# 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

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
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
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
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
what is the Synthesis, properties,and applications of carbon nano chemistry
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?
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.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
how did you get the value of 2000N.What calculations are needed to arrive at it
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