Years of wavelet developments have generated an inflation of
"wavelet-like" names. They are generally built in a diminutive form based on the suffix "-let" or "-lette".
Hence the term "starlet", from the "★let" wildcard combination,
and the ★-(star)-like status of wavelets in signal or image
processing, as well as in many other fields. More generally,
suffixes -et, -ette, -let, -ling, and -ule reffer to "little". A very tiny wavelet could then be baptised "lingulet".
And a generic one a starling, the globish form for the more common étourneau in French.
Étournellette, what a beautiful, beautiful name...
"WITS: Where is the starlet?" stands here for an approximate
translation of the basic French sentence "Où
est l'étoilette ?" In French again, many synonyms exist, such as
"le petit coin" (somewhat equivalent to "de la menue monnaie",
for the simple "change" in English). Now we have an approximation,
what are the details? What kinds of "★let" names exist? What do
they mean? A first (obvious) answer is provided by Wim Seldens, in the
introduction for his PhD thesis in 1994:
The following provides a quick reference to numerous wavelet
names and some of their contributors. Of course, it cannot be
exhaustive, and should be considered only as a starting point.
Some names are not exactly wavelets (but what is a wavelet
exactly?), but belong to this domain. Given properties are stated
in a very coarse sense, and should not be taken as 100% accurate.
However, corrections and especially additions are very
welcome (send a message to lcd (ad) ieee (dod) org).
| In short: |
The mother (wavelet) of them all (see below) |
| Etymology: |
The "-lette" (or "-let") suffix association generally means "petite" ("small"). "Ondelette" is built upon "onde" (French for "wave"). It thus means "small
wave", hence "wavelet". The "-let" suffix is somewhat about decay |
| Origin: |
It is often attributed to Jean Morlet, engineer at the (late)
French oil company Elf-Aquitaine, now merged within Total (personal note: ELF used to be associated (apocryphly) Essences et Lubrifiants de France). The
most famous references arise from the collaboration of Alex
Grossman and Jean Morlet, Decomposition of Hardy functions
into square integrable wavelets of constant shape , SIAM
Journal of Mathematical Analysis, vol. 15, no. 4, pp. 723-736,
July 1984.
Some earlier works need be mentioned:
- Morlet, J., Arens, G., Fourgeau, I. and Giard, D., Wave
propagation and sampling theory, Geophysics, 47, pp.
203-236, 1982
- Morlet, J., Sampling theory and wave propagation, in
NATO ASI Series, vol. 1, Issues in Acoustics signal/Image
processing and recognition, C. H. Chen (Ed.), Springer Verlag,
pp. 233-261, 1983
First mention of wavelets by Jean Morlet himself might have been given at a
geophysicist Conference (SEG) in 1975 in Denver, CO, USA, under the title Seismic
tomorrow, interferometry and Quantum Mechanics. A mere 25-line
abstract remains.
|
| Contributors: |
Probably too many to mention, with the great risk of
forgetting some of them. |
| Some properties: |
Basically, wavelets are basis functions that are localized both in time (or
spaces of higher dimension) and frequency. Wavelet atoms are generally related
by scale properties. |
| Anecdote: | The term wavelet is ubiquituous in the field on geosphysics, more
specifically in reflection seismology. It refers to the seismic pulse (once
called impulsion sismique in French) sent through the ground
subsurface in order to detect (after its reflections on interfaces) earth "
structures". Its accurate determination is thus crucial for the wavefield
deconvolution. The word wavelet is attested in early works
such as the one by N. Ricker, A note on the determination of the viscocity
of shale from the measurement of the wavelet breadth, Geophysics, Society
of Exploration Geophysicists, vol. 06, pp. 254-258, 1941. The Ricker wavelet
(aka the Mexican hat) is often used in geophysics modelling. The first known
wavelet basis (under a different name) is the Haar basis, for instance in Haar,
A., Zur Theorie der orthogonalen Funktionen-Systeme, Math.
Ann., vol. 69, pp. 331-371, 1910. Early nearly wavelets include Philip Franklin's
construction of piecewise polynomial orthonormal splines on a bounded
interval (1928), taken to its asymptotics on the whole line by J.-O. Strömberg (1981).
For other earlier wavelet bases (indeed including Haar, Franklin and Strömberg systems), read a nice paper by
Hans G. Feichtinger,
Precursors in
mathematics: early wavelet bases [pdf].
The concept of "wavelet" in the sense of a small light pulse also
appears in Christian Huygens's (Dutch physicist) light propagation theory. The
term was apparently introduced by Huygens in 1678, but this matter needs
further investigations.
It has been widely recognized that wavelets have aggregated
numerous works from the fields of harmonic analysis, coherent
states in quantum mechanics, electrical engineering or computer
vision.
2005/05/25: i have just discovered that many
french speaking people use "ondulette" instead of "ondelette". It probably comes
form the verb "onduler".
But some googling tells you quite fast that the term is also used for
certain types of "stores" ("Venetian Blind"). This deserves further investigation.
|
| Usage: |
Probably too many to mention, considering the great risk of
forgetting some of them. |
| See also: |
There are many information sources, either books, articles,
web sites or even bed-time stories. We shall mention here the DMOZ
Open Directory - Science: Math: Numerical Analysis: Wavelets,
the Wavelet Digest, which contributes a lot
to the diffusion of wavelet related information.
The Wikipedia: wavelet transform
provides useful links on wavelets.
A recent article, La surprenante ascension des ondelettes, in the
La Recherche monthly (number 383, Feb. 2005, p. 55--59) by Mathieu
Nowak and Yves Meyer recalls the early days of the wavelet and
its recent applications. |
| Comments: |
Sources for wavelet and wavelet packets code: Wavelab 850 (Matlab 6.x or 7), C++ Source Code for the Wavelet Packet Transform, WAILI - Wavelets with Integer Lifting, with WAILI.xl, an extension for very large images,
YAWTB: "Yet Another Wavelet Toolbox" (Matlab),
Computational Toolsmiths, WavBox (Matlab).
