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Seismic wavelet

WaveletFrom Wikipedia, the free encyclopedia

A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases, and then decreases back to zero.It can typically be visualized as a "brief oscillation" like one might see recorded by a seismograph or heart monitor.Generally, wavelets are purposefully crafted to have specific properties that make them useful for signal processing.Wavelets can be combined, using a "reverse, shift, multiply and integrate" technique called convolution, with portions ofa known signal to extract information from the unknown signal.

For example, a wavelet could be created to have a frequency of Middle Cand a short duration of roughly a 32nd note. If this wavelet was to beconvolved with a signal created from the recording of a song, then theresulting signal would be useful for determining when the Middle C note wasbeing played in the song. Mathematically, the wavelet will correlate with thesignal if the unknown signal contains information of similar frequency. Thisconcept of correlation is at the core of many practical applications ofwavelet theory.

As a mathematical tool, wavelets can be used to extract information frommany different kinds of data, including – but certainly not limited to – audiosignals and images. Sets of wavelets are generally needed to analyze datafully. A set of "complementary" wavelets will decompose data without gapsor overlap so that the decomposition process is mathematically reversible.Thus, sets of complementary wavelets are useful in wavelet basedcompression/decompression algorithms where it is desirable to recover theoriginal information with minimal loss.

In formal terms, this representation is a wavelet series representation of asquare-integrable function with respect to either a complete, orthonormal setof basis functions, or an overcomplete set or frame of a vector space, for theHilbert space of square integrable functions.

Contents

1 Name

2 Wavelet theory

2.1 Continuous wavelet transforms (continuous shift and

scale parameters)

2.2 Discrete wavelet transforms (discrete shift and scale

parameters)

2.3 Multiresolution based discrete wavelet transforms

3 Mother wavelet

4 Comparisons with Fourier transform (continuous-time)

5 Definition of a wavelet

5.1 Scaling filter

5.2 Scaling function

5.3 Wavelet function

6 History

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6.1 Timeline

7 Wavelet transforms

7.1 Generalized transforms

8 Applications of Wavelet Transform

8.1 As a representation of a signal

8.2 Wavelet Denoising

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

Name

The word wavelet has been used for decades in digital signal processing and exploration geophysics.[1] The equivalentFrench word ondelette meaning "small wave" was used by Morlet and Grossmann in the early 1980s.

Wavelet theory

Wavelet theory is applicable to several subjects. All wavelet transforms may be considered forms of time-frequencyrepresentation for continuous-time (analog) signals and so are related to harmonic analysis. Almost all practically usefuldiscrete wavelet transforms use discrete-time filterbanks. These filter banks are called the wavelet and scalingcoefficients in wavelets nomenclature. These filterbanks may contain either finite impulse response (FIR) or infiniteimpulse response (IIR) filters. The wavelets forming a continuous wavelet transform (CWT) are subject to theuncertainty principle of Fourier analysis respective sampling theory: Given a signal with some event in it, one cannotassign simultaneously an exact time and frequency response scale to that event. The product of the uncertainties of timeand frequency response scale has a lower bound. Thus, in the scaleogram of a continuous wavelet transform of thissignal, such an event marks an entire region in the time-scale plane, instead of just one point. Also, discrete waveletbases may be considered in the context of other forms of the uncertainty principle.

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

Continuous wavelet transforms (continuous shift and scale parameters)

In continuous wavelet transforms, a given signal of finite energy is projected on a continuous family of frequency bands

(or similar subspaces of the Lp function space L2(R) ). For instance the signal may be represented on every frequencyband of the form [f, 2f] for all positive frequencies f > 0. Then, the original signal can be reconstructed by a suitableintegration over all the resulting frequency components.

The frequency bands or subspaces (sub-bands) are scaled versions of a subspace at scale 1. This subspace in turn is in

most situations generated by the shifts of one generating function ψ in L2(R), the mother wavelet. For the example ofthe scale one frequency band [1, 2] this function is

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Meyer Morlet Mexican Hat

with the (normalized) sinc function. That, Meyer's, and two other examples of mother wavelets are:

The subspace of scale a or frequency band [1/a, 2/a] is generated by the functions (sometimes called child wavelets)

where a is positive and defines the scale and b is any real number and defines the shift. The pair (a, b) defines a pointin the right halfplane R+ × R.

