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Manifold models

Manifold models generalize the conciseness of sparsity-based signal models. In particular, in many situations where a signal isbelieved to have a concise description or “few degrees of freedom,” the result is that the signal will live on or near aparticular submanifold of the ambient signal space.

Parametric models

We begin with an abstract motivation for the manifold perspective. Consider a signal f (such as a natural image), and suppose that we can identify some single 1-D piece of information about thatsignal that could be variable; that is, other signals might rightly be called “similar” to f if they differ only in this piece of information. (For example, this 1-D parameter coulddenote the distance from some object in an image to the camera.) We let θ denote the variable parameter and write the signal as f θ to denote its dependence on θ . In a sense, θ is a single “degree of freedom” driving the generation of the signal f θ under this simple model. We let Θ denote the set of possible values of the parameter θ . If the mapping between θ and f θ is well-behaved, then thecollection of signals { f θ : θ Θ } forms a 1-D path in the ambient signal space.

More generally, when a signal has K degrees of freedom, we may model it as depending on some parameter θ that is chosen from a K -dimensional manifold Θ . (The parameter space Θ could be, for example, a subset of R K , or it could be a more general manifold such as SO(3).) We again let f θ denote the signal corresponding to a particular choice of θ , and we let F = { f θ : θ Θ } . Assuming the mapping f is continuous and injective over Θ (and its inverse is continuous), then by virtue of the manifold structure of Θ , its image F will correspond to a K -dimensional manifold embedded in the ambient signal space (see [link] (c)).

These types of parametric models arise in a number of scenarios in signal processing. Examples include: signals of unknowntranslation, sinusoids of unknown frequency (across a continuum of possibilities), linear radar chirps described by a starting andending time and frequency, tomographic or light field images with articulated camera positions, robotic systems with few physicaldegrees of freedom, dynamical systems with low-dimensional attractors  [link] , [link] , and so on.

In general, parametric signals manifolds are nonlinear (by which we mean non-affine as well); this can again be seen byconsidering the sum of two signals f θ 0 + f θ 1 . In many interesting situations, signal manifolds are non-differentiable as well.

Nonparametric models

Manifolds have also been used to model signals for which there is no known parametric model. Examples include images of faces andhandwritten digits [link] , [link] , which have been found empirically to cluster near low-dimensional manifolds.Intuitively, because of the configurations of human joints and muscles, it may be conceivable that there are relatively “few”degrees of freedom driving the appearance of a human face or the style of handwriting; however, this inclination is difficult orimpossible to make precise. Nonetheless, certain applications in face and handwriting recognition have benefitted from algorithmsdesigned to discover and exploit the nonlinear manifold-like structure of signal collections. Manifold Learning from Dimensionality Reduction discusses such methods for learning parametrizations and other information from data livingalong manifolds.

Much more generally, one may consider, for example, the set of all natural images. Clearly, this set has small volume with respect to the ambient signal space — generating an imagerandomly pixel-by-pixel will almost certainly produce an unnatural noise-like image. Again, it is conceivable that, at least locally,this set may have a low-dimensional manifold-like structure: from a given image, one may be able to identify only a limited numberof meaningful changes that could be performed while still preserving the natural look to the image. Arguably, most work insignal modeling could be interpreted in some way as a search for this overall structure.

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
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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, Concise signal models. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10635/1.4
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