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.
Frame analysis and synthesis
Suppose that a sparse family of vectors
${\left\{{\phi}_{p}\right\}}_{p\in \Lambda}$ has
been selected to approximate a signal
$\phantom{\rule{0.166667em}{0ex}}f$ .
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.
In a
dual-synthesis problem, the orthogonal projection
$\phantom{\rule{0.166667em}{0ex}}{f}_{\lambda}$ of
$\phantom{\rule{0.166667em}{0ex}}f$ in
V
_{λ} must be computed from dictionary coefficients,
${\{\u27e8\phantom{\rule{0.166667em}{0ex}}f,{\phi}_{p}\u27e9\}}_{p\in \lambda}$ , provided by an analysis operator.
This is the case when a signal transform
${\{\u27e8\phantom{\rule{0.166667em}{0ex}}f,{\phi}_{p}\u27e9\}}_{p\in \Gamma}$ 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.
In a
dual-analysis problem, the decomposition
coefficients of
$\phantom{\rule{0.166667em}{0ex}}{f}_{\lambda}$ must be computed
on a family of selected vectors
${\left\{{\phi}_{p}\right\}}_{p\in \Lambda}$ . 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
${\left\{{\phi}_{p}\right\}}_{p\in \Lambda}$ is a frame of the space
V it generatesif there exists
$B\ge A>0$ such that
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
${\left\{{\tilde{\phi}}_{p}\right\}}_{p\in \Lambda}$ that solves both the
dual-synthesis and dual-analysisproblems:
The frame bounds
A and
B are redundancy factors.
If the vectors
${\left\{{\phi}_{p}\right\}}_{p\in \Gamma}$ are normalized and
linearly independent, then
$A\le 1\le B$ . Such a dictionary
is called a
Riesz basis of
V and the dual frame is biorthogonal:
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?
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
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?
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
Google Play and the Google Play logo are trademarks of Google Inc.
Notification Switch
Would you like to follow the 'A wavelet tour of signal processing, the sparse way' conversation and receive update notifications?