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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.

Overview

In characterizing a given problem in signal processing, one is often able to specify a model for the signals to be processed. This model may distinguish (either statistically or deterministically)classes of interesting signals from uninteresting ones, typical signals from anomalies, information from noise, etc.

Very commonly, models in signal processing deal with some notion of structure, constraint, or conciseness. Roughly speaking, one often believes that a signal has “few degrees of freedom”relative to the size of the signal. This notion of conciseness is a very powerful assumption, and it suggests the potential for dramatic gains via algorithms that capture and exploit the true underlyingstructure of the signal.

In these modules, we survey three common examples of concise models: linear models, sparse nonlinear models, and manifold-based models. In each case, we discuss an important phenomenon:the conciseness of the model corresponds to a low-dimensional geometric structure along which the signals of interest tend to cluster. This low-dimensional geometry again has important implicationsin the understanding and the development of efficient algorithms for signal processing.

We discuss this low-dimensional geometry in several contexts, including projecting a signal onto the model class (i.e., forming a concise approximation to a signal), encoding such an approximation(i.e., data compression), and reducing the dimensionality of signals and data sets. We conclude with an important and emerging application area known as Compressed Sensing (CS), which is a novel methodfor data acquisition that relies on concise models and builds upon strong geometric principles. We discuss CS in its traditional, sparsity-based context and also discuss extensions of CS to otherconcise models such as manifolds.

General mathematical preliminaries

Signal notation

We will treat signals as real- or complex-valued functions having domains that are either discrete (and finite) or continuous (andeither compact or infinite). Each of these assumptions will be made clear as needed. As a generalrule, however, we will use x to denote a discrete signal in R N and f to denote a function over a continuousdomain D . We also commonly refer to these as discrete- or continuous- time signals, though the domain need not actually be temporal in nature.

Lp and lp norms

As measures for signal energy, fidelity, or sparsity, we will employ the L p and p norms. For continuous-time functions, the L p norm is defined as

f L p ( D ) = D | f | p 1 / p , p ( 0 , ) ,
and for discrete-time functions, the p norm is defined as
x p = ( i = 1 N | x ( i ) | p ) 1 / p , p ( 0 , ) , max i = 1 , , N | x ( i ) | , p = , i = 1 N 1 x ( i ) 0 , p = 0 ,
where 1 denotes the indicator function. (While we often refer to these measures as “norms,” they actually do not meetthe technical criteria for norms when p < 1 .)

Linear algebra

Let A be a real-valued M × N matrix. We denote the nullspace of A as N ( A ) (note that N ( A ) is a linear subspace of R N ), and we denote the transpose of A as A T .

We call A an orthoprojector from R N to R M if it has orthonormal rows. From such a matrix we call A T A the corresponding orthogonal projection operator onto the M -dimensional subspace of R N spanned by the rows of A .

Questions & Answers

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
learn
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
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
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Professor
I think
Professor
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
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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
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
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sciencedirect big data base
Ernesto
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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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