Matlab source code for the Ricker wavelet
|
| In short: |
Non-linear and non-parametric estimator of additive models
with wavelets |
| Etymology: |
Additive Model
wavelet estimator (also with a
Robust extension) |
| Origin: |
Sardy, Sylvain and Tseng, Paul,
AMlet and GAMlet: Automatic Nonlinear Fitting of Additive Models and Generalized Additive Models with Wavelets, Journal of Computational and Graphical Statistics,
2004, [ps] |
| Contributors: |
Sylvain Sardy, Paul Tseng |
| Some properties: |
Provides universal thresholding rules for Gaussian and Poisson
distributions |
| Anecdote: |
|
| Usage: |
Statistics, fitting of additive models |
| See also: |
Its generalization, called GAMlet |
| Comments: |
Not truly a wavelet by itself |
| In short: |
Orthogonal multiwavelet for which polynomial perturbation of
the input does not affect the wavelet decomposition with highpass
output |
| Etymology: |
Analysis Ready
Multiwavelet |
| Origin: |
Lian, J. A. and Chui, C. K. Analysis-Ready Multiwavelets
(Armlets) for processing scalar-valued signals , Signal
Processing Letters, vol. 11, no. 2, pp. 205-208, Feb. 2004 |
| Contributors: |
Jian-ao Lian, and Charles K. Chui |
| Some properties: |
Defined to satisfy the n th order wavelet
consistency requirement (n -WAC). More general than
n -balanced multiwavelets. Correspond to the
Daubechies orthogonal wavelets (daublets) in the scalar setting |
| Anecdote: |
|
| Usage: |
|
| See also: |
|
| Comments: |
|
| In short: |
2-D multiscale basis vectors adaptively elongated in the direction of (image)
geometric flows |
| Etymology: |
From bandelet, little stripes, generally
made of soft matter (in French bandelette), or
the ring-shaped molding one can find at the top of columns |
| Origin: |
Le Pennec, Erwan and Mallat, Stéphane, Image
compression with geometrical wavelets, International
Conference on Image Processing (ICIP), September 2000,
Vancouver |
| Contributors: |
Erwan Le Pennec, Stéphane Mallat |
| Some properties: |
Bandelets have a support parallel to flow lines in images.
Approximation rate: M -a for
images having discontinuities along Ca contours, and
being Ca away from the contours |
| Anecdote: |
According to one of the authors, most of the obvious names in
"let" were already taken at the time of its invention, making it
difficult to find this one |
| Usage: |
Image coding, denoising, deconvolution, 3D surface compression |
| See also: |
Charles Dossal, for further
bandelet developments, Gabriel Peyré, for the development of second
generation bandelets, and Let it wave, a start-up devoted to
bandelet applications, including low bit-rate id pictures |
| Comments: |
A second-generation Matlab bandelet toolbox is available from Gabriel Peyré at MatlabCentral |
| In short: |
An orthogonal or biorthogonal wavelet designed, through a balanced weighted uncertainty (time and frequency spread)
approach, to improve its coding capabilities |
| Etymology: |
From the University of Bath, School of
Electronic and Electrical Engineering, where the design has been
proposed |
| Origin: |
Orthonormal wavelets with balanced uncertainty, DM Monro, BE Bassil and GJ
Dickson, IEEE International Conference on Image Processing, 1996, Vol.2, pp.581-
584 (local copy).
Space-frequency balance in biorthogonal wavelets,
DM Monro and BG Sherlock, IEEE International Conference on Image Processing,
1997, Vol.1, pp.624-627 (local copy).
|
| Contributors: |
D. M. Monro, B. E. Bassil, G. J. Dickson |
| Some properties: |
Based on an Heisenberg uncertainty metric, efficient FIR
filters are designed to improve image coding, as compared to
maximum regularity filters, via the balancing of both the time
and frequency spread of the function. Provides apparently better
subjective quality than maximum regularity wavelets. |
| Anecdote: |
The word "bathlet" belongs to the Klingon vocabulary (from
the Star Trek space soap opera). It is a personal weapon that
every Klingon carries on with him. You never know! Notice (on the
right) the smoothness of the contours and the sharpness of the
edges. For others bathlet pictures... (Klingonwaffen in german,
what a beautiful, beautiful name)
Trivia: Colorado 7-eleven (7-
11 math problem here) stores fear a Klingon-weaponed robber threatening
clerks with the spiky, crescent shaped Star Trek inspired sword called bat'leth or Klingon's personal
sword of honor. Details at The Denver Channel.
|
|
| Usage: |
Image coding |
| See also: |
The Bath Wavelet Warehouse, for Bath wavelets
coefficient tables,
orthogonal
and biorthogonal wavelet
coefficients. A Where-Is-The-Starlet entry: WITS: Bathlet wavelets from
La vertu d'un LA.
