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Wavelet

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A wavelet is a wave-like oscillation with an amplitudethat starts out at zero, increases, and then decreases back to zero. Itcan typically be visualized as a "brief oscillation" like one might seerecorded by a seismograph or heart monitor. Generally, wavelets arepurposefully crafted to have specific properties that make them usefulfor signal processing. Wavelets can be combined, using a "shift,multiply and sum" technique called convolution,with portions of an unknown signal to extract information from theunknown signal.

For example, a wavelet could be created to have a frequency of Middle C and a short duration of roughly a 32nd note. If this wavelet were to beconvolved at periodic intervals with a signal created from the recordingof a song, then the results of these convolutions would be useful fordetermining when the Middle C note was being played in the song.Mathematically, the wavelet will resonate if the unknown signal containsinformation of similar frequency - just as a tuning fork physicallyresonates with sound waves of its specific tuning frequency. Thisconcept of resonance is at the core of many practical applications ofwavelet theory.

As wavelets are a mathematical tool they can be used to extractinformation from many different kinds of data, including - but certainlynot limited to - audio signals and images. Sets of wavelets aregenerally needed to analyze data fully. A set of "complementary"wavelets will deconstruct data without gaps or overlap so that thedeconstruction process is mathematically reversible. Thus, sets ofcomplementary wavelets are useful in wavelet basedcompression/decompression algorithms where it is desirable to recoverthe original information with minimal loss.

More technically, a wavelet is a mathematical function usedto divide a given function or continuous-time signal into different scale components.Usually one can assign a frequency range to each scale component. Eachscale component can then be studied with a resolution that matches itsscale. A wavelet transform is the representation of a function bywavelets. The wavelets are scaled and translated copies (known as "daughterwavelets") of a finite-length or fast-decaying oscillating waveform(known as the "mother wavelet"). Wavelet transforms have advantages overtraditional Fourier transforms for representing functions that havediscontinuities and sharp peaks, and for accurately deconstructing andreconstructing finite, non-periodic and/or non-stationary signals.

In formal terms, this representation is a wavelet series representation of a square-integrablefunction with respect to either a complete, orthonormal set of basis functions, or an overcompleteset or Frame of a vector space, for the Hilbert space of square integrable functions.

Wavelet transforms are classified into discrete wavelet transforms(DWTs) and continuous wavelet transforms(CWTs). Note that both DWT and CWT are continuous-time (analog)transforms. They can be used to represent continuous-time (analog)signals. CWTs operate over every possible scale and translation whereasDWTs use a specific subset of scale and translation values orrepresentation grid.

Look up wavelet in Wiktionary, the free dictionary.

The word wavelet is due to Morletand Grossmann in the early1980s. They used the French word ondelette, meaning "small wave". Soonit was transferred to English by translating "onde" into "wave", giving"wavelet".

Contents

  • 1 Wavelet theory
    • 1.1 Continuous wavelet transforms (Continuous Shift & Scale Parameters)
    • 1.2 Discrete wavelet transforms (Discrete Shift & Scale parameters)
    • 1.3 Multiresolution discrete wavelet transforms
  • 2 Mother wavelet
  • 3 Comparisons with Fourier Transform (Continuous-Time)
  • 4 Definition of a wavelet
    • 4.1 Scaling filter
    • 4.2 Scaling function
    • 4.3 Wavelet function
  • 5 Applications of Discrete Wavelet Transform
  • 6 History
    • 6.1 Timeline
  • 7 Wavelet Transforms
  • 8 Generalized Transforms
  • 9 List of wavelets
    • 9.1 Discrete wavelets
    • 9.2 Continuous wavelets
      • 9.2.1 Real valued
      • 9.2.2 Complex valued
  • 10 See also
  • 11 Notes
  • 12 References
  • 13 External links

