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WITS = Where Is The Starlet? (wavelet names ending with *let)

A Shorter WITS wavelet/starlet link: http://tinyurl.com/wits-wavelets-starlet
Let us start with two novel wavelet words:
  • Cymatiophile: a wavelet "amateur" (lover of small waves, forged from the greek by TB@HK)
  • Leptostatonymomane: a person eagerly collecting common names with a diminutive ending (-let, -lette, -ling), possibly related to wavelets
[Activelet] [AMlet] [Aniset*] [Armlet] [Bandlet] [Barlet] [Bathlet] [Beamlet] [Binlet] [Bumplet*] [Brushlet] [Camplet] [Caplet] [Chirplet] [Chordlet*] [Circlet] [Coiflet] [Contourlet] [Cooklet] [Coslet*] [Craplet] [Cubelet*] [CURElet] [Curvelet] [Daublet] [Directionlet] [Dreamlet*] [Edgelet] [ERBlet] [FAMlet*] [FLaglet*] [Flatlet] [Formlet] [Fourierlet*] [Framelet] [Fresnelet] [G-let] [Gaborlet] [Gabor shearlet*] [GAMlet] [Gausslet] [Graphlet] [Grouplet] [Haarlet] [Haardlet] [Heatlet] [Hutlet] [Hyperbolet] [Icalet (Icalette)] [Interpolet] [Lesslet (cf. Morelet)] [Loglet] [Marrlet*] [MIMOlet] [Monowavelet*] [Morelet] [Morphlet] [Multiselectivelet] [Multiwavelet] [Needlet] [Noiselet] [Ondelette/wavelet] [Ondulette] [Prewavelet*] [Phaselet] [Planelet] [Platelet] [Purelet] [Quadlet/q-Quadlet*] [QVlet] [Radonlet] [RAMlet] [Randlet] [Ranklet] [Ridgelet] [Riezlet*] [Ripplet (original, type-I and II)] [Scalet] [S2let*] [Seamlet] [Seislet] [Shadelet*] [Shapelet] [Shearlet] [Sinclet] [Singlet] [Sinlet*] [Slantlet] [Smoothlet] [Snakelet*] [SOHOlet] [Sparselet] [Speclet*] [Spikelet] [Splinelet] [Starlet*] [Steerlet] [Stokeslet*] [Subwavelet (Sub-wavelet)] [Superwavelet] [SURE-let (SURElet)] [Surfacelet] [Surflet] [Symlet/Symmlet] [S2let*] [Tetrolet] [Treelet] [Vaguelette] [Walet*] [Wavelet-Vaguelette] [Wavelet] [Warblet] [Warplet] [Wedgelet] [Xlet/X-let]
Starred [starlet*] wavelets, in the above index, do not have an full-fledged entry yet. Patience. A brief description is given in futurelets.

Published as: A panorama on Multiscale Geometric Representations, Signal Processing (special issue: advances in Multirate Filter Bank Structures and Multiscale Representations), Volume 91, Issue 12, December 2011, p. 2699-2730, Laurent Jacques, Laurent Duval, Caroline Chaux, Gabriel Peyré

An overview/review/tutorial paper on two-dimensional (2D) geometric wavelets, multiscale and multidirectional transforms: contourlets, surflets, beamlets, curvelets, directionlets, shearlets, starlets, und so weiter (pdf) and additional panoramas/tutorials/review on wavelets and a set of wavelets, contourlets, curvelets, shearlets toolboxes and references.

The Great Wave off Kanagawa, Hokusai Katsushika
The Great Wave off Kanagawa by Hokusai [Katsushika]
The vague sighings of a wind at even; That wakes the wavelets of the slumbering sea (Shelley, 1813)
A panorama on Multiscale Geometric Representations, intertwining spatial, directional and frequency selectivity
"Worth a bite... let", The Able Set (mixed)
"WITS: The * in *let (the star in starlet)"
An anonymous contributor
What is the starlet?
Define: star-let (/'starlit/)
Noun: A young actress with aspirations to become a star
Example: "a Hollywood starlet".
Synonyms: star

Mistakelets may occur in wavelet names below. Send YOUR correctionlets, additionlets and commentlets at: lcd (ad) ieee (dod) org. Next paper in mind: Wavelet without casualties, due to this strange weaponry related connection between Morlet (as a genealogist) and Wavelet (as a father name), bothers in arms

Otherlets: wavelet names not in *let Artlets: wavelet uses (and misuses) in art (music, painting,...) Forgottenlets: waiting for adoption Linklets: other star-let/wavelet pages
[Multiselective wavelets] [SOHO wavelets]
[AguaSonic Acoustics] [BIG Art Gallery] [Le Spy art]
[Arclet] [Beanlet] [Besselet] [Bricklet] [Cordlet] [Disklet] [Droplet] [Gauntlet] [Islet] [Multiplet] [Squarelet] [Stringlet] [Toylet] [Winglet] [& other future starlet]
[Xiaobo Qu] [Agnieszka Lisowska]

Futurelet(ter)

Some starlets are shining like crazy diamonds, on the dark side of the moon. Awaiting capture, here they are on the run:

Newslet(ter) [Lastest news about wavelets]

2014/01/07: addition and update on the icalet, a combination of Independent Component Analysis and wavelets
2013/11/08: addition on the
speclet: groups of Morlet atoms with the same scale and time support, atoms with bounded correlation, aimed at capturing a major part of timbre information in acoustic sources
2013/01/28: addition on the
quadlet and q-quadlet, references to quadlet papers; addition on the coslet and the sinlet
2013/01/11: addition on the warplet: references to warplet papers
2013/01/05: addition on the
ERBlet a linear and invertible time-frequency transformation adapted to human auditory perception, for masking and perceptual sparsity
2012/12/15: addition on the
ERBlet a linear and invertible time-frequency transformation adapted to human auditory perception, for masking and perceptual sparsity (implemented in the LTFAT: Large Time-Frequency Analysis Toolbox)
2012/09/03: addition on the wavelet (references and abstracts). There additionally are 617 words ending in let, yet only a hundred of starlets. Keep on working, girlets & boylets!
2012/09/02: addition on the
graphlet: PySGWT, a python code port for graphlet (aka Spectral Graph Wavelet Transform). PySGWT
2012/08/24: addition on the
binlet: references to binlet papers
2012/07/30: addition on the
ripplet: matlab codes for recents ripplet-I and ripplet-II
2012/04/15: addition on the
hyperbolet, closely related to the shearlets
2012/03/06: post on
Spherical Wavelets Code Release, version 1.2.2 available by B. T. Thomas Yeo.
2012/03/06: post on Shearlets, after MIA 2012 related to the shearlet entry
2012/02/21: addition on the
shearlet with a new FFST (Fast finite shearlet transform) matlab toolbox
2012/01/20: addition on the chordlet
2012/01/20: update on the
multiselectivelet (or Multiselective wavelets) by Laurent Jacques et al., SOHO wavelet (SOHOlets)
2011/12/20: update on the ripplet-II (or type-II ripplet) preprint by Xu et al., "Ripplet-II Transform for Feature Extraction"
2011/11/22: update on the multiwavelet, by Gabriel Peyré
2011/11/10: addition on the qvlet, class of fourband piecewise polynomials multiwavelets (with paper "Fourband multiwavelet series and orthogonal analysis for geometric shape")
2011/11/05: addition on the morphlet, a multiscale representation for diffeomorphisms (with paper The Morphlet Transform: A Multiscale Representation for Diffeomorphisms)
2011/11/05: addition on the curelet (with preprint A CURE for noisy magnetic resonance images: Chi-square unbiased risk estimation)
2011/10/04: addition on the interpolet (or interpolating wavelet transform)
2011/08/30: addition on the haarlet
2011/08/25: addition on the smoothlet
2011/08/25: update on the slantlet
2011/08/08: addition on the seislet
2011/06/26: addition on the seamlet
2011/05/27: That's some bad hat Harry: addition on the hutlet, with some more hat like wavelets
2011/05/26: addition on the hyperbolet
2011/04/04: addition on graphlet, a short name for wavelets on graphs
2011/03/09: update on directionlet: added link on the author webpage
2011/03/09: update on shearlet: added link on the ShearLab toolbox
2011/02/18: update on sinclet: added papers on sinclet
2010/12/11: update on ripplet: added paper on newborn type-I ripplet (akin to curvelets)
2010/08/27: update on haarlet: added references on H-transform and comments on performance
2010/08/27: update on noiselet: added references on incoherence (Tuma et al.) and Laurent Jacques matlab toolbox link
2010/06/09: update on platelet: changed platelet matlab toolbox link
2010/03/27: update on spikelet: added paper and spikelet illustration
2010/03/24: update on haarlet: A. Haar and F. Riesz Memorial Plaque at Szeged university (benchmark image)
2010/03/24: update on circlet: a circling wavelet for oceanographic applications
2009/09/20: update on loglet: added references
2009/09/20: update on spikelet: added references and homonym pictures
2009/09/20: update on steerlet: added homonym pictures
2009/06/16: addition on steerlet: added references
2009/04/12: addition on tetrolet: updated references and blog entry
2009/02/05: update on bathlet
2008/05/21: update on shapelet
2008/05/08: update on framelet
2008/04/11: update on spikelet
2008/04/11: update on noiselet (with Matlab code for building noiselet projections)
2008/04/11: addition on sparselet
2008/03/10: addition on treelet (thanks to Igor Carron)
2008/03/05: update on MIMOlet
2008/03/05: addition on needlet, thanks to (thanks to Laurent Jacques)
2008/03/05: update on noiselet, with a preprint by Allouche and Skordev
2008/03/05: update on multiwavelet, matlab multiwavelet toolbox made available
2007/12/16: update on multiwavelet
2007/12/01: update on activelet
2007/11/19: addition on surfacelet, link to the surfbox, surfacelet toolbox
2007/11/19: update on platelet
2007/09/04: addition on SURE-let: wavelets and real estate
2007/09/01: addition on craplet: where crap and wavelets meet
2007/09/01: addition on activelet
2007/07/12: addition on noiselet (thanks to Igor Carron)
2007/06/30: update on MIMOlet
2007/06/30: addition on loglet
2007/04/21: addition on spikelet
2007/04/02: addition on MIMOlet
2007/04/02: update on SURE-let (SURElet)
2007/03/07: addition on shearlet to the shearlet website (thanks to Igor Carron)
2007/01/12: addition on ranklet and grouplet
2006/12/16: addition on randlet, a family random basis.
2006/10/20: new category for wavelet in arts (music, painting) artlet
2006/10/10: addition for the not in *let multiselective wavelets (not: multiselectivelet)
2006/10/10: new category for wavelet "not in *let" names otherlet
2006/10/08: submitted journal paper on surfacelet
2006/10/04: preprint added on surflet
2006/08/21: update on seislet
2006/08/21: update on scalet
2006/07/07: update on flatlet, a family piecewise linear basis functions
2006/07/06: addition on ranklet, a family of multiscale rank features related to Haar wavelets.
2006/05/12: update on contourlets.
2006/05/12: addition on an historical wavelet toolbox WavBox.
2006/05/10: news on aptonyms (aptonymes), or more simply onomastics/anthroponymy: "Wavelet" is a surname which may be encountered in Nord-Pas-de-Calais (North of France). It is a diminutive form for the name "Wawel", from the german root waffan (waffen), which means "weapon" or "arm" (see arm-let). Funny enough, this information comes from a book by Marie-Thérèse Morlet, "Dictionnaire étymologique des noms de famille". A strange "linguistic" connection between Jean Morlet and wavelets (source). See also the bathlet on WITS, or a bathlet blog entry.
2006/04/08: icalet (icalette), a program for a wavelet based ICA (independent component analysis) contrast estimator
2006/02/28: update on surfacelet, thanks to Zoologist Yue Lu.
2005/11/14: shearlet, sparse representations based on anisotropic dilations and shear operators.
2005/11/14: warplet, a recent alternative by A. Bhalerao and R. Wilson (thanks to A. Bhalerao).
2005/11/01: barlet, quoted by Richard Baraniuk.
2005/09/25: ondulette, recently found on the Internet.
2005/07/15: link to a bandelet Matlab toolbox.
2005/06/20: curvelet Matlab and C++ toolbox.
2005/06/10: cooklet.
2005/02/10: news: Ten words have been proposed for the 10e semaine de la langue française et de la francophonie. This year's (2005) theme was science: "le français, langue de l’aventure scientifique" to celebrate the century of the death of Jules Verne. Michel Serres has proposed the word ondelette (wavelet). The nine other words were: variation, complexité, cristal, rayonnement, miroir, désenchevêtrement, hélice, icône, élémentaire, and... ordinateur.

Motivations

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. Étournelette, 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, yet) answer is provided by Wim Sweldens (twitter) in the introduction for his PhD thesis, Construction and Application of Wavelets in Numerical Analysis, in 1994:

Uit de wiskundige analyse volgde dat de integraal van deze functie nul moet zijn en dat deze functie naar nul moet convergeren als het argument naar oneindig gaat. M.a.w. deze functie moet een beetje "schommelen" en dan geleidelijk uitsterven; het is een soort "lokaal golfje".

CQFD/QED/USW

More seriously, one of my favorite, yet not very specific, definition is due to Wim Sweldens too: "Wavelet are building blocks that can quickly decorrelate data." (The Lifting Scheme: A New Philosophy in Biorthogonal Wavelet Constructions, 1995, Proc. SPIE Wavelet Applications in Signal and Image Processing). 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).

Wavelet (even Wavlet)/Ondelette (parfois ondulette) [back to the starlet list]

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 its decay. Wavelets by other names (in other languages): ondicula, golfje, ondeta/tes, pndosimilajo
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 (apocryphally) with 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 (pdf), SIAM Journal of Mathematical Analysis, vol. 15, no. 4, pp. 723-736, July 1984.
Abstract: An arbitrary square integrable real-valued function (or, equivalently, the associated Hardy function) can be conveniently analyzed into a suitable family of square integrable wavelets of constant shape, (i.e. obtained by shifts and dilations from anyone of them.) The resulting integral transform is isometric and self-reciprocal if the wavelets satisfy an "admissibility condition" given here. Explicit expressions are obtained in the case of a particular analyzing family that plays a role analogous to that of coherent states (Gabor wavelets) in the usual Lz-theory. They are written in terms of a modified f-function that is introduced and studied. From the point of view of group theory, this paper is concerned with square integrable coefficients of an irreducible representation of the nonunimodular ax +b-group

