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Nonlinear approximations operate in two stages. First, a linear operator approximates the analog signal f ¯ with N samples written f [ n ] = f ¯ φ ¯ s ( n s ) . Then, a nonlinear approximation of f [ n ] is computed to reduce the N coefficients f [ n ] to M N coefficients in a sparse representation.

The discrete signal f can be considered as a vector of C N . Inner products and norms in C N are written

f , g = n = 0 N - 1 f [ n ] g * [ n ] and f 2 = n = 0 N - 1 | f [ n ] | 2 .

To obtain a sparse representation with a nonlinear approximation, we choose a new orthonormal basis B = { g m [ n ] } m Γ of C N , which concentrates the signal energy as much as possibleover few coefficients. Signal coefficients { f , g m } m Γ are computed from the N input sample values f [ n ] with an orthogonal change of basis that takes N 2 operations in nonstructured bases. In a wavelet or Fourier bases, fastalgorithms require, respectively, O ( N ) and O ( N log 2 N ) operations.

Approximation by thresholding

For M < N , an approximation f M is computed by selecting the “best” M < N vectors within B . The orthogonal projectionof f on the space V λ generated by M vectors { g m } m Λ in B is

f λ = m λ f , g m g m .

Since f = m γ f , g m g m , the resulting error is

f - f λ 2 = m / λ | f , g m | 2 .

We write | λ | the size of the set λ . The best M = | λ | term approximation, which minimizes f - f λ 2 ,  is thus obtained by selecting the M coefficients of largest amplitude. These coefficients are above a threshold T that depends on M :

f M = f λ T = m λ T f , g m g m with λ T = { m γ : | f , g m | T } .

This approximation is nonlinear because the approximation set λ T changes with f . The resulting approximation error is:

ϵ n ( M , f ) = f - f M 2 = m / Λ T | f , g m | 2 .

[link] (b) shows that the approximation support λ T of an image in a wavelet orthonormal basis depends on the geometry of edges and textures.Keeping large wavelet coefficients is equivalent to constructing an adaptiveapproximation grid specified by the scale–space support λ T . It increases the approximation resolutionwhere the signal is irregular. The geometry of λ T gives the spatial distribution of sharp image transitions and edges, and their propagation across scales.Chapter 6 proves that wavelet coefficients give important information about singularities and local Lipschitz regularity.This example illustrates how approximation support provides “geometric”information on f , relative to a dictionary, that is a wavelet basis in this example.

[link] (d) gives the nonlinear wavelet approximation f M recovered from the M = N / 16 large-amplitude wavelet coefficients, with an error f - f M 2 / f 2 = 5 × 10 - 3 . This error is nearly three times smaller thanthe linear approximation error obtained with the same number of wavelet coefficients, and the imagequality is much better.

An analog signal can be recovered from the discrete nonlinear approxima-tion  f M :

f ¯ M ( x ) = n = 0 N - 1 f M [ n ] φ s ( x - n s ) .

Since all projections are orthogonal, the overall approximationerror on the original analog signal f ¯ ( x ) is the sum of the analog sampling error and the discrete nonlinear error:

f ¯ - f ¯ M 2 = f ¯ - f ¯ N 2 + f - f M 2 = ϵ l ( N , f ) + ϵ n ( M , f ) .

In practice, N is imposed by the resolution of the signal-acquisition hardware, and M is typically adjusted so that ϵ n ( M , f ) ϵ l ( N , f ) .

Questions & Answers

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Damian Reply
research.net
kanaga
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.
Bharti
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Damian Reply
absolutely yes
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Anassong
Do somebody tell me a best nano engineering book for beginners?
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there is no specific books for beginners but there is book called principle of nanotechnology
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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
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.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
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.
Harper
Do you know which machine is used to that process?
s.
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SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
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in general
s.
Graphene has a hexagonal structure
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China
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I start with an easy one. carbon nanotubes woven into a long filament like a string
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Source:  OpenStax, A wavelet tour of signal processing, the sparse way. OpenStax CNX. Sep 14, 2009 Download for free at http://cnx.org/content/col10711/1.3
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