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

with wavelet coefficients

See a list of some Continuous wavelets.

For the analysis of the signal x, one can assemble the wavelet coefficients into a scaleogram of the signal.

Discrete wavelet transforms (discrete shift and scale parameters)

It is computationally impossible to analyze a signal using all wavelet coefficients, so one may wonder if it is sufficient topick a discrete subset of the upper halfplane to be able to reconstruct a signal from the corresponding waveletcoefficients. One such system is the affine system for some real parameters a > 1, b > 0. The corresponding discrete

subset of the halfplane consists of all the points (am, namb) with m, n in Z. The corresponding baby wavelets are nowgiven as

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

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D4 wavelet

is that the functions form a tight frame of L2(R).

Multiresolution based discrete wavelet transforms

In any discretised wavelet transform, there are only a finite number ofwavelet coefficients for each bounded rectangular region in the upperhalfplane. Still, each coefficient requires the evaluation of an integral. Inspecial situations this numerical complexity can be avoided if the scaled andshifted wavelets form a multiresolution analysis. This means that there has to

exist an auxiliary function, the father wavelet φ in L2(R), and that a is aninteger. A typical choice is a = 2 and b = 1. The most famous pair of fatherand mother wavelets is the Daubechies 4-tap wavelet. Note that not everyorthonormal discrete wavelet basis can be associated to a multiresolutionanalysis; for example, the Journe wavelet admits no multiresolution

analysis.[2]

From the mother and father wavelets one constructs the subspaces

The mother wavelet keeps the time domain properties, while the father wavelets keeps the frequency domain

properties.

From these it is required that the sequence

forms a multiresolution analysis of L2 and that the subspaces are the orthogonal

"differences" of the above sequence, that is, Wm is the orthogonal complement of Vm inside the subspace Vm−1,

In analogy to the sampling theorem one may conclude that the space Vm with sampling distance 2m more or less covers

the frequency baseband from 0 to 2−m-1. As orthogonal complement, Wm roughly covers the band [2−m−1, 2−m].

From those inclusions and orthogonality relations, especially , follows the existence of sequences

and that satisfy the identities

so that and

so that

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

From the multiresolution analysis derives the orthogonal decomposition of the space L2 as

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For any signal or function this gives a representation in basis functions of the corresponding subspaces as

where the coefficients are

and

.

Mother wavelet

For practical applications, and for efficiency reasons, one prefers continuously differentiable functions with compactsupport as mother (prototype) wavelet (functions). However, to satisfy analytical requirements (in the continuous WT)and in general for theoretical reasons, 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 of zero 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 mustsatisfy an admissibility criterion (loosely speaking, a kind of half-differentiability) in order to get a stably invertibletransform.

For the discrete wavelet transform, one needs at least the condition that the wavelet series is a representation of the

identity in the space L2(R). Most constructions of discrete WT make use of the multiresolution analysis, which definesthe wavelet by a scaling function. This scaling function itself is solution to a functional equation.

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

The mother wavelet is scaled (or dilated) by a factor of a and translated (or shifted) by a factor of b to give (underMorlet's original formulation):

For the continuous WT, the pair (a,b) varies over the full half-plane R+ × R; 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 basis functions of the (continuous) transform. In fact, as in thecontinuous Fourier transform, there is no basis in the continuous wavelet transform. Time-frequency interpretation usesa subtly different formulation (after Delprat).

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Restriction：

(1) when a1 = a and b1 = b,

(2) has a finite time interval

Comparisons with Fourier transform (continuous-time)

The wavelet transform is often compared with the Fourier transform, in which signals are represented as a sum ofsinusoids. In fact, the Fourier transform can be viewed as a special case of the continuous wavelet transform with thechoice of the mother wavelet . The main difference in general is that wavelets are localized in both

time and frequency whereas the standard Fourier transform is only localized in frequency. The Short-time Fouriertransform (STFT) is similar to the wavelet transform, in that it is also time and frequency localized, but there are issueswith the frequency/time resolution trade-off.

