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This collection comprises Chapter 1 of the book A Wavelet Tour of Signal Processing, The Sparse Way(third edition, 2009) by Stéphane Mallat. The book's website at Academic Press ishttp://www.elsevier.com/wps/find/bookdescription.cws_home/714561/description#description The book's complementary materials are available athttp://wavelet-tour.com

In natural languages, large dictionaries are needed to refine ideas with short sentences, and they evolve with usage.Eskimos have eight different words to describe snow quality , whereas a single word is typically sufficient in a Parisian dictionary.Similarly, large signal dictionaries of vectors are needed to constructsparse representations of complex signals. However, computing and optimizing a signal approximation by choosingthe best M dictionary vectors is much more difficult.

A local cosine basis divides the time axis with smooth windows g p ( t ) and translates these windows into frequency.

Frame analysis and synthesis

Suppose that a sparse family of vectors { φ p } p Λ has been selected to approximate a signal f . An approximation can be recoveredas an orthogonal projection in the space V λ generated by these vectors. We then face one of the following twoproblems.

  1. In a dual-synthesis problem, the orthogonal projection f λ of f in V λ must be computed from dictionary coefficients, { f , φ p } p λ , provided by an analysis operator.  This is the case when a signal transform { f , φ p } p Γ is calculated in some large dictionary and a subset of inner products are selected. Such inner products may correspond tocoefficients above a threshold or local maxima values.
  2. In a dual-analysis problem, the decomposition coefficients of f λ must be computed on a family of selected vectors { φ p } p Λ . This problem appears when sparse representation algorithmsselect vectors as opposed to inner products. This is the case forpursuit algorithms, which compute approximation supports in highly redundant dictionaries.

The frame theory gives energy equivalence conditions to solve both problems with stable operators.A family { φ p } p Λ is a frame of the space V it generatesif there exists B A > 0 such that

h V , A h 2 m λ | h , φ p | 2 B h 2 .

The representation is stable since any perturbation of frame coefficientsimplies a modification of similar magnitude on h . Chapter 5 proves that the existence of a dual frame { φ ˜ p } p Λ that solves both the dual-synthesis and dual-analysisproblems:

f λ = p λ f , φ p φ ˜ p = p λ f , φ ˜ p φ p .

Algorithms are provided to calculate these decompositions.The dual frame is also stable:

f V , B - 1 f 2 m γ | f , φ ˜ p | 2 B - 1 f 2 .

The frame bounds A and B are redundancy factors. If the vectors { φ p } p Γ are normalized and linearly independent, then A 1 B . Such a dictionary is called a Riesz basis of V and the dual frame is biorthogonal:

( p , p ' ) λ 2 , φ p , φ ˜ p ' = δ [ p - p ' ] .

When the basis is orthonormal, then both bases are equal. Analysis and synthesis problems are then identical.

The frame theory is also used to construct redundant dictionaries that define complete, stable, and redundant signal representations,where V is then the whole signal space. The frame bounds measure the redundancy of such dictionaries.Chapter 5 studies the construction of windowed Fourier and wavelet frame dictionaries by samplingtheir time, frequency, and scaling parameters, while controlling frame bounds.In two dimensions, directional wavelet frames include wavelets sensitive to directional image structures such as textures or edges.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
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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