Some earlier works need be mentioned:
  • Morlet, J., Arens, G., Fourgeau, I. and Giard, D., Wave propagation and sampling theory (pdf as part I: Complex signal and scattering in multilayered media and part II: Sampling theory and complex waves ), Geophysics, 47, pp. 203-236, 1982
    Abstract: From experimental studies in digital processing of seismic reflection data, geophysicists know that a seismic signal does vary in amplitude, shape, frequency and phase, versus propagation time. To enhance the resolution of the seismic reflection method, we must investigate these variations in more detail.We present quantitative results of theoretical studies on propagation of plane waves for normal incidence, through perfectly elastic multilayered media.As wavelet shapes, we use zero-phase cosine wavelets modulated by a Gaussian envelope and the corresponding complex wavelets. A finite set of such wavelets, for an appropriate sampling of the frequency domain, may be taken as the basic wavelets for a Gabor expansion of any signal or trace in a two-dimensional (2-D) domain (time and frequency). We can then compute the wave propagation using complex functions and thereby obtain quantitative results including energy and phase of the propagating signals. These results appear as complex 2-D functions of time and frequency, i.e., as 'instantaneous frequency spectra.'Choosing a constant sampling rate on the logarithmic scale in the frequency domain leads to an appropriate sampling method for phase preservation of the complex signals or traces. For this purpose, we developed a Gabor expansion involving basic wavelets with a constant time duration/mean period ratio.For layered media, as found in sedimentary basins, we can distinguish two main types of series: (1) progressive series, and (2) cyclic or quasi-cyclic series. The second type is of high interest in hydrocarbon exploration.Progressive series do not involve noticeable distortions of the seismic signal. We studied, therefore, the wave propagation in cyclic series and, first, simple models made up of two components (binary media). Such periodic structures have a spatial period. We present synthetic traces computed in the time domain using the Goupillaud-Kunetz model of propagation for one-dimensional (1-D) synthetic seismograms.Three different cases appear for signal scattering, depending upon the value of the ratio wavelength of the signal/spatial period of the medium.(1) Large wavelengths The composite medium is fully transparent, but phase delaying. It acts like an homogeneous medium, with an 'effective velocity' and an 'effective impedance.'(2) Short wavelengths For wavelengths close to twice the spatial period of the medium, the composite medium strongly attenuates the transmission, and superreflectivity occurs as counterpart.(3) Intermediate wavelengths For intermediate values of the frequency, velocity dispersion versus frequency appears.All these phenomena are studied in the frequency domain, by analytic formulation of the transfer functions of the composite media for transmission and reflection. Such phenomena are similar to Bloch waves in crystal lattices as studied in solid state physics, with only a difference in scale, and we checked their conformity with laboratory measurements.Such models give us an easy way to introduce the use of effective velocities and impedances which are frequency dependent, i.e., complex. They will be helpful for further developments of 'complex deconvolution.'The above results can be extended to quasi-cyclic media made up of a random distribution of double layers. For signal transmission, quasi-cyclic series act as a high cut filter with possible time delay, velocity dispersion, and 'constant Q' type of law for attenuation. For signal reflection they act as a low cut filter, with possible superreflections.These studies could be extended to three-dimensional (3-D) binary models (grains and pores in a porous reservoir), in agreement with well-known acoustic properties of gas reservoirs (theory of bright spots).We present some applications to real well data. Velocity dispersion may explain: mistying between sonic logs and velocity surveys; mistying between synthetic seismograms and seismic sections.Effective velocities lower than mean velocities may explain: very low velocities for P- and S-waves, especially in weathered shallow layers; anomalies on the values of the ratio Vs/Vp ; mistying between P and S seismic sections. Finally, the Gabor expansion provides a tool to obtain sampled instantaneous frequency spectra, and to carry out a suitable recording and processing method in high-resolution seismic, especially to preserve the phase information. Such a processing will involve complex signals, complex traces, complex velocities and complex impedances.For practical purpose, this paper comprises two separate parts. Here, we present some interesting features on the scattering of seismic signals obtained by a simulation method using simple models. We show the usefulness of the notions of complex signals, complex velocities, and complex impedances, and overall of the Gabor expansion, by simulation on very simple models. Morlet et al, (1982, this issue) will be concerned with the development of fundamental notions useful to handle seismic data from the sampling method of recording to processing methods. There we give theoretical and practical tools to sample and handle these data in the time-frequency domain, using complex functions.
    Abstract: Morlet et al (1982, this issue) showed the advantages of using complex values for both waves and characteristics of the media. We simulated the theoretical tools we present here, using the Goupillaud-Kunetz algorithm.Now we present sampling methods for complex signals or traces corresponding to received waves, and sampling methods for complex characterization of multilayered or heterogeneous media.Regarding the complex signals, we present a two-dimensional (2-D) method of sampling in the time-frequency domain using a special or 'extended' Gabor expansion on a set of basic wavelets adapted to phase preservation. Such a 2-D expansion permits us to handle in a proper manner instantaneous frequency spectra. We show the differences between 'wavelet resolution' and 'sampling grid resolution.' We also show the importance of phase preservation in high-resolution seismic.Regarding the media, we show how analytical studies of wave propagation in periodic structured layers could help when trying to characterize the physical properties of the layers and their large scale granularity as a result of complex deconvolution. Analytical studies of wave propagation in periodic structures are well known in solid state physics, and lead to the so-called 'Bloch waves.'The introduction of complex waves leads to replacing the classical wave equation by a Schrodinger equation.Finally, we show that complex wave equations, Gabor expansion, and Bloch waves are three different ways of introducing the tools of quantum mechanics in high-resolution seismic (Gabor, 1946; Kittel, 1976, Morlet, 1975). And conversely, the Goupillaud-Kunetz algorithm and an extended Gabor expansion may be of some use in solid state physics.
  • 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
    Abstract: ###
First known mention of wavelets, as we know them now, 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.
Abstract: ###
Contributors: Probably too many to mention, with the great risk of forgetting some of them. See the lists by Andreas Klappenecker: Some Wavelet People, or Palle Jorgensen: Some Wavelet Researchers, with Their E-Mail Addresses
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.
Abstract: From the breadth of a wavelet for a given travel time, it is possible to calculate the viscosity of the formation through which the seismic disturbance has passed. This calculation has been carried out for the Cretaceous Shale of Eastern Colorado, and the value thus found ranges from 2.7 X 10 7 to 4.9 X 10 7 , with a mean value of 3.8 X 10 7 grams per cm. per second.
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 Alfred Haar, Zur Theorie der orthogonalen Funktionen-Systeme, Math. Ann., vol. 69, pp. 331-371, 1910 (english translation: On the Theory of Orthogonal Function Systems by Georg Zimmermann, with local copy)
Abstract: Die vorliegende Arbeit ist, bis auf unwesentliche Änderungen, ein Abdruck meiner im Juli 1909 erschienenen Göttinger Inauguraldissertation.
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
Abstract: The plain fact that wavelet families are very interesting orthonormal systems for L 2 (R) makes it natural to view them as an important contribution to the field of orthogonal expansions of functions. This classical field of mathematical analysis was particularly flourishing in the first 30 years of the 20th century, when detailed discussions of the convergence of orthogonal series, in particular of trigonometric series, were undertaken. Alfred Haar describes the situation in his 1910 paper in Math. Annalen appropriately as follows: for any given (family of) orthonormal system(s) of functions on the unit interval [0, 1] one has to ask the following questions: convergence theory (sufficient conditions that a series is convergent); divergence theory (in contrast to convergence theory it exhibits examples of relatively "decent" functions for which nevertheless no good convergence, e.g., at that time mostly in the pointwise or uniform sense, takes place); summability theory (to which extent can summation methods help to overcome the problems of divergence);
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. Here are a few short reviews or tutorials on wavelets:
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. Spherical Wavelets Code Release, version 1.2.2 available by B. T. Thomas Yeo