[edit] Wavelet theory

Wavelet theory is applicable to several subjects. All wavelettransforms may be considered forms of time-frequencyrepresentation for continuous-time (analog)signals and so are related to harmonic analysis. Almost all practically useful discretewavelet transforms use discrete-time filterbanks. These filter banks are called thewavelet and scaling coefficients in wavelets nomenclature. Thesefilterbanks may contain either finite impulse response (FIR) or infinite impulse response (IIR)filters. The wavelets forming a continuous wavelet transform(CWT) are subject to the uncertaintyprinciple of Fourier analysis respective sampling theory: Given asignal with some event in it, one cannot assign simultaneously an exacttime and frequency response scale to that event. The product of theuncertainties of time and frequency response scale has a lower bound.Thus, in the scaleogramof a continuous wavelet transform of this signal, such an event marksan entire region in the time-scale plane, instead of just one point.Also, discrete wavelet bases may be considered in the context of otherforms of the uncertainty principle.

Wavelet transforms are broadly divided into three classes:continuous, discrete and multiresolution-based.

[edit] Continuouswavelet transforms (Continuous Shift & Scale Parameters)

In continuous wavelet transforms, agiven signal of finite energy is projected on a continuous family offrequency bands (or similar subspaces of the Lpfunction space ).For instance the signal may be represented on every frequency band ofthe form [f,2f] for allpositive frequencies f>0. Then, the original signal can bereconstructed by a suitable integration over all the resulting frequencycomponents.

The frequency bands or subspaces (sub-bands) are scaled versions of asubspace at scale 1. This subspace in turn is in most situationsgenerated by the shifts of one generating function ,the mother wavelet. For the example of the scale one frequencyband [1,2] this function is

with the (normalized) sincfunction. Other example mother wavelets are:

Meyer Morlet Mexican Hat

The subspace of scale a or frequency band is generated by the functions (sometimes called child wavelets)

,

where a is positive and defines the scale and b is anyreal number and defines the shift. The pair (a,b) defines a pointin the right halfplane .

The projection of a function x onto the subspace of scale athen has the form

with wavelet coefficients

.

See a list of some Continuous wavelets.

For the analysis of the signal x, one can assemble the waveletcoefficients into a scaleogramof the signal.

[edit] Discretewavelet transforms (Discrete Shift & Scale parameters)

It is computationally impossible to analyze a signal using allwavelet coefficients, so one may wonder if it is sufficient to pick adiscrete subset of the upper halfplane to be able to reconstruct asignal from the corresponding wavelet coefficients. One such system isthe affine system for some real parameters a>1,b>0. The corresponding discrete subset of the halfplaneconsists of all the points with integers .The corresponding baby wavelets are now given as

ψm,n(t) = a− m / 2ψ(a − mt − nb).

A sufficient condition for the reconstruction of any signal xof finite energy by the formula

is that the functions form a tight frame of .

[edit]Multiresolutiondiscrete wavelet transforms

D4 wavelet

In any discretised wavelet transform, there are only a finite numberof wavelet coefficients for each bounded rectangular region in the upperhalfplane. Still, each coefficient requires the evaluation of anintegral. To avoid this numerical complexity, one needs one auxiliaryfunction, the father wavelet .Further, one has to restrict a to be an integer. A typicalchoice is a=2 and b=1. The most famous pair of father andmother wavelets is the Daubechies 4 tapwavelet.

From the mother and father wavelets one constructs the subspaces

,where φm,n(t) = 2− m / 2φ(2 − mt − n)

and

,where ψm,n(t) = 2− m / 2ψ(2 − mt − n).

From these one requires that the sequence

forms a multiresolution analysis of and that the subspaces are the orthogonal "differences" of the above sequence, that is, Wm is the orthogonalcomplement of Vminside the subspace Vm − 1.In analogy to the sampling theorem one may conclude thatthe space Vm withsampling distance 2m moreor less covers the frequency baseband from 0 to 2 − m − 1. As orthogonalcomplement, Wmroughly covers the band [2 − m − 1,2− m].

From those inclusions and orthogonality relations follows theexistence of sequences and that satisfy the identities

and

and

and .

The second identity of the first pair is a refinement equation for the father wavelet φ. Both pairs of identities form the basis forthe algorithm of the fast wavelet transform.

