Wavelet

A wavelet is a kind of mathematical function used to divide a given function or continuous-time signal into different frequency components and study each component with a resolution that matches its scale. A continuous signal or a continuous-time signal is a varying quantity (a signal) that is expressed as a function of a real-valued domain usually time 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"). In Euclidean geometry, uniform scaling or Isotropic scaling is a Linear transformation that enlarges or diminishes objects the Scale factor In Euclidean geometry, a translation is moving every point a constant distance in a specified direction 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. This article specifically discusses Fourier transformation of functions on the Real line; for other kinds of Fourier transformation see Fourier analysis and In Mathematics, a periodic function is a function that repeats its values after some definite period has been added to its Independent variable In the mathematical sciences, a stationary process (or strict(ly stationary process or strong(ly stationary process) is a Stochastic process

In formal terms, this representation is a wavelet series representation of a square-integrable function with respect to either a complete, orthonormal set of basis functions, or an overcomplete set of Frame of a vector space (also known as a Riesz basis), for the Hilbert space of square integrable functions. In Mathematics, a wavelet series is a representation of a Square-integrable ( real - or complex -valued function by a certain Orthonormal In Mathematics, an integrable function is a function whose Integral exists In Mathematical analysis, a Metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has In Linear algebra, two vectors in an Inner product space are orthonormal if they are orthogonal and both of unit length In Mathematics, particularly Numerical analysis, a basis function is an element of the basis for a Function space. In general an object is complete if nothing needs to be added to it In Mathematics, a frame of a Vector space can mean any Ordered basis for that vector space In Mathematics, a Sequence of vectors ( x n) in a Hilbert space ( H, &lang &rang is called a This article assumes some familiarity with Analytic geometry and the concept of a limit.

Wavelet transforms are classified into discrete wavelet transforms (DWTs) and continuous wavelet transforms (CWTs). In Numerical analysis and Functional analysis, a discrete wavelet transform (DWT is any Wavelet transform for which the Wavelets are discretely A continuous wavelet transform is used to divide a continuous-time function into wavelets Note that both DWT and CWT are of 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.

The word wavelet is due to Morlet and Grossmann in the early 1980s. Jean Morlet ( January 13 1931 - April 27 2007) is a French Geophysicist who pioneering work in the field of Wavelet Alexander Grossmann (Croatian Aleksandar Grossman) (born 5 August 1930 is a Croatian French Physicist at the Université de la Méditerranée The 1980s was the decade spanning from January 1 1980 to December 31 1989. They used the French word ondelette, meaning "small wave". French ( français,) is a Romance language spoken around the world by 118 million people as a native language and by about 180 to 260 million people Soon it was transferred to English by translating "onde" into "wave", giving "wavelet".

Wavelet theory

Wavelet theory is applicable to several subjects. All wavelet transforms may be considered forms of time-frequency representation for continuous-time (analog) signals and so are related to harmonic analysis. A time-frequency representation ( TFR) is a view of a signal (taken to be a function of time represented over both time and Frequency. A continuous signal or a continuous-time signal is a varying quantity (a signal) that is expressed as a function of a real-valued domain usually time Harmonic analysis is the branch of Mathematics that studies the representation of functions or signals as the superposition of basic Waves It investigates and generalizes Almost all practically useful discrete wavelet transforms use discrete-time filterbanks. A discrete signal or discrete-time signal is a Time series, perhaps a signal that has been sampled from a continuous-time signal. A filter bank is an array of band-pass filters that separates the input signal into several components each one carrying a single Frequency Subband These filter banks are called the wavelet and scaling coefficients in wavelets nomenclature. These filterbanks may contain either finite impulse response (FIR) or infinite impulse response (IIR) filters. A finite impulse response (FIR filter is a type of a Digital filter. IIR redirects here For the conference company IIR see Informa. The wavelets forming a CWT are subject to the uncertainty principle of Fourier analysis respective sampling theory: Given a signal with some event in it, one cannot assign simultaneously an exact time and frequency resp. In Quantum physics, the Heisenberg uncertainty principle states that locating a particle in a small region of space makes the Momentum of the particle uncertain scale to that event. The product of the uncertainties of time and frequency resp. scale has a lower bound. Thus, in the scaleogram 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. This is related to Heisenberg's uncertainty principle of quantum physics and has a similar derivation. Werner Heisenberg (5 December 1901 in Würzburg &ndash1 February 1976 in Munich) was a German theoretical physicist best known for enunciating the Also, discrete wavelet bases may be considered in the context of other forms of the uncertainty principle.