Wavelets in 2D: anisotropic, hyperbolic, square and rectangular

When applied in two dimensions, there are essentially two possibilities in processing the rows and columns of an image in a separable way with a discrete wavelet transform. First consists applying one level of dyadic wavelet on the rows, then on the colums, and to iterate the process. This gives the standard, familiar embedded squares images. The second applies r level of DWT on the rows, then c levels on the columns. The image is then splitted into rectangles. There are independent AND interleaved dyadic wavelet cascades. Recent works advocate the interest of the second approach for anisotropic data. The latter is not so common, and is often refered to with different names. For instance, the literature comes up with the following adjectives: They could also be termed "combined/separated".
Matching entries: 0
settings...
AuthorTitleYearJournal/ProceedingsReftypeDOI/URL
Abry, P., Clausel, M., Jaffard, S., Roux, S. and Vedel, B. Hyperbolic wavelet transform: an efficient tool for multifractal analysis of anisotropic textures 2012 PREPRINT  article  
Abstract: Global and local regularities of functions are analyzed in anisotropic function spaces, under a common framework, that of hyperbolic wavelet bases. Local and directional regularity features are characterized by means of global quantities constructed upon the coefficients of hyperbolic wavelet decompositions. A multifractal analysis is introduced, that jointly accounts for scale invariance and anisotropy. Its properties are studied in depth.
BibTeX:
@article{Abry_P_2012_PREPRINT_hyperbolic_wtetmaat,
  author = {Abry, P. and Clausel, M. and Jaffard, S. and Roux, S. and Vedel, B.},
  title = {Hyperbolic wavelet transform: an efficient tool for multifractal analysis of anisotropic textures},
  journal = {PREPRINT},
  year = {2012}
}
Averbuch, A., Beylkin, G., Coifman, R., Fischer, P. and Israeli, M. Adaptive Solution of Multidimensional PDEs via Tensor Product Wavelet Decomposition 2008 Intern. J. of Pure and Applied Mathematics
Vol. 44(1), pp. 75-115 
article  
BibTeX:
@article{Averbuch_A_2008_j-inter-j-pure-appl-math_adaptive_smpdetpwd,
  author = {A. Averbuch and G. Beylkin and R. Coifman and P. Fischer and M. Israeli},
  title = {Adaptive Solution of Multidimensional PDEs via Tensor Product Wavelet Decomposition},
  journal = {Intern. J. of Pure and Applied Mathematics},
  year = {2008},
  volume = {44},
  number = {1},
  pages = {75--115}
}
Ben Slimane, M. and Ben Braiek, H. Directional and Anisotropic Regularity and Irregularity Criteria in Triebel Wavelet Bases 2012 J. Fourier Anal. Appl.
Vol. 18, pp. 893-914 
article URL 
Abstract: Many natural mathematical objects, as well as many multi-dimensional signals and images from real physical problems, need to distinguish local directional behaviors (for tracking contours in image processing for example). Using some results of Jaffard and Triebel, we obtain criteria of directional and anisotropic regularities by decay conditions on Triebel anisotropic wavelet coefficients (resp. wavelet leaders).
BibTeX:
@article{BenSlimane_B_2012_j-four-anal-appl_directional_arictwb,
  author = {Ben Slimane, Mourad and Ben Braiek, Hnia},
  title = {Directional and Anisotropic Regularity and Irregularity Criteria in Triebel Wavelet Bases},
  journal = {J. Fourier Anal. Appl.},
  publisher = {Birkhäuser Boston},
  year = {2012},
  volume = {18},
  pages = {893--914},
  note = {10.1007/s00041-012-9226-5},
  url = {http://dx.doi.org/10.1007/s00041-012-9226-5}
}
Beylkin, G. Wavelets and fast numerical algorithms 1993
Vol. 47#p-symp-appl-math# 
inproceedings  
Abstract: Wavelet based algorithms in numerical analysis are similar to other transform methods in that vectors and operators are expanded into a basis and the computations take place in this new system of coordinates. However, due to the recursive definition of wavelets, their controllable localization in both space and wave number (time and frequency) domains, and the vanishing moments property, wavelet based algorithms exhibit new and important properties.For example, the multiresolution structure of the wavelet expansions brings about an efficient organization of transformations on a given scale and of interactions between different neighbouring scales. Moreover, wide classes of operators which naively would require a full (dense) matrix for their numerical description, have sparse representations in wavelet bases. For these operators sparse representations lead to fast numerical algorithms, and thus address a critical numerical issue.We note that wavelet based algorithms provide a systematic generalization of the Fast Multipole Method (FMM) and its descendents.These topics will be the subject of the lecture. Starting from the notion of multiresolution analysis, we will consider the so-called non-standard form (which achieves decoupling among the scales) and the associated fast numerical algorithms. Examples of non-standard forms of several basic operators (e.g. derivatives) will be computed explicitly.
BibTeX:
@inproceedings{Beylkin_G_1993_p-symp-appl-math_wavelets_fna,
  author = {Beylkin, G.},
  title = {Wavelets and fast numerical algorithms},
  booktitle = {#p-symp-appl-math#},
  year = {1993},
  volume = {47}
}
Beylkin, G., Coifman, R. and Rokhlin, V. Fast wavelet transforms and numerical algorithms I 1991 Comm. Pure Appl. Math.
Vol. 44(2), pp. 141-183 
article DOI URL 
Abstract: A class of algorithms is introduced for the rapid numerical application of a class of linear operators to arbitrary vectors. Previously published schemes of this type utilize detailed analytical information about the operators being applied and are specific to extremely narrow classes of matrices. In contrast, the methods presented here are based on the recently developed theory of wavelets and are applicable to all Calderon-Zygmund and pseudo-differential operators. The algorithms of this paper require order O(N) or O(N log N) operations to apply an N $ N matrix to a vector (depending on the particular operator and the version of the algorithm being used), and our numerical experiments indicate that many previously intractable problems become manageable with the techniques presented here.
BibTeX:
@article{Beylkin_G_1991_j-comm-pure-appl-math_fast_wtna1,
  author = {Beylkin, G. and Coifman, R. and Rokhlin, V.},
  title = {Fast wavelet transforms and numerical algorithms I},
  journal = {Comm. Pure Appl. Math.},
  publisher = {Wiley Subscription Services, Inc., A Wiley Company},
  year = {1991},
  volume = {44},
  number = {2},
  pages = {141--183},
  url = {http://dx.doi.org/10.1002/cpa.3160440202},
  doi = {http://dx.doi.org/10.1002/cpa.3160440202}
}
Dahlke, S., Friedrich, U., Maaß, P., Raasch, T. and Ressel, R.A. An adaptive wavelet solver for a nonlinear parameter identification problem for a parabolic differential equation with sparsity constraints 2012 J. Inv. Ill-Posed Problems  article  
Abstract: In this paper, we combine concepts from two different mathematical research topics: adaptive wavelet techniques for well-posed problems and regularization theory for nonlinear inverse problems with sparsity constraints. We are concerned with identifying certain parameters in a parabolic reaction-diffusion equation from measured data. Analytical properties of the related parameter-to-state operator are summarized, which justify the application of an iterated soft shrinkage algorithm for minimizing a Tikhonov functional with sparsity constraints. The forward problem is treated by means of a new adaptive wavelet algorithm which is based on tensor wavelets. In its general form, the underlying PDE describes gene concentrations in embryos at an early state of development. We implemented an algorithm for the related nonlinear parameter identification problem and numerical results are presented for a simplified test equation.
BibTeX:
@article{Dahlke_S_2012_j-inv-ill-posed-problems_adaptive_wsnpippdesc,
  author = {Dahlke, Stephan and Friedrich, Ulrich and Maaß, Peter and Raasch, Thorsten and Ressel, Rudolf A.},
  title = {An adaptive wavelet solver for a nonlinear parameter identification problem for a parabolic differential equation with sparsity constraints},
  journal = {J. Inv. Ill-Posed Problems},
  year = {2012}
}
Dahlke, S., Friedrich, U., Maaß, P., Raasch, T. and Ressel, R.A. An adaptive wavelet method for parameteridentification problems in parabolic partialdifferential equations 2011 PREPRINT  article  
BibTeX:
@article{Dahlke_S_2011_PREPRINT_adaptive_wmpipppde,
  author = {S. Dahlke and U. Friedrich and P. Maaß and T. Raasch and R. A. Ressel},
  title = {An adaptive wavelet method for parameteridentification problems in parabolic partialdifferential equations},
  journal = {PREPRINT},
  year = {2011}
}
Davis, A.B., Marshak, A. and Clothiaux, E.E. Anisotropic multiresolution analysis in 2D: application to long-range correlations in cloud millimeter-radar fields 1999
Vol. 3723Proc. SPIE Wavelet Applications VI, pp. 194-207 
inproceedings DOI  
Abstract: Taking a wavelet standpoint, we survey on the one hand various approaches to multifractal analysis, as a means of characterizing long-range correlations in data, and on the other hand various ways of statistically measuring anisotropy in 2Dfields. In both instances, we present new and related techniques: (i) a simple multifractal analysis methodology based onDiscrete Wavelet Transforms (DWTs), and (ii) a specific DWT adapted to strongly anisotropic fields sampled on rectangular grids with large aspect ratios. This DWT uses a tensor product of the standard dyadic Haar basis (dividing ratio 2) and a nonstandardiriadic counterpart (dividing ratio 3) which includes the famous "French top-hat" wavelet. The new DWT is amenableto an anisotropic version of Multi-Resolution Analysis (MRA) in image processing where the natural support of the field is2z pixels (vertically) by Y' pixels (horizontally), n being the number of levels in the MRA. The complete 2D basis has onescaling function and five wavelets. The new MRA is used in synthesis mode to generate random multifractal fields thatmimic quite realistically the structure and distribution of boundary-layer clouds even though only a few parameters are used tocontrol statistically the wavelet coefficients of the liquid water density field.
BibTeX:
@inproceedings{Davis_A_1999_p-spie-wa_anisotropic_ma2dalrccmrf,
  author = {Davis, A. B. and A. Marshak and Clothiaux, E. E.},
  title = {Anisotropic multiresolution analysis in 2D: application to long-range correlations in cloud millimeter-radar fields},
  booktitle = {Proc. SPIE Wavelet Applications VI},
  year = {1999},
  volume = {3723},
  pages = {194--207},
  doi = {http://dx.doi.org/10.1117/12.342928}
}
DeVore, R., Konyagin, S.V. and Temlyakov, V.N. Hyperbolic wavelet approximation 1998 Constructive Approximation
Vol. 14, pp. 1-26 
article  
Abstract: We study the multivariate approximation by certain partial sums (hyperbolicwavelet sums) of wavelet bases formed by tensor products of univariate wavelets.We characterize spaces of functions which have a prescribed approximation error byhyperbolicwavelet sums in terms of a K-functional and interpolation spaces. The resultsparallel those for hyperbolic trigonometric cross approximation of periodic functions[DPT].
BibTeX:
@article{DeVore_R_1998_j-const-approx_hyperbolic_wa,
  author = {DeVore, R. and Konyagin, S. V. and Temlyakov, V. N.},
  title = {Hyperbolic wavelet approximation},
  journal = {Constructive Approximation},
  year = {1998},
  volume = {14},
  pages = {1--26}
}
Duarte, M.F. and Baraniuk, R.G. Kronecker Compressive Sensing 2012 IEEE Trans. Instrum. Meas.
Vol. 21(2), pp. 494-504 
article  
Abstract: Compressive sensing (CS) is an emerging approachfor the acquisition of signals having a sparse or compressible representationin some basis.While the CS literature has mostly focusedon problems involving 1-D signals and 2-D images, many importantapplications involve multidimensional signals; the constructionof sparsifying bases and measurement systems for such signalsis complicated by their higher dimensionality. In this paper,wepropose the use of Kronecker product matrices in CS for two purposes.First, such matrices can act as sparsifying bases that jointlymodel the structure present in all of the signal dimensions. Second,such matrices can represent themeasurement protocols used in distributedsettings. Our formulation enables the derivation of analyticalbounds for the sparse approximation of multidimensionalsignals and CS recovery performance, as well as a means of evaluatingnovel distributed measurement schemes.
BibTeX:
@article{Duarte_M_2012_j-ieee-tim_kronecker_cs,
  author = {Duarte, M. F. and Baraniuk, R. G.},
  title = {Kronecker Compressive Sensing},
  journal = {IEEE Trans. Instrum. Meas.},
  year = {2012},
  volume = {21},
  number = {2},
  pages = {494--504}
}
Fournier, A. Wavelets and their Applications in Computer Graphics 1995 SIGGRAPH’95 Course Notes  misc  
BibTeX:
@misc{Fournier_A_1995_lect_wavelets_acg,
  author = {Alain Fournier},
  title = {Wavelets and their Applications in Computer Graphics},
  year = {1995}
}
Fridman, J. and Manolakos, E.S. On the Scalability of 2-D Discrete Wavelet Transform Algorithms 1997 Multidimension. Syst. Signal Process.
Vol. 8, pp. 185-217 
article DOI URL 
Abstract: TBC: The ability of a parallel algorithm to make efficient use of increasing computational resources is known as its scalability. In this paper, we develop four parallel algorithms for the 2-dimensional Discrete Wavelet Transform algorithm (2-D DWT), and derive their scalability properties on Mesh and Hypercube interconnection networks. We consider two versions of the 2-D DWT algorithm, known as the Standard (S) and Non-standard (NS) forms, mapped onto P processors under two data partitioning schemes, namely checkerboard (CP) and stripped (SP) partitioning. The two checkerboard partitioned algorithms M2=?(PlogP) (Non-standard form, NS-CP), and as M2=?(Plog2P) (Standard form, S-CP); while on the store-and-forward-routed (SF-routed) Mesh and Hypercube they are scalable as 3?? (NS-CP), and as 2?? (S-CP), respectively, where M 2 is the number of elements in the input matrix, and ? ? (0,1) is a parameter relating M to the number of desired octaves J as J=??logM? . On the CT-routed Hypercube, scalability of the NS-form algorithms shows similar behavior as on the CT-routed Mesh. The Standard form algorithm with stripped partitioning (S-SP) is scalable on the CT-routed Hypercube as M 2 = ?(P 2), and it is unscalable on the CT-routed Mesh. Although asymptotically the stripped partitioned algorithm S-SP on the CT-routed Hypercube would appear to be inferior to its checkerboard counterpart S-CP, detailed analysis based on the proportionality constants of the isoefficiency function shows that S-SP is actually more efficient than S-CP over a realistic range of machine and problem sizes. A milder form of this result holds on the CT- and SF-routed Mesh, where S-SP would, asymptotically, appear to be altogether unscalable.
Review: Could be applied to TRAN Huy-Quan for processing speed-up, or in scalable data compression, for seismic data
BibTeX:
@article{Fridman_J_1997_j-mult-syst-sp_scalability_2ddwta,
  author = {Fridman, J. and Manolakos, E. S.},
  title = {On the Scalability of 2-D Discrete Wavelet Transform Algorithms},
  journal = {Multidimension. Syst. Signal Process.},
  publisher = {Kluwer Academic Publishers},
  year = {1997},
  volume = {8},
  pages = {185--217},
  url = {http://dx.doi.org/10.1023/A%3A1008229209464},
  doi = {http://dx.doi.org/10.1023/A:1008229209464}
}
Grohs, P. Tree approximation with anisotropic decompositions 2012 Appl. Comp. Harm. Analysis
Vol. 33, pp. 44-57 
article  
Abstract: In recent years anisotropic transforms like the shearlet or curvelet transform have received a considerable amount of interest due to their ability to efficiently capture anisotropic features in terms of nonlinear N-term approximation. In this paper we study treeapproximation properties of such transforms where the N-term approximant has to satisfy the additional constraint that the set of kept indices possesses a tree structure. The main result of this paper is that for shearlet- and related systems, this additional constraint does not deteriorate the approximation rate. As an application of our results we construct (almost) optimal encoding schemes for cartoon images.
BibTeX:
@article{Grohs_P_2012_j-acha_tree_aad,
  author = {Grohs, P.},
  title = {Tree approximation with anisotropic decompositions},
  journal = {Appl. Comp. Harm. Analysis},
  year = {2012},
  volume = {33},
  pages = {44-57}
}
Hochmuth, R. Anisotropic wavelet bases and thresholding 2007 Math. Nachr.
Vol. 280(5-6), pp. 523-533 
article DOI URL 
Abstract: We consider thresholding with respect to anisotropic wavelet bases measuring the approximation error in anisotropic Hardy spaces $ H^a_p $ for $p > 0$, which are known to be equal to $L_p$ for $p >$ 1. In particular, we characterize those functions that provide a specific convergence rate by intrinsic smoothness properties. To this end we apply restricted nonlinear approximation, see [3], which is a generalization of $n$-term approximation in which a weight function is used to control the terms of the approximations.
BibTeX:
@article{Hochmuth_R_2007_j-math-nachr_anisotropic_wbt,
  author = {Hochmuth, Reinhard},
  title = {Anisotropic wavelet bases and thresholding},
  journal = {Math. Nachr.},
  publisher = {WILEY-VCH Verlag},
  year = {2007},
  volume = {280},
  number = {5-6},
  pages = {523--533},
  url = {http://dx.doi.org/10.1002/mana.200410500},
  doi = {http://dx.doi.org/10.1002/mana.200410500}
}
Hochmuth, R. $N$-term Approximation in Anisotropic Function Spaces 2002 Math. Nachr.
Vol. 244(1), pp. 131-149 
article DOI URL 
Abstract: In $L_2((0, 1)^2)$ infinitely many different biorthogonal wavelet bases may be introduced by taking tensor products of one-dimensional biorthogonal wavelet bases on the interval $(0, 1)$. Most well-known are the standard tensor product bases and the hyperbolic bases. In [23, 24] further biorthogonal wavelet bases are introduced, which provide wavelet characterizations for functions in anisotropic Besov spaces. Here we address the following question: Which of those biorthogonal tensor product wavelet bases is the most appropriate one for approximating nonlinearly functions from anisotropic Besov spaces? It turns out, that the hyperbolic bases lead to nonlinear algorithms which converge as fast as the corresponding schemes with respect to specific anisotropy adapted bases.
BibTeX:
@article{Hochmuth_R_2002_j-math-nachr_n-term_aafs,
  author = {Hochmuth, Reinhard},
  title = {$N$-term Approximation in Anisotropic Function Spaces},
  journal = {Math. Nachr.},
  publisher = {WILEY-VCH Verlag},
  year = {2002},
  volume = {244},
  number = {1},
  pages = {131--149},
  url = {http://dx.doi.org/10.1002/1522-2616(200210)244:1<131::AID-MANA131>3.0.CO;2-G},
  doi = {http://dx.doi.org/10.1002/1522-2616(200210)244:1}
}
Hochmuth, R. Wavelet Characterizations for AnisotropicBesov Spaces 2002 Appl. Comp. Harm. Analysis
Vol. 12, pp. 179-208 
article DOI  
Abstract: The goal of this paper is to provide wavelet characterizations for anisotropicBesov spaces. Depending on the anisotropy, appropriate biorthogonal tensorproduct bases are introduced and Jackson and Bernstein estimates are provedfor two-parameter families of finite-dimensional spaces. These estimates leadto characterizations for anisotropic Besov spaces by anisotropy-dependent linearapproximation spaces and lead further on to interpolation and embedding results.Finally, wavelet characterizations for anisotropic Besov spaces with respect to $L_p$-spaceswith $0
BibTeX:
@article{Hochmuth_R_2012_j-acha_wavelet_cabs,
  author = {Hochmuth, R.},
  title = {Wavelet Characterizations for AnisotropicBesov Spaces},
  journal = {Appl. Comp. Harm. Analysis},
  year = {2002},
  volume = {12},
  pages = {179--208},
  doi = {http://dx.doi.org/10.1006/acha.2001.0377}
}
Neumann, M.H. MULTIVARIATE WAVELET THRESHOLDING IN ANISOTROPIC FUNCTION SPACES 2000 Statist. Sinica
Vol. 10, pp. 399-431 
article  
Abstract: It is well known that multivariate curve estimation under standard (isotropic) smoothness conditions suffers from the ``curse of dimensionality''. This is reflected by rates of convergence that deteriorate seriously in standard asymptotic settings. Better rates of convergence than those corresponding to isotropic smoothness priors are possible if the curve to be estimated has different smoothness properties in different directions and the estimation scheme is capable of making use of a lower complexity in some of the directions. We consider typical cases of anisotropic smoothness classes and explore how appropriate wavelet estimators can exploit such restrictions on the curve that require an adaptation to different smoothness properties in different directions. It turns out that nonlinear thresholding with an anisotropic multivariate wavelet basis leads to optimal rates of convergence under smoothness priors of anisotropic type. We derive asymptotic results in the model ``signal plus Gaussian white noise'', where a decreasing noise level mimics the standard asymptotics with increasing sample size.
BibTeX:
@article{Neumann_M_2000_j-statist-sinica_multivariate_wtafs,
  author = {Neumann, M. H.},
  title = {MULTIVARIATE WAVELET THRESHOLDING IN ANISOTROPIC FUNCTION SPACES},
  journal = {Statist. Sinica},
  year = {2000},
  volume = {10},
  pages = {399--431}
}
Neumann, M.H. and von Sachs, R. Wavelet thresholding in anisotropic function classes and application to adaptive estimation of evolutionary spectra 1997 Ann. Stat.
Vol. 25(1), pp. 38-76 
article  
Abstract: We derive minimax rates for estimation in anisotropic smoothness classes. These rates are attained by a coordinatewise thresholded wavelet estimator based on a tensor product basis with separate scale parameter for every dimension. It is shown that this basis is superior to its one-scale multiresolution analog, if different degrees of smoothness in different directions are present. As an important application we introduce a new adaptive waveletestimator of the time-dependent spectrum of a locally stationary time series. Using this model which was recently developed by Dahlhaus, we show that the resulting estimator attains nearly the rate, which is optimal in Gaussian white noise, simultaneously over a wide range of smoothness classes. Moreover, by our new approach we overcome the difficulty of how to choose the right amount of smoothing, that is, how to adapt to the appropriate resolution, for reconstructing the local structure of the evolutionary spectrum in the time-frequency plane.
BibTeX:
@article{Neumann_M_1997_j-ann-stat_wavelet_tafcaaees,
  author = {M. H. Neumann and von Sachs, R.},
  title = {Wavelet thresholding in anisotropic function classes and application to adaptive estimation of evolutionary spectra},
  journal = {Ann. Stat.},
  year = {1997},
  volume = {25},
  number = {1},
  pages = {38--76}
}
Nowak, R.D. and Baraniuk, R.G. Wavelet-based transformations for nonlinear signal processing 1999 IEEE Trans. Signal Process.
Vol. 47(7), pp. 1852-1865 
article DOI  
Abstract: Nonlinearities are often encountered in the analysis and processing of real-world signals. We introduce two new structures for nonlinear signal processing. The new structures simplify the analysis, design, and implementation of nonlinear filters and can be applied to obtain more reliable estimates of higher order statistics. Both structures are based on a two-step decomposition consisting of a linear orthogonal signal expansion followed by scalar polynomial transformations of the resulting signal coefficients. Most existing approaches to nonlinear signal processing characterize the nonlinearity in the time domain or frequency domain; in our framework any orthogonal signal expansion can be employed. In fact, there are good reasons for characterizing nonlinearity using more general signal representations like the wavelet expansion. Wavelet expansions often provide very concise signal representations and thereby can simplify subsequent nonlinear analysis and processing. Wavelets also enable local nonlinear analysis and processing in both time and frequency, which can be advantageous in nonstationary problems. Moreover, we show that the wavelet domain offers significant theoretical advantages over classical time or frequency domain approaches to nonlinear signal analysis and processing
BibTeX:
@article{Nowak_R_1999_j-ieee-tsp_wavelet-based_tnsp,
  author = {Nowak, R. D. and Baraniuk, R. G.},
  title = {Wavelet-based transformations for nonlinear signal processing},
  journal = {IEEE Trans. Signal Process.},
  year = {1999},
  volume = {47},
  number = {7},
  pages = {1852--1865},
  doi = {http://dx.doi.org/10.1109/78.771035}
}
Rosiene, C.P. and Nguyen, T.Q. Tensor-product wavelet vs. Mallat decomposition: a comparative analysis 1999
Vol. 3Proc. Int. Symp. Circuits Syst., pp. 431-434 
inproceedings DOI  
Abstract: The two-dimensional tensor product wavelet transform is compared to the Mallat representation for the purpose of data compression. It is shown that the tensor product wavelet transform will always provide a coding gain greater than or equal to that of the Mallat representation. Further, the costs of obtaining the tensor product wavelet transform are outlined
BibTeX:
@inproceedings{Rosiene_C_1999_p-iscas_ten_pwmdca,
  author = {Rosiene, C. P. and Nguyen, T. Q.},
  title = {Tensor-product wavelet vs. Mallat decomposition: a comparative analysis},
  booktitle = {Proc. Int. Symp. Circuits Syst.},
  year = {1999},
  volume = {3},
  pages = {431--434},
  doi = {http://dx.doi.org/10.1109/ISCAS.1999.778877}
}
Roux, S., Clausel, M., Vedel, B., Jaffard, S. and Abry, P. Wavelet analysis for anisotropic images 2012 INPREPARATION  article  
BibTeX:
@article{Roux_S_2012_INPREPARATION_wavelet_aai,
  author = {Roux, S. and Clausel, M. and Vedel, B. and Jaffard, S. and Abry, P.},
  title = {Wavelet analysis for anisotropic images},
  journal = {INPREPARATION},
  year = {2012}
}
Roux, S.G., Clausel, M., Vedel, B., Jaffard, S. and Abry, P. Self-Similar Anisotropic Texture Analysis: the Hyperbolic Wavelet Transform Contribution 2013 PREPRINT  article  
Abstract: Textures in images can often be well modeled using self-similar processes while they may at the same time display anisotropy. The present contribution thus aims at studying jointly selfsimilarity and anisotropy by focusing on a specific classical class of Gaussian anisotropic selfsimilar processes. It will first be shown that accurate joint estimates of the anisotropy and selfsimilarity parameters are performed by replacing the standard 2D-discrete wavelet transform by the hyperbolic wavelet transform, which permits the use of different dilation factors along the horizontal and vertical axis. Defining anisotropy requires a reference direction that needs not a priori match the horizontal and vertical axes according to which the images are digitized, this discrepancy defines a rotation angle. Second, we show that this rotation angle can be jointly estimated. Third, a non parametric bootstrap based procedure is described, that provides confidence interval in addition to the estimates themselves and enables to construct an isotropy test procedure, that can be applied to a single texture image. Fourth, the robustness and versatility of the proposed analysis is illustrated by being applied to a large variety of different isotropic and anisotropic self-similar fields. As an illustration, we show that a true anisotropy built-in self-similarity can be disentangled from an isotropic self-similarity to which an anisotropic trend has been superimposed.
BibTeX:
@article{Roux_S_2013_PREPRINT_self-similar_atahwtc,
  author = {Roux, S. G. and Clausel, M. and Vedel, B. and Jaffard, S. and Abry, P.},
  title = {Self-Similar Anisotropic Texture Analysis: the Hyperbolic Wavelet Transform Contribution},
  journal = {PREPRINT},
  year = {2013}
}
Schremmer, C. Decomposition strategies for wavelet-based image coding 2001
Vol. 2Signal Processing and its Applications, Sixth International, Symposium on. 2001, pp. 529 -532 
inproceedings DOI  
Abstract: The wavelet transform has become the most interesting new algorithm for still image compression. Yet there are many parameters within a wavelet analysis and synthesis which govern the quality of a decoded image. In this paper, we discuss different decomposition strategies of a two-dimensional signal and their implications for the decoded image: a pool of gray-scale images has been wave let-transformed with different settings of the wavelet filter bank, quantization threshold and decomposition method. Contrary to the new standard JPEG-2000, where nonstandard decomposition is implemented, our investigation proposes standard decomposition for low-bitrate coding
BibTeX:
@inproceedings{Schremmer_C_2001_p-isspa_decomposition_swbic,
  author = {Schremmer, C.},
  title = {Decomposition strategies for wavelet-based image coding},
  booktitle = {Signal Processing and its Applications, Sixth International, Symposium on. 2001},
  year = {2001},
  volume = {2},
  pages = {529 --532},
  doi = {http://dx.doi.org/10.1109/ISSPA.2001.950197}
}
Triebel, H. Wavelet bases in anisotropic function spaces 2005 Proc. Conf. "Function spaces, differential operators and nonlinear analysis", Milovy, 2004, pp. 370-387  inproceedings  
Abstract: The paper deals with wavelet characterisations for anisotropicspaces $B_pq^s,alpha$ and $F_pq^s,alpha$ in $R^n$ for all parameters $s$, $p$, $q$ and all anisotropies $.Some applications are given.
BibTeX:
@inproceedings{Triebel_H_2005_PREPRINT_wavelet_bafs,
  author = {Triebel, H.},
  title = {Wavelet bases in anisotropic function spaces},
  booktitle = {Proc. Conf. "Function spaces, differential operators and nonlinear analysis", Milovy, 2004},
  year = {2005},
  pages = {370--387}
}
Velisavljević, V. Directionlets: Anisotropic Multi-directional Representation with Separable Filtering 2005 School: EPFL  phdthesis  
Abstract: Efficient representation of geometrical information in images is very important in many image processing areas, including compression, denoising and feature extraction. However, the design of transforms that can capture these geometrical features and represent them with a sparse description is very challenging. Recently, the separable wavelet transform achieved a great success providing a computationally simple tool and allowing for a sparse representation of images. However, in spite of the success, the efficiency of the representation is limited by the spatial isotropy of the wavelet basis functions built in the horizontal and vertical directions as well as the lack of directionality. One-dimensional discontinuities in images (edges and contours), which are very important elements in visual perception, intersect with too many wavelet basis functions leading to a non-sparse representation. To capture efficiently these anisotropic geometrical structures characterized by many more than the horizontal and vertical directions, more flexible multi-directional and anisotropic transforms are required. We present a new lattice-based perfect reconstruction and critically sampled anisotropic multi-directional wavelet transform. The transform retains the separable filtering, subsampling and simplicity of computations and filter design from the standard two-dimensional wavelet transform, unlike in the case of some other existing directional transform constructions (e.g. curvelets, contourlets or edgelets). The corresponding anisotropic basis functions, which we call directionlets, have directional vanishing moments along any two directions with rational slopes. Furthermore, we show that this novel transform provides an efficient tool for non-linear approximation of images, achieving the decay of mean-square errorO(N -1.55), which, while slower than the optimal rate O(N-2), is much better than O(N-1) achieved with wavelets, but at similar complexity. Owing to critical sampling, directionlets can easily be applied to image compression since it is possible to use Lagrange optimization as opposed to the case of overcomplete expansions. The compression algorithms based on directionlets outperform the methods based on the standard wavelet transform achieving better numerical results and visual quality of the reconstructed images. Moreover, we have adapted image denoising algorithms to be used in conjunction with an undecimated version of directionlets obtaining results that are competitive with the current state-of-the-art image denoising methods while having lower computational complexity.
BibTeX:
@phdthesis{Velisavljevic_V_2005_phd_dir_amdrsf,
  author = {Velisavljević, V.},
  title = {Directionlets: Anisotropic Multi-directional Representation with Separable Filtering},
  school = {EPFL},
  year = {2005}
}
Velisavljević, V., Beferull-Lozano, B., Vetterli, M. and Dragotti, P.L. Directionlets: Anisotropic multi-directional representation with separable filtering 2006 IEEE Trans. Image Process.
Vol. 15(7), pp. 1916-1933 
article DOI  
Abstract: In spite of the success of the standard wavelet transform (WT) in image processing in recent years, the efficiency of its representation is limited by the spatial isotropy of its basis functions built in the horizontal and vertical directions. One-dimensional (1-D) discontinuities in images (edges and contours) that are very important elements in visual perception, intersect too many wavelet basis functions and lead to a nonsparse representation. To efficiently capture these anisotropic geometrical structures characterized by many more than the horizontal and vertical directions, a more complex multidirectional (M-DIR) and anisotropic transform is required. We present a new lattice-based perfect reconstruction and critically sampled anisotropic M-DIR WT. The transform retains the separable filtering and subsampling and the simplicity of computations and filter design from the standard two-dimensional WT, unlike in the case of some other directional transform constructions (e.g., curvelets, contourlets, or edgelets). The corresponding anisotropic basis functions (directionlets) have directional vanishing moments along any two directions with rational slopes. Furthermore, we show that this novel transform provides an efficient tool for nonlinear approximation of images, achieving the approximation power O(N-1.55), which, while slower than the optimal rate O(N-2), is much better than O(N-1) achieved with wavelets, but at similar complexity.
BibTeX:
@article{Velisavljevic_V_2006_tip_dir_amdrsf,
  author = {Velisavljević, V. and Beferull-Lozano, B. and Vetterli, M. and Dragotti, P. L.},
  title = {Directionlets: Anisotropic multi-directional representation with separable filtering},
  journal = {IEEE Trans. Image Process.},
  year = {2006},
  volume = {15},
  number = {7},
  pages = {1916--1933},
  doi = {http://dx.doi.org/10.1109/TIP.2006.877076}
}
Velisavljević, V., Beferull-Lozano, B., Vetterli, M. and Dragotti, P.L. Directionlets: anisotropic multi-directional representation with separable filtering 2006   misc  
BibTeX:
@misc{Velisavljevic_V_2006_pp_dir_amdrsf,
  author = {Velisavljević, V. and Beferull-Lozano, B. and Vetterli, M. and Dragotti, P. L.},
  title = {Directionlets: anisotropic multi-directional representation with separable filtering},
  year = {2006},
  note = {Preprint, better reference teVelisavljevic_V_2006_tip_dir_amdrsf}
}
Wegmann, B. and Zetzsche, C. Efficient image sequence coding by vector quantization of spatiotemporal bandpass outputs 1992 Proc. SPIE 1818, Visual Communications and Image Processing, pp. 1146-1154  inproceedings DOI URL 
Abstract: A coding scheme for image sequences is designed in analogy to human visual information processing. We propose a feature-specific vector quantization method applied to multi-channel representation of image sequences. The vector quantization combines the corresponding local/momentary amplitude coefficients of a set of three-dimensional analytic band-pass filters being selective for spatiotemporal frequency, orientation, direction and velocity. Motion compensation and decorrelation between successive frames is achieved implicitly by application of a non-rectangular subsampling to the 3D-bandpass outputs. The nonlinear combination of the outputs of filters which are selective for constantly moving one- dimensional (i.e. spatial elongated) image structures allows a classification of the local/momentary signal features with respect to their intrinsic dimensionality. Based on statistical investigations a natural hierarchy of signal features is provided. This is then used to construct an efficient encoding procedure. Thereby, the different sensitivity of the human vision to the various signal features can be easily incorporated. For a first example, all multi- dimensional vectors are mapped to constantly moving 1D-structures.
BibTeX:
@inproceedings{Wegmann_B_1992_p-spie-vcip_efficient_iscvqsbo,
  author = {Wegmann, Bernhard and Zetzsche, Christoph},
  title = {Efficient image sequence coding by vector quantization of spatiotemporal bandpass outputs},
  booktitle = {Proc. SPIE 1818, Visual Communications and Image Processing},
  year = {1992},
  pages = {1146--1154},
  url = {http://dx.doi.org/10.1117/12.131386},
  doi = {http://dx.doi.org/10.1117/12.131386}
}
Welk, M., Weickert, J. and Steidl, G. From Tensor-Driven Diffusion to Anisotropic Wavelet Shrinkage 2006 Proc. Eur. Conf. Comput. Vis.  inproceedings  
BibTeX:
@inproceedings{Welk_M_2006_p-eccv_ten_ddaws,
  author = {Welk, M. and Weickert, J. and Steidl, G.},
  title = {From Tensor-Driven Diffusion to Anisotropic Wavelet Shrinkage},
  booktitle = {Proc. Eur. Conf. Comput. Vis.},
  year = {2006}
}
Westerink, P.H. Subband coding of images 1989 School: Delft University of Technology  phdthesis  
BibTeX:
@phdthesis{Westerink_P_1989_phd_subband_ci,
  author = {P. H. Westerink},
  title = {Subband coding of images},
  school = {Delft University of Technology},
  year = {1989}
}
Xu, D. and Do, M.N. Anisotropic 2D wavelet packets and rectangular tiling: theory and algorithms 2003 Proc. SPIE, Wavelets: Appl. Signal Image Process., pp. 619-630  inproceedings DOI  
Abstract: We propose a new subspace decomposition scheme called anisotropic wavelet packets which broadens the existing definition of 2-D wavelet packets. By allowing arbitrary order of row and column decompositions, this scheme fully considers the adaptivity, which helps find the best bases to represent an image. We also show that the number of candidate tree structures in the anisotropic case is much larger than isotropic case. The greedy algorithm and double-tree algorithm are then presented and experimental results are shown.
BibTeX:
@inproceedings{Xu_D_2003_p-spie-wasip_ani_2dwprtta,
  author = {Xu, D. and Do, M. N.},
  title = {Anisotropic 2D wavelet packets and rectangular tiling: theory and algorithms},
  booktitle = {Proc. SPIE, Wavelets: Appl. Signal Image Process.},
  year = {2003},
  pages = {619--630},
  doi = {http://dx.doi.org/10.1117/12.506601}
}
Zavadsky, V. Image Approximation by Rectangular Wavelet Transform 2007 J. Math. Imaging Vis.
Vol. 27, pp. 129-138 
article URL 
Abstract: We study image approximation by a separable wavelet basis $$ 2^k_1x-i)2^k_2y-j), x-i)2^k_2y-j), 2^k_1(x-i)y-j), x-i)y-i),$ where $k_1, k_2 in Z_+; i,jinZ; $$ and ?,? are elements of a standard biorthogonal wavelet basis in L 2 (?). Because k 1 ? k 2 , the supports of the basis elements are rectangles, and the corresponding transform is known as the rectangular wavelet transform . We provide a self-contained proof that if one-dimensional wavelet basis has M dual vanishing moments then the rate of approximation by N coefficients of rectangular wavelet transform is $$ O(N^-M) $$ for functions with mixed derivative of order M in each direction. These results are consistent with optimal approximation rates for such functions. The square wavelet transform yields the approximation rate is $$ O(N^-M/2) $$ for functions with all derivatives of the total order M . Thus, the rectangular wavelet transform can outperform the square one if an image has a mixed derivative. We provide experimental comparison of image approximation which shows that rectangular wavelet transform outperform the square one.
BibTeX:
@article{Zavadsky_V_2007_j-math-imaging-vis_image_arwt,
  author = {Zavadsky, Vyacheslav},
  title = {Image Approximation by Rectangular Wavelet Transform},
  journal = {J. Math. Imaging Vis.},
  publisher = {Springer Netherlands},
  year = {2007},
  volume = {27},
  pages = {129--138},
  note = {10.1007/s10851-007-0777-z},
  url = {http://dx.doi.org/10.1007/s10851-007-0777-z}
}