[edit] Mother wavelet

For practical applications, and for efficiency reasons, one preferscontinuously differentiable functions with compact support as mother(prototype) wavelet (functions). However, to satisfy analyticalrequirements (in the continuous WT) and in general for theoreticalreasons, one chooses the wavelet functions from a subspace of the space This is the space of measurable functions that are absolutely and square integrable:

and

Being in this space ensures that one can formulate the conditions ofzero mean and square norm one:

is the condition for zero mean, and
is the condition for square norm one.

For ψ to be a wavelet for the continuous wavelet transform(see there for exact statement), the mother wavelet must satisfy anadmissibility criterion (loosely speaking, a kind ofhalf-differentiability) in order to get a stably invertible transform.

For the discrete wavelet transform, oneneeds at least the condition that the wavelet series is a representation ofthe identity in the space .Most constructions of discrete WT make use of the multiresolution analysis, whichdefines the wavelet by a scaling function. This scaling function itselfis solution to a functional equation.

In most situations it is useful to restrict ψto be a continuous function with a higher number M of vanishingmoments, i.e. for all integer m

Some example mother wavelets are:

Meyer Morlet Mexican Hat

The mother wavelet is scaled (or dilated) by a factor of a and translated (or shifted) by a factorof b to give (under Morlet'soriginal formulation):

For the continuous WT, the pair (a,b) varies over the fullhalf-plane ;for the discrete WT this pair varies over a discrete subset of it,which is also called affine group.

These functions are often incorrectly referred to as the basisfunctions of the (continuous) transform. In fact, as in the continuousFourier transform, there is no basis in the continuous wavelettransform. Time-frequency interpretation uses a subtly differentformulation (after Delprat).

[edit] Comparisonswith Fourier Transform (Continuous-Time)

The wavelet transform is often compared with the Fourier transform, in which signals are represented as asum of sinusoids. The main difference is that wavelets are localized inboth time and frequency whereas the standard Fourier transform is only localized in frequency.The Short-time Fourier transform(STFT) is more similar to the wavelet transform, in that it is also timeand frequency localized, but there are issues with the frequency/timeresolution trade-off. Wavelets often give a better signal representationusing Multiresolution analysis, withbalanced resolution at any time and frequency.

The discrete wavelet transform is also less computationally complex, taking O(N) time as compared to O(N log N)for the fast Fourier transform. This computationaladvantage is not inherent to the transform, but reflects the choice of alogarithmic division of frequency, in contrast to the equally spacedfrequency divisions of the FFT.[citation needed][clarification needed] Itis also important to note that this complexity only applies when thefilter size has no relation to the signal size. A wavelet withoutcompact support such as the Shannon wavelet would require O(N^2). (For instance,a logarithmic Fourier Transform also exists with O(N)complexity, but the original signal must be sampled logarithmically intime, which is only useful for certain types of signals.[1])

[edit] Definition of a wavelet

There are a number of ways of defining a wavelet (or a waveletfamily).

[edit] Scaling filter

An orthogonal wavelet is entirely defined by the scaling filter - alow-pass finite impulse response (FIR) filterof length 2N and sum 1. In biorthogonal wavelets, separatedecomposition and reconstruction filters are defined.

For analysis with orthogonal wavelets the high pass filter iscalculated as the quadrature mirror filter of the lowpass, and reconstruction filters are the time reverse of thedecomposition filters.

Daubechies and Symlet wavelets can be defined by the scaling filter.

[edit] Scaling function

Wavelets are defined by the wavelet function ψ(t)(i.e. the mother wavelet) and scaling function φ(t)(also called father wavelet) in the time domain.

The wavelet function is in effect a band-pass filter and scaling itfor each level halves its bandwidth. This creates the problem that inorder to cover the entire spectrum, an infinite number of levels wouldbe required. The scaling function filters the lowest level of thetransform and ensures all the spectrum is covered. See [1]

 for a detailedexplanation.

 

For a wavelet with compact support, φ(t)can be considered finite in length and is equivalent to the scalingfilter g.

Meyer wavelets can be defined by scaling functions

[edit] Wavelet function

The wavelet only has a time domain representation as the waveletfunction ψ(t).

