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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 .

Theorem

[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

find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
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Abhi
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
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ninjadapaul
20/(×-6^2)
Salomon
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ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
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Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
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Abhi
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salma
Commplementary angles
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Sherica
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Uday
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salma
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a perfect square v²+2v+_
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Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
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or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
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Differences Between Laspeyres and Paasche Indices
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No. 7x -4y is simplified from 4x + (3y + 3x) -7y
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Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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what's the easiest and fastest way to the synthesize AgNP?
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China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
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Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
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what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
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Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
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
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
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele 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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