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

Continuous wavelet transforms (Continuous Shift & 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 L^p function space $L^2(\R)$ ). A continuous wavelet transform is used to divide a continuous-time function into wavelets In Mathematics, the Lp and ℓp spaces are spaces of p-power integrable functions, and corresponding In Mathematics, a function space is a set of functions of a given kind from a set X to a set Y. For instance the signal may be represented on every frequency band of the form $[f,2 f]$ 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 $\psi \in L^2(\R)$ , the mother wavelet. For the example of the scale one frequency band $[1,2]$ this function is

$\psi(t)=2\,\operatorname{sinc}(2t)-\,\operatorname{sinc}(t)=\frac{\sin(2\pi t)-\sin(\pi t)}{\pi t}$

with the (normalized) sinc function. In Mathematics, the sinc function, denoted by \scriptstyle\mathrm{sinc}(x\ and sometimes as \scriptstyle\mathrm{Sa}(x\ has two definitions sometimes Other example mother wavelets are:

Meyer

Morlet

Mexican Hat

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

$\psi_{a,b} (t) = \frac1{\sqrt a }\psi \left( \frac{t - b}{a} \right)$ ,

where a is positive and defines the scale and b is any real number and defines the shift. The pair (a,b) defines a point in the right halfplane $\R_+\times \R$ .

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

$x_a(t)=\int_\R WT_\psi\{x\}(a,b)\cdot\psi_{a,b}(t)\,db$

with wavelet coefficients

$WT_\psi\{x\}(a,b)=\langle x,\psi_{a,b}\rangle=\int_\R x(t)\overline{\psi_{a,b}(t)}\,dt$ .

See a list of some Continuous wavelets. In Numerical analysis, continuous Wavelets are functions used by the Continuous wavelet transform.

For the analysis of the signal x, one can assemble the wavelet coefficients into a scaleogram of the signal. In Signal processing, a scaleogram is a visual method of displaying a Wavelet transform.

Discrete wavelet transforms (Discrete Shift & Scale parameters)

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 $(a^m, n\,a^m b)$ with integers $m,n\in\Z$ . The corresponding baby wavelets are now given as

ψ m, n (t) = a - m / 2 ψ(a - m t - n b)

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

$x(t)=\sum_{m\in\Z}\sum_{n\in\Z}\langle x,\,\psi_{m,n}\rangle\cdot\psi_{m,n}(t)$

is that the functions $\{\psi_{m,n}:m,n\in\Z\}$ form a tight frame of $L^2(\R)$ . In Mathematics, a frame of a Vector space can mean any Ordered basis for that vector space

Multiresolution-based discrete wavelet transforms

D4 wavelet

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. To avoid this numerical complexity, one needs one auxiliary function, the father wavelet $\phi\in L^2(\R)$ . Further, one has to restrict a to be an integer. A typical choice is a=2 and b=1. The most famous pair of father and mother wavelets is the Daubechies 4 tap wavelet. Named after Ingrid Daubechies, the Daubechies wavelets are a family of Orthogonal wavelets defining a Discrete wavelet transform and characterized by a maximal

From the mother and father wavelets one constructs the subspaces

$V_m=\operatorname{span}(\phi_{m,n}:n\in\Z)$ , where

φ m, n (t) = 2 - m / 2 φ(2 - m t - n)

and

$W_m=\operatorname{span}(\psi_{m,n}:n\in\Z)$ , where

ψ m, n (t) = 2 - m / 2 ψ(2 - m t - n)

From these one requires that the sequence

$\{0\}\subset\dots\subset V_1\subset V_0\subset V_{-1}\subset\dots\subset L^2(\R)$

forms a multiresolution analysis of $L^2(\R)$ and that the subspaces $\dots,W_1,W_0,W_{-1},\dots\dots$ are the orthogonal "differences" of the above sequence, that is, $W m$ is the orthogonal complement of $V m$ inside the subspace $V m - 1$ . A multiresolution analysis (MRA or multiscale approximation (MSA is the design method of most of the practically relevant Discrete wavelet transforms (DWT and the In analogy to the sampling theorem one may conclude that the space $V m$ with sampling distance $2 m$ more or less covers the frequency baseband from 0 to $2 - m - 1$ . The Nyquist–Shannon sampling theorem is a fundamental result in the field of Information theory, in particular Telecommunications and Signal processing As orthogonal complement, $W m$ roughly covers the band $[2 - m - 1,2 - m]$ .

