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This collection reviews fundamental concepts underlying the use of concise models for signal processing. Topics are presented from a geometric perspective and include low-dimensional linear, sparse, and manifold-based signal models, approximation, compression, dimensionality reduction, and Compressed Sensing.

A new theory known as Compressed Sensing (CS) has recently emerged that can also be categorized as a type of dimensionalityreduction. Like manifold learning, CS is strongly model-based (relying on sparsity in particular).However, unlike many of the standard techniques in dimensionality reduction (such as manifold learning or the JL lemma), the goal ofCS is to maintain a low-dimensional representation of a signal x from which a faithful approximation to x can be recovered. In a sense, this more closely resembles the traditional problem ofdata compression (see Compression ). In CS, however, the encoder requires no a priori knowledge of thesignal structure. Only the decoder uses the model (sparsity) to recover the signal. Wejustify such an approach again using geometric arguments.

Motivation

Consider a signal x R N , and suppose that the basis Ψ provides a K -sparse representation of x

x = Ψ α ,
with α 0 = K . (In this section, we focus on exactly K -sparse signals, though many of the key ideas translate to compressible signals  [link] , [link] . In addition, we note that the CS concepts are also extendable totight frames.)

As we discussed in Compression , the standard procedure for compressing sparse signals, known as transformcoding, is to (i) acquire the full N -sample signal x ; (ii) compute the complete set of transform coefficients α ; (iii) locate the K largest, significant coefficients and discard the (many) small coefficients; (iv) encode the values and locations of the largest coefficients.

This procedure has three inherent inefficiencies: First, for a high-dimensional signal, we must start with a large number ofsamples N . Second, the encoder must compute all N of the transform coefficients α , even though it will discard all but K of them. Third, the encoder must encode the locations of the large coefficients, which requiresincreasing the coding rate since the locations change with each signal.

Incoherent projections

This raises a simple question: For a given signal, is it possible to directly estimate the set of large α ( n ) 's that will not be discarded? While this seems improbable, Candès, Romberg,and Tao  [link] , [link] and Donoho [link] have shown that a reduced set of projections can contain enoughinformation to reconstruct sparse signals. An offshoot of this work, often referred to as Compressed Sensing (CS) [link] , [link] , [link] , [link] , [link] , [link] , [link] , has emerged that builds on this principle.

In CS, we do not measure or encode the K significant α ( n ) directly. Rather, we measure and encode M < N projections y ( m ) = < x , φ m T > of the signal onto a second set of functions { φ m } , m = 1 , 2 , ... , M . In matrix notation, we measure

y = Φ x ,
where y is an M × 1 column vector and the measurement basis matrix Φ is M × N with each row a basis vector φ m . Since M < N , recovery of the signal x from the measurements y is ill-posed in general; however the additional assumption of signal sparsity makes recovery possible and practical.

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
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
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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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