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
A closed circulatory system is a closed-loop system, in which blood is not free in a cavity. Blood is separate from the bodily interstitial fluid and contained within blood vessels. In this type of system, blood circulates unidirectionally from the heart around the systemic circulatory route, and th
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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