|
| Comments: |
|
| In short: |
Collection of dyadically-organized line segments, occupying a
range of dyadic locations and scales, and occuring at a range of
orientations |
| Etymology: |
From beam a piece of timber used for
construction, or directly beamlet, a small beam
of light |
| Origin: |
Donoho, David and Huo, Xiaoming, Beamlets and Multiscale
Image Processing, 2001, Stanford, Research report |
| Contributors: |
David Donoho, Xiaoming
Huo |
| Some properties: |
|
| Anecdote: |
Beamlet is also the name of a single-beam laser |
| Usage: |
Filament or object boundary extraction in noise. Analysis of
large-scale structures of the Universe, esp. in 3D |
| See also: |
Wedgelets, which share a similar
dyadic recursive decomposition |
| Comments: |
Beamlab: a Matlab (TM) toolbox code for the
implementation of various feature oriented transforms |
| In short: |
Wavelet with "binary" coefficients or generated by "binary"
coefficients filter bank |
| Etymology: |
From the contraction binary (symmetric)
wavelet |
| Origin: |
Strang, G. and Nguyen, T., Wavelets and filter
banks, pp. 217, Wellesley-Cambridge Presss, 1996 |
| Contributors: |
Gilbert Strang, Truong Nguyen, and many others under the name
of reversible wavelets. |
| Some properties: |
DSP-friendly wavelet filter banks with integer coefficients
(like the Haar wavelet) or with the form
c = n/2k (with
n and k integers), up to a normalization
scaling coefficient (sometimes irrational). Such transforms are
easily computed by adds or binary shifts |
| Anecdote: |
Apparently, a 9/7 wavelet filter pair was found by Gilbert
Strang by solving the halfband equation, and discovered later
that Wim Sweldens created earlier a whole family of binary
symmetric filters in 1995. One of them, an integer reversible 5/3
filter bank is used for lossless compression in the JPEG 2000 standard. The
binary 9/7 filters are [1 0 -8 16 46 ...]/64 [-1 0 9 16
...]/64. The Le Gall 5/3 analysis filters [-1 2 6 2 -1]/8 and [-1 2 -1]/3 |
| Usage: |
Binlet are especially useful for finite arithmetic reversible transforms, esp. for lossless compression |
| See also: |
Some other integer-to-integer transforms (Generalized S Transform) have been developed
by Michael Adams, who develops the JPEG 2000
JasPer project |
| Comments: |
Often used in "the 9/7 binlet" expression. Also used for the Haar
wavelet, some biorthogonal spline wavelets; also used for the S+P
transform from A. Said and W. Pearlman SPIHT image compression
and other (NB: the S+P transform is non-linear). Thus, binlet is a relatively ill-defined term. "Binary" structures may be
generated by the lifting scheme, developed by Wim
Sweldens in 1995 |
| In short: |
Biorthogonal basis with good spatial localization and precise
localization, providing a decomposition with different
orientations, frequencies, sizes and positions |
| Etymology: |
From brush, from the brush stroke aspect of
the 2-D tensor products |
| Origin: |
Meyer, François G. and Coifman, Ronald R.,
Brushlets: a tool for directional image analysis and image
compression, Applied and Computational Harmonic Analysis,
vol. 4, pp. 147-187, 1997 |
| Contributors: |
François G. Meyer, Ronald R. Coifman
Lasse
Borup |
| Some properties: |
Works directly in the Fourier domain |
| Anecdote: |
|
| Usage: |
Image coding (esp. for highly textured images) |
| See also: |
|
| Comments: |
Applied for denoising and segmentation of cardiac ultrasound |
| In short: |
A blend of standard MRA (multiresolution analysis), framelets and hierarchical bases, based of a set
of three filters, a lowpass decomposition, a lowpass prediction
and an alignment filter |
| Etymology: |
From the contraction of Coarsification,
Alignment, Prediction (in the
first papers). More recent works use CAP for
Compression, Alignment,
Prediction, and CAMP for
Compression, Alignment,
Modified Prediction |
| Origin: |
Ron, A. Caplets: wavelets without wavelets, 29th
Annual Spring Lecture Series, Recent Developments in Applied
Harmonic Analyis, Multiscale Geometric Analysis, April 15-17,
2004 |
| Contributors: |
Amos Ron, Youngmi Hur
(Univ. Wisconsin) |
| Some properties: |
Caplet coefficients provide characterization of function
spaces analogous to wavelet's. Redundant description, with
redundancy decreasing with the spatial dimension. |
| Anecdote: |
Caplet information is hard to find on the Internet, since it
is often mixed with advertising on medicines (tablets), especially
on Amazon web pages. See for instance the answer for a Google
search on wavelet and caplet, performed on
2005/02/02.
Amazon.com: Editorial Reviews: Multirate and Wavelet Signal ...
... Customers interested in Multirate and Wavelet Signal Processing ...
in ... Aleve All Day Strong Pain Reliever, Fever Reducer, Caplet, 100-pack ...
www.amazon.com/exec/obidos/tg/ detail/-/0126775605?v=glance&vi=reviews ...
|
| Usage: |
|
| See also: |
Hur, Yougmi and Ron, Amos, CAPlets: wavelet representations without wavelets [pdf] |
| Comments: |
|
| In short: |
A windowed portion of a chirp |
| Etymology: |
From chirp, an oscillating function whose
"period" varies with the variable (e.g. time) position |
| Origin: |
Mann, Steve and Haykin, Simon, The chirplet transform: a
generalization of Gabor's Logon transform, Proc. Vision
Interface'91, June 3-7, pp. 205-212, 1991.
Mihovilovic, D. and Bracewell, R., Adaptive chirplet
representation of signals on time-frequency plane,
Electronic Letters, 27(13), pp. 1159-1161, June 1991. |
| Contributors: |
Steve
Mann and Simon Haykin
Domingo Mihovilovic and Ronald Bracewell | ((wiki))
| Some properties: |
Offers a mapping from a continuous function of one real
variable to a continuous function of 5-6 real variables.
Quadratic (aot. linear) chirplets are also of interest for radar
applications. Adaptive or even |
| Anecdote: |
The chirplet formulation was motivated by the discovery that
the Doppler radar backscatter from a small piece of ice floating
in an ocean environment is chirp-like. Examples of chirps are the
sounds made by birds where the resonant cavity changes size while
oscillating |
| Usage: |
Radar applications, projective geometry acting on a periodic
structure (e.g. arcanes in a perspective
picture) |
| See also: |
Several publications on chirplets on
Steve Mann's page |
| Comments: |
The "independent" discovery and naming controversy of
chirplets by two groups at about the same time is not even
discussed here |
| In short: |
Orthogonal compactly supported wavelet with vanishing moments
equally distributed for the scaling function and the wavelet |
| Etymology: |
Contraction from the name of R. R.
Coifman |
| Origin: |
Daubechies, Ingrid, Orthonormal bases of compactly
supported wavelet II. Variations on a theme, SIAM, J. Math.
