Wavelet
A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases or decreases, and then returns to zero one or more times. Wavelets are termed a "brief oscillation". A taxonomy of wavelets has been established, based on the number and direction of its pulses. Wavelets are imbued with specific properties that make them useful for signal processing.
For example, a wavelet could be created to have a frequency of
As a mathematical tool, wavelets can be used to extract information from many kinds of data, including
In formal terms, this representation is a
In
Etymology
The word wavelet has been used for decades in digital signal processing and exploration geophysics.[2] The equivalent French 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
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 frequency band of the form [f, 2f] for all positive frequencies f > 0. Then, the original signal can be reconstructed by a suitable integration 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 of the scale one frequency band [1, 2] this function is
The subspace of scale a or frequency band [1/a, 2/a] is generated by the functions (sometimes called child wavelets)
The projection of a function x onto the subspace of scale a then has the form
For the analysis of the signal x, one can assemble the wavelet coefficients into a
See a list of some
Discrete wavelet transforms (discrete shift and scale parameters, continuous in time)
It is computationally impossible to analyze a signal using all wavelet coefficients, so one may wonder if it is sufficient to pick a discrete subset of the upper halfplane to be able to reconstruct a signal from the corresponding wavelet coefficients. 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, nb am) with m, n in Z. The corresponding child wavelets are now given as
A sufficient condition for the reconstruction of any signal x of finite energy by the formula
Multiresolution based discrete wavelet transforms (continuous in time)
In any discretised wavelet transform, there are only a finite number of wavelet coefficients for each bounded rectangular region in the upper halfplane. Still, each coefficient requires the evaluation of an integral. In special situations this numerical complexity can be avoided if the scaled and shifted wavelets form a
From the mother and father wavelets one constructs the subspaces
From these it is required that the sequence
In analogy to the
From those inclusions and orthogonality relations, especially , follows the existence of sequences and that satisfy the identities
From the multiresolution analysis derives the orthogonal decomposition of the space L2 as
Time-causal wavelets
For processing temporal signals in real time, it is essential that the wavelet filters do not access signal values from the future as well as that minimal temporal latencies can be obtained. Time-causal wavelets representations have been developed by Szu et al [8] and Lindeberg,[9] with the latter method also involving a memory-efficient time-recursive implementation.
Mother wavelet
For practical applications, and for efficiency reasons, one prefers continuously differentiable functions with compact support 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 Lebesgue measurable functions that are both absolutely integrable and square integrable in the sense that
Being in this space ensures that one can formulate the conditions of zero mean and square norm one:
For ψ to be a wavelet for the continuous wavelet transform (see there for exact statement), the mother wavelet must satisfy an admissibility criterion (loosely speaking, a kind of half-differentiability) in order to get a stably invertible transform.
For the
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 (under Morlet'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 the continuous Fourier transform, there is no basis in the continuous wavelet transform. Time-frequency interpretation uses a subtly different formulation (after Delprat).
Restriction:
- when a1 = a and b1 = b,
- 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 of sinusoids. In fact, the Fourier transform can be viewed as a special case of the continuous wavelet transform with the choice 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 Fourier transform (STFT) is similar to the wavelet transform, in that it is also time and frequency localized, but there are issues with 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 different kernel
and the square of the spectral support of the window acting on a frequency
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 the continuous-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
A given resolution cell's time-bandwidth product may not be exceeded with the STFT. All STFT basis elements maintain a uniform spectral and temporal support for all temporal shifts or offsets, thereby attaining an equal resolution in time for lower and higher frequencies. The resolution is purely determined by the sampling width.
In contrast, the wavelet transform's multiresolutional properties enables large temporal supports for lower frequencies while 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.[10]
The discrete wavelet transform is less computationally
Definition of a wavelet
A wavelet (or a wavelet family) can be defined in various ways:
Scaling filter
An orthogonal wavelet is entirely defined by the scaling filter – a low-pass finite impulse response (FIR) filter of length 2N 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 father wavelet) in the time domain.
The wavelet function is in effect a band-pass filter and scaling that for each level halves its bandwidth. This creates the problem that in order to cover the entire spectrum, an infinite number of levels would be required. The scaling function filters the lowest level of the transform and ensures all the spectrum is covered. See[13] 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,
History
The development of wavelets can be linked to several separate trains of thought, starting with Haar's work in the early 20th 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 since then can be attributed to
The
Timeline
- First wavelet (Haar Wavelet) by Alfréd Haar(1909)
- Since the 1970s: George Zweig, Jean Morlet, Alex Grossmann
- Since the 1980s: Yves Meyer, Didier Le Gall, Ali J. Tabatabai, Stéphane Mallat, Ingrid Daubechies, Ronald Coifman, Ali Akansu, Victor Wickerhauser
- Since the 1990s: Nathalie Delprat, Newland, Amir Said, William A. Pearlman, Touradj Ebrahimi, JPEG 2000
Wavelet transforms
A wavelet is a mathematical function used to divide a given function or
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 whereas DWTs use a specific 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)
- 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, Yosef Joseph Segman introduced scale into the Heisenberg group, giving rise to a continuous transform space that is a function of time, scale, and frequency. The CWT is a two-dimensional slice through the resulting 3d time-scale-frequency volume.