For instance, Mexican hat wavelets can be defined by a waveletfunction. See a list of a few Continuous wavelets.

[edit]Applications of DiscreteWavelet Transform

Generally, an approximation to DWT is used for data compression if signal is already sampled, and theCWT for signal analysis. Thus, DWTapproximation is commonly used in engineering and computer science, andthe CWT in scientific research.

Wavelet transforms are now being adopted for a vast number ofapplications, often replacing the conventional Fourier Transform. Many areas ofphysics have seen this paradigm shift, including molecular dynamics, abinitio calculations, astrophysics,density-matrix localisation, seismology, optics, turbulence and quantum mechanics. This change has also occurred in image processing, blood-pressure,heart-rate and ECGanalyses, DNA analysis, proteinanalysis, climatology,general signal processing, speech recognition, computer graphics and multifractal analysis.In computer vision and image processing, the notion of scale-space representation and Gaussianderivative operators is regarded as a canonical multi-scalerepresentation.

One use of wavelet approximation is in data compression. Like someother transforms, wavelet transforms can be used to transform data, thenencode the transformed data, resulting in effective compression. Forexample, JPEG2000 is an image compression standard that uses biorthogonalwavelets. This means that although the frame is overcomplete, it is a tightframe (see types of Frame of a vector space), and the same framefunctions (except for conjugation in the case of complex wavelets) areused for both analysis and synthesis, i.e., in both the forward andinverse transform. For details see wavelet compression.

A related use is that of smoothing/denoising data based on waveletcoefficient thresholding, also called wavelet shrinkage. By adaptivelythresholding the wavelet coefficients that correspond to undesiredfrequency components smoothing and/or denoising operations can beperformed.

Wavelet transforms are also starting to be used for communicationapplications. Wavelet OFDM is the basic modulation scheme used in HD-PLC(a powerline communications technology developed by Panasonic), and in one of the optional modesincluded in the IEEEP1901 draft standard. The advantage of Wavelet OFDM over traditional FFTOFDM systems is that Wavelet canachieve deeper notches and that it does not require a Guard Interval(which usually represents significant overhead in FFT OFDM systems)[2].

[edit] History

The development of wavelets can be linked to several separate trainsof thought, starting with Haar's work in the early 20th century. Notablecontributions to wavelet theory can be attributed to Zweig’sdiscovery of the continuous wavelet transform in 1975 (originallycalled the cochlear transform and discovered while studying the reactionof the ear to sound)[3],Pierre Goupillaud, Grossmann and Morlet'sformulation of what is now known as the CWT (1982), Jan-OlovStrömberg's early work on discrete wavelets (1983), Daubechies' orthogonal wavelets with compact support(1988), Mallat's multiresolution framework(1989), Nathalie Delprat's time-frequency interpretation of the CWT(1991), Newland's Harmonic wavelet transform (1993)and many others since.

[edit] Timeline

  • First wavelet (Haar wavelet) by Alfred Haar (1909)
  • Since the 1970s: George Zweig, Jean Morlet, Alex Grossmann
  • Since the 1980s: Yves Meyer, Stéphane Mallat, Ingrid Daubechies, Ronald Coifman, Victor Wickerhauser,

[edit] Wavelet Transforms

There are a large number of wavelet transforms each suitable fordifferent applications. For a full list see list of wavelet-relatedtransforms but the common ones are listed below:

  • Continuous wavelet transform (CWT)
  • Discrete wavelet transform (DWT)
  • Fast wavelet transform (FWT)
  • Lifting scheme & Generalized Lifting Scheme
  • Wavelet packet decomposition (WPD)
  • Stationary wavelet transform (SWT)
  • Fractional wavelet transform (?)

[edit] Generalized Transforms

There are a number of generalized transforms of which the wavelettransform is a special case. For example, Joseph Segman introduced scaleinto the Heisenberg group, giving rise to a continuous transform spacethat is a function of time, scale, and frequency. The CWT is atwo-dimensional slice through the resulting 3d time-scale-frequencyvolume.

Another example of a generalized transform is the chirplet transform in which the CWT is also a twodimensional slice through the chirplet transform.