Anal., vol. 24, no. 2, pp. 499-519, March 1993 |
| Contributors: |
Ingrid Daubechies |
| Some properties: |
For p vanishing moments, the minimum support size
of the wavelet is 3p-1 (instead of 2p-1 for
Daubechies wavelets). Scaling functions with vanishing moments
help establish precise quadrature formulas |
| Anecdote: |
In 1989, R. Coifman proposed the idea of constructing
orthogonal wavelets with vanishing moments equally distributed
for the scaling function and wavelet |
| Usage: |
Numerical analysis |
| See also: |
Other classical compactly supported orthogonal Daubechies
wavelets (aka daublet),
with minimum phase property or the nearly symmetric symmlets. |
| Comments: |
|
| In short: |
A discrete domain wavelet-like expansion allowing contour
description, based on a Laplacian pyramid and a directional
filter bank |
| Etymology: |
|
| Origin: |
Do, M. N. and Vetterli, M. Contourlets: A Directional
Multiresolution Image Representation, Proc. of IEEE
International Conference on Image Processing ( ICIP), Rochester,
September 2002 |
| Contributors: |
Minh N. Do, Martin
Vetterli, with Arthur L. Cunha and
Jianping Zhou for the contourlet nonsubsampled version, and
Yue Lu for the critically sampled CRISP-contourlet |
| Some properties: |
Approximation rate: M -2(log
M)3 for images having discontinuities along
C2 curves. Slightly redundant due to the Laplacian
pyramid. |
| Anecdote: |
|
| Usage: |
Image coding, denoising |
| See also: |
The CRISP-contourlet, a critically sampled avatar (by Y. Lu
and M. N. Do, SPIE 2003) |
| Comments: |
Contourlet toolbox Matlab code available at
www.ifp.uiuc.edu/~minhdo/software/, with a
Nonsubsampled Contourlet Transform Matlab toolbox at MatlabCentral
|
| In short: |
Multiscale elongated and rotated functions that defines (bases
or) frames in L2(R2) |
| Etymology: |
Simply from curved wavelets |
| Origin: |
Candès, E. J. and Donoho, D. L., Curvelets --- a
surprinsingly effective nonadative representation for objects
with edges, in Curve and Surface fitting, A. Cohen, C. Rabut
and L. L. Schumaker (Eds.), 1999 |
| Contributors: |
Emmanuel Candès, David
Donoho, Jean-Luc Starck
Laurent Demanet
|
| Some properties: |
Approximation rate: M -2(log
M)3 for images having discontinuities along
C2 curves |
| Anecdote: |
|
| Usage: |
|
| See also: |
|
| Comments: |
Curvelets have evolved both in concept and
implemetation since the earlier works, dealing with what's now
called "curvelets 99", which relied to some extend on ridgelets.
Second generation curvelet code is available at http://www.curvelet.org, with version 2.0 |
| In short: |
Element for a collection of edgels (small line segments)
connecting vertices on the boundary of a dyadic square |
| Etymology: |
From edge or edgel, an
edge element in the computer vision
literature |
| Origin: |
David L. Donoho, Manuscript, Stanford
University, Fast edgelet transform and applications, Manuscript,
September 1998 |
| Contributors: |
David Donoho |
| Some properties: |
|
| Anecdote: |
|
| Usage: |
|
| See also: |
|
| Comments: |
Edgelets might be combined with wavelet for an overcomplete
image representation, as in Donoho, D. and Huo, X., Combined
Image representation using edgelets and wavelets ??? |
|
In short:
|
A basis made of M adjacent box function scalets (scaling functions)
and $M$ piecewise constant functions with $M$ vanishing moments
|
|
Etymology:
|
From flat, meaning... flat, and again, let |
|
Origin:
|
Steven J. Gortler, Peter Schröder, Michael F. Cohen, Pat Hanrahan
Wavelet radiosity, Computer Graphics, SIGGRAPH 1993
|
|
Contributors:
|
Steven J. Gortler,
Peter Schröder,
Michael F. Cohen,
Pat Hanrahan,
|
|
Some properties:
|
For the given example, 2 rows of the two-scale relationship
are orthogonal to constant and linear variations |
|
Anecdote:
|
|
|
Usage:
|
Sparse basis for hierarchical radiosity formulation,
to solve the global illumination problem |
|
See also:
|
|
|
Comments:
|
|
| In short: |
Element of a wavelet frame or the wavelet frame by itself |
| Etymology: |
From frame, an extension from the (vector) base concept |
| Origin: |
Ingrid Daubechies, Bin Han, Amos Ron, Zuowei Shen,
Framelets: MRA-Based Constructions of Wavelet Frames
(local copy), 2000
|
| Contributors: |
Ramesh A. Gopinath (phaselets of
framelets) |
| Some properties: |
|
| Anecdote: |
The framelet term was also introduced in the field of
software framework to designate non-overlapping groups of
logically related design patterns and interfaces. Those
interested could take a look at Alessandro Pasetti homepage. |
| Usage: |
|
| See also: |
Many developments on framelets (inpainting, deconvolution, restoration, missing samples recovery) by Zuowei Shen and co-authors, for instance in Jianfeng Cai, Raymond Chan, Lixin Shen, Zuowei Shen,
Convergence analysis of tight framelet approach for missing data recovery, Advances in Computational Mathematics,xx (200x)
or in Anwei Chai, Zuowei Shen,
Deconvolution: A wavelet frame approach, Numerische Mathematik, 106 (2007), 529-587
|
| Comments: |
|
| In short: |
Wavelet-like basis made of a wavelet basis combined with a
unitary Fresnel transform. |
| Etymology: |
From the Fresnel transform, after the name
of physicist Augustin Jean Fresnel (MacTutor History) |
| Origin: |
Liebling, M., Blu, T., Unser, M., Fresnelets — A
New Wavelet Basis for Digital Holography, Proceedings of the
SPIE Conference on Mathematical Imaging: Wavelet Applications in
Signal and Image Processing IX, San Diego CA, USA, July 29-
August 1, 2001, vol. 4478, pp. 347-352 |
| Contributors: |
Michael Liebling, Thierry Blu, Michael
Unser |
| Some properties: |
|
| Anecdote: |
|
| Usage: |
Reconstruction and processing of optically generated Fresnel holograms recorded on CCD-arrays |
| See also: |
Liebling, M., Blu, T., Unser, M., Fresnelets: New Multiresolution Wavelet bases for digital holography, Proceedings of the
IEEE Transactions on Image processing, vol. 12, no. 1, January 2003
[pdf] |
| Comments: |
|
| In short: |
Complex exponentials modulated by a "smooth" function,
originally a Gaussian |
| Etymology: |
From the name of the godfather Denis Gabor,
and especially his Theory of Communication paper,
Journal of the IEE, vol. 93, pp. 429-457, 1946 |
| Origin: |
Not clear, but named in some papers, esp. by Bruno
Torrésani, Time-frequency and time-scale
analysis, Signal Processing for multimedia, J. S. Byrnes
(Ed.), IOS Press, 1999 |
| Contributors: |
Bruno Torrésani |
| Some properties: |
|
| Anecdote: |
|
| Usage: |
|
| See also: |
|
| Comments: |
|
| In short: |
Non linear and non-parametric estimator of generalized
additive models with wavelets |
| Etymology: |
Generalized Additive
Model wavelet estimator |
| Origin: |
Sardy, Sylvain and Tseng, Paul, Automatic Nonlinear
Fitting of Additive Models and Generalized Additive Models with
Wavelets, Journal of Computational and Graphical Statistics,
2004 (submitted) |
| Contributors: |
Sylvain Sardy, Paul Tseng |
| Some properties: |
Universal thresholding rule for Gaussian and Poisson
distributions |
| Anecdote: |
|
| Usage: |
Fitting of generalized additive models |
| See also: |
Its simpler version, called AMlet |
| Comments: |
Not truly a wavelet by itself |
| In short: |
|
| Etymology: |
From the famous mathematician Johann Carl Friedrich
Gauss (MacTutor History), and the ubiquituous bell
curve named after him. Gauss is also believed to have discovered
the Fast Fourier Transform (FFT algorithm) |
| Origin: |
Triebel H. Towards a Gausslet analysis : Gaussian
representations of functions. In M. Cwikel, M. Englis, A.