Another example of a generalized transform is the chirplet transform in which the CWT is also a two dimensional slice through the chirplet transform.
An important application area for generalized transforms involves systems in which high frequency resolution is crucial. For example,
Fractional wavelet transform (FRWT) is a generalization of the classical wavelet transform in the fractional Fourier transform domains. This transform is capable of providing the time- and fractional-domain information simultaneously and representing signals in the time-fractional-frequency plane.[28]
Applications
Generally, an approximation to DWT is used for
Like some other transforms, wavelet transforms can be used to transform data, then encode the transformed data, resulting in effective compression. For example,
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 smoothing and/or denoising operations can be performed.
Wavelet transforms are also starting to be used for communication applications. Wavelet
As a representation of a signal
Often, signals can be represented well as a sum of sinusoids. However, consider a non-continuous signal with an abrupt discontinuity; this signal can still be represented as a sum of sinusoids, but requires an infinite number, which is an observation known as Gibbs phenomenon. This, then, requires an infinite number of Fourier coefficients, which is not practical for many applications, such as compression. Wavelets are more useful for describing these signals with discontinuities 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 be non-sparse in the Fourier domain, but very sparse in the wavelet domain. This is particularly useful in signal reconstruction, especially in the recently popular field of compressed sensing. (Note that the short-time Fourier transform (STFT) is also localized in time and frequency, but there are often problems with the frequency-time resolution 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 the conventional
Wavelet denoising
Suppose we measure a noisy signal , where represents the signal and represents the noise. Assume has a sparse representation in a certain wavelet basis, and
Let the wavelet transform of be , where is the wavelet transform of the signal component and is the wavelet transform of the noise component.
Most elements in are 0 or close to 0, and
Since is orthogonal, the estimation problem amounts to recovery of a signal in iid Gaussian noise. As is sparse, one method is to apply a Gaussian mixture model for .
Assume a prior , where 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 . By setting coefficients that fall below a shrinkage threshold to zero, once the inverse transform is applied, an expectedly small amount of signal is lost due to the sparsity assumption. The larger coefficients are expected to primarily represent signal due to sparsity, and statistically very little of the signal, albeit the majority of the noise, is expected to be represented in such lower magnitude coefficients... therefore the zeroing-out operation is expected to remove most of the noise and not much signal. Typically, the above-threshold coefficients are not modified during this process. Some algorithms for wavelet-based denoising may attenuate larger coefficients as well, based on a statistical estimate of the amount of noise expected to be removed by such an attenuation.
At last, apply the inverse wavelet transform to obtain
Multiscale climate network
Agarwal et al. proposed wavelet based advanced linear [39] and nonlinear [40] methods to construct and investigate Climate as complex networks at different timescales. Climate networks constructed using SST datasets at different timescale averred that wavelet based multi-scale analysis of climatic processes holds the promise of better understanding the system dynamics that may be missed when processes are analyzed at one timescale only [41]
List of wavelets
Discrete wavelets
- Beylkin (18)
- Moore Wavelet
- Biorthogonal nearly coiflet (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[42]
Continuous wavelets
Real-valued
- Beta wavelet
- Hermitian wavelet
- Meyer wavelet
- Mexican hat wavelet
- Poisson wavelet
- Shannon wavelet
- Spline wavelet
- Strömberg wavelet
Complex-valued
See also
- Chirplet transform
- Curvelet
- Digital cinema
- Dimension reduction
- Filter banks
- Fourier-related transforms
- Fractal compression
- Fractional Fourier transform
- Gabor wavelet § Wavelet space[43]
- Huygens–Fresnel principle (physical wavelets)
- JPEG 2000
- Least-squares spectral analysis for computing periodicity in any including unevenly spaced data
- Multiresolution analysis
- Noiselet
- Non-separable wavelet
- Scale space
- Scaled correlation
- Shearlet
- Short-time Fourier transform
- Spectrogram
- Ultra widebandradio – transmits wavelets
References
- ^ Wireless Communications: Principles and Practice, Prentice Hall communications engineering and emerging technologies series, T. S. Rappaport, Prentice Hall, 2002, p. 126.
- .
- ISBN 0-521-42000-8
- ISBN 0-12-174584-8
- ISBN 978-0-89871-274-2
- ISBN 978-0-12-047141-6
- ^ Larson, David R. (2007), Wavelet Analysis and Applications (See: Unitary systems and wavelet sets), Appl. Numer. Harmon. Anal., Birkhäuser, pp. 143–171
- doi:10.1117/12.59911.
- PMID 36689001.
- ^ Mallat, Stephane. "A wavelet tour of signal processing. 1998." 250-252.
- ^ 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
- S2CID 9720759.