Starlet/wavelet names

Activelet [back to the starlet list]

In short: Wavelets inspired by the shape of canonical hemodynamic response functions
Etymology: Active wavelet
Origin: Khalidov, Ildar and Van De Ville, Dimitri and Fadili, Jalal M. and Unser, Michael A. Activelets and sparsity: a new way to detect brain activation from fMRI data, SPIE Optics and Photonics, Wavelets XII Conference 6701 - Proceedings of SPIE Volume 6701, 26-29 August 2007 [(pdf)]
Abstract: FMRI time course processing is traditionally performed using linear regression followed by statistical hypothesis testing. While this analysis method is robust against noise, it relies strongly on the signal model. In this paper, we propose a non-parametric framework that is based on two main ideas. First, we introduce a problem-specific type of wavelet basis, for which we coin the term "activelets". The design of these wavelets is inspired by the form of the canonical hemodynamic response function. Second, we take advantage of sparsity-pursuing search techniques to find the most compact representation for the BOLD signal under investigation. The non-linear optimization allows to overcome the sensitivity-specificity trade-off that limits most standard techniques. Remarkably, the activelet framework does not require the knowledge of stimulus onset times; this property can be exploited to answer to new questions in neuroscience.

Activelets: Wavelets for Sparse Representation of Hemodynamic Responses (DOI:10.1016/j.sigpro.2011.03.008), Ildar Khalidov and Jalal Fadili and François Lazeyras and Dimitri Van De Ville and Michael Unser (Related work)
Abstract: We propose a new framework to extract the activity-related component in the BOLD functional Magnetic Resonance Imaging (fMRI) signal. As opposed to traditional fMRI signal analysis techniques, we do not impose any prior knowledge of the event timing. Instead, our basic assumption is that the activation pattern is a sequence of short and sparsely-distributed stimuli, as is the case in slow event-related fMRI. We introduce new wavelet bases, termed ``activelets'', which sparsify the activity-related BOLD signal. These wavelets mimic the behavior of the differential operator underlying the hemodynamic system. To recover the sparse representation, we deploy a sparse-solution search algorithm. The feasibility of the method is evaluated using both synthetic and experimental fMRI data. The importance of the activelet basis and the non-linear sparse recovery algorithm is demonstrated by comparison against classical B-spline wavelets and linear regularization, respectively.
Contributors: Ildar Khalidov, Dimitri Van De Ville Jalal Fadili Michael Unser
Some properties:
Anecdote:
Usage: Detect brain activation from fMRI data
See also:
Comments:

AMlet, RAMlet [back to the starlet list]

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, [local AMlet GAMlet copy in pdf and ps]
Abstract: A simple and yet powerful method is presented to estimate nonlinearly and nonparametrically the components of additive models using wavelets. The estimator enjoys the good statistical and computational properties of the Waveshrink scatterplot smoother and it can be efficiently computed using the block coordinate relaxation optimization technique. A rule for the automatic selection of the smoothing parameters, suitable for data mining of large datasets, is derived. The wavelet-based method is then extended to estimate generalized additive models. A primal-dual log-barrier interior point algorithm is proposed to solve the corresponding convex programming problem. Based on an asymptotic analysis, a rule for selecting the smoothing parameters is derived, enabling the estimator to be fully automated in practice. We illustrate the finite sample property with a Gaussian and a Poisson simulation.
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

Armlet [back to the starlet list]

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
Abstract: The notion of armlets is introduced in this letter as a precise formulation of orthonormal multiwavelets that guarantee wavelet decomposition with highpass output not being effected by polynomial perturbation of the input. A mathematical scheme for constructing armlets is given, and it is shown that the notions of armlets and balanced multiwavelets are different. In particular, while balanced wavelets are armlets, the converse is false in general. One advantage of armlets is that the weaker assumption provides flexibility to facilitate wavelet and filter construction.
Contributors: Jian-ao Lian, and Charles K. Chui
Some properties: Defined to satisfy the th order wavelet consistency requirement (-WAC). More general than -balanced multiwavelets. Correspond to the Daubechies orthogonal wavelets (daublets) in the scalar setting
Anecdote:
Usage:
See also:
Comments:

Bandlet/Bandelet/Bandelette [back to the starlet list]

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
Abstract: We introduce a sparse image representation that takes advantage of the geometrical regularity of edges in images. A new class of one-dimensional wavelet orthonormal bases, called foveal wavelets, are introduced to detect and reconstruct singularities. Foveal wavelets are extended in two dimensions, to follow the geometry of arbitrary curves. The resulting two dimensional “bandelets” define orthonormal families that can restore close approximations of regular edges with few non-zero coefficients. A double layer image coding algorithm is described. Edges are coded with quantized bandelet coefficients, and a smooth residual image is coded in a standard two-dimensional wavelet basis
Bandlet Image Estimation with Model Selection (DOI:10.1016/j.sigpro.2011.01.013) [back to the starlet list] Charles Dossal and Stéphane Mallat and Erwan Le Pennec
Abstract: To estimate geometrically regular images in the white noise model and obtain an adaptive near asymptotic minimaxity result, we consider a model selection based bandlet estimator. This bandlet estimator combines the best basis selection behaviour of the model selection and the approximation properties of the bandlet dictionary. We derive its near asymptotic minimaxity for geometrically regular images as an example of model selection with general dictionary of orthogonal bases. This paper is thus also a self contained tutorial on model selection with orthogonal bases dictionary.
Contributors: Erwan Le Pennec, Stéphane Mallat, Charles Dossal, Gabriel Peyré
Some properties: Bandelets have a support parallel to flow lines in images. Approximation rate: -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 (Zoran), a start-up devoted to bandelet applications, including low bit-rate identity pictures
Comments: A second-generation Matlab bandelet toolbox is available from Gabriel Peyré at MatlabCentral