Kufner, L.-E. Persson, and G. Sparr, editors, Function Spaces,
Interpolation Theory and Related Topics. Proc. Conf. Lund, August
2000, 425-450, de Gruyter Proceedings, 2002. |
| Contributors: |
Hans Triebel |
| Some properties: |
|
| Anecdote: |
|
| Usage: |
|
| See also: |
|
| Comments: |
|
| In short: |
Biorthogonal wavelet with the Hut function
as the father wavelet |
| Etymology: |
From Hut, German for hat |
| Origin: |
Meyer-Bäse, Uwe Die Hutlets - eine
biorthogonale Wavelet-Familie: Effiziente Realisierung durch
multipliziererfreie, perfekt rekonstruierende Quadratur Mirror
Filter , Frequenz., vol; 51, p. 39-49, 1997, also in
Meyer-Bäse, Uwe and Taylor, F., The Hutlets - a biorthogonal
wavelet family and their high speed implementation with RNS,
multiplier-free, perfect reconstruction QMF |
| Contributors: |
|
| Some properties: |
The Hut function has an asymptotically
fast decrease in amplitude. Multiplier-free implementation with
the residue number system (RNS). Synthesis filters are IIR |
| Anecdote: |
Notice the first author name; is Meyer-Bäse related to
the Meyer wavelet basis? |
| Usage: |
Envelope discontinuity detection in amplitude modulation |
| See also: |
|
| Comments: |
The Hut function was defined by W. Hilberg, Impulse und Impulsfolgen, die durch Integration oder
Differentiation in einem veränderten Zeitmasstab
reproduziert werden, Arch. für Eltr. Übertr.
(AEÜ), vol. 25, pp. 39-48, 1971. It results from the
infinite convolution of rectangles with area one
(2k/T)r(T/2
k), k varying from 1 to infinity |
| In short: |
Interpolating wavelet transform |
| Etymology: |
let |
| Origin: |
Apparently, Donoho, D. L. (once again), Interpolating wavelet transforms 1992, preprint, although the name "interpolet" itself has been coined later. |
| Contributors: |
David Donoho |
| Some properties: |
Loosely speaking, based on the autocorrelation of some scaling function or interpolating filter |
| Anecdote: |
Early mention of interpolets is found in "Savior of the Nations, Come"
by St. Ambrose, (340-397). Seventh verse:
Praesepe iam fulget tuum,
lumenque nox spirat suum,
quod nulla nox interpolet
fideque iugi luceat.
|
| Usage: |
|
| See also: |
|
| Comments: |
|
| In short: |
Properties of filter sets used in local structure estimation that are the most important
are provided via the introduction of a number of fundamental invariances. Mathematical formulations
corresponding to the required invariances leads up to the introduction of a new class of filter sets termed loglets. Loglets
are polar separable and have excellent uncertaintyproperties. The directional part uses a spherical harmonics basis.
Using loglets it is shown how the concepts of quadrature and phase can be defined in n-dimensions. It is also shown how
a reliable measure of the certainty of the estimate can be obtained byfinding the deviation from the signal model
manifold. |
| Etymology: |
From
Logarithmic wavelets |
| Origin: |
Knutsson, Hans and Andersson, Mats, Loglets - Generalized Quadrature and Phase for
Local Spatio-temporal Structure Estimation, 2003, Scandinavian Conference on Image Analysis
Knutsson, Hans and Andersson, Mats, Implications of invariance and uncertaintyfor
local structure analysis filter sets, 2005, Signal Processing: Image Communication |
| Contributors: |
Hans Knutsson
Mats Andersson |
| Some properties: |
Polar separable filter banks in the Fourier domain |
| Anecdote: |
| Usage: |
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| See also: |
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| Comments: |
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| In short: |
A sort of M-band wavelet |
| Etymology: |
From
MIMO (Multiple-input/Multiple-ouput) systems generating wavelets |
| Origin: |
The netherlands, the other cheese country |
| Contributors: |
Will remain anonymous (none of the famous dutch wavelet school) |
| Some properties: |
Wavelets with frequencies in the orange tones. |
| Anecdote: |
| Usage: |
Tasteful for RAClet and TARTIFlet recomposition (pun borrowed from "TB from CH", aka "TB from HK").
M-band wavelets (such as the dual-tree wavelets, see M-band dual-tree and discrete complex wavelets, a blog entry: PhD thesis award on M-band dual-tree wavelets
or Wikipedia, Complex Wavelet) in filter bank form, since they are related to the LOT (Lapped Orthogonal Transform),
may be called "bancs de
LOT(tes)" ("lote/lotte" the fish, not the transform) in french |
| See also: |
A recent MIMOlet preprint |
| Comments: |
Still waiting for SISOlets, MISOlets and SIMOlets |
| In short: |
Short name for the Morlet wavelet |
| Etymology: |
A clever combinaison of the father Jean
Morlet and the mother
wavelet |
| Origin: |
Misprint found in a preprint |
| Contributors: |
Will remain anonymous |
| Some properties: |
No parent-child dependency known to date. Its dual basis (the
lesslet?) remains to be described (or even defined). |
| Anecdote: |
Morelet is also the name of a crocodile, Crocodylus moreletii,
from the French naturalist P. M. A. Morelet (1809-1892), who
discovered this species in 1850 in Mexico |
| Usage: |
The Morelet wavelet is becoming increasingly popular due to
three typical wavelet phenomena:
- keyboard aliasing (with typing frequency and proximity of the
keys "e" and "r" on Azerty and Qwerty keyboards),
- dilation of the number of people working with wavelets,
- translation (and replication) due to (erroneous)
citations.
