Awavelet is awave-likeoscillation with anamplitude 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 forsignal processing.
For example, a wavelet could be created to have a frequency ofmiddle C and a short duration of roughly one tenth of a second. If this wavelet were to beconvolved with a signal created from the recording of a melody, then the resulting signal would be useful for determining when the middle C note appeared in the song. Mathematically, a wavelet correlates with a signal if a portion of the signal is similar.Correlation is at the core of many practical wavelet applications.
As a mathematical tool, wavelets can be used to extract information from many kinds of data, includingaudio signals and images. Sets of wavelets are needed to analyze data fully. "Complementary" wavelets decompose a signal without gaps or overlaps so that the decomposition process is mathematically reversible. Thus, sets of complementary wavelets are useful inwavelet-based compression/decompression algorithms, where it is desirable to recover the original information with minimal loss.
In formal terms, this representation is awavelet series representation of asquare-integrable function with respect to either acomplete,orthonormal set ofbasis functions, or anovercomplete set orframe of a vector space, for theHilbert space of square-integrable functions. This is accomplished throughcoherent states.
Inclassical physics, the diffraction phenomenon is described by theHuygens–Fresnel principle that treats each point in a propagatingwavefront as a collection of individual spherical wavelets.[1] The characteristic bending pattern is most pronounced when a wave from acoherent source (such as a laser) encounters a slit/aperture that is comparable in size to itswavelength. This is due to the addition, orinterference, of different points on the wavefront (or, equivalently, each wavelet) that travel by paths of different lengths to the registering surface. Multiple,closely spaced openings (e.g., adiffraction grating), can result in a complex pattern of varying intensity.
The wordwavelet has been used for decades in digital signal processing and exploration geophysics.[2] The equivalentFrench wordondelette meaning "small wave" was used byJean Morlet andAlex Grossmann in the early 1980s.
 | This articleneeds additional citations forverification. Please helpimprove this article byadding citations to reliable sources. Unsourced material may be challenged and removed. Find sources: "Wavelet" – news ·newspapers ·books ·scholar ·JSTOR(May 2025) (Learn how and when to remove this message) |
Wavelet theory is applicable to several subjects. All wavelet transforms may be considered forms oftime-frequency representation forcontinuous-time (analog) signals and so are related toharmonic analysis.[3] Discrete wavelet transform (continuous in time) of adiscrete-time (sampled) signal by usingdiscrete-timefilterbanks of dyadic (octave band) configuration is a wavelet approximation to that signal. The coefficients of such a filter bank are called the shift and scaling coefficients in wavelets nomenclature. These filterbanks may contain eitherfinite impulse response (FIR) orinfinite impulse response (IIR) filters. The wavelets forming acontinuous wavelet transform (CWT) are subject to theuncertainty principle of Fourier analysis respective sampling theory:[4] given a signal with some event in it, one cannot assign simultaneously an exact time and frequency response scale to that event. The product of the uncertainties of time and frequency response scale has a lower bound. Thus, in thescaleogram of a continuous wavelet transform of this signal, such an event marks an entire region in the time-scale plane, instead of just one point. Also, discrete wavelet bases may be considered in the context of other forms of the uncertainty principle.[5][6][7][8]
Wavelet transforms are broadly divided into three classes: continuous, discrete and multiresolution-based.
Incontinuous wavelet transforms, a given signal of finite energy is projected on a continuous family of frequency bands (or similar subspaces of theLpfunction spaceL2(R) ). For instance the signal may be represented on every frequency band of the form [f, 2f] for all positive frequenciesf > 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 inL2(R), themother wavelet. For the example of the scale one frequency band [1, 2] this function iswith the (normalized)sinc function. That, Meyer's, and two other examples of mother wavelets are:
 |
 |
 |
The subspace of scalea or frequency band [1/a, 2/a] is generated by the functions (sometimes calledchild wavelets)wherea is positive and defines the scale andb is any real number and defines the shift. The pair (a,b) defines a point in the right halfplaneR+ ×R.
The projection of a functionx onto the subspace of scalea then has the formwithwavelet coefficients
For the analysis of the signalx, one can assemble the wavelet coefficients into ascaleogram of the signal.
See a list of someContinuous wavelets.
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 theaffine system for some real parametersa > 1,b > 0. The corresponding discrete subset of the halfplane consists of all the points (am,nb am) withm,n inZ. The correspondingchild wavelets are now given as
A sufficient condition for the reconstruction of any signalx of finite energy by the formulais that the functions form anorthonormal basis ofL2(R).