From those inclusions and orthogonality relations follows the existence of sequences $h=\{h_n\}_{n\in\Z}$ and $g=\{g_n\}_{n\in\Z}$ that satisfy the identities

$h_n=\langle\phi_{0,0},\,\phi_{1,n}\rangle$ and $\phi(t)=\sqrt2 \sum_{n\in\Z} h_n\phi(2t-n)$

and

$g_n=\langle\psi_{0,0},\,\phi_{1,n}\rangle$ and $\psi(t)=\sqrt2 \sum_{n\in\Z} g_n\phi(2t-n)$ .

The second identity of the first pair is a refinement equation for the father wavelet $φ$ . In Mathematics, in the area of Wavelet analysis a refinable function is a function which fulfills some kind of self-similarity Both pairs of identities form the basis for the algorithm of the fast wavelet transform. The Fast Wavelet Transform is a mathematical Algorithm designed to turn a Waveform or signal in the Time domain into a Sequence of

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 $L^1(\R)\cap L^2(\R)$ . In Mathematics, the Lp and ℓp spaces are spaces of p-power integrable functions, and corresponding This is the space of measurable functions that are absolutely and square integrable:

$\int_{-\infty}^{\infty} |\psi (t)|\, dt <\infty$ and $\int_{-\infty}^{\infty} |\psi (t)|^2 \, dt <\infty$ . In Mathematics, the Integral of a non-negative function can be regarded in the simplest case as the Area between the graph of In Mathematics, an integrable function is a function whose Integral exists

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

$\int_{-\infty}^{\infty} \psi (t)\, dt = 0$ is the condition for zero mean, and

$\int_{-\infty}^{\infty} |\psi (t)|^2\, dt = 1$ is the condition for square norm one.

For $ψ$ to be a wavelet for the continuous wavelet transform (see there for exact statement), the mother wavelet must satisfy an admissibility criterion (loosely speaking, a kind of half-differentiability) in order to get a stably invertible transform. A continuous wavelet transform is used to divide a continuous-time function into wavelets

For the discrete wavelet transform, one needs at least the condition that the wavelet series is a representation of the identity in the space $L^2(\R)$ . In Numerical analysis and Functional analysis, a discrete wavelet transform (DWT is any Wavelet transform for which the Wavelets are discretely In Mathematics, a wavelet series is a representation of a Square-integrable ( real - or complex -valued function by a certain Orthonormal In Mathematics, the Lp and ℓp spaces are spaces of p-power integrable functions, and corresponding Most constructions of discrete WT make use of the multiresolution analysis, which defines the wavelet by a scaling function. A multiresolution analysis (MRA or multiscale approximation (MSA is the design method of most of the practically relevant Discrete wavelet transforms (DWT and the 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

$\int_{-\infty}^{\infty} t^m\,\psi (t)\, dt = 0$

Some example mother wavelets are:

Meyer

Morlet

Mexican Hat

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

$\psi _{a,b} (t) = {1 \over {\sqrt a }}\psi \left( {{{t - b} \over a}} \right)$ .

For the continuous WT, the pair (a,b) varies over the full half-plane $\R_+\times\R$ ; 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).

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. This article specifically discusses Fourier transformation of functions on the Real line; for other kinds of Fourier transformation see Fourier analysis and The main difference is that wavelets are localized in both time and frequency whereas the standard Fourier transform is only localized in frequency. This article specifically discusses Fourier transformation of functions on the Real line; for other kinds of Fourier transformation see Fourier analysis and Frequency is a measure of the number of occurrences of a repeating event per unit Time. The Short-time Fourier transform (STFT) is also time and frequency localized but there are issues with the frequency time resolution and wavelets often give a better signal representation using Multiresolution analysis. The short-time Fourier transform ( STFT) or alternatively short-term Fourier transform, is a Fourier-related transform used to determine the sinusoidal A multiresolution analysis (MRA or multiscale approximation (MSA is the design method of most of the practically relevant Discrete wavelet transforms (DWT and the

The discrete wavelet transform is also less computationally complex, taking O(N) time as compared to O(N log N) for the fast Fourier transform. In general usage complexity often tends to be used to characterize something with many parts in intricate arrangement In mathematics big O notation (so called because it uses the symbol O) describes the limiting behavior of a function for very small or very large arguments 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.