The present page proudly adds its conscious contribution. |
| See also: |
The future invention of the lesslet |
| Comments: |
(Relative) fun exists in DSP, as in the invention of Softy
space (see Hardy spaces), or in company names like Let it
wave |
|
In short:
|
A breed of spherical wavelets |
|
Etymology:
|
From their needle shape + let |
|
Origin:
|
P. Baldi, G. Kerkyacharian, D. Marinucci, D. Picard, Asymptotics for Spherical Needlets
Also in D. Marinucci, D. Pietrobon, A. Balbi, P. Baldi, P. Cabella,
G. Kerkyacharian, P. Natoli, D. Picard, N. Vittorio,
Spherical Needlets for CMB Data Analysis
(arXiv page)
|
|
Contributors:
|
|
|
Some properties:
|
Do not rely on any tangent plane approximation. Computationally attractive.
Same needlets functions are present in the direct and the inverse transform. Quasi-exponentially cocentrated (hence, the needle shape). Random needlets coefficients can be shown to be
asymptotically uncorrelated
|
|
Anecdote:
|
|
|
Usage:
|
Cosmic Microwave Background (CMB) analysis
|
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See also:
|
|
|
Comments:
|
|
|
In short:
|
Sort of twisted wavelet packets, maximally incoherent system with respect to the Haar wavelet |
|
Etymology:
|
From the signal-processing-ubiquitous noise + let |
|
Origin:
|
R. Coifman, F. Geshwind, and Y. Meyer,
Noiselets, Appl. Comp. Harmonic Analysis, 10:27-44, 2001 (local copy) |
| Contributors: | Ronald Coifman, F. Geshwind, Yves Meyer |
| Some properties: |
Perfectly incoherent with the Haar basis (similar to the perfect incoherence
of the canonical basis with respect to the Fourier basis). Can be decomposed as
a multirate filter bank. Binary valued real and imaginery parts |
| Anecdote: | Mark Noiselet is a make-up artist.
Have a look at this interesting page by artist Michael Thieke:
Very sparse. Very minimal. These musicians make sounds with their instruments
that may not have been intended by the original inventors. They do this in a
way that at first seems to be a very random. After a longer listen, the
inspirations soak through. These “noiselets and sounduals” (my words entirely)
may be improvised, but they are very potent in their expressive capability.
|
| Usage: |
Compressed sensing |
| See also: | The
noiselets have been recently mentioned in a paper by J.-P. Allouche and G.
Skordev, Von Koch and Thue-Morse revisited (arXiv page), which links fractal objects and automatic
sequences, focused on the Thue-Morse sequence and the Von Koch curve. See also:
Sparsity and Incoherence in Compressive Sampling by Emmanuel
Candès and Justin Romberg
|
|
Comments:
|
Basic Noiselet Matlab code for building orthogonal noiselet bases (or Zipped Matlab code (or finally there Zipped Matlab code)) |