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 amultiresolution analysis. This means that there has to exist anauxiliary function, thefather wavelet φ inL2(R), and thata is an integer. A typical choice isa = 2 andb = 1. The most famous pair of father and mother wavelets is theDaubechies 4-tap wavelet. Note that not every orthonormal discrete wavelet basis can be associated to a multiresolution analysis; for example, the Journe wavelet admits no multiresolution analysis.[9]
From the mother and father wavelets one constructs the subspacesThe father wavelet keeps the time domain properties, while the mother wavelets keeps the frequency domain properties.
From these it is required that the sequenceforms amultiresolution analysis ofL2 and that the subspaces are the orthogonal "differences" of the above sequence, that is,Wm is the orthogonal complement ofVm inside the subspaceVm−1,
In analogy to thesampling theorem one may conclude that the spaceVm with sampling distance 2m more or less covers the frequency baseband from 0 to 1/2m-1. As orthogonal complement,Wm roughly covers the band [1/2m−1, 1/2m].
From those inclusions and orthogonality relations, especially, follows the existence of sequences and that satisfy the identities so that and so thatThe second identity of the first pair is arefinement equation for the father wavelet φ. Both pairs of identities form the basis for the algorithm of thefast wavelet transform.
From the multiresolution analysis derives the orthogonal decomposition of the spaceL2 asFor any signal or function this gives a representation in basis functions of the corresponding subspaces aswhere the coefficients are and
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[10] and Lindeberg,[11] with the latter method also involving a memory-efficient time-recursive implementation.
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 thespace This is the space ofLebesgue measurable functions that are bothabsolutely integrable andsquare integrable in the sense that 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.
Forψ to be a wavelet for thecontinuous 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 thediscrete wavelet transform, one needs at least the condition that thewavelet series is a representation of the identity in thespaceL2(R). Most constructions of discrete WT make use of themultiresolution analysis, which defines the wavelet by a scaling function. This scaling function itself is a solution to a functional equation.
In most situations it is useful to restrict ψ to be a continuous function with a higher numberM of vanishing moments, i.e. for all integerm <M
The mother wavelet is scaled (or dilated) by a factor ofa and translated (or shifted) by a factor ofb to give (under Morlet's original formulation):
For the continuous WT, the pair (a,b) varies over the full half-planeR+ ×R; for the discrete WT this pair varies over a discrete subset of it, which is also calledaffine 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：
The wavelet transform is often compared with theFourier transform, in which signals are represented as a sum of sinusoids. In fact, theFourier 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 standardFourier transform is only localized infrequency. Theshort-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 kernelwhere can often be written as, where andu respectively denote the length and temporal offset of the windowing function. UsingParseval's theorem, one may define the wavelet's energy asFrom this, the square of the temporal support of the window offset by timeu is given by
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 spuriousringing 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 otherapodizing filter, such as aGaussian. 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 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'smultiresolutional 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.[12]
The discrete wavelet transform is less computationallycomplex, takingO(N) time as compared to O(N log N) for thefast Fourier transform (FFT). This computational advantage is not inherent to the transform, but reflects the choice of a logarithmic division of frequency, in contrast to the equally spaced frequency divisions of the FFT which uses the same basis functions as the discrete Fourier transform (DFT).[13] This complexity only applies when the filter size has no relation to the signal size. A wavelet withoutcompact support such as theShannon wavelet would require O(N2). (For instance, a logarithmic Fourier Transform also exists with O(N) complexity, but the original signal must be sampled logarithmically in time, which is only useful for certain types of signals.[14])
A wavelet (or a wavelet family) can be defined in various ways:
An orthogonal wavelet is entirely defined by the scaling filter – a low-passfinite impulse response (FIR) filter of length 2N and sum 1. Inbiorthogonal wavelets, separate decomposition and reconstruction filters are defined.
For analysis with orthogonal wavelets the high pass filter is calculated as thequadrature 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.
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[15] for a detailed explanation.
For a wavelet with compact support, φ(t) can be considered finite in length and is equivalent to the scaling filterg.
Meyer wavelets can be defined by scaling functions
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 fewcontinuous wavelets.
The development of wavelets can be linked to several separate trains of thought, starting withAlfréd Haar's work in the early 20th century. Later work byDennis Gabor yieldedGabor atoms (1946), which are constructed similarly to wavelets, and applied to similar purposes.
