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| [[File:Jpeg2000 2-level wavelet transform-lichtenstein.png|thumb|300px|An example of the 2D [[discrete wavelet transform]] that is used in [[JPEG2000]].]]
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| In [[mathematics]], a '''wavelet series''' is a representation of a [[square-integrable]] ([[real number|real]]- or [[complex number|complex]]-valued) [[function (mathematics)|function]] by a certain [[orthonormal]] [[series (mathematics)|series]] generated by a [[wavelet]]. Nowadays, wavelet transformation is one of the most popular candidates of the time-frequency-transformations. This article provides a formal, mathematical definition of an '''orthonormal wavelet''' and of the '''integral wavelet transform'''.
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| ==Formal definition==
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| A function <math>\psi\in L^2(\mathbb{R})</math> is called an '''orthonormal wavelet''' if it can be used to define a [[Hilbert space#Orthonormal bases|Hilbert basis]], that is a [[complete space|complete]] [[Orthonormality|orthonormal system]], for the [[Hilbert space]] <math>L^2(\mathbb{R})</math> of [[Square-integrable function|square integrable]] functions.
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| The Hilbert basis is constructed as the family of functions <math>\{\psi_{jk}: j, k \in \Z\}</math> by means of [[Dyadic transformation|dyadic]] [[translation (geometry)|translation]]s and [[dilation (operator theory)|dilation]]s of <math>\psi\,</math>,
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| :<math>\psi_{jk}(x) = 2^\frac{j}{2} \psi(2^jx - k)\,</math>
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| for integers <math>j, k \in \mathbb{Z}</math>.
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| This family is an orthonormal system if it is orthonormal under the standard [[inner product]] <math>\langle f, g\rangle = \int_{-\infty}^\infty f(x)\overline{g(x)}dx</math> on <math>L^2(\mathbb{R}).</math>
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| :<math>\langle\psi_{jk},\psi_{lm}\rangle = \delta_{jl}\delta_{km}</math>
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| where <math>\delta_{jl}\,</math> is the [[Kronecker delta]].
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| Completeness is satisfied if every function <math>h \in L^2(\mathbb{R})</math> may be expanded in the basis as
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| :<math>h(x) = \sum_{j, k=-\infty}^\infty c_{jk} \psi_{jk}(x)</math>
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| with convergence of the series understood to be [[norm_(mathematics)#Properties|convergence in norm]]. Such a representation of a function ''f'' is known as a '''wavelet series'''. This implies that an orthonormal wavelet is [[dual wavelet|self-dual]].
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| ==Wavelet transform==
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| The '''integral wavelet transform''' is the [[integral transform]] defined as
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| :<math>\left[W_\psi f\right](a, b) = \frac{1}{\sqrt{|a|}} \int_{-\infty}^\infty \overline{\psi\left(\frac{x-b}{a}\right)}f(x)dx\,</math>
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| The '''wavelet coefficients''' <math>c_{jk}</math> are then given by
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| :<math>c_{jk} = \left[W_\psi f\right]\left(2^{-j}, k2^{-j}\right)</math>
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| Here, <math>a = 2^{-j}</math> is called the '''binary dilation''' or '''dyadic dilation''', and <math>b = k2^{-j}</math> is the '''binary''' or '''dyadic position'''.
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| === Basic idea ===
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| The fundamental idea of wavelet transforms is that the transformation should allow only changes in time extension, but not shape. This is effected by choosing suitable basis functions that allow for this.{{how|date=July 2013}} Changes in the time extension are expected to conform to the corresponding analysis frequency of the basis function. Based on the [[Uncertainty_principle#Signal_processing|uncertainty principle]] of signal processing,
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| :<math>\Delta t * \Delta \omega \geqq \frac{1}{2}</math>
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| where t represents time and ω angular velocity (ω = 2*Pi*frequency).
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| The higher the required resolution in time, the lower the resolution in frequency has to be. The larger the extension of the analysis windows is chosen, the larger is the value of <math>\Delta t</math>.
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| [[File:Basis function with compression factor.jpg|right|430px]]
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| When Δt is large,
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| # Bad time resolution
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| # Good frequency resolution
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| # Low frequency, large scaling factor
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| When Δt is small
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| # Good time resolution
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| # Bad frequency resolution
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| # High frequency, small scaling factor
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| In other words, the basis function Ψ can be regarded as an impulse response of a system with which the function x(t) has been filtered. The transformed signal provides information about the time and the frequency. Therefore, wavelet-transformation contains information similar to the [[Short-time Fourier transform|short-time-Fourier-transformation]], but with additional special properties of the wavelets, which show up at the resolution in time at higher analysis frequencies of the basis function. The difference in time resolution at ascending frequencies for the [[Fourier transform]] and the wavelet transform is shown below.