In particular, assuming a rectangular window region, one may think of the STFT as a transform with a slightly differentkernel

where can often be written as , where and u respectively denote the length and

temporal offset of the windowing function. Using Parseval’s theorem, one may define the wavelet’s energy as

=

From this, the square of the temporal support of the window offset by time u is given by

and the square of the spectral support of the window acting on a frequency

As stated by the Heisenberg uncertainty principle, the product of the temporal and spectral supports

for any given time-frequency atom, or resolution cell. The STFT windows restrict the resolution cells to spectral andtemporal supports determined by .

Multiplication with a rectangular window in the time domain corresponds to convolution with a function

in the frequency domain, resulting in spurious ringing artifacts for short/localized temporal windows. With thecontinuous-time Fourier Transform, and this convolution is with a delta function in Fourier space, resulting

in the true Fourier transform of the signal . The window function may be some other apodizing filter, such as a

Gaussian. The choice of windowing function will affect the approximation error relative to the true Fourier transform.

A given resolution cell’s time-bandwidth product may not be exceeded with the STFT. All STFT basis elementsmaintain a uniform spectral and temporal support for all temporal shifts or offsets, thereby attaining an equal resolutionin time for lower and higher frequencies. The resolution is purely determined by the sampling width.

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STFT time-frequency atoms (left)

and DWT time-scale atoms (right).

The time-frequency atoms are four

different basis functions used for the

STFT (i.e. four separate Fourier

transforms required). The time-

scale atoms of the DWT achieve

small temporal widths for high

frequencies and good temporal widths

for low frequencies with a single

transform basis set.

In contrast, the wavelet transform’s multiresolutional properties enables large temporal supports for lower frequencieswhile maintaining short temporal widths for higher frequencies by the scaling properties of the wavelet transform. This

property extends conventional time-frequency analysis into time-scale analysis.[3]

The discrete wavelet transform is less computationally complex, taking O(N)time as compared to O(N log N) for the fast Fourier transform. Thiscomputational advantage is not inherent to the transform, but reflects thechoice of a logarithmic division of frequency, in contrast to the equallyspaced frequency divisions of the FFT (Fast Fourier Transform) which uses

the same basis functions as DFT (Discrete Fourier Transform).[4] It is alsoimportant to note that this complexity only applies when the filter size has norelation to the signal size. A wavelet without compact support such as the

Shannon wavelet would require O(N2). (For instance, a logarithmic FourierTransform also exists with O(N) complexity, but the original signal must besampled logarithmically in time, which is only useful for certain types of

signals.[5])

Definition of a wavelet

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

Scaling filter

An orthogonal wavelet is entirely defined by the scaling filter – a low-pass finite impulse response (FIR) filter of length2N and sum 1. In biorthogonal wavelets, separate decomposition and reconstruction filters are defined.

For analysis with orthogonal wavelets the high pass filter is calculated as the quadrature mirror filter of the low pass,and reconstruction filters are the time reverse of the decomposition filters.

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

Scaling function

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

The wavelet function is in effect a band-pass filter and scaling it for each level halves its bandwidth. This creates theproblem that in order to cover the entire spectrum, an infinite number of levels would be required. The scaling functionfilters the lowest level of the transform and ensures all the spectrum is covered. See [1](http://www.polyvalens.com/blog/?page_id=15#7.+The+scaling+function+%5B7%5D) for a detailed explanation.

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

Meyer wavelets can be defined by scaling functions

Wavelet function

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

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

History

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The development of wavelets can be linked to several separate trains of thought, starting with Haar's work in the early20th century. Later work by Dennis Gabor yielded Gabor atoms (1946), which are constructed similarly to wavelets,and applied to similar purposes. Notable contributions to wavelet theory can be attributed to Zweig’s discovery of thecontinuous wavelet transform in 1975 (originally called the cochlear transform and discovered while studying the

reaction of the ear to sound),[6] Pierre Goupillaud, Grossmann and Morlet's formulation of what is now known as theCWT (1982), Jan-Olov Strömberg's early work on discrete wavelets (1983), Daubechies' orthogonal wavelets withcompact support (1988), Mallat's multiresolution framework (1989), Akansu's Binomial QMF (1990), NathalieDelprat's time-frequency interpretation of the CWT (1991), Newland's harmonic wavelet transform (1993) and manyothers since.