Definition of a wavelet

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

Scaling filter

The wavelet is entirely defined by the scaling filter - a low-pass finite impulse response (FIR) filter of length 2N and sum 1. A finite impulse response (FIR filter is a type of a Digital filter. In biorthogonal wavelets, separate decomposition and reconstruction filters are defined.

For analysis the high pass filter is calculated as the quadrature mirror filter of the low pass, and reconstruction filters the time reverse of the decomposition. In Digital signal processing, a quadrature mirror filter is a filter most commonly used to implement a Filter bank that splits an input signal into two

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 it 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 [1] 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. In Mathematics and Numerical analysis, the Mexican hat wavelet \psi(t = {1 \over {\sqrt {2\pi}\sigma^3}} \left( 1 - {t^2 \over \sigma^2} \right See a list of a few Continuous wavelets. In Numerical analysis, continuous Wavelets are functions used by the Continuous wavelet transform.

Applications of Discrete Wavelet Transform

Generally, an approximation to DWT is used for data compression if signal is already sampled, and the CWT for signal analysis. Signal processing is the analysis interpretation and manipulation of signals Signals of interest include sound, images, biological signals such as Thus, DWT approximation is commonly used in engineering and computer science, and the CWT in scientific research.

Wavelet transforms are now being adopted for a vast number of applications, often replacing the conventional Fourier Transform. This article specifically discusses Fourier transformation of functions on the Real line; for other kinds of Fourier transformation see Fourier analysis and Many areas of physics have seen this paradigm shift, including molecular dynamics, ab initio calculations, astrophysics, density-matrix localisation, seismic geophysics, optics, turbulence and quantum mechanics. Molecular dynamics ( MD) is a form of Computer simulation in which atoms and molecules are allowed to interact for a period of time by approximations of Ab Initio Software Corporation was founded in the mid 1990's by the former CEO of Thinking Machines Corporation Sheryl Handler, and several other former employees Astrophysics is the branch of Astronomy that deals with the Physics of the Universe, including the physical properties ( Luminosity, In Fluid dynamics, turbulence or turbulent flow is a fluid regime characterized by chaotic Stochastic property changes Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons This change has also occurred in image processing, blood-pressure, heart-rate and ECG analyses, DNA analysis, protein analysis, climatology, general signal processing, speech recognition, computer graphics and multifractal analysis. Image processing is any form of Signal processing for which the input is an image such as photographs or frames of video the output of image processing can be either an image Deoxyribonucleic acid ( DNA) is a Nucleic acid that contains the genetic instructions used in the development and functioning of all known Proteins are large Organic compounds made of Amino acids arranged in a linear chain and joined together by Peptide bonds between the Carboxyl Climatology (from Greek grc κλίμα klima, "region zone" and grc -λογία -logia) is the study of Climate, scientifically Signal processing is the analysis interpretation and manipulation of signals Signals of interest include sound, images, biological signals such as Speech recognition (also known as automatic speech recognition or computer speech recognition) converts spoken words to machine-readable input (for example to keypresses Computer graphics are Graphics created by Computers and more generally the Representation and Manipulation of Pictorial Data A multifractal system is a generalization of a Fractal system in which a single exponent (the Fractal dimension) is not enough to describe its dynamics instead a In computer vision and image processing, the notion of scale-space representation and Gaussian derivative operators is regarded as a canonical multi-scale representation. Computer vision is the science and technology of machines that see Image processing is any form of Signal processing for which the input is an image such as photographs or frames of video the output of image processing can be either an image Scale-space theory is a framework for multi-scale signal representation developed by the Computer vision, Image processing and

One use of wavelet approximation is in data compression. 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. JPEG 2000 is a Wavelet -based Image compression standard It was created by the Joint Photographic Experts Group committee in the year 2000 with the intention This means that although the frame is overcomplete, it is a tight frame (see types of Frame 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. In Mathematics, a frame of a Vector space can mean any Ordered basis for that vector space e. , in both the forward and inverse transform. For details see wavelet compression. Wavelet compression is a form of Data compression well suited for Image compression (sometimes also Video compression and Audio compression)

A related use is that of 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.

