# English

## Noun

- a small wave; a ripple
- A fast-decaying oscillation

## Derived terms

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 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.

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.

Wavelet transforms are classified into discrete
wavelet transforms (DWTs) and
continuous wavelet transforms (CWTs). 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. They used the
French
word ondelette, meaning "small wave". 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. Almost all practically useful discrete wavelet
transforms use discrete-time
filterbanks. 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. 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. 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. 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 Lp function space L^2(\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 \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(2t)-\,\operatorname(t)=\frac

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

- \psi_ (t) = \frac1\psi \left( \frac \right),

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

- x_a(t)=\int_\R WT_\psi\(a,b)\cdot\psi_(t)\,db

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

See a list of some Continuous
wavelets.

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

## Discrete wavelet transforms (Discrete Shift & 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- \psi_(t)=a^\psi(a^t-nb).

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

- x(t)=\sum_\sum_\langle x,\,\psi_\rangle\cdot\psi_(t)

## Multiresolution-based discrete wavelet transforms

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.

From the mother and father wavelets one
constructs the subspaces

- V_m=\operatorname(\phi_:n\in\Z), where \phi_(t)=2^\phi(2^t-n)

- W_m=\operatorname(\psi_:n\in\Z), where \psi_(t)=2^\psi(2^t-n).

- \\subset\dots\subset V_1\subset V_0\subset V_\subset\dots\subset L^2(\R)

From those inclusions and orthogonality relations
follows the existence of sequences h=\_ and g=\_ that satisfy the
identities

- h_n=\langle\phi_,\,\phi_\rangle and \phi(t)=\sqrt2 \sum_ h_n\phi(2t-n)

- g_n=\langle\psi_,\,\phi_\rangle and \psi(t)=\sqrt2 \sum_ g_n\phi(2t-n).

# 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). This is the space of measurable functions that are absolutely and square integrable:- \int_^ |\psi (t)|\, dt and \int_^ |\psi (t)|^2 \, dt .

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

- \int_^ \psi (t)\, dt = 0 is the condition for zero mean, and
- \int_^ |\psi (t)|^2\, dt = 1 is the condition for square norm one.

For \psi 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 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).
Most constructions of discrete WT make use of the multiresolution
analysis, which defines the wavelet by a scaling function. This
scaling function itself is solution to a functional equation.

In most situations it is useful to restrict \psi
to be a continuous function with a higher number M of vanishing
moments, i.e. for all integer ''m\int_^ t^m\,\psi (t)\, dt =
0

Some example mother wavelets are:

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 _ (t) = \psi \left( \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. The main difference 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 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 discrete wavelet transform is also less
computationally complex, taking O(N)
time as compared to O(N log N) for the fast
Fourier transform. 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. 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.

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

## Scaling function

Wavelets are defined by the wavelet function \psi (t) (i.e. the mother wavelet) and scaling function \phi (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
http://perso.wanadoo.fr/polyvalens/clemens/wavelets/wavelets.html#note7
for a detailed explanation.

For a wavelet with compact support, \phi (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 \psi (t).For instance, Mexican
hat wavelets can be defined by a wavelet function. See a list
of a few Continuous
wavelets.

# 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. 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. 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. 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. In computer
vision and image
processing, the notion of scale-space
representation and Gaussian derivative operators is regarded as a
canonical multi-scale representation.

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. 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.e., in both the forward and inverse
transform. For details see wavelet
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. 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), 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.## 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. 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, 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. 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 and strain/metrology
applications for intermediate transforms with high frequency
resolution (like brushlets and ridgelets) is growing rapidly.

# 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

# See also

# 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. 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

# 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

wavelet in German: Wavelet

wavelet in Spanish: Wavelet

wavelet in Finnish: Wavelet-muunnos

wavelet in French: Ondelette

wavelet in Indonesian: Wavelet

wavelet in Italian: Wavelet

wavelet in Polish: Falki

wavelet in Portuguese: Wavelet

wavelet in Russian: Вейвлет

wavelet in Swedish: Vågelement

wavelet in Ukrainian: Вейвлет

wavelet in Chinese: 小波分析

wavelet in Japanese: ウェーブレット