An important application area for generalized transforms involvessystems in which high frequency resolution is crucial. For example, darkfield electronoptical transforms intermediate between direct and reciprocal space have been widely used inthe harmonic analysis of atom clustering, i.e. in the study ofcrystals and crystal defects[4].Now that transmissionelectron microscopes are capable of providing digital images withpicometer-scale information on atomic periodicity in nanostructureof all sorts, the range of pattern recognition[5]and strain[6]/metrology[7]applications for intermediate transforms with high frequency resolution(like brushlets[8]and ridgelets[9])is growing rapidly.

Fractional wavelet transforms are based on scaling functions which,contrary to the "integer" wavelet transform, may have infinite support.Nonetheless, they allow non truncation of basis functions, exacttreatment of boundaries, and perfect reconstruction.[10]

[edit] List of wavelets

[edit] Discrete wavelets

  • Beylkin (18)
  • BNC wavelets
  • Coiflet (6, 12, 18, 24, 30)
  • Cohen-Daubechies-Feauveau wavelet (Sometimes referred to as CDF N/P or Daubechies biorthogonal wavelets)
  • Daubechies wavelet (2, 4, 6, 8, 10, 12, 14, 16, 18, 20)
  • Binomial-QMF (Also referred to as Daubechies wavelet)
  • Haar wavelet
  • Mathieu wavelet
  • Legendre wavelet
  • Villasenor wavelet
  • Symlet

[edit] Continuous wavelets

[edit] Real valued

  • Beta wavelet
  • Hermitian wavelet
  • Hermitian hat wavelet
  • Mexican hat wavelet
  • Shannon wavelet

[edit] Complex valued

  • Complex mexican hat wavelet
  • Morlet wavelet
  • Shannon wavelet
  • Modified Morlet wavelet

[edit] See also

  • Chirplet transform
  • Curvelet
  • Filter banks
  • Fractional Fourier transform
  • Multiresolution analysis
  • Scale space
  • Short-time Fourier transform
  • Ultra wideband radio- transmits wavelets
  • Wave packet
  • Gabor wavelet#Wavelet space
  • Dimension reduction
  • Fourier-related transforms

[edit] Notes

  1. ^ http://homepages.dias.ie/~ajones/papers/28.pdf  
  2. ^ Recent Developments in the Standardization of Power Line Communications within the IEEE   , (Galli, S. and Logvinov, O - IEEE Communications Magazine, July 2008)
  3. ^ http://scienceworld.wolfram.com/biography/Zweig.html   Zweig, George Biography on Scienceworld.wolfram.com
  4. ^ P. Hirsch, A. Howie, R. Nicholson, D. W. Pashley and M. J. Whelan (1965/1977) Electron microscopy of thin crystals (Butterworths, London/Krieger, Malabar FLA) ISBN 0-88275-376-2
  5. ^ P. Fraundorf, J. Wang, E. Mandell and M. Rose (2006) Digital darkfield tableaus, Microscopy and Microanalysis 12:S2, 1010-1011 (cf. arXiv:cond-mat/0403017   )
  6. ^ M. J. Hÿtch, E. Snoeck and R. Kilaas (1998) Quantitative measurement of displacement and strain fields from HRTEM micrographs, Ultramicroscopy 74:131-146.
  7. ^ Martin Rose (2006) Spacing measurements of lattice fringes in HRTEM image using digital darkfield decomposition (M.S. Thesis in Physics, U. Missouri - St. Louis)
  8. ^ F. G. Meyer and R. R. Coifman (1997) Applied and Computational Harmonic Analysis 4:147.
  9. ^ A. G. Flesia, H. Hel-Or, A. Averbuch, E. J. Candes, R. R. Coifman and D. L. Donoho (2001) Digital implementation of ridgelet packets (Academic Press, New York).
  10. ^ Michael Unser, Thierry Blu, "Fractional Splines and Wavelets", SIAM Review, Vol. 42, No. 1 (Mar., 2000), pp. 43-67, Society for Industrial and Applied Mathematics, Stable URL: http://www.jstor.org/stable/2653376  