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| [[File:STFT and WT.jpg|center|500px]]
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| This shows that wavelet transformation is good in time resolution of high frequencies, while for slowly varying functions, the frequency resolution is remarkable.
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| Another example: The analysis of three superposed sinusoidal signals <math>y(t) = \sin(2 \pi f_0 t) + \sin(4 \pi f_0 t) + \sin(8 \pi f_0 t)</math> with STFT and wavelet-transformation.
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| [[File:Analysis of three superposed sinusoidal signals.jpg|center|500px]]
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| ==Wavelet compression==
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| '''Wavelet compression''' is a form of [[data compression]] well suited for [[image compression]] (sometimes also [[video compression]] and [[audio compression (data)|audio compression]]). Notable implementations are [[JPEG 2000]], [[DjVu]] and [[ECW (file format)|ECW]] for still images, [[REDCODE]], [[CineForm]], the BBC's [[Dirac (codec)|Dirac]], and Ogg [[Tarkin (codec)|Tarkin]] for video. The goal is to store image data in as little space as possible in a [[Computer file|file]]. Wavelet compression can be either [[lossless data compression|lossless]] or [[lossy data compression|lossy]].<ref>[[JPEG 2000]], for example, may use a 5/3 wavelet for lossless (reversible) transform and a 9/7 wavelet for lossy (irreversible) transform.</ref>
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| Using a wavelet transform, the wavelet compression methods are adequate for representing [[Transient (acoustics)|transient]]s, such as percussion sounds in audio, or high-frequency components in two-dimensional images, for example an image of stars on a night sky. This means that the transient elements of a data signal can be represented by a smaller amount of information than would be the case if some other transform, such as the more widespread [[discrete cosine transform]], had been used.
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| Wavelet compression is not good for all kinds of data: transient signal characteristics mean good wavelet compression, while smooth, periodic signals are better compressed by other methods, particularly traditional harmonic compression (frequency domain, as by Fourier transforms and related).
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| See [http://x264dev.multimedia.cx/?p=317 Diary Of An x264 Developer: The problems with wavelets] (2010) for discussion of practical issues of current methods using wavelets for video compression.
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| ===Method===
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| First a wavelet transform is applied. This produces as many [[coefficient]]s as there are [[pixel]]s in the image (i.e., there is no compression yet since it is only a transform). These [[coefficient]]s can then be compressed more easily because the information is statistically concentrated in just a few coefficients. This principle is called [[transform coding]]. After that, the [[coefficient]]s are [[Quantization (signal processing)|quantized]] and the quantized values are [[entropy encoding|entropy encoded]] and/or [[run-length encoding|run length encoded]].
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| A few 1D and 2D applications of wavelet compression use a technique called "wavelet footprints".<ref>
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| N. Malmurugan, A. Shanmugam, S. Jayaraman and V. V. Dinesh Chander.
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| [http://www.acadjournal.com/2005/V14/part6/p1/ "A New and Novel Image Compression Algorithm Using Wavelet Footprints"]
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| </ref><ref>
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| Ho Tatt Wei and Jeoti, V.
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| "A wavelet footprints-based compression scheme for ECG signals".
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| {{cite doi|10.1109/TENCON.2004.1414412}}
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| </ref>
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| ==Comparison with wavelet transformation, Fourier transformation and time-frequency analysis==
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| {| class="wikitable"
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| |-
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| ! Transformation !! Representation !! Output
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| |-
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| | [[Fourier transform]] || <math>f(\xi) = \int_{-\infty}^{\infty} f(x)e^{-2 \pi ix \xi}\, dx</math> || ''ξ'', frequency
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| |-
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| | [[Time-frequency analysis]] || <math>X(t, f)</math> || ''t'', time; ''f'', frequency
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| |-
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| | Wavelet transform || <math> X(a,b) = \frac{1}{\sqrt{a}}\int_{-\infty}^{\infty}\overline{\Psi\left(\frac{t - b}{a}\right)} x(t)\, dt </math> || ''a'', scaling; ''b'', time
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| |}
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| ==Other practical applications==
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| The wavelet transform can provide us with the frequency of the signals and the time associated to those frequencies, making it very convenient for its application in numerous fields. For instance, signal processing of accelerations for gait analysis,<ref>[http://www.youtube.com/watch?v=DTpEVQSEBBk "Novel method for stride length estimation with body area network accelerometers"], ''IEEE BioWireless 2011'', pp. 79-82</ref> for fault detection,<ref>{{ cite journal | title=Shannon wavelet spectrum analysis on truncated vibration signals for machine incipient fault detection| journal=Measurement Science and Technology| year=2012 | last=Liu | volume=23 | issue=5 | pages=1–11 | first1=Jie }}</ref> for design of low power pacemakers and also in ultra-wideband (UWB) wireless communications.<ref>A.N. Akansu, W.A. Serdijn and I.W. Selesnick, [http://web.njit.edu/~akansu/PAPERS/ANA-IWS-WAS-ELSEVIER%20PHYSCOM%202010.pdf Emerging applications of wavelets: A review], Physical Communication, Elsevier, vol. 3, issue 1, pp. 1-18, March 2010.</ref>