Timeline

First wavelet (Haar wavelet) by Alfréd Haar (1909)

Since the 1970s: George Zweig, Jean Morlet, Alex Grossmann

Since the 1980s: Yves Meyer, Stéphane Mallat, Ingrid Daubechies, Ronald Coifman, Ali Akansu, Victor

Wickerhauser,

Wavelet transforms

A wavelet is a mathematical function used to divide a given function or continuous-time signal into different scalecomponents. Usually one can assign a frequency range to each scale component. Each scale component can then bestudied with a resolution that matches its scale. A wavelet transform is the representation of a function by wavelets. Thewavelets are scaled and translated copies (known as "daughter wavelets") of a finite-length or fast-decaying oscillatingwaveform (known as the "mother wavelet"). Wavelet transforms have advantages over traditional Fourier transformsfor representing functions that have discontinuities and sharp peaks, and for accurately deconstructing andreconstructing finite, non-periodic and/or non-stationary signals.

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 representcontinuous-time (analog) signals. CWTs operate over every possible scale and translation whereas DWTs use aspecific subset of scale and translation values or representation grid.

There are a large number of wavelet transforms each suitable for different applications. For a full list see list of wavelet-related transforms 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 Fourier transform (FRFT)

Fractional wavelet transform (FRWT)

Generalized transforms

There are a number of generalized transforms of which the wavelet transform is a special case. For example, JosephSegman introduced scale into the Heisenberg group, giving rise to a continuous transform space that is a function oftime, scale, and frequency. The CWT is a two-dimensional slice through the resulting 3d time-scale-frequency volume.

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Another example of a generalized transform is the chirplet transform in which the CWT is also a two dimensional slicethrough the chirplet transform.

An important application area for generalized transforms involves systems in which high frequency resolution is crucial.For example, darkfield electron optical transforms intermediate between direct and reciprocal space have been widely

used in the harmonic analysis of atom clustering, i.e. in the study of crystals and crystal defects.[7] Now thattransmission electron microscopes are capable of providing digital images with picometer-scale information on atomic

periodicity in nanostructure of all sorts, the range of pattern recognition[8] and strain[9]/metrology[10] applications for

intermediate transforms with high frequency resolution (like brushlets[11] and ridgelets[12]) is growing rapidly.

Fractional wavelet transform (FRWT) is a generalization of the classical wavelet transform in the fractional Fouriertransform domains. This transform is capable of providing the time- and fractional-domain information simultaneously

and representing signals in the time-fractional-frequency plane.[13]

Applications of Wavelet Transform

Generally, an approximation to DWT is used for data compression if a signal is already sampled, and the CWT for

signal analysis.[14] Thus, DWT approximation is commonly used in engineering and computer science, and the CWT inscientific research.

Like some other transforms, wavelet transforms can be used to transform data, then encode the transformed data,resulting in effective compression. For example, JPEG 2000 is an image compression standard that uses biorthogonalwavelets. This means that although the frame is overcomplete, it is a tight frame (see types of Frame of a vectorspace), and the same frame functions (except for conjugation in the case of complex wavelets) are used for bothanalysis and synthesis, i.e., in both the forward and inverse transform. For details see wavelet compression.

A related use is for smoothing/denoising data based on wavelet coefficient thresholding, also called wavelet shrinkage.By adaptively thresholding the wavelet coefficients that correspond to undesired frequency components smoothingand/or denoising operations can be performed.

Wavelet transforms are also starting to be used for communication applications. Wavelet OFDM is the basicmodulation scheme used in HD-PLC (a power line communications technology developed by Panasonic), and in oneof the optional modes included in the IEEE 1901 standard. Wavelet OFDM can achieve deeper notches thantraditional FFT OFDM, and wavelet OFDM does not require a guard interval (which usually represents significant

overhead in FFT OFDM systems).[15]

As a representation of a signal

Often, signals can be represented well as a sum of sinusoids. However, consider a non-continuous signal with anabrupt discontinuity; this signal can still be represented as a sum of sinusoids, but requires an infinite number, which isan observation known as Gibbs phenomenon. This, then, requires an infinite number of Fourier coefficients, which isnot practical for many applications, such as compression. Wavelets are more useful for describing these signals withdiscontinuities because of their time-localized behavior (both Fourier and wavelet transforms are frequency-localized,but wavelets have an additional time-localization property). Because of this, many types of signals in practice may benon-sparse in the Fourier domain, but very sparse in the wavelet domain. This is particularly useful in signalreconstruction, especially in the recently popular field of compressed sensing. (Note that the Short-time Fouriertransform (STFT) is also localized in time and frequency, but there are often problems with the frequency-timeresolution trade-off. Wavelets are better signal representations because of multiresolution analysis.)