[edit] References

  • Paul S. Addison, The Illustrated Wavelet Transform Handbook, Institute of Physics, 2002, ISBN 0-7503-0692-0
  • Ingrid Daubechies, Ten Lectures on Wavelets, Society for Industrial and Applied Mathematics, 1992, ISBN 0-89871-274-2
  • A. N. Akansu and R. A. Haddad, Multiresolution Signal Decomposition: Transforms, Subbands, Wavelets, Academic Press, 1992, ISBN 0-12-047140-X
  • P. P. Vaidyanathan, Multirate Systems and Filter Banks, Prentice Hall, 1993, ISBN 0-13-605718-7
  • Mladen Victor Wickerhauser, Adapted Wavelet Analysis From Theory to Software, A K Peters Ltd, 1994, ISBN 1-56881-041-5
  • Gerald Kaiser, A Friendly Guide to Wavelets, Birkhauser, 1994, ISBN 0-8176-3711-7
  • Haar A., Zur Theorie der orthogonalen Funktionensysteme, Mathematische Annalen, 69, pp 331–371, 1910.
  • Ramazan Gençay, Faruk Selçuk and Brandon Whitcher, An Introduction to Wavelets and Other Filtering Methods in Finance and Economics, Academic Press, 2001, ISBN 0-12-279670-5
  • Donald B. Percival and Andrew T. Walden, Wavelet Methods for Time Series Analysis, Cambridge University Press, 2000, ISBN 0-5216-8508-7
  • Tony F. Chan and Jackie (Jianhong) Shen, Image Processing and Analysis - Variational, PDE, Wavelet, and Stochastic Methods, Society of Applied Mathematics, ISBN 089871589X (2005)
  • Stéphane Mallat, "A wavelet tour of signal processing" 2nd Edition, Academic Press, 1999, ISBN 0-12-466606-x
  • Barbara Burke Hubbard, "The World According to Wavelets: The Story of a Mathematical Technique in the Making", AK Peters Ltd, 1998, ISBN 1568810725, ISBN 978-1568810720

[edit] External links

Wikimedia Commons has media related to: Wavelet
  • Wavelets   by Gilbert Strang, American Scientist 82 (1994) 250-255. (A very short and excellent introduction)
  • Wavelet Digest  
  • NASA Signal Processor featuring Wavelet methods   Description of NASA Signal & Image Processing Software and Link to Download
  • 1st NJIT Symposium on Wavelets (April 30, 1990) (First Wavelets Conference in USA)  
  • Binomial-QMF Daubechies Wavelets  
  • Wavelets made Simple  
  • Course on Wavelets given at UC Santa Barbara, 2004  
  • The Wavelet Tutorial by Polikar   (Easy to understand when you have some background with fourier transforms!)
  • OpenSource Wavelet C++ Code  
  • An Introduction to Wavelets  
  • Wavelets for Kids (PDF file)   (Introductory (for very smart kids!))
  • Link collection about wavelets  
  • Gerald Kaiser's acoustic and electromagnetic wavelets  
  • A really friendly guide to wavelets  
  • Wavelet-based image annotation and retrieval  
  • Very basic explanation of Wavelets and how FFT relates to it  
  • A Practical Guide to Wavelet Analysis   is very helpful, and the wavelet software in FORTRAN, IDL and MATLAB are freely available online. Note that the biased wavelet power spectrum needs to be rectified   .
  • Where Is The Starlet?   A dictionary of wavelet names.
  • Python Wavelet Transforms Package   OpenSource code for computing 1D and 2D Discrete wavelet transform, Stationary wavelet transform and Wavelet packet transform.
  • Wavelet Library   GNU/GPL library for n-dimensional discrete wavelet/framelet transforms.
  • The Fractional Spline Wavelet Transform   describes a fractional wavelet transform based on fractional b-Splines.
Retrieved from "http://en.wikipedia.org/wiki/Wavelet"Categories: WaveletsHidden categories:All articleswith unsourced statements | Articleswith unsourced statements from July 2009 | All pages needing cleanup| Wikipedia articles needing clarification from July 2009
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