<< Chapter < Page Chapter >> Page >

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

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
I'm not sure why it wrote it the other way
I got X =-6
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
Idrissa Reply
im all ears I need to learn
right! what he said ⤴⤴⤴
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
Rachael Reply
I'm not good at math so would you help me
what is the problem that i will help you to self with?
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris 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?
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
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
Privacy Information Security Software Version 1.1a
Got questions? Join the online conversation and get instant answers!
QuizOver.com Reply

Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Concise signal models. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10635/1.4
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Concise signal models' conversation and receive update notifications?