Barlet [back to the starlet list]

In short: A fat edgelet/beamlet
Etymology: From bar (a solid, more or less rigid object with a uniform cross-section smaller than its length) and the ewig let
Origin: Multiscale Geometric Image Compression using Wavelets and Wedgelets, Richard Baraniuk, Hyeokho Choi, Justin Romberg, Mike Wakin. [pdf]
Contributors:
Some properties:
Anecdote:
Usage:
See also:
Comments:

Bathlet [back to the starlet list]

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, D. M. Monro, B. E. Bassil and G. J. Dickson, IEEE International Conference on Image Processing, 1996, Vol.2, pp.581- 584 (local copy).
Abstract: This paper addresses the question: "What makes a good wavelet for image compression?", by considering objective and subjective measurements of quality. A new metric is proposed for the design of the Finite Impulse Response (FIR) filters used in the Discrete Wavelet Transform (DWT). The metric is the diagonal of the Heisenberg uncertainty rectangle, with time weighted by a factor k relative to frequency. Minimization of the metric balances the time and frequency spreads of the filter response. The metric can be computed directly from the filter coefficients, so it can be used to optimize wavelets for image compression without the cost of repeatedly compressing and decompressing images. A psychovisual evaluation carried out with 24 subjects demonstrates that orthonormal FIR filters designed this way give good subjective results with zerotree image compression.With suitably chosen k, both better subjective quality and lower RMS error are achived than with wavelets chosen for maximum regularity.
Space-frequency balance in biorthogonal wavelets, D. M. Monro and B. G. Sherlock, IEEE International Conference on Image Processing, 1997, Vol.1, pp.624-627 (local copy).
Abstract: This paper shows how to design good biorthogonal FIR filters for wavelet image compression by balancing the space and frequency dispersions of analysis and synthesis lowpass filters. A quality metric is proposed which can be computed directly from the filter coefficients. By optimizing over the space of FIR filter coefficients, a filter bank can be found which minimizes the metric in about 60 seconds on a high performance workstation. The metric contains three parameters which weight the space and frequency dispersions of the low pass analysis and synthesis filters. A series of biorthogonal, symmetric wavelet filters of length 10 was found, each optimized for different weightings. Each of these filter banks was then evaluated by compressing and decompressing five test images at three compression ratios. Selecting each optimum provides fifteen sets of parameters corresponding to filter banks which maximize the PSNR in each case. The average of these parameters was used to define a ‘mean’ filter bank, which was then evaluated on the test images. Individual images can produce substantially different weightings of the time dispersion at the optimum, but the PSNR of the mean filter is normally close to the optimum. The mean filter also compares favourably with a maximum regularity biorthogonal filter of the same length.
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" (the correct spelling is bat'leth, but the mistake is quite common, perhaps due to the analogy with a small "battle") 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)

The Klingon Bathlet, a personal weapon


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:

Beamlet [back to the starlet list]

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 Analysis, 2001, Stanford, Research report
Abstract: We describe a framework for multiscale image analysis in which line segments play a role analogous to the role played by points in wavelet analysis. The framework has 5 key components. The beamlet dictionary is a dyadically- organized collection of line segments, occupying a range of dyadic locations and scales, and occurring at a range of orientations. The beamlet transform of an image f(x, y) is the collection of integrals of f over each segment in the beamlet dictionary; the resulting information is stored in a beamlet pyramid. The beamlet graph is the graph structure with pixel corners as vertices and beamlets as edges; a path through this graph corresponds to a polygon in the original image. By exploiting the ?rst four components of the beamlet framework, we can formulate beamlet-based algorithms which are able to identify and extract beamlets and chains of beamlets with special properties. In this paper we describe a four-level hierarchy of beamlet algorithms. The ?rst level consists of simple procedures which ignore the structure of the beamlet pyra- mid and beamlet graph; the second level exploits only the parent-child dependence between scales; the third level incorporates collinearity and co-curvity relationships; and the fourth level allows global optimization over the full space of polygons in an image. These algorithms can be shown in practice to have suprisingly powerful and apparently unprecedented capabilities, for example in detection of very faint curves in very noisy data. We compare this framework with important antecedents in image processing (Brandt and Dym; Horn and collaborators; G¨otze and Druckenmiller) and in geo- metric measure theory (Jones; David and Semmes; and Lerman).
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. Also recent chordlets.
Comments: Beamlab: a Matlab (TM) toolbox code for the implementation of various feature oriented transforms

Binlet [back to the starlet list]

In short: A wavelet with "binary filter" coefficients or generated by "binary" wavelet coefficients filter bank
Etymology: From the contraction binary filter (symmetric) wavelet
Origin: Le Gall, D. and Tabatabai, A. Sub-band coding of digital images using symmetric short kernel filters and arithmetic coding techniques, Proc. ICASSP 1988
Abstract: A simple and efficient method of subband coding of digital images is reported. First, a technique for designing symmetric short tap filters is presented, and it is shown that such filters can be easily implemented by using simple arithmetic operations (e.g. addition and multiplication). By applying the above filters, the input image is decomposed into four bands, which are then coded by using arithmetic coding in combination with discrete PCM coding of the lowest band and PCM coding of higher bands. Simulation results demonstrate that by using the method mentioned above good quality pictures can be obtained in the range of 0.7 to 0.8 bits/pel
Strang, G. and Nguyen, T., Wavelets and filter banks, p. 217 or p. 249, Wellesley-Cambridge Presss, 1996

A. R. Calderbank and Ingrid Daubechies and Wim Sweldens and Boon-Lock Yeo Wavelet Transforms That Map Integers to Integers, ACHA, 1998
Abstract: Invertible wavelet transforms that map integers to integers have important applications in lossless coding. In this paper we present two approaches to build integer to integer wavelet transforms. The first approach is to adapt the precoder of Laroiaet al.,which is used in information transmission; we combine it with expansion factors for the high and low pass band in subband filtering. The second approach builds upon the idea of factoring wavelet transforms into so-called lifting steps. This allows the construction of an integer version of every wavelet transform. Finally, we use these approaches in a lossless image coder and compare the results to those given in the literature.
Contributors: Gilbert Strang, Truong Nguyen, and many others, sometimes under the name of reversible wavelets.
Some properties: DSP-friendly wavelet filter banks with integer coefficients (like the Haar wavelet) or integers divided by powers of 2, 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. Related works mention reversible ITI-wavelets (integer-to-inter wavelets, or filterbanks in general), multiplierless transforms, SOPOT (sum-of-powers-of-two) coefficients.
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 biorthogonal reversible 5/3 filter bank (known as the 5/3 Le Gall-Tabatabai filter bank) is used for lossless compression in the JPEG 2000 standard, with coefficients [1 2 1}/2 and [-1 2 6 2 -1]/8. The binary 9/7 filters are [1 0 -8 16 46 16 -8 0 1]/64 and [-1 0 9 16 9 0 -1]/32. The Le Gall 5/3 analysis filters [-1 2 6 2 -1]/8 and [-1 2 -1]/3
Usage: Binlets are especially useful for finite arithmetic reversible transforms, especially 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.

Brushlet [back to the starlet list]

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
Abstract: We construct a new adaptive basis of functions which is reasonably well localized with only one peak in frequency. We develop a compression algorithm that exploits this basis to obtain the most economical representation of the image in terms of textured patterns with different orientations, frequencies, sizes, and positions. The technique directly works in the Fourier domain and has potential applications for compression of highly textured images, texture analysis, etc.
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

Caplet, Camplet [back to the starlet list]

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 CAP, from 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 (CAPlet local copy)
Abstract: Wavelet decompositions are implemented and inverted by fast algorithms, the socalled fast wavelet transform (FWT). The FWT is the primary reason for the popularity of wavelet-based methods in so many different scientific and engineering disciplines. The second most important reason for the popularity of wavelets is their mathematical theory: that theory shows that the wavelet coefficients record faithfully the precise smoothness class of the underlying dataset/function. These characterizations are instrumental for the mathematical analysis of wavelet-based algorithms in the areas of image and signal analysis. The third most important reason for the popularity of wavelets (which is closely related to the first one) is the vehicle of MultiResolution Analysis (MRA) which allows for the construction of a wide variety of wavelet systems. This approach is epitomized in the univariate Mallat's algorithm. The effective construction of wavelet systems is more cumbersome in higher dimensions. For example, in 4D (and dyadic downsampling) one employs (at least) 15 different highpass filters in any MRA-based wavelet system. And the struggle in higher dimensions to balance optimally between time localization (short filters) and frequency localization is hampered by the need to adhere to the MRA-based construction principles.
Contributors: Amos Ron (University of Wisconsin), Youngmi Hur (Johns Hopkins University)
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.
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Usage:
See also: Hur, Yougmi and Ron, Amos, CAP representations (The mathematical theory of pyramidal algorithms), Wavelet Theory and Applications: Singapore, August 2004 (CAP representations (The mathematical theory of pyramidal algorithms))

Hur, Yougmi and Ron, Amos, CAPlets: wavelet representations without wavelets (CAPlet local copy)
Abstract: MultiResolution (MR) is among the most effective and the most popular approaches for data representation. In that approach, the given data are organized into a sequence of resolution layers, and then the "difference" between each two consecutive layers is recorded in terms of detail coefficients. Wavelet decomposition is the best known representation methodology in the MR category. The major reason for the popularity of wavelet decompositions is their implementation and inversion by a fast algorithm, the so-called fast wavelet transform (FWT). Another central reason for the success of wavelets is that the wavelet coefficients capture very accurately the smoothness class of the function hidden behind the data. This is essential for the understanding of the performance of key wavelet-based algorithms in compression, in denoising, and in other applications. On the downside, constructing wavelets with good space-frequency localization properties becomes involved as the spatial dimension grows. An alternative to the sometime-hard-to-construct wavelet representations is the always-easy-to-construct (and slightly older) non-orthogonal pyramidal algorithms. Similar to wavelets, the (linear, regular, isotropic) pyramidal representations are based on some method for linear coarsening (by a decomposition filter) of their data, and a complementary method for linear prediction (by a prediction filter) of the original data from the coarsened one. The first step creates the resolution layers and the second allows for trivial extractions of suitable detail coefficients. The decomposition and reconstruction algorithms in the pyramidal approach are as fast as those of wavelets. In contrast with orthonormal wavelets, the representation is redundant, viz. the total number of detail coefficients exceeds the original size of the data: denoting by s the ratio between the size of the data at two consecutive resolution layers, the “redundancy ratio” in the pyramidal representation is s/(s - 1). In this paper, we introduce and study a general class of pyramidal representations that we refer to as Compression- Alignment-Prediction (CAP) representations. The CAP representation is based on the selection of three filters: the low-pass decomposition filter, the low-pass prediction filter, and the full-pass alignment filter. Like previous pyrami- dal algorithms, CAP are implemented by a simple, fast, wavelet-like decomposition and a trivial reconstruction. The primary goal of this paper is to establish the precise way in which the CAP representations encode the smoothness class of the underlying function. Remarkably, the CAP coefficients provide the same characterizations of Triebel- Lizorkin spaces and Besov spaces as the wavelet coefficients do, provided that the three CAP filters satisfy certain requirements. This means, at least in principle, that the performance of CAP-based algorithms should be similar to their wavelet counterparts, despite of the fact that, when compared with wavelets, it is much easier to develop CAP representations with “customized” or “optimal” properties. Moreover, upon assuming the prediction filter to be interpolatory, we extract from the CAP representation a sister CAMP representation (“M” for “modified”). Those CAMP representations strike a phenomenal balance between performance (viz., smoothness characterization) and space localization. Our analysis of the CAP representations is based on the existing theory of framelet (redundant wavelet) representations.
Comments:

Chirplet [back to the starlet list]

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 (local copy), Proc. Vision Interface'91, June 3-7, pp. 205-212, 1991.
Abstract: We propose a novel transform, an expansion of an arbitrary function onto a basis of multi-scale chirps (swept frequency wave packets). We apply this new transform to a practical problem in marine radar: the detection of floating objects by their "acceleration signature" (the "chirpyness" of their radar backscatter), and obtain results far better than those previously obtained by other current Doppler radar methods. Each of the chirplets essentially models the underlying physics of motion of a floating object. Because it so closely captures the essence of the physical phenomena, the transform is near optimal for the problem of detecting floating objects. Besides applying it to our radar image processing interests, we also found the transform provided a very good analysis of actual sampled sounds, such as bird chirps and police sirens, which have a chirplike nonstationarity, as well as Doppler sounds from people entering a room, and from swimmers amid sea clutter.
Mihovilovic, D. and Bracewell, R., Adaptive chirplet representation of signals on time-frequency plane (local copy), Electronic Letters, 27(13), pp. 1159-1161, June 1991.
Abstract: Dynamic spectra, which exhibit the spectral content of a signal as time elapses, are based on subdivision of the time-frequency plane into minimum-area rectangular cells. The cell dimensions in time and frequency are usually held constant throughout. A more general spectral analysis would allow the cells to change aspect ratio with time. Elementary cells assuming oblique forms (chirplets) are proposed, together with an adaptive method for selecting their aspect ratio and obliquity to suit the data.
The chirplet transform: physical considerations (local copy), Mann, S. and Haykin, S., IEEE Trans. Signal Processing, 1995
Abstract: We consider a multidimensional parameter space formed by inner products of a parameterizable family of chirp functions with a signal under analysis. We propose the use of quadratic chirp functions (which we will call q-chirps for short), giving rise to a parameter space that includes both the time-frequency plane and the time-scale plane as 2-D subspaces. The parameter space contains a “time-frequency-scale volume” and thus encompasses both the short-time Fourier transform (as a slice along the time and frequency axes) and the wavelet transform (as a slice along the time and scale axes). In addition to time, frequency, and scale, there are two other coordinate axes within this transform space: shear in time (obtained through convolution with a q-chirp) and shear in frequency (obtained through multiplication by a q-chirp). Signals in this multidimensional space can be obtained by a new transform, which we call the “q-chirplet transform” or simply the “chirplet transform”. The proposed chirplets are generalizations of wavelets related to each other by 2-D affine coordinate transformations (translations, dilations, rotations, and shears) in the time-frequency plane, as opposed to wavelets, which are related to each other by 1-D affine coordinate transformations (translations and dilations) in the time domain only
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 (as opposed to linear) chirplets are also of interest for radar applications. Adaptive or even
Wave, wavelet, chirp, chirplet
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. arcades in a perspective picture)
Chirplets for image processing
See also: Several publications on chirplets on Steve Mann's page, and a wikipedia page chirplets with a reference to w-chirplets as warblets
Comments: The "independent" discovery and naming controversy of chirplets by two groups at about the same time is not even discussed here

Chordlet [back to the starlet list]