Notable contributions to wavelet theory since then can be attributed toGeorge Zweig's discovery of thecontinuous wavelet transform (CWT) in 1975 (originally called the cochlear transform and discovered while studying the reaction of the ear to sound),[16] Pierre Goupillaud,Alex Grossmann andJean Morlet's formulation of what is now known as the CWT (1982), Jan-Olov Strömberg's early work ondiscrete wavelets (1983), the Le Gall–Tabatabai (LGT) 5/3-taps non-orthogonal filter bank with linear phase (1988),[17][18][19]Ingrid Daubechies' orthogonal wavelets with compact support (1988),Stéphane Mallat's non-orthogonal multiresolution framework (1989),Ali Akansu'sbinomial QMF (1990), Nathalie Delprat's time-frequency interpretation of the CWT (1991), Newland'sharmonic wavelet transform (1993), andset partitioning in hierarchical trees (SPIHT) developed by Amir Said with William A. Pearlman in 1996.[20]
TheJPEG 2000 standard was developed from 1997 to 2000 by aJoint Photographic Experts Group (JPEG) committee chaired by Touradj Ebrahimi (later the JPEG president).[21] In contrast to the DCT algorithm used by the originalJPEG format, JPEG 2000 instead usesdiscrete wavelet transform (DWT) algorithms. It uses theCDF 9/7 wavelet transform (developed by Ingrid Daubechies in 1992) for itslossy compression algorithm, and the Le Gall–Tabatabai (LGT) 5/3 discrete-time filter bank (developed by Didier Le Gall and Ali J. Tabatabai in 1988) for itslossless compression algorithm.[22]JPEG 2000 technology, which includes theMotion JPEG 2000 extension, was selected as thevideo coding standard fordigital cinema in 2004.[23]
A wavelet is a mathematical function used to divide a given function orcontinuous-time signal into different scale components. Usually one can assign a frequency range to each scale component. Each scale component can then be studied with a resolution that matches its scale. A wavelet transform is the representation of a function by wavelets. The wavelets arescaled andtranslated copies (known as "daughter wavelets") of a finite-length or fast-decaying oscillating waveform (known as the "mother wavelet"). Wavelet transforms have advantages over traditionalFourier transforms for representing functions that have discontinuities and sharp peaks, and for accurately deconstructing and reconstructing finite, non-periodic and/or non-stationary signals.
Wavelet transforms are classified intodiscrete wavelet transforms (DWTs) andcontinuous 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 seelist of wavelet-related transforms but the common ones are listed below:
There are a number of generalized transforms of which the wavelet transform is a special case. For example, Yosef Joseph Segman introduced scale into theHeisenberg 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 thechirplet 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,darkfield electron optical transforms intermediate between direct andreciprocal space have been widely used in theharmonic analysis of atom clustering, i.e. in the study ofcrystals andcrystal defects.[24] Now thattransmission electron microscopes are capable of providing digital images with picometer-scale information on atomic periodicity innanostructure of all sorts, the range ofpattern recognition[25] andstrain[26]/metrology[27] applications for intermediate transforms with high frequency resolution (like brushlets[28] and ridgelets[29]) is growing rapidly.
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.[30]
Generally, an approximation to DWT is used fordata compression if a signal is already sampled, and the CWT forsignal analysis.[31][32] Thus, DWT approximation is commonly used in engineering and computer science,[33] and the CWT in scientific research.[34]
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 biorthogonal wavelets. This means that although the frame is overcomplete, it is atight frame (see types offrames of a vector space), and the same frame functions (except for conjugation in the case of complex wavelets) are used for both analysis and synthesis, i.e., in both the forward and inverse transform. For details seewavelet 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 smoothing and/or denoising operations can be performed.
Wavelet transforms are also starting to be used for communication applications. WaveletOFDM is the basic modulation scheme used inHD-PLC (apower line communications technology developed byPanasonic), and in one of the optional modes included in theIEEE 1901 standard. Wavelet OFDM can achieve deeper notches than traditionalFFT OFDM, and wavelet OFDM does not require a guard interval (which usually represents significant overhead in FFT OFDM systems).[35]
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 asGibbs 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 ofcompressed sensing. (Note that theshort-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 ofmultiresolution analysis.)
This motivates why wavelet transforms are now being adopted for a vast number of applications, often replacing the conventionalFourier transform. Many areas of physics have seen this paradigm shift, includingmolecular dynamics,chaos theory,[36]ab initio calculations,astrophysics,gravitational wave transient data analysis,[37][38]density-matrix localisation,seismology,optics,turbulence andquantum mechanics. This change has also occurred inimage processing,EEG,EMG,[39]ECG analyses,brain rhythms,DNA analysis,protein analysis,climatology, human sexual response analysis,[40] generalsignal processing,speech recognition, acoustics, vibration signals,[41]computer graphics,multifractal analysis, andsparse coding. Incomputer vision andimage processing, the notion ofscale space representation and Gaussian derivative operators is regarded as a canonical multi-scale representation.
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 iidGaussian 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
Agarwal et al. proposed wavelet based advanced linear[42] and nonlinear[43] methods to construct and investigateClimate as complex networks at different timescales. Climate networks constructed usingSST 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[44]
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.
