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In the case where the class of signals of interest corresponds to a low dimensional subspace, a truncated, simplified sparse approximation can be applied as a detection algorithm; this has been dubbed as IDEA  [link] . In simple terms, the algorithm will mark a detection when a large enough amount of energy from the measurements lies in the projected subspace. Since this problem does not require accurate estimation of the signal values, but rather whether it belongs in the subspace of interest or not, the number of measurements necessary is much smaller than that required for reconstruction, as shown in [link] .

Performance for IDEA. (Top) Sample wideband chirp signal and same chirp embedded in strong narrowband interference. (Bottom) Probability of error to reconstruct and detect chirpsignals embedded in strong sinusoidal interference ( SIR = - 6 dB) using greedy algorithms. In this case, detection requires 3 × fewer measurements and 4 × fewer computations than reconstruction for an equivalent probability of success. Taken from  [link] .


Similarly, random projections have long been used for a variety of classification and clustering problems. The Johnson-Lindenstrauss Lemma is often exploited in this setting to compute approximate nearest neighbors, which is naturally related to classification. The key result that randomprojections result in an isometric embedding allows us to generalize this work to several new classification algorithms and settings  [link] .

Classification can also be performed when more elaborate models are used for the different classes. Suppose the signal/image class of interest can be modeled as a low-dimensional manifold in the ambient space. In such case it can be shown that, even under random projections, certain geometric properties of the signal class are preserved up to a small distortion; for example, interpoint Euclidean ( 2 ) distances are preserved  [link] . This enables the design of classification algorithms in the projected domain. One such algorithm is known as the smashed filter  [link] . As an example, under equal distribution among classes and a gaussian noise setting, the smashed filter is equivalent to building a nearest-neighbor (NN) classifier in the measurement domain. Further, it has been shown that for a K - dimensional manifold, M = O ( K log N ) measurements are sufficient to perform reliable compressive classification. Thus, the number of measurements scales as the dimension of the signal class, as opposed to the sparsity of the individual signal. Some example results are shown in [link] (a).

Results for smashed filter image classification and parameter estimation experiments. (a) Classification rates and(b) average estimation error for varying number of measurements M and noise levels σ for a set of images of several objects under varying shifts. As M increases, the distances between the manifolds increase as well, thus increasing the noise tolerance and enabling more accurate estimation and classification. Thus, the classification and estimation performances improve as σ decreases and M increases in all cases. Taken from  [link] .


Consider a signal x R N , and suppose that we wish to estimate some function f ( x ) but only observe the measurements y = Φ x , where Φ is again an M × N matrix. The data streaming community has previously analyzed this problem for many common functions, such as linear functions, p norms, and histograms. These estimates are often based on so-called sketches , which can be thought of as random projections.

As an example, in the case where f is a linear function, one can show that the estimation error (relative to the norms of x and f ) can be bounded by a constant determined by M . This result holds for a wide class of random matrices, and can be viewed as a straightforward consequence of the same concentration of measure inequality that has proven useful for CS and in proving the JL Lemma  [link] .

Parameter estimation can also be performed when the signal class is modeled as a low-dimensional manifold. Suppose an observed signal x can be parameterized by a K - dimensional parameter vector θ , where K N . Then, it can be shown that with 0 ( K log N ) measurements, the parameter vector can be obtained via multiscale manifold navigation in the compressed domain  [link] . Some example results are shown in [link] (b).

Questions & Answers

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
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
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
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, An introduction to compressive sensing. OpenStax CNX. Apr 02, 2011 Download for free at http://legacy.cnx.org/content/col11133/1.5
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