In short: Multiscale arc-based dictionary with constrainted curvature and endpoints
Etymology: From chord. It ought to be straight line connecting two points on a curve. Here a chord (reminiscent of a beamlet) subtends a set of arcs
Origin: He, Z. and Bystrom, M. The chordlet transform with an application to shape compression, Signal Processing: Image Communication, 2012. (chordlet local copy)
Due to their abilities to succinctly capture features at different scales and directions, wavelet-based decomposition or representation methods have found wide use in image analysis, restoration, and compression. While there has been a drive to increase the representation ability of these methods via directional filters or elongated basis functions, they still have been focused on essentially piecewise linear representation of curves in images. We propose to extend the line-based dictionary of the beamlet framework to one that includes sets of arcs that are quantized in height. The proposed chordlet dictionary has elements that are constrained at their endpoints and limited in curvature by system rate or distortion constraints. This provides a more visually natural representation of curves in images and, furthermore, it is shown that for a class of images the chordlet representation is more efficient than the beamlet representation under tight distortion constraints. A data structure, the fat quadtree and an algorithm for determining an optimal chordlet representation of an image are proposed. Codecs have been implemented to illustrate applications to both lossy and lossless low bitrate compressions of binary edge images, and better rate or rate–distortion performance over the JBIG2 standard and a beamlet-based compression method are demonstrated.
He, Z. Texture- and structure-based image representation with applications to image retrieval and compression, PhD Thesis, Boston university, 2007. (chordlet local copy)
The design of an efficient image representation methods using small numbers of features can facilitate image processing tasks such as compression of images and content-based retrieval of images from databases. In this dissertation, three methods for capturing and concisely representing two distinguishing characteristics of images, namely texture and structure, are developed. Applications of these compact representations of image characteristics to image compression as well as retrieval of images and hand-sketches of images from databases are given and performance is compared with other compression and retrieval methods. The first method to be introduced is a directional, hidden-Markov-model-based method for succinctly describing image texture using a small number of features. This method employs the well known, multi-scale contourlet and steerable-pyramid transforms to isolate in different subbands the edges that comprise the image texture. Statistical inter- and intrasubband dependencies are captured via hidden Markov models, and model parameters are used to represent texture in small feature sets. Application of this method to content-based retrieval of images with homogeneous textures from database is shown. At the similar computation cost, about 10% higher retrieval rates over comparable methods are demonstrated; when approximately one third fewer features are used, similar retrieval rates can be obtained using the proposed method. A method for concisely describing large image structures, that is, significant image edges, is then proposed. This method decomposes an image using the contourlet transform into directional subbands which contain edges of different orientations. Each subband is then projected onto its associated primary and orthogonal directions and the resulting projections are filtered and then modeled using piece-wise linear approximations or Gaussian mixture models. The model parameters then form the concise feature sets used to represent the image's structure. An application of this image-representation method to retrieval of images from databases based on users' sketches of the images is shown. An retrieval rate increase of 13% using the proposed method is demonstrated over a current spatial-histogram-based method. Finally, a new multi-scale curve representation framework, the chordlet, is constructed for succinct curve-based image structure representation. This framework can be viewed as an extension to curves of the well known beamlet transform, a multi-scale line representation system. In this dissertation, the representation efficiency, in terms of bits versus distortion, of the chordlet transform is compared with that of the beamlet transform. An algorithm for performing a fast chordlet transform has been developed. A chordlet-based coding system is constructed for application of the chordlet transform to compression of image shapes. By using the proposed method increased compression is obtained at lower distortion when compared with two well known methods.
Contributors: Zhihua He and M. Bystrom
Some properties: Uses a fat quadtree
Chorlets at differents scales
Anecdote:
Usage: Image compression, especially contour/shape compression (JBIG2, JBEAM)
See also: Chordlets extends beamlet dictionary. Directionlets and bandlets do not stand afar.
Comments:

Circlet [back to the starlet list]

In short: The result of a convolution between a limited width circular shape and a wavelet
Etymology: Wavelet in circles
Origin: Chauris, H., Karoui, I., Garreau, P., Wackernagel, H., Craneguy, P. and Bertino, L., The circlet transform: A robust tool for detecting features with circular shapes (local copy), Computers & Geosciences, 2011-03, Vol. 37, N. 3, P. 331-342
Hervé Chauris et al., Ocean eddy tracking with circlets, GeoInformatics for Environmental Surveillance (StatDIS 2009)
Contributors: Hervé Chauris
Some properties:
Anecdote: The Circlet (wikipedia), a.k.a. stephanos is a ancient type of crown without arches or cap, often used as a bridal or fairy attributes (aren't they the same?) According to Medieval Bridal Fashions, "It will work with any hairstyle." With any Haar Styl too?
Usage: Coastal oceanography and ocean eddy tracking
See also:
Comments:

Coiflet [back to the starlet list]

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 wavelets II. Variations on a theme (local copy), 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. The cooklet stands for a biorthogonal nearly coiflet
Comments:

Contourlet [back to the starlet list]

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

Cooklet [back to the starlet list]

In short: Biorthogonal nearly coiflet
Etymology: Named after Dr. T. Cooklev for his construction of the odd-length biorthogonal coiflets, and the let< /td>
Origin: Winger, L. L. and Venetsanopoulos, A. N. Biorthogonal nearly coiflet wavelets for image compression, Signal Processing: Image Communication, Volume 16, Issue 9, June 2001, Pages 859-869, see also an early version: Winger, L. L. and Venetsanopoulos, A. N. Biorthogonal modified coiflet filters for image compression
Contributors: Lowell L. Winger, Anastasios (Tas) Venetsanopoulos
Some properties:
Anecdote:
Usage: Image compression
See also:
Comments:

Craplet [back to the starlet list]

In short: Crap stuff in the wavelet domain, esp. broken wavelet code
Etymology: Simply from crap
Origin: Meerwald, Peter, The craplet page (assorted broken Wavelet code)
Contributors: Peter Meerwald
Some properties: Searches for crappy wavelet code
Anecdote:
Usage: For clean wavelet code. See Craplets by Peter Meerwald for examples
See also:
Comments: Akin to Sturgeon's Law: Ninety percent of everything is crap (or crude)

Curvelet [back to the starlet list]

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

Daublet [back to the starlet list]

In short: Orthogonal compactly supported wavelet with a maximal number of vanishing moments for some given (finite) support. A Daublet is each member of Daubechies's extremal phase family.
Etymology: Nickname for orthogonal Daubechies wavelets
Origin: Contraction from the name of Ingrid Daubechies
Contributors: ()
Some properties:
Anecdote:
Usage:
See also: Other classical compactly supported orthogonal Daubechies wavelets with approximate symmetry, the symmlets, or with vanishing moments equally distributed on the scaling function and of the wavelet, the coiflets. Armlets are multiwavelets that restrict to Daubechies wavelets in the scalar case
Comments:

Directionlet [back to the starlet list]

In short:
Etymology:
Origin: Velisavljevic, Vladan and Beferull-Lozano, Baltasar and Vetterli, Martin and Dragotti, Pier Luigi, Directionlets: Anisotropic multi-directional representation with separable filtering, submitted to IEEE Transactions on Image Processing (Dec. 2004)
Contributors: Vladan Velisavljevic, Baltasar Beferull-Lozano, Martin Vetterli, Pier Luigi Dragotti
Some properties:
Anecdote:
Usage:
See also: No public toolbox available, but additional details on Vladan Velisavljevic webpage
Comments:

Edgelet [back to the starlet list]

In short: Element for a collection of edgels (small line segments forming an edge) 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 ???

ERBlet [back to the starlet list]

In short: A linear and invertible time-frequency transformation adapted to human auditory perception, for masking and perceptual sparsity
Etymology: From the ERB scale or Equivalent Rectangular Bandwidth filter banks, devised for auditory based-representation, following the philosophy of third-octave filter banks. See also Frequency Analysis and Masking - MIT, Brian C. J. Moore, 1995 and Bark and ERB Bilinear Transforms - Stanford University, by J. O. Smith III
Origin: Thibaud Necciari, Design and implementation of the ERBlet transform, FLAME 12 (Frames and Linear Operators for Acoustical Modeling and Parameter Estimation), 2012
Time-frequency representations are widely used in audio applications involving sound analysis-synthesis. For such applications, obtaining a time-frequency transform that accounts for some aspects of human auditory perception is of high interest. To that end, we exploit the theory of non-stationary Gabor frames to obtain a perception-based, linear, and perfectly invertible time-frequency transform. Our goal is to design a non-stationary Gabor transform (NSGT) whose time-frequency resolution best matches the time-frequency analysis properties by the ear. The peripheral auditory system can be modeled in a first approximation as a bank of bandpass filters whose bandwidth increases with increasing center frequency. These so-called “auditory filters” are characterized by their equivalent rectangular bandwidths (ERB) that follow the ERB scale. Here, we use a NSGT with resolution evolving across frequency to mimic the ERB scale, thereby naming the resulting paradigm "ERBlet transform". Preliminary results will be presented. Following discussion shall focus on finding the "best" transform settings allowing to achieve perfect reconstruction while minimizing redundancy.
Thibaud Necciari with P. Balazs, B. Laback, P. Soendergaard, R. Kronland-Martinet, S. Meunier, S. Savel, and S. Ystad, The ERBlet transform, auditory time-frequency masking and perceptual sparsity, 2nd SPLab Workshop, October 24–26, 2012, Brno
The ERBlet transform, time-frequency masking and perceptual sparsity Time-frequency (TF) representations are widely used in audio applications involving sound analysis-synthesis. For such applications, obtaining an invertible TF transform that accounts for some aspects of human auditory perception is of high interest. To that end, we combine results of non-stationary signal processing and psychoacoustics. First, we exploit the theory of non-stationary Gabor frames to obtain a linear and perfectly invertible non-stationary Gabor transform (NSGT) whose TF resolution best matches the TF analysis properties by the ear. The peripheral auditory system can be modeled in a first approximation as a bank of bandpass filters whose bandwidth increases with increasing center frequency. These so-called “auditory filters” are characterized by their equivalent rectangular bandwidths (ERB) that follow the ERB scale. Here, we use a NSGT with resolution evolving across frequency to mimic the ERB scale, thereby naming the resulting paradigm “ERBlet transform”. Second, we exploit recent psychoacoustical data on auditory TF masking to find an approximation of the ERBlet that keeps only the audible components (perceptual sparsity criterion). Our long-term goal is to obtain a perceptually relevant signal representation, i.e., as close as possible to “what we see is what we hear”. Auditory masking occurs when the detection of a sound (referred to as the “target” in psychoacoustics) is degraded by the presence of another sound (the “masker”). To accurately predict auditory masking in the TF plane, TF masking data for masker and target signals with a good localization in the TF plane are required. To our knowledge, these data are not available in the literature. Therefore, we conducted psychoacoustical experiments to obtain a measure of the TF spread of masking produced by a Gaussian TF atom. The ERBlet transform and the psychoacoustical data on TF masking will be presented. The implementation of the perceptual sparsity criterion in the ERBlet will be discussed.
Contributors: Thibaud Necciari with P. Balazs, B. Laback, P. Soendergaard, R. Kronland-Martinet, S. Meunier, S. Savel, and S. Ystad
Some properties: Develops a non-stationary Gabor transform (NSGT) [Theory, Implementation and Application of Nonstationary Gabor Frames, P. Balazs et al., J. Comput. Appl. Math., 2011] with resolution evolving over frequency to mimic the ERB scale (Equivalent Rectangular Bandwidth, after B. C. J. Moore and B. R. Glasberg, "Suggested formulae for calculating auditory-filter bandwidths and excitation patterns", J. Acoustical Society of America 74:750-753, 1983). Linear and invertible time-frequency transform adapted to human auditory perception.
Anecdote:
Usage: A few ERBlet Matlab scripts for ICASSP 2013 are downloadable at the ERBlet transform project listing. An implementation of the ERBlet transform is available in the excellent The Large Time-Frequency Analysis Toolbox, also known as the LTFAT toolbox ("All your frame are belong to us")
See also:
Comments:

Flatlet [back to the starlet list]

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:

Framelet [back to the starlet list]

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:

Fresnelet [back to the starlet list]

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:

Gaborlet [back to the starlet list]

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:

GAMlet [back to the starlet list]

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

Gausslet [back to the starlet list]

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:

Graphlet [back to the starlet list]

In short: A nickname for wavelets on graphs
Etymology: From the Graph structure (as introduced by Sylvester in Nature, 1878) and let
Origin: Wavelets on Graphs via Spectral Graph Theory, Applied and Computational Harmonic Analysis, 2011 (local copy, DOI) David K. Hammond, Pierre Vandergheynst Rémi Gribonval
Abstract: We propose a novel method for constructing wavelet transforms of functions defined on the vertices of an arbitrary finite weighted graph. Our approach is based on defining scaling using the graph analogue of the Fourier domain, namely the spectral decomposition of the discrete graph Laplacian L. Given a wavelet generating kernel g and a scale parameter t, we define the scaled wavelet operator Ttg = g(tL). The spectral graph wavelets are then formed by localizing this operator by applying it to an indicator function. Subject to an admissibility condition on g, this procedure defines an invertible transform. We explore the localization properties of the wavelets in the limit of fine scales. Additionally, we present a fast Chebyshev polynomial approximation algorithm for computing the transform that avoids the need for diagonalizing L. We highlight potential applications of the transform through examples of wavelets on graphs corresponding to a variety of different problem domains.
Contributors: David K. Hammond, Pierre Vandergheynst Reacute;mi Gribonval
Some properties:
Anecdote: http://en.wikipedia.org/wiki/Graphlets: The name "graphlet" for "wavelets on graphs" steals from another network technology.
Graphlets are small connected non-isomorphic induced subgraphs of a large network.[1][2] Graphlets differ from network motifs, since they must be induced subgraphs, whereas motifs are partial subgraphs. An induced subgraph must contain all edges between its nodes that are present in the large network, while a partial subgraph may contain only some of these edges. Moreover, graphlets do not need to be over-represented in the data when compared with randomized networks, while motifs do.
Usage:
See also: The Spectral Graph Wavelets Matlab Toolbox page is now available, with a direct link to sgwt_toolbox-1.01.zip (local copy). PySGWT, a python code port for graphlet (aka Spectral Graph Wavelet Transform). PySGWT

Narang, S. K. and Ortega, A.:
Perfect Reconstruction Two-Channel Wavelet Filter-Banks for Graph Structured Data, 2012, 32 pages double spaced 12 Figures, to appear in IEEE Transactions of Signal Processing
Abstract: In this work we propose the construction of two-channel wavelet filterbanks for analyzing functions defined on the vertices of any arbitrary finite weighted undirected graph. These graph based functions are referred to as graph-signals as we build a framework in which many concepts from the classical signal processing domain, such as Fourier decomposition, signal filtering and downsampling can be extended to graph domain. Especially, we observe a spectral folding phenomenon in bipartite graphs which occurs during downsampling of these graphs and produces aliasing in graph signals. This property of bipartite graphs, allows us to design critically sampled two-channel filterbanks, and we propose quadrature mirror filters (referred to as graph-QMF) for bipartite graph which cancel aliasing and lead to perfect reconstruction. For arbitrary graphs we present a bipartite subgraph decomposition which produces an edge-disjoint collection of bipartite subgraphs. Graph-QMFs are then constructed on each bipartite subgraph leading to "multi-dimensional" separable wavelet filterbanks on graphs. Our proposed filterbanks are critically sampled and we state necessary and sufficient conditions for orthogonality, aliasing cancellation and perfect reconstruction. The filterbanks are realized by Chebychev polynomial approximations.

Yue M. Lu: Spectral graph wavelet frames with compact supports, Wavelets and Sparsity, Proc. SPIE 2011
Comments: The Spectral Graph Wavelets Toolbox page (SGWT) is not to be mistaken with the SGWT = Second Generation Wavelet Transform. Also different from other Graphlets which are small connected non-isomorphic induced subgraphs of a large network

Grouplet [back to the starlet list]

In short: Multiscale grouped coefficients through association fields
Etymology: From a grouping of (wavelet) coefficients)
Origin: Mallat, Stéphane, Geometrical Grouplets, submitted to ACHA - Applied and Computational Harmonic Analysis (Oct. 2006)
Contributors: Stéphane Mallat
Some properties:
Anecdote:
Usage:
See also:
Comments:

Haarlet [back to the starlet list]

In short: A not-so-common nickname for the Haar wavelet
Etymology: From hungarian mathematician Alfréd Haar (MacTutor History)
Origin: Haar, Alfréd, Zur Theorie der orthogonalen Funktionen-Systeme, Math. Ann., vol. 69, pp. 331-371, 1910 (On the Theory of Orthogonal Function Systems, translated for the magnificent collection of papers in Fundamental Papers in Wavelet Theory edited by Christopher Heil and David F. Walnut)
In Real-Time Body Pose Recognition Using 2D or 3D Haarlets (Internation Journal on Computer Vision, 2009), Van den Bergh et al. abbreviate a combination of Average Neighborhood Margin Maximization (ANMM) and (Viola and Jones 2001) Haar wavelet-like features as "Haarlets".
Contributors: Alfred Haar
Some properties: A Schauder basis, unconditional for Lp spaces, p > 1. Discontinuous
Anecdote: Celebrate Haar wavelet centenary with the following Memorial plaque in honor of A. Haar and F. Riesz found at Szeged University: the inscription says: "A szegedi matematikai iskola világhírű megalapítói (The worldwide famous founders of the mathematical school in Szeged)" [picture courtesy of Professor Károly Szatmáry]. The picture is a natural testbench for directional/textural analysis.