| In short: |
Compactly supported basis functions ressembling planar
structures, for the representations of locally planar structures
found in video sequences |
| Etymology: |
From plane, the common name for a flat
surface |
| Origin: |
Rajpoot, N. and Wilson, R. and Yao, Zhen Planelets: A new
analysis tool for planar feature extraction, International
Workshop on Image Analysis for Multimedia Interactive Services
(WIAMIS), 2004 |
| Contributors: |
Nasir Rajpoot, Roland
Wilson, Zhen Yao |
| Some properties: |
Non orthogonal basis and redundant by less than 14% (see the
paper: can a basis really be redundant?) |
| Anecdote: |
|
| Usage: |
Video sequence denoising |
| See also: |
|
| Comments: |
|
| In short: |
Partition based on a recursive, dyadic squares, allowing
wedge-shaped final nodes (instead of squares), with piece-wise
planar value |
| Etymology: |
From Plate, accounting for the piece-wise
planar value |
| Origin: |
Willett, R. M. and Nowak, R. D. Platelets: A Multiscale
Approach for Recovering Edges and Surfaces in Photon-Limited
Medical Imaging, preprint ??? |
| Contributors: |
Rebecca M. Willett, Robert D. Nowak |
| Some properties: |
Well suited for the approximation of images consisting in
smooth regions separated by smooth contours, especially in the
case of Poisson distributions |
| Anecdote: |
Platelets used to be a major component of blood. They are not anymore |
| Usage: |
Analysis, denoising, reconstruction of images, esp. Poisson
distributed (medical imaging) |
| See also: |
The wedgelet, which it
generalizes upon |
| Comments: |
A platelet and wedgelet Matlab toolbox |
| In short: |
Local line whose family is a basis for discrete signals |
| Etymology: |
From the Radon transform (which is performed
along lines), after the Austrian mathematician Johann Radon |
| Origin: |
Do, M. N. and Vetterli, M. The contourlet transform: an
efficient directional multiresolution image representation,
IEEE Transactions Image Processing, vol. 14, no. 12, pp. 2091-2106, Dec. 2005 |
| Contributors: |
Minh N. Do, Martin
Vetterli
|
| Some properties: |
Element of a family having almost linear support and
different orientations, defined by translating some filters over
some sampling lattices |
| Anecdote: |
The radonlet concept represents only an item on the above paper |
| Usage: |
|
| See also: |
|
| Comments: |
|
| In short: |
|
| Etymology: |
Contraction from ripple and
let |
| Origin: |
Tentative origin: Goodman, T. N. T. and Micchelli, C. A.,
On refinement equations determined by Pólya frequency
sequence , SIAM J. Math. Anal., vol. 23, pp. 766-784,
1992 |
| Contributors: |
T. N. T. Goodman, C.
A. Micchelli |
| Some properties: |
|
| Anecdote: |
The concept of a ripplets is an intermediate between concepts
on refinable functions (satisfying a refinement or scaling
equation) and the positivity of the coolocation matrices of their
shifts. The stronger concept is known as a Pólya frequency
function |
| Usage: |
Ripplets are used to build pre-wavelets by F. Pitolli,
Refinement masks of Hurwitz type in the cardinal
interpolation problem, Rendicondi di Matematica, Serie VII,
vol. 18, pp. 473-287, Roma 1998. Ripplet properties are also
valuable in computer-aided geometric design, for instance in
Goodman, T. N. T. Total positivity and the shape of
curves, in Total positivity and its applications,
M. Gasca and C. A. Micchelli (Eds.), p. 157-186, 1996 |
| See also: |
Goodman, T. N. T. and Sun, Q., Total positivity and
refinable functions with general dilation, 2004,
preprint |
| Comments: |
|
| In short: |
2-D set of functions based on the product of a gaussian with
a Hermite (or Laguerre) polynomial (tensor product of 1-D
function) |
| Etymology: |
From shape |
| Origin: |
Refregier, Alexandre and Chang and Bacon, David,
Shapelets: A New Method to Measure Galaxy Shapes.
Proceedings of the Workshop "The Shapes of Galaxies and their
Halos", Yale, May 2001 |
| Contributors: |
Alexandre Refregier, David Bacon |
| Some properties: |
Possess 4 degrees of freedom. Standard image operations are
possible in the shapelet space: translations, scaling, small
angle rotations, convolutions, shear estimation,
flux/radius/centroid measurements |
| Anecdote: |
Same functions arise in the solution of the quantum harmonic
oscillator |
| Usage: |
Useful for the representation (and compression) of
astronomical objects, object classification or galaxy
morphology |
| See also: |
Shapelets webpage by Richard Massey and Alexandre Refregier, much pointers to papers, IDL shapelets software, animations
Links
on shapelets by Christopher Spitzer |
| Comments: |
Not yet public Matlab and C++ code available from Christopher Spitzer.
Shapelets are also cited in programs by P. Kovesi for Computer
Vision, IDL shapelet software by Massey and Refregier |
| In short: |
Steerable wavelets in 3D |
| Etymology: |
From steer for the "directional" prefix |
| Origin: |
Papadakis, Azencott and Bodmann at Univ. Houston Three dimensional steerlets: a novel tool for
extractiong textural and structural features in 3D images, SPIE Wavelet
XIII, August 2009
Azencott, Bodmann, Papadakis at Univ. Houston Steerlets: A novel approach to rigid-motion covariant
multiscale transforms, preprint
|
| Contributors: |
Manos Papadakis,
Robert Azencott,
Bernhard G. Bodmann,
|
| Some properties: |
Steerlets form a new class of wavelets suitable for extracting structural and textural features from 3D-images. These wavelets extend the framework of Isotropic Multiresolution Analysis and allow a wide variety of design characteristics ranging from isotropy, that is the full insensitivity to orientations, to directional and orientational selectivity. The primary characteristic of steerlets is that any 3D-rotation of a steerlet is expressed as a linear combination of other steerlets associated with the same IMRA, yielding 3D-rotation covariant fast wavelet transforms.
Resulting subband decompositions covariant under the action of rotations.
|
| Anecdote: |
A steer is also a young male of ox type, which is nice from Ol'Texas contributors.
|
| Usage: |
|
| See also: |
A Where-Is-The-Starlet entry: WITS: Steerlet wavelets from
La vertu d'un LA
|
| Comments: |
|
|
In short:
|
A SURE (Stein's Unbiased Risk Estimate) method for wavelet denoising |
|
Etymology:
|
From SURE, acronym for "Stein's Unbiased Risk Estimate" and
LET for "Linear Expansion of Thresholds" |
|
Origin:
|
Luisier, F. and Blu, T. and Unser, M.,
A New SURE Approach to Image
Denoising: Inter-Scale Orthonormal Wavelet Thresholding, IEEE Transactions on Image Processing, vol. 16, no. 3, pp. 593-606, March 2007.
[pdf]
|
|
Contributors:
|
Florian Luisier,
Thierry Blu,
Michael Unser
|
|
Some properties:
|
|
|
Anecdote:
|
|
|
Usage:
|
Denoising, see bigwww.epfl.ch/demo/suredenoising/
|
|
See also:
|
|
|
Comments:
|
SURE-let are also related to property rental services, funnily enough related to the word Kingsbury (not Nick)
Surelet - Property Rental Services
Surelet 'to let' branding image (top) To Let - Thinking of Letting your Property or ... Gloucester, Hatfield, Hemel Hempstead, Kingsbury, Oldham, Reading ...
www.surelet.co.uk/kingsbury/
as in the Activelet case.
|
| In short: |
A 3-D directional multiresolution analysis,
combining a 3-D directional filter bank and a Laplacian pyramid |
| Etymology: |
From surface, obviously |
| Origin: |
Lu, Yue and Do, Minh N.
Multidimensional Directional Filter Banks and Surfacelets
IEEE Transactions on Image Processing, , vol. 16, no. 4, April 2007
(pdf)
Lu, Yue and Do, Minh N.
3-D directional filter banks and surfacelets
Proc. of SPIE Conference on Wavelet Applications in Signal and Image Processing XI, San Diego, USA, Jul. 2005, invited paper
(pdf) |
| Contributors: |
Yue Lu,
Minh N. Do
|
| Some properties: |
Redundancy factor up to 24/7 in 3-D for the 2005 SPIE version, about 4.05 for the 2006 preprint |
| Anecdote: |
|
| Usage: |
|
| See also: |
|
| Comments: |
SurfBox: : MATLAB and C++ toolbox implementing the NDFB and the surfacelet
transform as described in the paper Multidimensional directional filter banks
and surfacelets
|
| In short: |
Representation for approximation and compression of
Horizon-class functions containing a C K
smooth discontinuity in N-1 dimensions |
| Etymology: |
From surface |
| Origin: |
Chandrasekaran, V. Compression of higher dimensional
functions containing smooth discontinuities, 29th Annual
Spring Lecture Series, Recent Developments in Applied Harmonic
Analyis, Multiscale Geometric Analysis, April 15-17, 2004
Chandrasekaran, V. and Wakin, M. B. and Baron, D. and Baraniuk, R. G.
