# 0.3 Low-dimensional signal models  (Page 3/3)

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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 $\theta$ denote the variable parameter and write the signal as ${f}_{\theta }$ to denote its dependence on $\theta$ . In a sense, $\theta$ is a single “degree of freedom” driving the generation of the signal ${f}_{\theta }$ under this simple model. We let $\Theta$ denote the set of possible values of the parameter $\theta$ . If the mapping between $\theta$ and ${f}_{\theta }$ is well-behaved, then thecollection of signals $\left\{{f}_{\theta }:\theta \in \Theta \right\}$ 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 $\theta$ that is chosen from a $K$ -dimensional manifold $\Theta$ . (The parameter space $\Theta$ could be, for example, a subset of ${\mathbb{R}}^{K}$ , or it could be a more general manifold such as SO(3).) We again let ${f}_{\theta }$ denote the signal corresponding to a particular choice of $\theta$ , and we let $\mathcal{F}=\left\{{f}_{\theta }:\theta \in \Theta \right\}$ . Assuming the mapping $f$ is continuous and injective over $\Theta$ (and its inverse is continuous), then by virtue of the manifold structure of $\Theta$ , its image $\mathcal{F}$ 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 ${f}_{{\theta }_{0}}+{f}_{{\theta }_{1}}$ . 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.

what is the stm
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?
what is a peer
What is meant by 'nano scale'?
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
what is Nano technology ?
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?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
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?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
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.
Tarell
what is the actual application of fullerenes nowadays?
Damian
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.
Tarell
how did you get the value of 2000N.What calculations are needed to arrive at it
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