Szeged university Haar Riesz
Usage: Often considered of poor performance in "real life" applications, the Haar wavelet may prove very efficient if used cleverly (for instance Fast Haar-wavelet denoising of multidimensional fluorescence microscopy data, F. Luisier et al., ISBI 2009). Much sooner, an avatar of the 2-D Haar transform, under the name of "H-Transform" (at MathWorld), as been used for astronomical image compression (Hcompress Image Compression Software ), originated in Fritze, K.; Lange, M.; Möstle, G.; Oleak, H.; and Richter, G. M. "A Scanning Microphotometer with an On-Line Data Reduction for Large Field Schmidt Plates." Astron. Nachr. 298, 189-196, 1977.
See also: Wikipedia: Haar wavelet or the Multi-level Haar Transform at Connexions (Rice University)
Comments:

Haardlet [back to the starlet list]

In short:
Etymology: From a pun on mathematicians Alfréd Haar and Jacques Hadamard: Ha(dam)ard. Reminicent to the Waleymard transform, build upon J. L. Walsh, Raymond E.A.C. Paley and Jacques Hadamard, depending on the basis ordering (resp. sequency, dyadic or natural), see Wolfram Walsh page for instance
Origin:
Contributors: Grand-Admiral Petry
Some properties:
Anecdote:
Usage:
See also:
Comments:

Heatlet [back to the starlet list]

In short: The heat evolution of an initial wavelet state
Etymology: From the heat equation and the diminutive let
Origin: On Wavelet Fundamental Solutions to the Heat Equation---Heatlets, Shen, Jianhong and Strang, Gilbert, Journal of Differential Equations, 2000
We present an application of wavelet theory in partial differential equations. We study the wavelet fundamental solutions to the heat equation. The heat evolution of an initial wavelet state is called a heatlet. Like wavelets for the L2 space, heatlets are "atomic'' heat evolutions in the sense that any general heat evolution can be "assembled'' from a heatlet according to some simple rules. We study the basic properties and algorithms of heatlets and related functions
Contributors: Jackie (Jianhong) Shen
Gilbert Strang
Some properties:
Anecdote:
Usage:
See also: A few lines from Image Compression and Wavelet Applications at UCLA
Comments:

Hutlet [back to the starlet list]

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: Uwe Meyer-Bäse
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?
Other wavelets reveal a similar kind of hat trick: the Mexican hat wavelet (also known as the Ricker wavelet) and the
Usage: Envelope discontinuity detection in amplitude modulation
See also: A scaling function in the hutlet may be view as an instance of a binlet
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
Comments:

Hyperbolet [back to the starlet list]

In short: An example of multi-composite wavelets with hyperbolic scaling law
Etymology: From the hyperbola (wiki entry), with a potential reference (article no available on 2011/05/26) to the parabolic scaling law of the shearlets
Origin: Glenn R. Easley, Demetrio Labate, Vishal M. Patel: Multi-composite wavelet estimation, Proceedings of SPIE Volume 8138, Wavelets and Sparsity XIV, Aug. 2011 (local copy)
Abstract: In this work, we present a new approach to image denoising by using a general representation known as wavelets with composite dilations. These representations allow for waveforms to be defined not only at various scales and locations but also at various orientations. For this talk, we present many new representations such as hyperbolets and propose combining multiple estimates from various representations to form a unique denoised image. In particular, we can take advantage of different representations to sparsely represent important features such as edges and texture independently and then use these estimates to derive an improved estimate.
The hyperbolet construction is further refined in:
G. R. Easley, D. Labate and V. M. Patel, Hyperbolic shearlets, IEEE International Conference on Image Processing (ICIP), Orlando, FL, 2012, submitted (local copy)
G. R. Easley, D. Labate, and V. M. Patel, Directional multiscale processing of images using wavelets with composite dilations, submitted 2011 (local copy)
Contributors: Glenn R. Easley (no personal page), Demetrio Labate, Vishal M. Patel
Some properties:
Tiling of the frequency domain associated with an hyperbolic system of wavelets with composite dilations.
Closely related to shearlets
Anecdote:
Usage:
See also: The above work might be related to Glenn R. Easley, Demetrio Labate: Critically Sampled Wavelets with Composite Dilations (local copy), preprint, 2011, which develops interesting critically sampled directional wavelet schemes (DWTShear, CShear, QDWTShear)
Comments: See also: Hyperbolets (on WITS: Where is the Starlet)

Icatlet (icalette) [back to the starlet list]

In short: Independent Component Analysis by Wavelets
Etymology: Concatenation of ICA, a standard method for blind source separation, and let
Origin: Independent Component Analysis by Wavelets, Pascal Barbedor, Preprint, 2005, published in Test, 2009
This paper introduces a new approach in solving the ICA problem using a method that fits in the contrast and minimize paradigm, mostly found in the ICA literature. In our case, the contrast is a L_2 norm dependence measure, which constitutes an alternative to the usual criteria, based on mutual information. We propose a non parametric evaluation of the L_2 contrast, using a wavelet projection estimator. The mean square error of the procedure is bounded under Besov assumptions. Finally, we provide a set of simulations to show how the method performs in practice.
Independent component analysis and estimation of a quadratic functional, Pascal Barbedor, Preprint, 2006
Independent component analysis (ICA) is linked up with the problem of estimating a non linear functional of a density, for which optimal estimators are well known. The precision of ICA is analyzed from the viewpoint of functional spaces in the wavelet framework. In particular, it is shown that, under Besov smoothness conditions, parametric rate of convergence is achieved by a U-statistic estimator of the wavelet ICA contrast, while the previously introduced plug-in estimator C2j, with moderate computational cost, has a rate in n-4s/(4s+d).
Independent component analysis by wavelets, Pascal Barbedor, PhD thesis, 2006
Independent component analysis (ICA) is a form of multivariate analysis that emerged as a concept in the eighties/nineties. It is a type of inverse problem where one observes a variable X whose components are linear mixtures of an unobservable variable S. The components of S are mutually independent. The relation between both variables is expressed by X=AS, where A is an unknown mixing matrix. The main problem in ICA is to estimate the matrix A, seeing an i.i.d. sample of X, to reach S which constitutes a better explicative system than X, in the study of some phenomena. The problem is generally resolved through the minimization of a criteria coming from some dependence measure. ICA looks like principal component analysis (PCA) in the formulation. In PCA, one seeks after uncorrelated components, that is to say pairwise independent at order 2 ; as for ICA, one seeks after mutually independent components, which is much more constraining, and there is not any more a simple algebraic solution in the general case. The main problems in the identification of A are removed by restrictions imposed in the classical ICA model. The approach which is proposed in this thesis adopts a non parametric point of view. Under Besov assumptions, we study several estimators of an exact dependence criteria given by the L2 norm between a density and the product of its marginals. This criteria constitutes an alternative to mutual information which represented so far the exact criteria of reference for the majority of ICA methods. We give an upper bound of the mean squared error of different estimators of the L2 contrast. This bound takes into account the approximation bias between the Besov space and the projection space which, here, stems from a multiresolution analysis (MRA) generated by the tensorial product of Daubechies wavelets. This type of bound, taking into account the approximation bias, is generally absent from recent non parametric methods in ICA (kernel methods, mutual information). The L2 norm criteria makes it possible to get closer to well-known problems in the statistical literature, estimation of integral of squared f, L2 norm homogeneity tests, convergence rates for estimators adopting block thresholding. We propose estimators of the L2 contrast that reach the optimal minimax rate of the problem integral of squared f. These estimators, of U-statistic type, have numerical complexities quadratic in n, which can be a problem for the contrast minimization to follow, to obtain a concrete estimation of matrix A. However these estimators also admit a block-thresholded version, where knowledge of the regularity s of the underlying multivariate density is useless to obtain an optimal rate. We propose a plug-in type estimator whose convergence rate is sub-optimal but with a numerical complexity linear in n. The plug-in estimator also admits a term by term thresholded version, which dampens the convergence rate but yields an adaptive criteria. In its linear version, the plug-in estimator already seems auto-adaptive in facts, that is to say under the constraint 2^{jd} &<; n, where d is the dimension of the problem and n the number of observations, the majority of resolutions j allow to estimate A after minimization. To obtain these results, we had to develop specific combinatorial tools, that allow to bound the rth moment of a U-statistic or a V-statistic. Standard results on U-statistics are indeed not directly usable and not easily adaptable in the context of study of the thesis. The tools that were developed are usable in other contexts. The wavelet method builds upon the usual paradigm, estimation of an independence criteria, then minimization. So we study in the thesis the elements useful for minimization. In particular we give filter aware formulations of the gradient and the hessian of the contrast estimator, that can be computed with a complexity equivalent to that of the estimator itself. Simulations proposed in the thesis confirm the applicability of the method and give excellent results. All necessary information for the implementation of the method, and the commented code of key parts of the program (notably d-dimensional algorithms) also appear in the document.
Contributors: Pascal Barbedor (old page: http://www.proba.jussieu.fr/pageperso/barbedor/)
Some properties:
Anecdote:
Usage:
See also:
Comments: Fortran source code and Mac OS X Pascal Barbedor icalet binaries

Interpolet [back to the starlet list]

In short: Interpolating wavelet transform
Etymology: Combination of Interpolation and let
Origin: Apparently, Donoho, D. L. (once again), Interpolating wavelet transforms 1992, technical report, Stanford university, although the name "interpolet" itself has been coined later (local pdf copy).
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:

Loglet [back to the starlet list]

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:
See also:
Comments:

MIMOlet [back to the starlet list]

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

Morelet [back to the starlet list]

In short: Short name for the Morlet wavelet
Etymology: A clever combination, child of the father Jean Morlet and the mother wavelet
Origin: Misprint found in several preprints and wavelet papers
Contributors: Will remain anonymous, as long as you don't search for "Morelet wavelet"
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, or Crocodylus moreletii, from the French naturalist P. M. A. Morelet (1809-1892), who discovered this species in 1850 in Mexico.
Funnily enough, the Morelet crocodile is also called the Mexican crocodile. I suspect P. M. A. Morelet came back from Mexico with a Mexican hat (or Sombrero), which is one of the most famous wavelet shape, known as the sombrero wavelet, mexican hat, Ricker wavelet (in Geophysics) or Marrlet (from the work of David Marr). It is built with a normalized second derivative of a Gaussian function, related to the second Hermite function (cf. Hermite polynomials). It is a special case of the family of continuous wavelets (wavelets used in a continuous wavelet transform) known as Hermitian wavelets. It generalizes in higher dimensions to the Laplacian of Gaussians. It is sometimes approximated in practice by the difference of Gaussians function, or by derivatives of cardinal B-splines. The actual Morlet wavelet is not really admissible. It is a Gaussian modulated by a sine/cosine or a cisoid (for the complex Morlet wavelet), while the (Morelet) Mexican hat wavelet is a sort of Gaussian modulated by a (weakly) polynomial function.
Complex Morlet wavelet with real and imaginary approximate Hilbert pair parts.
Usage: The false appellation "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 Digital Signal Processing, as in the invention of Softy space (cf. Hardy spaces), or in company names like Let it wave (now Zoran)

Morphlet [back to the starlet list]

In short: A multiscale representation for diffeomorphisms
Etymology: A contraction of both Morphing or Morphism, and the mother wavelet
Origin: Jonathan R. Kaplan and David L. Donoho, The Morphlet Transform: A Multiscale Representation for Diffeomorphisms (local copy), Workshop on Image Registration in Deformable Environments, 2006
Contributors: Jonathan R. Kaplan and David L. Donoho
Some properties:
Anecdote:
Usage:
See also:
Comments:

Multiwavelet, Multi-wavelet [back to the starlet list]

In short: Sort of vector extension to the standard "scalar wavelet" based on multiple scaling functions and wavelet functions rather than a single pair
Etymology: Multi+let
Origin:
Contributors:
Some properties:
Anecdote:
Usage:
See also: Multiwavelet Alpert Transform Matlab toolbox by Gabriel Peyré, Multiwavelet matlab code by Vasily Strela (refering page not available, but a local copy of the matlab multiwavelet toolbox is made available), another Multiwavelet MATLAB Package at MatlabCentral, a last wavelet and multiwavelet Matlab package by Fritz Keinert at CRC Press.
Comments:

Needlet [back to the starlet list]

In short: A breed of spherical wavelets, with needle shape
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 concentrated (hence, the needle shape). Random needlets coefficients can be shown to be asymptotically uncorrelated
Anecdote:
Usage: Cosmic Microwave Background (CMB) analysis, cosmological data processing
See also:
Comments:

Noiselet [back to the starlet list]

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
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), cf. T. Tuma and P. Hurley, On the incoherence of noiselet and Haar bases, Proc. SAMPTA 2009 (local copy) . Can be decomposed as a multirate filter bank. Binary valued real and imaginery parts (see the recent discussion Some comments on noiselets by Laurent Jacques, mentioned Yves Meyer: Compressed Sensing, Quasi-crystals, Wavelets and Noiselets.)
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.

In Art is Arp - When art (noiselets) meets wavelets and compressive sensing, paintings by François Morellet vaguely ressemble noiselets noiselets.
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 eventually there Zipped Matlab code)). Other more interesting (faster, higher, stronger) codes are provided at Compressive Imaging Code by Justin Romberg, and especially at A Fast (1-D and 2-D) Noiselet Transform by Laurent Jacques.

Phaselet [back to the starlet list]

In short: An approximately shift-invariant redundant dyadic wavelet transform
Etymology:
Origin: Gopinath, Ramesh A. The phaselet transform - an integral redundancy nearly shift-invariant wavelet transform
Contributors: Ramesh A. Gopinath
Some properties:
Anecdote:
Usage:
See also:
Comments:

Planelet [back to the starlet list]

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:

Platelet [back to the starlet list]

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 Matlab toolbox (for Mac, Unix, Windows) by Rebecca Willett and Robert Nowak. See also platelets for photon-limited image reconstruction

Radonlet [back to the starlet list]

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:

Randlet [back to the starlet list]

In short: Randlets are randomly-chosen basis functions
Etymology: From random
Origin: Malkin, Michael and Venkatesan, Ramarathnam, The randlet transform, Allerton 2004,
Contributors: Michael Malkin, Ramarathnam Venkatesan
Some properties:
Anecdote:
Usage: Universal Perceptual Hashing, image verification, watermarking
See also:
Comments:

Ranklet [back to the starlet list]

In short: Ranklets are a complete family of multiscale rank features characterized by Haar-wavelet style orientation selectivity
Etymology: From rank, since they are related to Wilcoxon rank sum test
Origin: Smeraldi, F. Ranklets: orientation selective non-parametric features applied to face detection, Proceedings of the 16th International Conference on Pattern Recognition, Quebec QC, vol. 3, pages 379-382, August 2002
Contributors: Fabri Smeraldi
Some properties:
Anecdote:
Usage: Face detection
See also:
Comments: The ranklet page, software available upon request

Ridgelet [back to the starlet list]

In short:
Etymology:
Origin: ()
Contributors: David Donoho, E. Candès
Minh N. Do, Martin Vetterli, Image denoising using orthonormal finite ridgelet transform, Proc. of SPIE Conference on Wavelet Applications in Signal and Image Processing VIII, San Diego, USA, August 2000
Some properties:
Anecdote: In the La Recherche montly (Number 383, Feb. 2005, p. 55--59), Mathieu Nowak and Yves Meyer propose the translation arêtelette
Usage:
See also:
Comments:

Ripplet [back to the starlet list]

In short: A kind pre-wavelet or, more recently, some generalizations to curvelets and ridgelets
Etymology: Contraction from ripple and let
Origin: Tentative name 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

And now for completely different rippling ones: ripplet-I and ripplet-II (or type-1 and type-2 ripplets), generalizations of curvelets and ridgelets, respectively.
Jun Xu, Lei Yang, Dapeng Wu Ripplet: A new transform for image processing (local copy), Journal of Visual Communication and Image Representation, Oct. 2010, and Ripplet transform II Transform for Feature Extraction (local copy), IET Image processing, June 2012. matlab codes are available below.
Contributors: Tim N. T. Goodman, Charles A. Micchelli, Dapeng Oliver Wu
Some properties:
Anecdote: The idea behind the original or first generation ripplet (Goodman and Micchelli) 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.
In earth sciences, a ripplet denotes a tiny ripple. It is also, in computer science,
"A powerful, high available, flexible, enterprise-level load/stress test framework" (Java code).
Usage: First generation 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.

Applications of second generation ripplets to image fusion: Home > Vol. 30 > pp. 355-370 Medical image fusion based on ripplet transform type-I By S. Das, M. Chowdhury, and M. K. Kundu, Progress In Electromagnetics Research B, 2011.
See also: Goodman, T. N. T. and Sun, Q., Total positivity and refinable functions with general dilation, 2004, preprint
Comments: Example of an everyday-day-life ripplet: a B-spline with integer knots determined by a constant coefficient differential operator, whose characteristic polynomial only has real zeros (of first type).

