sábado, 11 de junio de 2022

Wavelet Definition

 https://www.wikiwand.com/en/Wavelet

Wavelet 

From Wikipedia, the free encyclopedia


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.

......

.....

.....

https://www.wikiwand.com/en/Wavelet

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.






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[11] 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

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 Zweig’s discovery of the continuous wavelet transform (CWT) in 1975 (originally called the cochlear transform and discovered while studying the reaction of the ear to sound),[12] Pierre Goupillaud, Grossmann and Morlet's formulation of what is now known as the CWT (1982), Jan-Olov Strömberg's early work on discrete wavelets (1983), the LeGall-Tabatabai (LGT) 5/3 wavelet developed by Didier Le Gall and Ali J. Tabatabai (1988),[13][14][15] Ingrid Daubechies' orthogonal wavelets with compact support (1988), Mallat's multiresolution framework (1989), Ali Akansu's Binomial QMF (1990), Nathalie Delprat's time-frequency interpretation of the CWT (1991), Newland's harmonic wavelet transform (1993), and set partitioning in hierarchical trees (SPIHT) developed by Amir Said with William A. Pearlman in 1996.[16]

The JPEG 2000 standard was developed from 1997 to 2000 by a Joint Photographic Experts Group (JPEG) committee chaired by Touradj Ebrahimi (later the JPEG president).[17] In contrast to the DCT algorithm used by the original JPEG format, JPEG 2000 instead uses discrete wavelet transform (DWT) algorithms. It uses the CDF 9/7 wavelet transform (developed by Ingrid Daubechies in 1992) for its lossy compression algorithm, and the LeGall-Tabatabai (LGT) 5/3 wavelet transform (developed by Didier Le Gall and Ali J. Tabatabai in 1988) for its lossless compression algorithm.[18] JPEG 2000 technology, which includes the Motion JPEG 2000 extension, was selected as the video coding standard for digital cinema in 2004.[19]

Timeline

Wavelet transforms

Main article: Wavelet transform

A wavelet is a mathematical function used to divide a given function or continuous-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 are scaled and translated copies (known as "daughter wavelets") of a finite-length or fast-decaying oscillating waveform (known as the "mother wavelet"). Wavelet transforms have advantages over traditional Fourier 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 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:

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, 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.[20] Now that transmission 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[21] and strain[22]/metrology[23] applications for intermediate transforms with high frequency resolution (like brushlets[24] and ridgelets[25]) 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.[26]

Applications

Generally, an approximation to DWT is used for data compression if a signal is already sampled, and the CWT for signal analysis.[27] Thus, DWT approximation is commonly used in engineering and computer science, and the CWT in scientific 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 biorthogonal wavelets. This means that although the frame is overcomplete, it is a tight frame (see types of frames 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 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 smoothing and/or denoising operations can be performed.

Wavelet transforms are also starting to be used for communication applications. Wavelet OFDM is the basic modulation scheme used in HD-PLC (a power line communications technology developed by Panasonic), and in one of the optional modes included in the IEEE 1901 standard. Wavelet OFDM can achieve deeper notches than traditional FFT OFDM, and wavelet OFDM does not require a guard interval (which usually represents significant overhead in FFT OFDM systems).[28]

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 Fourier transform. Many areas of physics have seen this paradigm shift, including molecular dynamics, chaos theory,[29] ab initio calculations, astrophysics, gravitational wave transient data analysis,[30][31] density-matrix localisation, seismology, optics, turbulence and quantum mechanics. This change has also occurred in image processing, EEG, EMG,[32] ECG analyses, brain rhythms, DNA analysis, protein analysis, climatology, human sexual response analysis,[33] general signal processing, speech recognition, acoustics, vibration signals,[34] computer graphics, multifractal analysis, and sparse coding. In computer vision and image processing, the notion of scale space representation and Gaussian derivative operators is regarded as a canonical multi-scale representation.

Wavelet denoising

Signal denoising by wavelet transform thresholding
Signal denoising by wavelet transform thresholding

Suppose we measure a noisy signal , where s represents the signal and v represents the noise. Assume s has a sparse representation in a certain wavelet basis, and 

Let the wavelet transform of  be . , the wavelet transform of the signal component. , the wavelet transform of the noise component.

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 . 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 [35] and nonlinear [36] 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 [37]

List of wavelets

Discrete wavelets

Continuous wavelets

Real-valued

Complex-valued

See also

References

Further reading

External links

Look up wavelet in Wiktionary, the free dictionary.
Wikimedia Commons has media related to Wavelet.
Wikiquote has quotations related to Wavelet.
Statistics
Categories
Publicado por en

No hay comentarios:

Publicar un comentario

Suscribirse a: Enviar comentarios (Atom)