This motivates why wavelet transforms are now being adopted for a vast number of applications, often replacing theconventional Fourier Transform. Many areas of physics 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, EEG, EMG,[16] ECG analyses, brain rhythms, DNA analysis,

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protein analysis, climatology, human sexual response analysis,[17] general signal processing, speech recognition,

acoustics, vibration signals,[18] computer graphics, multifractal analysis, and sparse coding. In computer vision andimage processing, the notion of scale space representation and Gaussian derivative operators is regarded as acanonical multi-scale representation.

Wavelet Denoising

Suppose we measure a noisy signal . Assume s has a sparse representation in a certain wavelet bases,and

So .

Most elements in p are 0 or close to 0, and

Since W is orthogonal, the estimation problem amounts to recovery of a signal in iid Gaussian noise. As p is sparse,one method is to apply a gaussian mixture model for p.

Assume a prior , is the variance of "significant" coefficients, and

is the variance of "insignificant" coefficients.

Then , is called the shrinkage factor, which depends on the prior variances and

. The effect of the shrinkage factor is that small coefficients are set early to 0, and large coefficients are unaltered.

Small coefficients are mostly noises, and large coefficients contain actual signal.

At last, apply the inverse wavelet transform to obtain

List of wavelets

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, etc.)

Binomial-QMF (Also referred to as Daubechies wavelet)

Haar wavelet

Mathieu wavelet

Legendre wavelet

Villasenor wavelet

Symlet[19]

Continuous wavelets

Real-valued

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Beta wavelet

Hermitian wavelet

Hermitian hat wavelet

Meyer wavelet

Mexican hat wavelet

Shannon wavelet

Complex-valued

Complex Mexican hat wavelet

fbsp wavelet

Morlet wavelet

Shannon wavelet

Modified Morlet wavelet

See also

Chirplet transform

Curvelet

Digital cinema

Filter banks

Fractal compression

Fractional Fourier transform

JPEG 2000

Multiresolution analysis

Noiselet

Scale space

Shearlet

Short-time Fourier transform

Ultra wideband radio- transmits wavelets

Wave packet

Gabor wavelet#Wavelet space[20]

Dimension reduction

Fourier-related transforms

Spectrogram

Notes

1. ^ Ricker, Norman (1953). "WAVELET CONTRACTION, WAVELET EXPANSION, AND THE CONTROL OF

SEISMIC RESOLUTION". Geophysics 18 (4). doi:10.1190/1.1437927 (http://dx.doi.org/10.1190%2F1.1437927).

2. ^ Larson, David R. (2007). "Unitary systems and wavelet sets". Wavelet Analysis and Applications. Appl. Numer.

Harmon. Anal. Birkhäuser. pp. 143–171.

3. ^ Mallat, Stephane. "A wavelet tour of signal processing. 1998." 250-252.

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4. ^ The Scientist and Engineer's Guide to Digital Signal Processing By Steven W. Smith, Ph.D. chapter 8 equation 8-1:

http://www.dspguide.com/ch8/4.htm

5. ^ http://homepages.dias.ie/~ajones/publications/28.pdf

6. ^ http://scienceworld.wolfram.com/biography/Zweig.html Zweig, George Biography on Scienceworld.wolfram.com

7. ^ 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

8. ^ 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 (http://arxiv.org/abs/cond-mat/0403017))

9. ^ M. J. Hÿtch, E. Snoeck and R. Kilaas (1998) Quantitative measurement of displacement and strain fields from

HRTEM micrographs, Ultramicroscopy 74:131-146.