For second generation ripplets, Matlab codes are available at Dapeng Oliver Wu publications page: Matlab code for type I Ripplet (local copy) and Matlab code for type II Ripplet (local copy)

Scalet [back to the starlet list]

In short: A short name for the scaling function or father wavelet
Etymology: From scale or the scaling function
Origin:
Contributors:
Some properties:
Anecdote: The scalet as nothing to do with the scarlet fever; a person can become infected with streptococcus either by touching or through airborne droplets, and may get some tablets (none of the aforementionned droplets and tablets are wavelets upto date)
Usage: Used in the scalet-Wigner transform, see C. R. Handy and H. A. Brooks, 2001, Phys. A: Math. Gen, vol. 34, pp. 3577 sq.
See also: http://www.wu.ece.ufl.edu/publications.html J. Xu, D. Wu, ``Ripplet-II Transform for Feature Extraction,'' accepted by IET Image Processing. [pdf] [Matlab code] http://www.wu.ece.ufl.edu/mypapers/rippletII_JNL_r1.pdf http://www.wu.ece.ufl.edu/SourceCode/rippletII.zip J. Xu, L. Yang, and D. Wu, ``Ripplet: A new transform for image processing,'' Journal of Visual Communication and Image Representation, vol. 21, no. 7, pp. 627--639, October 2010. [pdf] [Matlab code] http://www.wu.ece.ufl.edu/mypapers/ripplet_JVCIR_v1.3.pdf http://www.wu.ece.ufl.edu/SourceCode/rippletI.zip
Comments:

Seamlet [back to the starlet list]

In short: Efficient multi-resolution representation for retargeting applications
Etymology: seam and wavelet for a generalization of the DWT and seam carving function
Origin: Seamlets: Content-Aware Nonlinear Wavelet Transform, David D. Conger, Hayder Radha, Mrityunjay Kumar, ICASSP 2010
Contributors:
Some properties:
Anecdote:
Usage:
See also:
Comments:

Seislet [back to the starlet list]

In short: A form of wavelet decomposition based on seismic wavefield and data properties
Etymology: From seismic, one of the origin of the wavelet transform
Origin: Fomel, Sergey, Towards the seislet transform SEG (Society of Exploration Geophysicists) Annual Conference (2006) or Seislet transform and seislet frame Geophysics 75, V25 (2010).
Contributors: Sergey Fomel, with a baptism by Huub Douma
Some properties: The seislet provides a multiscale transform aligned along seismic event slopes in seismic data. Definition based on the wavelet lifting scheme combined with local plane-wave destruction.
Anecdote: The name "seislet" was, according to Sergey Fomel, suggested by Huub Douma
Usage:
See also: Page on the seislet transform at www.reproducibility.org or the Madagascar development blog
Comments:

Shapelet [back to the starlet list]

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

Shearlet [back to the starlet list]

In short: Non-separable wavelets built out of parabolic scaling, shear, and translation operations
Etymology: From shear, a sheer distorsion
Origin: Labate, Demetrio and Lim, W-Q. and Kutyniok, Gitta and Weiss Guido, Sparse multidimensional representation using shearlets (local copy) A handful lot of papers is available here: shealet papers. A first overview is given in Shearlets. The First Five Year (Oberwolfach Report, 2010, local copy).
Contributors: Guido Weiss
Gitta Kutyniok
Demetrio Labate
Kanghui Guo (Scholar)
Gabriele Steidl
Sören Häuser
Some properties: Unlike curvelets, shearlets form an affine system with a single generating mother shearlet function parameterized by a scaling, a shear, and a translation parameter. Provides the same approximation properties as curvelets, albeit with a different directional sensitivity. Exist in band-limited or compact support flavors. Possess natural, canonical smoothness spaces, called shearlet coorbit spaces, similar to Besov spaces for wavelets. Apparently extend to arbitrary any dimensions.
Anecdote:
Usage: Image denoising, restoration, Morphological component analysis
See also: The shearlet website, recently updated with ShearLab (... a rationally designed digital shearlet transform) For discrete implementation, there exists for instance a Digital Shearlet Transforms or Development of a Digital Shearlet Transform Based on Pseudo-Polar FFT. Shearlet Matlab toolboxes are available at ShearLab matlab toolboxes, local shearlet toolbox by G. Easley and FSST:
Comments: Potential a hard competitor, for years to come, to the quite oversold curvelets (IMHOlet: In My (little) Humble Opinion). See also Shearlets, after MIA 2012

Sinclet [back to the starlet list]

In short: A short name for the wavelet function associated with the cardinal sine (aka sinc function) scaling function
Etymology: From sine cardinal function
Origin: Unknown, but cited in some papers, such as Mammogram enhancement using a class of smooth wavelets, by Shi, Z. Zhang, D., Wang, H., Kouri, D. and Hoffman, D., (local pdf), submitted to IEEE 33rd Asilomar Conference on Signals, Systems, and Computers, 1999), or Generalized symmetric interpolating wavelets , by Shi, Z., Kouri, D., Wei G. W. and Hoffman, D., Computer Physics Communications, 1999 (local pdf)
Contributors:
Some properties:
Anecdote:
Usage:
See also:
Comments:

Singlet [back to the starlet list]

In short: A single-sided dampled Laplace wavelet transform for modal analysis
Etymology: A single mode subsystem related wavelet
Origin: Named in Real-Time Identification of Flutter Boundaries Using the Discrete Wavelet Transform, J. D. Johnson, Jun Lu, Atam P. Dhawan, Journal of guidance, control and dynamics, Vol. 25, N. 2, 2002, pages 334-339. Appeared in Correlation filtering of modal dynamics using the Laplace wavelet, L. Freudinger, R. Lind and M. Brenner, International Modal Analysis Conference, Santa Barbara, CA, February 1998, pp. 868-877. See also the NANSA report:
Contributors:
Some properties: Complex, analytic, single-sided damped exponential
Anecdote: A singlet is also the name of the attire worn by competitors in the sport of wrestling
Usage: Modal analtsis; Flutter identification
See also:
Comments:

Slantlet [back to the starlet list]

In short: A piece-wise linear (but discontinuous) wavelet basis reminiscent of the slant transform
Etymology: Using slant, "to strike obliquely" (against something), alteration of slenten "slip sideways" (see etymology and modern meaning clever of superior)
Origin: I. W. Selesnick, The slantlet transform, IEEE Trans on Signal Processing, vol 47, no 5, pp 1304-1313, May 1999
Contributors: Ivan Selesnick
Some properties: Piece-wise linear basis with two zero moments, orthogonal, based on the iteration of different filter banks at each scale
Anecdote: Ivan Selesnick's page for slantlet
Usage: Image denoising
See also: Matlab Source code available at http://taco.poly.edu/selesi/slantlet
Comments:

Smoothlet [back to the starlet list]

In short: Continuous generalization of (scond order) wedgelets
Etymology: From the smooth, "free from roughness, not harsh" (with interesting etymology and modern meaning clever of superior), and the diminutive let of the wavelet
Origin: Agnieszka Lisowska, Smoothlets - Multiscale Functions for Adaptive Representation of Images, IEEE Trans on Signal Processing, July 2001, Volume: 20 Issue: 7, 1777-1787 (local copy)
In this paper a special class of functions called smoothlets is presented. They are defined as a generalization of wedgelets and second-order wedgelets. Unlike all known geometrical methods used in adaptive image approximation, smoothlets are continuous functions. They can adapt to location, size, rotation, curvature, and smoothness of edges. The M-term approximation of smoothlets is O(M^3) . In this paper, an image compression scheme based on the smoothlet transform is also presented. From the theoretical considerations and experiments, both described in the paper, it follows that smoothlets can assure better image compression than the other known adaptive geometrical methods, namely, wedgelets and second-order wedgelets.
Contributors: Agnieszka Lisowska
Some properties: Adaptive geometrical decomposition. Adapt to location, size, rotation, curvature and smoothness of edges. The M-term approximation of smoothlets is O(M^3)
Anecdote:
Usage: Compression
See also:
Comments:

Sparselet [back to the starlet list]

In short: A set of mother wavelets, replicated at the different positions and scales of the pyramid and which allow for a translation and scale invariant representation of images
Etymology: From the sparse nature of some wavelet representations (and the let)
Origin: Laurent Perrinet Dynamical Neural Networks: modeling low-level vision at short latencies, The European Physical Journal, 2007 (local copy)
Contributors: Laurent Perrinet
Some properties:
Anecdote:
Usage:
See also:
Comments: To be discussed: a seemingly abusive use of sparsity

Spikelet [back to the starlet list]

In short: A wavelet transform matching a specified discrete-time signal
Etymology: From experimental spikes that need to be matched in a signal
Origin: Rodrigo Capobianco Guido, Jan Frans Willem Slaets, Roland Kouml;berle, Lírio Onofre Batista Almeida and José Carlos Pereira A new technique to construct a wavelet transform matching a specified signal with applications to digital, real time, spike, and overlap pattern recognition, Digital Signal Processing, 2006 (local copy)
Contributors:
Some properties:
Anecdote: A spikelet is also a kind of raceme, a small or secondary spike, characteristic of grasses and sedges, having a varying number of reduced flowers each subtended by one or two scalelike bracts.

Usage:
See also:
Comments:

Splinelet [back to the starlet list]

In short: A not so-common nickname for B-spline wavelets
Etymology: From sline+let, obviously
Origin:
Contributors:
Some properties:
Anecdote:
Usage:
See also:
Comments:

Steerlet [back to the starlet list]

In short: Steerable wavelets in 3D
Etymology: From steer for the "directional" prefix (as in "cyber-" from Greek kubernete)
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:

SURE-let [back to the starlet list]

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/ and bigwww.epfl.ch/research/projects/denoising.html for SURE/PURE-LET and CURE-LET (A CURE for noisy magnetic resonance images: Chi-square unbiased risk estimation) denoising. On the page Signal and processing (Matlab) codes, a Sure-LET denoising toolbox for oversampled complex filter banks is offered.
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. And the PURELET case as well:
http://www.purelet.co.uk/
Welcome to Purelet Letting Agency

Surfacelet [back to the starlet list]

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

Surflet [back to the starlet list]

In short: Representation for approximation and compression of Horizon-class functions containing a 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:

Symlet or Symmlet [back to the starlet list]

In short: Orthogonal wavelet with maximum symmetry and compact support. A Symmlet is each member of Daubechies's least assymetric family.
Etymology:
Origin: ()
Contributors: ()
Some properties:
Anecdote:
Usage:
See also:
Comments:

Tetrolet [back to the starlet list]

In short: Tetromino-based Haar like wavelet
Etymology: From the tetro-structured representation and the ubiquitous let
Origin: Jens Krommweh, Tetrolet Transform: A New Adaptive Haar Wavelet Algorithm for Sparse Image Representation (local copy), J. Vis. Commun. Image R., Vol. 21, No. 4 (2010) 364-374.
Contributors: Jens Krommweh
Some properties:
Anecdote:
Usage:
See also: A Where-Is-The-Starlet entry: WITS: Tetrolet wavelets from La vertu d'un LA
Comments: Tetrolet matlab Toolbox by Jens Krommweh

Treelet [back to the starlet list]

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.

Vaguelette [back to the starlet list]

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
  1. fast decay/localization
  2. null mean value/oscillation
  3. 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:

Wavelet-Vaguelette [back to the starlet list]

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:

Warblet [back to the starlet list]

In short:
Etymology: let
Origin:
Contributors:
Some properties:
Anecdote:
Usage:
See also:
Comments:

Warplet [back to the starlet list]

In short: An affine deformation of the Gabor wavelet (aka gaborlet) for Clerc and Mallat, or image-dependent patch-like wavelet representations based on PCA for Bhalerao and Wilson
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 (local copy).
Abstract: Studies the recovery of shape from texture under perspective projection. We regard shape from texture as a statistical estimation problem, the texture being the realization of a stochastic process. We introduce warplets, which generalize wavelets over the 2D affine group. At fine scales, the warpogram of the image obeys a transport equation, called texture gradient equation. In order to recover the 3D shape of the surface, one must estimate the deformation gradient, which measures metric changes in the image. This is made possible by imposing a notion of homogeneity for the original texture, according to which the deformation gradient is equal to the velocity of the texture gradient equation. By measuring the warplet transform of the image at different scales, we obtain a deformation gradient estimator

Bhalerao, Abhir and Wilson, Roland, Warplets: An image-dependent wavelet representation, IEEE International Conference on Image Processing (ICIP 2005) (local copy, poster).
Abstract: A novel image-dependent representation, warplets, based on self-similarity of regions is introduced. The representation is well suited to the description and segmentation of images containing textures and oriented patterns, such as fingerprints. An affine model of an image as a collection of self-similar image blocks is developed and it is shown how textured regions can be represented by a single prototype block together with a set of transformation coefficients. Images regions are alligned to a set of dictionary blocks and their variability captured by PCA analysis. The block-to-block transformations are found by Gaussian mixture modelling of the block spectra and a least-squares estimation. Clustering in the Warplet domain can be used to determine a warplet dictionary. Experimental results on a variety of images demonstrate the potential of the use of warplets for segmentation and coding, Proc. IEEE International Conference on Image Processing (ICIP) 2005, September 2005.
Contributors: Maureen Clerc, Stéphane Mallat
Abhir Bhalerao and Roland Wilson
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 (2004, 2005) on a somewhat different warplets by Abhir Bhalerao and Roland Wilson, 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

Wedgelet [back to the starlet list]

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:

Xlet or X-let [back to the starlet list]

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 (as well to all the animals, long time ago)
Usage:
See also:
Comments: Java-type Xlet
An Xlet is very similar to a Java applet and is originally introduced in Sun's Java TV specification to support applications for Digital TV. Though Xlet looks superficially different from other application models in Java such as applet and MIDlet, it is actually meant to be a generalization of such models.

Otherlets names

Multiselective wavelet (not: multiselectivelet) [back to the starlet list]

In short: (Linear) frame of directional wavelets with variable angular selectivity
Etymology: Multiselective wavelet
Origin: Jacques, Laurent and Antoine, Jean-Pierre, Multiselective Pyramidal Decomposition of Images: Wavelets with Adaptive Angular Selectivity, International Journal of Wavelets, Multiresolution and Information Processing, 2007 [pdf][pdf][paper]
Contributors: Laurent Jacques,
Some properties:
Anecdote:
Usage:
See also:
Comments:

SOHO wavelets (not: soholet) [back to the starlet list]

SOHO: Orthogonal and Symmetric Haar Wavelets on the Sphere
In short: The "first spherical Haar wavelet": Orthogonal and Symmetric Haar Wavelets on the Sphere (and extensions)
Etymology: Symmetric Orthogonal Haar wavelet
Origin: Lessig, Christian Orthogonal and Symmetric Haar Wavelets on the Sphere, Master of Science thesis 2007, [pdf][local copy]
Abstract: We propose the SOHO wavelet basis. To our knowledge this is the first spherical Haar wavelet basis that is both orthogonal and symmetric, making it particularly well suited for the approximation and processing of all-frequency signals on the sphere. The key to the derivation of the basis is a novel spherical subdivision scheme that defines a partition acting as domain of the basis functions. The construction of the SOHO wavelets refutes earlier claims doubting the existence of such a basis. We also investigate how signals represented in our new basis can be rotated. Experimental results for the representation of spherical signals verify that the superior theoretical properties of the SOHO wavelet basis are also relevant in practice.
Lessig, Christian and Fiume, E. Orthogonal and Symmetric Haar Wavelets on the Sphere, ACM Transactions of Graphics, SIGGRAPH 2008, [pdf][local copy]
Abstract: We propose the SOHO wavelet basis – the first spherical Haar wavelet basis that is both orthogonal and symmetric, making it particularly well suited for the approximation and processing of all- frequency signals on the sphere. We obtain the basis with a novel spherical subdivision scheme that defines a partition acting as the domain of the basis functions. Our construction refutes earlier claims doubting the existence of a basis that is both orthogonal and symmetric. Experimental results for the representation of spherical signals verify that the superior theoretical properties of the SOHO wavelet basis are also relevant in practice.
Chow, Andy. Orthogonal and Symmetric Haar Wavelets on the Three-Dimensional Ball, Master's thesis, 2010, University of Toronto, Toronto, [pdf][local copy]
Abstract: 3D SOHO is the first Haar wavelet basis on the three-dimensional ball that is both orthogonal and symmetric. These theoretical properties allow for a fast wavelet transform, optimal approximation and perfect reconstruction.
Contributors: Christian Lessig
Eugene Fiume
Andy Chow
Some properties:
Anecdote: SOHO denotes many things. Among which the SoHo neighborhood in Manhattan (for South of Houston Street), New York and the SOlar and Heliospheric Observatory. The latter may likely be the motivation for wavelets on the sphere.
Usage:
See also:
Comments:

Artlets

AguaSonic Acoustics [back to the starlet list]

Domain: Painting (and music)
Description: Cetacean Stills or Shape of the Sound, still paintings based on continuous wavelet transform diagrams of dolphins and whale recording (whalets?).
Comments:

BIG Art Gallery [back to the starlet list]

Domain: Painting
Description: Art gallery inspired by wavelets (esp. splines), by Annette Unser.
Comments: Example for a subset of fractional splines:

Le Spy art [back to the starlet list]

Domain: Art authentication
Description: Le Spy art or ArtSpy, an algorithm to detected the artist of the painting with the discrete wavelet transform, by a team at Rice University, Houston, TX, USA. Tests on Rembrandt, Monet, or Picasso.
Comments: