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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

Do somebody tell me a best nano engineering book for beginners?
s. Reply
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Abhijith Reply
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s. Reply
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s. Reply
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Sanket Reply
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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
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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
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Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
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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