Representation and Compression of Multi-Dimensional
Piecewise Functions Using Surflets, Preprint (pdf)
|
| Contributors: |
Venkat Chandrasekaran,
Mike Wakin,
Dror Baron,
Richard G. Baraniuk
|
| Some properties: |
|
| Anecdote: |
|
| Usage: |
|
| See also: |
|
| Comments: |
|
|
In short:
|
An adaptive method combining multi-scale representation and eigenanalysis |
|
Etymology:
|
From the tree-structured representation and the ubiquitous let |
|
Origin:
|
Ann B. Lee, Boaz Nadler, and Larry Wasserman
Treelets - An Adaptive Multi-Scale Basis for Sparse
Unordered Data (local copy), to appear in Annals of Applied Statistics
|
|
Contributors:
|
|
|
Some properties:
|
Dimensionality reduction and feature selection tool; Based on the Jacobi method, it groups together a each level of the tree, the most similar variables and
replace them by a coarse-grained "sum variable" and a residual "difference variable" computed by a local PCA |
|
Anecdote:
|
The treelet is a small tree
|
|
Usage:
|
Blocked covariance models; Hyperspectral Analysis and Classification of Biomedical Tissue; Internet Advertisement Data Set
|
|
See also:
|
Treelet Matlab code |
|
Comments:
|
The term was coined before by
people at Microsoft:
Chris Quirk, Arul Menezes and Colin Cherry,
Dependency Treelet Translation: Syntactically Informed Phrasal SMT, July 2005. |
| In short: |
A "wavelet-like" bounded, continuous function which, under
the action of a specific standardization operator (dilation +
translation), satisfies a set of axioms related to
- fast decay/localization
- null mean value/oscillation
- miminal regularity
|
| Etymology: |
Vague means "wave" in French, as far as liquids
are concerned (especially the sea), but also in a more vague
sense. Vaguelette could be described
as a moderate size ripple, a small wave vanishing on the shore.
It could thus be read as "wavelet" or ondelette in a limited sense. Or more precisely,
"les vaguelettes sont de vagues ondelettes" |
| Origin: |
Meyer, Yves, Ondelettes et opérateurs: II.
Opérateurs de Calderón Zygmund, 1990, p. 270,
Hermann et Cie, Paris |
| Contributors: |
Yves Meyer |
| Some properties: |
|
| Anecdote: |
|
| Usage: |
|
| See also: |
Wavelet-Vaguelette |
| Comments: |
|
| In short: |
Often described as a wavelet analogue to the singular value
decomposition. Wavelets and vaguelettes act like "reciprocal"
under the action of an linear operator (and its transpose) |
| Etymology: |
Composition of wavelet and vaguelette. The
resulting acronym (WVD for wavelet-vaguelette decomposition) is
reminiscent of that of the SVD (singular value
decomposition) |
| Origin: |
David L. Donoho, Nonlinear solution of linear problems by
wavelet-vaguelette decomposition, 1992, Stanford, Research
report (also in App. and Comp. Harmonic Analysis, 2, 1995) |
| Contributors: |
David L. Donoho |
| Some properties: |
This decomposition exists for a class of special linear
inverse problems of homogeneous type (numerical differentiation,
Radon transform, inversion of Abel-type transforms). Improves
upon SVD inversion for the recovery of spatially inhomogeneous
objets |
| Anecdote: |
|
| Usage: |
Solution of Nonlinear PDEs via adaptive Wavelet-Vaguelette
decomposition, (by J. Fröhlich and K. Schneider,
Konrad-Zuse-Zentrum Berlin, Preprint SC 95-28) |
| See also: |
Vaguelette |
| Comments: |
|
| In short: |
Affine deformation of the Gabor wavelet (aka gaborlet) |
| Etymology: |
From warp, for a twist or
distorsion (of a shape) |
| Origin: |
Clerc, Maureen and Mallat, Stéphane, The Texture
Gradient Equation for recovering Shape from Texture IEEE
Transactions on Pattern Analysis and Matching Intelligence, pp. 536-549, vol. 24, no. 4, April 2002. |
| Contributors: |
Maureen Clerc, Stéphane Mallat |
| Some properties: |
A four scale operator related to a transport equation called
the "texture gradient equation". Addresses the problem known as
"shape to texture", i.e. the retrieval of 3D shapes from a
textured perspective image |
| Anecdote: |
For a stochastic process, the variance of the warplets
coefficients is called a warpogram |
| Usage: |
Texture and shape problems |
| See also: |
Recent works (2005) by Abhir Bhalerao and Roland Wilson on other kinds of warplets, thought as image-dependent patch-like wavelet representations based on PCA (principal component analysis, see the following
tutorial on PCA)
|
| Comments: |
Also associated with the names of R. Baraniuk and D. L. Jones
in a talk by X. Huo, 1999, but no accurate reference found to date |
| In short: |
Partition based on a recursive, dyadic squares, allowing
wedge-shaped final nodes (instead of squares), with piece-wise
constant value |
| Etymology: |
|
| Origin: |
David L. Donoho, Wedgelets: Nearly-minimax estimations of
edges, Ann. Statist., vol. 27, pp. 353-382, 1999 |
| Contributors: |
David Donoho |
| Some properties: |
Nearly-Minimax estimation of edges. The analysis performance
is controlled by a key parameter d (the wedgelet
resolution), which accounts for the spacing between nodes of the
square perimeter |
| Anecdote: |
|
| Usage: |
A software package for image segmentation is distributed on www.wedgelet.de |
| See also: |
The platelet
generalization |
| Comments: |
|
| In short: |
A generic name for a wannabee wavelet (before it actually
gets its name or waiting to be invented) |
| Etymology: |
|
| Origin: |
Probably diffuse, but attested in: Do, M. N. and Vetterli, M.
The contourlet transform: an efficient directional
multiresolution image representation, IEEE
Transactions Image Processing, 2005, [pdf]
and several other talks by
these authors |
| Contributors: |
Minh N. Do, Martin
Vetterli |
| Some properties: |
|
| Anecdote: |
Man gave names to all the x-lets, in the beginning, long time
ago |
| Usage: |
|
| See also: |
|
| Comments: |
|
