵ ) K random projections, ϵ > 0 , converges to zero as N .

In an illuminating series of recent papers, Donoho and Tanner [link] , [link] , [link] have characterized the oversampling factor c ( S ) precisely (see also "The geometry of Compressed Sensing" ). With appropriate oversampling, reconstruction via Basis Pursuit is also provably robust tomeasurement noise and quantization error [link] .

We often use the abbreviated notation c to describe the oversampling factor required in various settings even though c ( S ) depends on the sparsity K and signal length N .

A CS recovery example on the Cameraman test image is shown in [link] . In this case, with M = 4 K we achieve near-perfect recovery of the sparse measured image.

Compressive sensing reconstruction of the nonlinear approximation Cameraman image from [link] (b). Using M = 16384 random measurements of the K -term nonlinear approximation image (where K = 4096 ), we solve an 1 -minimization problem to obtain the reconstruction shown above. The MSE with respect to the measured image is 0.08 , so the reconstruction is virtually perfect.

Questions & Answers

How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
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
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
for teaching engĺish at school how nano technology help us
How can I make nanorobot?
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
how can I make nanorobot?
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?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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