# 0.3 Approximation and processing in bases  (Page 4/5)

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## Sparsity with regularity

Sparse representations are obtained in a basis that takes advantage of some form of regularity of the input signals,creating many small-amplitude coefficients. Since wavelets have localized support,functions with isolated singularities produce few large-amplitude wavelet coefficients in the neighborhoodof these singularities. Nonlinear wavelet approximation produces a small error over spaces of functions that do not have “too many”sharp transitions and singularities. Chapter 9 shows that functionshaving a bounded total variation norm are useful models for images with nonfractal (finite length) edges.

Edges often define regular geometric curves. Waveletsdetect the location of edges but their square support cannot take advantage of their potential geometric regularity.More sparse representations are defined in dictionaries of curvelets or bandlets, which have elongated support in multiple directions,that can be adapted to this geometrical regularity. In such dictionaries, the approximation support λ T is smaller but provides explicit information about edges' local geometricalproperties such as their orientation. In this context, geometry does not just apply to multidimensional signals.Audio signals, such as musical recordings, also have a complex geometric regularity in time-frequency dictionaries.

## Compression

Storage limitations and fast transmission through narrow bandwidth channels require compression ofsignals while minimizing degradation. Transform codes compress signals by coding a sparse representation.Chapter 10 introduces the information theory needed to understand these codes and to optimize their performance.

In a compression framework, the analog signal has already been discretized intoa signal $\phantom{\rule{0.166667em}{0ex}}f\left[n\right]$ of size N . This discrete signal isdecomposed in an orthonormal basis $\mathcal{B}={\left\{{g}_{m}\right\}}_{m\in \Gamma }$ of ${\mathbb{C}}^{N}$ :

$f=\sum _{m\in \Gamma }⟨\phantom{\rule{0.166667em}{0ex}}f,{g}_{m}⟩\phantom{\rule{0.166667em}{0ex}}{g}_{m}.$

Coefficients $⟨\phantom{\rule{0.166667em}{0ex}}f,{g}_{m}⟩$ are approximated by quantized values $Q\left(⟨\phantom{\rule{0.166667em}{0ex}}f,{g}_{m}⟩\right)$ . If Q is auniform quantizer of step Δ , then $|x-Q\left(x\right)|\le \Delta /2$ ; and if $|x|<\Delta /2$ , then $Q\left(x\right)=0$ . The signal $\phantom{\rule{0.166667em}{0ex}}\stackrel{˜}{f}$ restored from quantized coefficients is

$\stackrel{˜}{f}=\sum _{m\in \Gamma }Q\left(⟨\phantom{\rule{0.166667em}{0ex}}f,{g}_{m}⟩\right)\phantom{\rule{0.166667em}{0ex}}{g}_{m}.$

An entropy code records these coefficients with R bits. The goal is to minimize the signal-distortion rate $d\left(R,f\right)=\parallel \phantom{\rule{0.166667em}{0ex}}\stackrel{˜}{f}{-f\parallel }^{2}$ .

The coefficients not quantized to zero correspond to the set ${\lambda }_{T}=\left\{m\in \gamma :|⟨\phantom{\rule{0.166667em}{0ex}}f,{g}_{m}⟩|\ge T\right\}$ with $T=\Delta /2$ . For sparse signals, Chapter 10shows that the bit budget R is dominated by the number of bitsto code λ T in γ , which is nearly proportional to its size $|{\lambda }_{T}|$ . This means that the “information” about a sparse representationismostly geometric. Moreover, the distortion isdominated by the nonlinear approximation error $\parallel \phantom{\rule{0.166667em}{0ex}}f-{f}_{{\Lambda }_{T}}{\parallel }^{2}$ , for $\phantom{\rule{0.166667em}{0ex}}{f}_{{\Lambda }_{T}}={\sum }_{m\in {\lambda }_{T}}⟨\phantom{\rule{0.166667em}{0ex}}f,{g}_{m}⟩{g}_{m}$ . Compression is thus a sparse approximation problem. For a given distortion $d\left(R,f\right)$ , minimizing R requires reducing $|{\lambda }_{T}|$ and thus optimizing the sparsity.

The number of bits to code Λ T can take advantage of any prior information on the geometry. [link] (b) shows that large wavelet coefficients are not randomly distributed. They have a tendency to be aggregated towardlarger scales, and at fine scales they are regrouped along edge curves or in texture regions. Using suchprior geometric models is a source of gain in coders such as JPEG-2000.

#### Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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
Google
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
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
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
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
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
what is a peer
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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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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 By Brooke Delaney By Michael Nelson By OpenStax By Samuel Madden By Cath Yu By Jugnu Khan By Frank Levy By Robert Murphy By OpenStax By Jonathan Long