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| (1) Discretizing of the c-τ-axis
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| Applied the following discretization of frequency and time:
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| :<math>\begin{align}
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| c_n &= c_0^n \\
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| \tau_m &= m \cdot T \cdot c_0^n
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| \end{align}</math>
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| Leading to wavelets of the form, the discrete formula for the basis wavelet:
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| :<math>\Psi(k, n, m) = \frac{1}{\sqrt{c_0^n}}\cdot\Psi\left[\frac{k - m c_0^n}{c_0^n}T\right] = \frac{1}{\sqrt{c_0^n}}\cdot\Psi\left[\left(\frac{k}{c_0^n} - m\right)T\right]</math>
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| Such discrete wavelets can be used for the transformation:
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| :<math>Y_{DW}(n, m) = \frac{1}{\sqrt{c_0^n}}\cdot\sum_{k=0}^{K - 1} y(k)\cdot\Psi\left[\left(\frac{k}{c_0^n} - m\right)T\right]</math>
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| (2) Implementation via the FFT (fast Fourier transform)
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| As apparent from wavelet-transformation representation (shown below)
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| :<math>Y_W(c, \tau) = \frac{1}{\sqrt{c}}\cdot\int_{-\infty}^{\infty} y(k) \cdot \Psi\left(\frac{t - \tau}{c}\right)\, dt </math>
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| where c is scaling factor, τ represents time shift factor
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| and as already mentioned in this context, the wavelet-transformation corresponds to a convolution of a function y(t) and a wavelet-function. A convolution can be implemented as a multiplication in the frequency domain. With this the following approach of implementation results into:
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| * Fourier-transformation of signal y(k) with the FFT
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| * Selection of a discrete scaling factor <math>c_n</math>
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| * Scaling of the wavelet-basis-function by this factor <math>c_n</math> and subsequent FFT of this function
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| * Multiplication with the transformed signal YFFT of the first step
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| * Inverse transformation of the product into the time domain results in ''Y<sub>W</sub><math>(c, \tau)</math>'' for different discrete values of τ and a discrete value of <math>c_n</math>
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| * Back to the second step, until all discrete scaling values for <math>c_n</math>are processed
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| There are large different types of wavelet transforms for specific purposes. See also a full [[list of wavelet-related transforms]] but the common ones are listed below: [[Mexican hat wavelet]], [[Haar Wavelet]], [[Daubechies wavelet]], triangular wavelet.
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| ==See also==
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| * [[Continuous wavelet transform]]
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| * [[Discrete wavelet transform]]
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| * [[Complex wavelet transform]]
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| * [[Dual wavelet]]
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| * [[Multiresolution analysis]]
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| * [[ECW (file format)|ECW]], a wavelet-based [[geospatial]] image format designed for speed and processing efficiency
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| * [[JPEG 2000]], a wavelet-based [[image compression]] standard
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| * Some people generate [[spectrogram]]s using wavelets, called [[scaleogram]]s. Other people generate spectrograms using a [[short-time Fourier transform]]
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| * [[Chirplet transform]]
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| * [[Time-frequency representation]]
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| * [[S transform]]
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| * [[Short-time Fourier transform]]
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| ==References==
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| * {{cite book |first=Charles K. |last=Chui |title=An Introduction to Wavelets |year=1992 |publisher=Academic Press |location=San Diego |isbn=0-12-174584-8}}
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| * {{cite book |first1=Ali N. |last1=Akansu |first2=Richard A. |last2=Haddad|title=Multiresolution Signal Decomposition: Transforms, Subbands, Wavelets |year=1992 |publisher=Academic Press |location=San Diego |isbn=978-0-12-047141-6}}
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| {{Reflist}}
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| ==External links==
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| {{Commons category|Wavelets}}
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| * {{cite web
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| | author = Amara Graps
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| | date=
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| | title = An Introduction to Wavelets
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| | url = http://dl.acm.org/citation.cfm?id=615342
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| }}
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| * {{cite web
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| | author = Robi Polikar
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| | date= 2001-01-12
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| | title = The Wavelet Tutorial
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| | url = http://users.rowan.edu/~polikar/WAVELETS/WTtutorial.html
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| }}
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| [[Category:Wavelets| ]]
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| [[Category:Functional analysis]]
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| [[Category:Image compression]]
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