- ^ "A Really Friendly Guide To Wavelets – PolyValens". www.polyvalens.com.
- ^ Weisstein, Eric W. "Zweig, George -- from Eric Weisstein's World of Scientific Biography". scienceworld.wolfram.com. Retrieved 2021-10-20.
- ^ Sullivan, Gary (8–12 December 2003). "General characteristics and design considerations for temporal subband video coding". ITU-T. Video Coding Experts Group. Retrieved 13 September 2019.
- ISBN 9780080922508.
- S2CID 109186495.
- ISSN 1051-8215.
- ISBN 9781461507994.
- S2CID 2765169. Archived from the original(PDF) on 2019-10-13.
- ISBN 9780240806174.
- ISBN 0-88275-376-2
- ^ 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)
- .
- ^ Martin Rose (2006) Spacing measurements of lattice fringes in HRTEM image using digital darkfield decomposition (M.S. Thesis in Physics, U. Missouri – St. Louis)
- ^ F. G. Meyer and R. R. Coifman (1997) Applied and Computational Harmonic Analysis 4:147.
- ^ 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).
- S2CID 255201598.
- ^ A.N. Akansu, W.A. Serdijn and I.W. Selesnick, Emerging applications of wavelets: A review, Physical Communication, Elsevier, vol. 3, issue 1, pp. 1-18, March 2010.
- ISSN 2227-7390.
Wavelets are actively used to solve a wide range of image processing problems in various fields of science and technology, e.g., image denoising, reconstruction, analysis, and video analysis and processing. Wavelet processing methods are based on the discrete wavelet transform using 1D digital filtering.
- S2CID 263317973.
- S2CID 2650873. An overview of P1901 PHY/MAC proposal.
- PMID 19072712.
- S2CID 119313566.
- .
- ^ J. Rafiee et al. Feature extraction of forearm EMG signals for prosthetics, Expert Systems with Applications 38 (2011) 4058–67.
- ^ J. Rafiee et al. Female sexual responses using signal processing techniques, The Journal of Sexual Medicine 6 (2009) 3086–96. (pdf)
- .
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- PMID 31217490.
- ^ Matlab Toolbox – URL: http://matlab.izmiran.ru/help/toolbox/wavelet/ch06_a32.html
- ^ Erik Hjelmås (1999-01-21) Gabor Wavelets URL: http://www.ansatt.hig.no/erikh/papers/scia99/node6.html
Further reading
- Haar A., Zur Theorie der orthogonalen Funktionensysteme, Mathematische Annalen, 69, pp. 331–371, 1910.
- ISBN 0-89871-274-2.
- ISBN 0-12-047140-X.
- ISBN 0-13-605718-7.
- Gerald Kaiser, A Friendly Guide to Wavelets, Birkhauser, 1994, ISBN 0-8176-3711-7.
- 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.
- ISBN 978-1-56881-072-0.
- 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.
- 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.
- Paul S. Addison, The Illustrated Wavelet Transform Handbook, ISBN 0-7503-0692-0.
- B. Boashash, editor, "Time-Frequency Signal Analysis and Processing – A Comprehensive Reference", Elsevier Science, Oxford, 2003, ISBN 0-08-044335-4.
- ISBN 0-89871-589-X(2005).
- Press, W. H.; Teukolsky, S. A.; Vetterling, W. T.; Flannery, B. P. (2007), "Section 13.10. Wavelet Transforms", Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press, ISBN 978-0-521-88068-8, archived from the originalon 2011-08-11, retrieved 2011-08-13.
- "How Wavelets Allow Researchers to Transform — and Understand — Data". Quanta Magazine. 2021-10-13. Retrieved 2021-10-20.
External links
This section's use of external links may not follow Wikipedia's policies or guidelines. (July 2016) |
- "Wavelet analysis", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
- 1st NJIT Symposium on Wavelets (April 30, 1990) (First Wavelets Conference in USA)
- Binomial-QMF Daubechies Wavelets
- Wavelets by Gilbert Strang, American Scientist 82 (1994) 250–255. (A very short and excellent introduction)
- Course on Wavelets given at UC Santa Barbara, 2004
- Wavelets for Kids (PDF file) (Introductory (for very smart kids!))
- WITS: Where Is The Starlet? A dictionary of tens of wavelets and wavelet-related terms ending in -let, from activelets to x-lets through bandlets, contourlets, curvelets, noiselets, wedgelets.
- The Fractional Spline Wavelet Transform describes a fractional wavelet transform based on fractional b-Splines.
- A Panorama on Multiscale Geometric Representations, Intertwining Spatial, Directional and Frequency Selectivity provides a tutorial on two-dimensional oriented wavelets and related geometric multiscale transforms.
- Concise Introduction to Wavelets by René Puschinger
- A Really Friendly Guide To Wavelets by Clemens Valens
- "How Wavelets Allow Researchers to Transform — and Understand — Data". Quanta Magazine. 2021-10-13. Retrieved 2021-10-20.