0.3 Approximation and processing in bases  (Page 3/5)

 Page 3 / 5

Nonlinear approximations operate in two stages. First, a linear operator approximates the analog signal $\phantom{\rule{0.166667em}{0ex}}\overline{f}$ with N samples written $\phantom{\rule{0.166667em}{0ex}}f\left[n\right]=\phantom{\rule{0.166667em}{0ex}}\overline{f}☆{\overline{\phi }}_{s}\left(ns\right)$ . Then, a nonlinear approximation of $\phantom{\rule{0.166667em}{0ex}}f\left[n\right]$ is computed to reduce the N coefficients $\phantom{\rule{0.166667em}{0ex}}f\left[n\right]$ to $M\ll N$ coefficients in a sparse representation.

The discrete signal $\phantom{\rule{0.166667em}{0ex}}f$ can be considered as a vector of ${\mathbb{C}}^{N}$ . Inner products and norms in ${\mathbb{C}}^{N}$ are written

$⟨\phantom{\rule{0.166667em}{0ex}}f,g⟩=\sum _{n=0}^{N-1}f\left[n\right]\phantom{\rule{0.166667em}{0ex}}{g}^{*}{\left[n\right]\phantom{\rule{1.em}{0ex}}\text{and}\phantom{\rule{1.em}{0ex}}\parallel \phantom{\rule{0.166667em}{0ex}}f\parallel }^{2}=\sum _{n=0}^{N-1}{|\phantom{\rule{0.166667em}{0ex}}f\left[n\right]|}^{2}.$

To obtain a sparse representation with a nonlinear approximation, we choose a new orthonormal basis $\mathcal{B}={\left\{{g}_{m}\left[n\right]\right\}}_{m\in \Gamma }$ of ${\mathbb{C}}^{N}$ , which concentrates the signal energy as much as possibleover few coefficients. Signal coefficients ${\left\{⟨\phantom{\rule{0.166667em}{0ex}}f,{g}_{m}⟩\right\}}_{m\in \Gamma }$ are computed from the N input sample values $\phantom{\rule{0.166667em}{0ex}}f\left[n\right]$ with an orthogonal change of basis that takes N 2 operations in nonstructured bases. In a wavelet or Fourier bases, fastalgorithms require, respectively, $O\left(N\right)$ and $O\left(N{log}_{2}N\right)$ operations.

Approximation by thresholding

For $M , an approximation $\phantom{\rule{0.166667em}{0ex}}{f}_{M}$ is computed by selecting the “best” $M vectors within $\mathcal{B}$ . The orthogonal projectionof $\phantom{\rule{0.166667em}{0ex}}f$ on the space V λ generated by M vectors ${\left\{{g}_{m}\right\}}_{m\in \Lambda }$ in $\mathcal{B}$ is

$\begin{array}{c}\hfill {f}_{\lambda }=\sum _{m\in \lambda }⟨\phantom{\rule{0.166667em}{0ex}}f\phantom{\rule{-0.5pt}{0ex}},{g}_{m}⟩\phantom{\rule{0.166667em}{0ex}}{g}_{m}.\end{array}$

Since $\phantom{\rule{0.166667em}{0ex}}f={\sum }_{m\in \gamma }⟨\phantom{\rule{0.166667em}{0ex}}f,{g}_{m}⟩\phantom{\rule{0.166667em}{0ex}}{g}_{m}$ , the resulting error is

$\begin{array}{c}\hfill \parallel \phantom{\rule{0.166667em}{0ex}}f-{f}_{\lambda }{\parallel }^{2}=\sum _{m\in \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}/\lambda }{|⟨\phantom{\rule{0.166667em}{0ex}}f,{g}_{m}⟩|}^{2}.\end{array}$

We write $|\lambda |$ the size of the set λ . The best $M=|\lambda |$ term approximation, which minimizes $\parallel \phantom{\rule{0.166667em}{0ex}}f-{f}_{\lambda }{\parallel }^{2}$ ,  is thus obtained by selecting the M coefficients of largest amplitude. These coefficients are above a threshold T that depends on M :

$\begin{array}{c}\hfill {f}_{M}\phantom{\rule{0.222222em}{0ex}}=\phantom{\rule{0.222222em}{0ex}}{f}_{{\lambda }_{T}}=\sum _{m\in {\lambda }_{T}}⟨\phantom{\rule{0.166667em}{0ex}}f,{g}_{m}⟩\phantom{\rule{0.166667em}{0ex}}{g}_{m}\phantom{\rule{1.em}{0ex}}\text{with}\phantom{\rule{1.em}{0ex}}{\lambda }_{T}=\left\{m\in \gamma \phantom{\rule{3.33333pt}{0ex}}:\phantom{\rule{3.33333pt}{0ex}}|⟨\phantom{\rule{0.166667em}{0ex}}f,{g}_{m}⟩|\ge T\right\}.\end{array}$

This approximation is nonlinear because the approximation set λ T changes with $\phantom{\rule{0.166667em}{0ex}}f$ . The resulting approximation error is:

$\begin{array}{c}\hfill {ϵ}_{n}\left(M,f\right)=\parallel \phantom{\rule{0.166667em}{0ex}}f-{f}_{M}{\parallel }^{2}=\sum _{m\in \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}/{\Lambda }_{T}}{|⟨\phantom{\rule{0.166667em}{0ex}}f,{g}_{m}⟩|}^{2}.\end{array}$

[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 $\phantom{\rule{0.166667em}{0ex}}f$ , relative to a dictionary, that is a wavelet basis in this example.

[link] (d) gives the nonlinear wavelet approximation $\phantom{\rule{0.166667em}{0ex}}{f}_{M}$ recovered from the $M=N/16$ large-amplitude wavelet coefficients, with an error $\parallel \phantom{\rule{0.166667em}{0ex}}f-{f}_{M}{\parallel }^{2}/{\parallel \phantom{\rule{0.166667em}{0ex}}f\parallel }^{2}=5\phantom{\rule{0.45pt}{0ex}}×{10}^{-3}$ . 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  $\phantom{\rule{0.166667em}{0ex}}{f}_{M}$ :

${\overline{f}}_{M}\left(x\right)=\sum _{n=0}^{N-1}{f}_{M}\left[n\right]\phantom{\rule{0.166667em}{0ex}}{\phi }_{s}\left(x-ns\right).$

Since all projections are orthogonal, the overall approximationerror on the original analog signal $\phantom{\rule{0.166667em}{0ex}}\overline{f}\left(x\right)$ is the sum of the analog sampling error and the discrete nonlinear error:

$\parallel \phantom{\rule{0.166667em}{0ex}}\overline{f}-\phantom{\rule{0.166667em}{0ex}}{\overline{f}}_{M}{\parallel }^{2}=\parallel \phantom{\rule{0.166667em}{0ex}}\overline{f}-\phantom{\rule{0.166667em}{0ex}}{\overline{f}}_{N}{\parallel }^{2}+{\parallel \phantom{\rule{0.166667em}{0ex}}f-\phantom{\rule{0.166667em}{0ex}}{f}_{M}\parallel }^{2}={ϵ}_{l}\left(N,f\right)+{ϵ}_{n}\left(M,f\right).$

In practice, N is imposed by the resolution of the signal-acquisition hardware, and M is typically adjusted so that ${ϵ}_{n}\left(M,f\right)\phantom{\rule{0.222222em}{0ex}}\ge \phantom{\rule{0.222222em}{0ex}}{ϵ}_{l}\left(N,f\right)$ .

are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
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
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
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
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 ?
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Got questions? Join the online conversation and get instant answers!