History

The development of wavelets can be linked to several separate trains of thought, starting with Haar's work in the early 20th century. Alfréd Haar ( October 11, 1885 - March 16, 1933) was a Hungarian mathematician. Notable contributions to wavelet theory can be attributed to Zweig’s discovery of the continuous wavelet transform in 1975 (originally called the cochlear transform and discovered while studying the reaction of the ear to sound)^[1], 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), Daubechies' orthogonal wavelets with compact support (1988), Mallat's multiresolution framework (1989), Nathalie Delprat's time-frequency interpretation of the CWT (1991), Newland's Harmonic wavelet transform (1993) and many others since. George Zweig (born 1937 in Moscow Russia into a Jewish family was originally trained as a Particle physicist under Richard Feynman and later turned his attention Alexander Grossmann (Croatian Aleksandar Grossman) (born 5 August 1930 is a Croatian French Physicist at the Université de la Méditerranée Jean Morlet ( January 13 1931 - April 27 2007) is a French Geophysicist who pioneering work in the field of Wavelet Ingrid Daubechies (born August 17, 1954) (approximate pronunciation "Dobe-uh-shee" is a Belgian Physicist and Mathematician Stéphane G Mallat made some fundamental contributions to the development of Wavelet theory in the late 1980s and early 1990s In the Mathematics of Signal processing, the harmonic wavelet transform, introduced by David Edward Newland in 1993 is a Wavelet -based linear

Timeline

First wavelet (Haar wavelet) by Alfred Haar (1909)
Since the 1950s: George Zweig, Jean Morlet, Alex Grossmann
Since the 1980s: Yves Meyer, Stéphane Mallat, Ingrid Daubechies, Ronald Coifman, Victor Wickerhauser,

Wavelet Transforms

There are a large number of wavelet transforms each suitable for different applications. The Haar wavelet is the first known Wavelet and was proposed in 1909 by Alfréd Haar. Alfréd Haar ( October 11, 1885 - March 16, 1933) was a Hungarian mathematician. George Zweig (born 1937 in Moscow Russia into a Jewish family was originally trained as a Particle physicist under Richard Feynman and later turned his attention Jean Morlet ( January 13 1931 - April 27 2007) is a French Geophysicist who pioneering work in the field of Wavelet Alexander Grossmann (Croatian Aleksandar Grossman) (born 5 August 1930 is a Croatian French Physicist at the Université de la Méditerranée Yves F Meyer (b July 19, 1939) is a French Mathematician and Scientist and a foremost expert on Wavelets He is the author Stéphane G Mallat made some fundamental contributions to the development of Wavelet theory in the late 1980s and early 1990s Ingrid Daubechies (born August 17, 1954) (approximate pronunciation "Dobe-uh-shee" is a Belgian Physicist and Mathematician Ronald Coifman is the Phillips Professor of Mathematics at Yale University. Mladen Victor Wickerhauser, born in Zagreb, Croatia, in 1959. 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. A list of Wavelet related transforms Continuous wavelet transform (CWT Multiresolution analysis (MRA Discrete wavelet A continuous wavelet transform is used to divide a continuous-time function into wavelets In Numerical analysis and Functional analysis, a discrete wavelet transform (DWT is any Wavelet transform for which the Wavelets are discretely The Fast Wavelet Transform is a mathematical Algorithm designed to turn a Waveform or signal in the Time domain into a Sequence of The lifting scheme is a technique for both designing Wavelets and performing the Discrete wavelet transform. Wavelet packet decomposition (WPD (sometimes known as just wavelet packets) is a Wavelet transform where the signal is passed through more filters than the DWT The Stationary wavelet transform (SWT is similar to the DWT except the signal is never subsampled and instead the filters are upsampled at each level of decomposition For example, Joseph Segman introduced scale into the Heisenberg group, giving rise to a continuous transform space that is a function of time, scale, and frequency. In Mathematics, the term Heisenberg group, named after Werner Heisenberg, refers to the group of 3×3 upper triangular matrices of the 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. In Signal processing, the chirplet transform is an Inner product of an input signal with a family of analysis primitives called chirplets.