10. ^ Martin Rose (2006) Spacing measurements of lattice fringes in HRTEM image using digital darkfield decomposition

(M.S. Thesis in Physics, U. Missouri – St. Louis)

11. ^ F. G. Meyer and R. R. Coifman (1997) Applied and Computational Harmonic Analysis 4:147.

12. ^ 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).

13. ^ J. Shi, N.-T. Zhang, and X.-P. Liu, "A novel fractional wavelet transform and its applications," Sci. China Inf. Sci.,

vol. 55, no. 6, pp. 1270–1279, June 2012. URL: http://www.springerlink.com/content/q01np2848m388647/

14. ^ A.N. Akansu, W.A. Serdijn and I.W. Selesnick, Emerging applications of wavelets: A review

(http://web.njit.edu/~akansu/PAPERS/ANA-IWS-WAS-ELSEVIER%20PHYSCOM%202010.pdf), Physical

Communication, Elsevier, vol. 3, issue 1, pp. 1-18, March 2010.

15. ^ Stefano Galli, O. Logvinov (July 2008). "Recent Developments in the Standardization of Power Line

Communications within the IEEE". IEEE Communications Magazine 46 (7): 64–71.

doi:10.1109/MCOM.2008.4557044 (http://dx.doi.org/10.1109%2FMCOM.2008.4557044). An overview of P1901

PHY/MAC proposal.

16. ^ J. Rafiee et al. Feature extraction of forearm EMG signals for prosthetics, Expert Systems with Applications 38

(2011) 4058–67.

17. ^ J. Rafiee et al. Female sexual responses using signal processing techniques, The Journal of Sexual Medicine 6

(2009) 3086–96. (pdf) (http://rafiee.us/files/JSM_2009.pdf)

18. ^ J. Rafiee and Peter W. Tse, Use of autocorrelation in wavelet coefficients for fault diagnosis, Mechanical Systems

and Signal Processing 23 (2009) 1554–72.

19. ^ Matlab Toolbox – URL: http://matlab.izmiran.ru/help/toolbox/wavelet/ch06_a32.html

20. ^ Erik Hjelmås (1999-01-21) Gabor Wavelets URL: http://www.ansatt.hig.no/erikh/papers/scia99/node6.html

References

Paul S. Addison, The Illustrated Wavelet Transform Handbook, Institute of Physics, 2002, ISBN 0-7503-

0692-0

Ali Akansu and Richard Haddad, Multiresolution Signal Decomposition: Transforms, Subbands, Wavelets,

Academic Press, 1992, ISBN 0-12-047140-X

B. Boashash, editor, "Time-Frequency Signal Analysis and Processing – A Comprehensive Reference", Elsevier

Science, Oxford, 2003, ISBN 0-08-044335-4.

Tony F. Chan and Jackie (Jianhong) Shen, Image Processing and Analysis – Variational, PDE, Wavelet,

and Stochastic Methods, Society of Applied Mathematics, ISBN 0-89871-589-X (2005)

Ingrid Daubechies, Ten Lectures on Wavelets, Society for Industrial and Applied Mathematics, 1992, ISBN

0-89871-274-2

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Look up wavelet inWiktionary, the freedictionary.

Wikimedia Commons hasmedia related to Wavelet.

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

Haar A., Zur Theorie der orthogonalen Funktionensysteme, Mathematische Annalen, 69, pp 331–371,

1910.

Barbara Burke Hubbard, "The World According to Wavelets: The Story of a Mathematical Technique in the

Making", AK Peters Ltd, 1998, ISBN 1-56881-072-5, ISBN 978-1-56881-072-0

Gerald Kaiser, A Friendly Guide to Wavelets, Birkhauser, 1994, ISBN 0-8176-3711-7

Stéphane Mallat, "A wavelet tour of signal processing" 2nd Edition, Academic Press, 1999, ISBN 0-12-

466606-X

Donald B. Percival and Andrew T. Walden, Wavelet Methods for Time Series Analysis, Cambridge

University Press, 2000, ISBN 0-521-68508-7

Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007), "Section 13.10. Wavelet Transforms"

(http://apps.nrbook.com/empanel/index.html#pg=699), Numerical Recipes: The Art of Scientific Computing

(3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8

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

Martin Vetterli and Jelena Kovačević, "Wavelets and Subband Coding", Prentice Hall, 1995, ISBN 0-13-