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. Dark field microscopy (dark ground microscopy describes microscopy methods in both light and electron microscopy which exclude the unscattered beam from the image In Crystallography, the reciprocal lattice of a Bravais lattice is the set of all vectors K such that e^{i\mathbf{K}\cdot\mathbf{R}}=1 Harmonic analysis is the branch of Mathematics that studies the representation of functions or signals as the superposition of basic Waves It investigates and generalizes e. in the study of crystals and crystal defects^[2]. In Materials science, a crystal is a Solid in which the constituent Atoms Molecules or Ions are packed in a regularly ordered repeating Crystalline solids have a very regular atomic structure that is the local positions of atoms with respect to each other are repeated at the atomic scale 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^[3] and strain^[4]/metrology^[5] applications for intermediate transforms with high frequency resolution (like brushlets^[6] and ridgelets^[7]) is growing rapidly. A nanostructure is an object of intermediate size between Molecular and Microscopic ( micrometer -sized Structures In describing nanostructures Pattern recognition is a sub-topic of Machine learning. It is "the act of taking in raw data and taking an action based on the category of the data" Metrology (from Ancient Greek metron (measure and logos (study of is the Science of Measurement.

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)
Binomial-QMF
Haar wavelet
Mathieu wavelet
Legendre wavelet
Villasenor wavelet
Symlet

Continuous wavelets

Real valued

Complex valued

References

Paul S. Addison, The Illustrated Wavelet Transform Handbook, Institute of Physics, 2002, ISBN 0-7503-0692-0
Ingrid Daubechies, Ten Lectures on Wavelets, Society for Industrial and Applied Mathematics, 1992, ISBN 0-89871-274-2
A. The Institute of Physics (IOP is the UK and Ireland 's main professional body for Physicists It was founded as the Physical Society Ingrid Daubechies (born August 17, 1954) (approximate pronunciation "Dobe-uh-shee" is a Belgian Physicist and Mathematician N. Akansu and R. A. Haddad, Multiresolution Signal Decomposition: Transforms, Subbands, Wavelets, Academic Press, 1992, ISBN 0-12-047140-X
P. P. Vaidyanathan, Multirate Systems and Filter Banks, Prentice Hall, 1993, ISBN 0-13-605718-7
Mladen Victor Wickerhauser, Adapted Wavelet Analysis From Theory to Software, A K Peters Ltd, 1994, ISBN 1-56881-041-5
Gerald Kaiser, A Friendly Guide to Wavelets, Birkhauser, 1994, ISBN 0-8176-3711-7
Haar A. , Zur Theorie der orthogonalen Funktionensysteme, Mathematische Annalen, 69, pp 331-371, 1910.
Ramazan Gençay, Faruk Selçuk and Brandon Whitcher, An Introduction to Wavelets and Other Filtering Methods in Finance and Economics, Academic Press, 2001, ISBN 0-12-279670-5
Donald B. Percival and Andrew T. Walden, Wavelet Methods for Time Series Analysis, Cambridge University Press, 2000, ISBN 0-5216-8508-7
Tony F. Chan and Jackie (Jianhong) Shen, Image Processing and Analysis - Variational, PDE, Wavelet, and Stochastic Methods, Society of Applied Mathematics, ISBN 089871589X (2005)
Stéphane Mallat, "A wavelet tour of signal processing" 2nd Edition, Academic Press, 1999, ISBN 0-12-466606-x
Barbara Burke Hubbard, "The World According to Wavelets: The Story of a Mathematical Technique in the Making", AK Peters Ltd, 1998, ISBN 1568810725, ISBN-13 978-1568810720

Footnotes

^ http://scienceworld.wolfram.com/biography/Zweig.html Zweig, George Biography on Scienceworld. wolfram. com
^ 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
^ 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)
^ M. J. Hÿtch, E. Snoeck and R. Kilaas (1998) Quantitative measurement of displacement and strain fields from HRTEM micrographs, Ultramicroscopy 74:131-146.
^ 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).

External links

Wavelet Digest
NASA Signal Processor featuring Wavelet methods Description of NASA Signal & Image Processing Software and Link to Download
1st NJIT Symposium on Wavelets (April 30, 1990) (First Wavelets Conference in USA)
Binomial-QMF Daubechies Wavelets
Wavelets made Simple
Course on Wavelets given at UC Santa Barbara, 2004
Wavelet Posting Board
The Wavelet Tutorial by Polikar (Easy to understand when you have some background with fourier transforms!)
OpenSource Wavelet C++ Code
An Introduction to Wavelets
Wavelets for Kids (PDF file) (Introductory (for very smart kids!))
Link collection about wavelets
Wavelet forums (French) Wavelet forum (English)
Gerald Kaiser's acoustic and electromagnetic wavelets
A really friendly guide to wavelets
Wavelet-based image annotation and retrieval
Very basic explanation of Wavelets and how FFT relates to it

Dictionary

-noun

A small wave; a ripple.
(mathematics) A fast-decaying oscillation.

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