097080-8

External links

Hazewinkel, Michiel, ed. (2001), "Wavelet analysis"

(http://www.encyclopediaofmath.org/index.php?title=p/w097160),

Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-

4

OpenSource Wavelet C# Code (http://www.waveletstudio.net/)

JWave – Open source Java implementation of several orthogonal and

non-orthogonal wavelets (https://code.google.com/p/jwave/)

Wavelet Analysis in Mathematica (http://reference.wolfram.com/mathematica/guide/Wavelets.html) (A very

comprehensive set of wavelet analysis tools)

1st NJIT Symposium on Wavelets (April 30, 1990) (First Wavelets Conference in USA)

(http://web.njit.edu/~ali/s1.htm)

Binomial-QMF Daubechies Wavelets

(http://web.njit.edu/~ali/NJITSYMP1990/AkansuNJIT1STWAVELETSSYMPAPRIL301990.pdf)

Wavelets (http://www-math.mit.edu/~gs/papers/amsci.pdf) by Gilbert Strang, American Scientist 82 (1994)

250–255. (A very short and excellent introduction)

Wavelet Digest (http://www.wavelet.org)

NASA Signal Processor featuring Wavelet methods

(http://www.grc.nasa.gov/WWW/OptInstr/NDE_Wave_Image_ProcessorLab.html) Description of NASA

Signal & Image Processing Software and Link to Download

19/8/2014 Wavelet - Wikipedia, the free encyclopedia

http://en.wikipedia.org/wiki/Wavelet 14/14

Course on Wavelets given at UC Santa Barbara, 2004

(http://wavelets.ens.fr/ENSEIGNEMENT/COURS/UCSB/index.html)

The Wavelet Tutorial by Polikar (http://users.rowan.edu/~polikar/WAVELETS/WTtutorial.html) (Easy to

understand when you have some background with fourier transforms!)

OpenSource Wavelet C++ Code (http://herbert.the-little-red-haired-girl.org/en/software/wavelet/)

Wavelets for Kids (PDF file) (http://www.isye.gatech.edu/~brani/wp/kidsA.pdf) (Introductory (for very smart

kids!))

Link collection about wavelets (http://www.cosy.sbg.ac.at/~uhl/wav.html)

Gerald Kaiser's acoustic and electromagnetic wavelets (http://wavelets.com/pages/center.html)

A really friendly guide to wavelets (http://perso.wanadoo.fr/polyvalens/clemens/wavelets/wavelets.html)

Wavelet-based image annotation and retrieval (http://www.alipr.com)

Very basic explanation of Wavelets and how FFT relates to it

(http://www.relisoft.com/Science/Physics/sampling.html)

A Practical Guide to Wavelet Analysis (http://paos.colorado.edu/research/wavelets/) 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 (http://ocgweb.marine.usf.edu/~liu/wavelet.html).

WITS: Where Is The Starlet? (http://www.laurent-duval.eu/siva-wits-where-is-the-starlet.html) A dictionary of

tens of wavelets and wavelet-related terms ending in -let, from activelets to x-lets through bandlets, contourlets,

curvelets, noiselets, wedgelets.

Python Wavelet Transforms Package (http://www.pybytes.com/pywavelets/) OpenSource code for computing

1D and 2D Discrete wavelet transform, Stationary wavelet transform and Wavelet packet transform.

Wavelet Library (http://pages.cs.wisc.edu/~kline/wvlib) GNU/GPL library for n-dimensional discrete

wavelet/framelet transforms.

The Fractional Spline Wavelet Transform (http://bigwww.epfl.ch/publications/blu0001.pdf) describes a

fractional wavelet transform based on fractional b-Splines.

A Panorama on Multiscale Geometric Representations, Intertwining Spatial, Directional and Frequency

Selectivity (http://dx.doi.org/10.1016/j.sigpro.2011.04.025) provides a tutorial on two-dimensional oriented

wavelets and related geometric multiscale transforms.

HD-PLC Alliance (http://www.hd-plc.org/)

Signal Denoising using Wavelets (http://tx.technion.ac.il/~rc/SignalDenoisingUsingWavelets_RamiCohen.pdf)

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Categories: Wavelets Time–frequency analysis Signal processing

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