In mathematics, a wavelet series is a representation of a squareintegrable (real or complexvalued) function by a certain orthonormal series generated by a wavelet. Nowadays, wavelet transformation is one of the most popular candidates of the timefrequencytransformations. This article provides a formal, mathematical definition of an orthonormal wavelet and of the integral wavelet transform.
Formal definition
A function $\backslash psi\backslash in\; L^2(\backslash mathbb\{R\})$ is called an orthonormal wavelet if it can be used to define a Hilbert basis, that is a complete orthonormal system, for the Hilbert space $L^2(\backslash mathbb\{R\})$ of square integrable functions.
The Hilbert basis is constructed as the family of functions $\backslash \{\backslash psi\_\{jk\}:\; j,\; k\; \backslash in\; \backslash Z\backslash \}$ by means of dyadic translations and dilations of $\backslash psi\backslash ,$,
 $\backslash psi\_\{jk\}(x)\; =\; 2^\backslash frac\{j\}\{2\}\; \backslash psi(2^jx\; \; k)\backslash ,$
for integers $j,\; k\; \backslash in\; \backslash mathbb\{Z\}$.
This family is an orthonormal system if it is orthonormal under the standard inner product $\backslash langle\; f,\; g\backslash rangle\; =\; \backslash int\_\{\backslash infty\}^\backslash infty\; f(x)\backslash overline\{g(x)\}dx$ on $L^2(\backslash mathbb\{R\}).$
 $\backslash langle\backslash psi\_\{jk\},\backslash psi\_\{lm\}\backslash rangle\; =\; \backslash delta\_\{jl\}\backslash delta\_\{km\}$
where $\backslash delta\_\{jl\}\backslash ,$ is the Kronecker delta.
Completeness is satisfied if every function $h\; \backslash in\; L^2(\backslash mathbb\{R\})$ may be expanded in the basis as
 $h(x)\; =\; \backslash sum\_\{j,\; k=\backslash infty\}^\backslash infty\; c\_\{jk\}\; \backslash psi\_\{jk\}(x)$
with convergence of the series understood to be convergence in norm. Such a representation of a function f is known as a wavelet series. This implies that an orthonormal wavelet is selfdual.
Wavelet transform
The integral wavelet transform is the integral transform defined as
 $\backslash left[W\_\backslash psi\; f\backslash right](a,\; b)\; =\; \backslash frac\{1\}\{\backslash sqrt\{a\}\}\; \backslash int\_\{\backslash infty\}^\backslash infty\; \backslash overline\{\backslash psi\backslash left(\backslash frac\{xb\}\{a\}\backslash right)\}f(x)dx\backslash ,$
The wavelet coefficients $c\_\{jk\}$ are then given by
 $c\_\{jk\}\; =\; \backslash left[W\_\backslash psi\; f\backslash right]\backslash left(2^\{j\},\; k2^\{j\}\backslash right)$
Here, $a\; =\; 2^\{j\}$ is called the binary dilation or dyadic dilation, and $b\; =\; k2^\{j\}$ is the binary or dyadic position.
Basic idea
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. Changes in the time extension are expected to be conform to the corresponding analysis frequency of the basis function. Based on the uncertainty principle of signal processing,
 $\backslash Delta\; t\; *\; \backslash Delta\; \backslash omega\; \backslash geqq\; \backslash frac\{1\}\{2\}$
where t represents time and ω angular velocity (ω = 2*Pi*frequency).
The higher the resolution in time is required, the lower resolution in frequency has to be. The larger the extension of the analysis windows is chosen, the larger is the value of $\backslash Delta\; t$.
When Δt is large,
 Bad time resolution
 Good frequency resolution
 Low frequency, large scaling factor
When Δt is small
 Good time resolution
 Bad frequency resolution
 High frequency, small scaling factor
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, wavelettransformation contains information similar to the shorttimeFouriertransformation, 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.
This shows that wavelet transformation is good in time resolution of high frequencies, while for slowly varying functions, the frequency resolution is remarkable.
An other example: The analysis of three superposed sinusoidal signals $y(t)\; =\; \backslash sin(2\; \backslash pi\; f\_0\; t)\; +\; \backslash sin(4\; \backslash pi\; f\_0\; t)\; +\; \backslash sin(8\; \backslash pi\; f\_0\; t)$ with STFT and wavelettransformation.
Wavelet compression
Wavelet compression is a form of data compression well suited for image compression (sometimes also video compression and audio compression). Notable implementations are JPEG 2000, DjVu and ECW for still images, REDCODE, CineForm, the BBC's Dirac, and Ogg Tarkin for video. The goal is to store image data in as little space as possible in a file. Wavelet compression can be either lossless or lossy.^{[1]}
Using a wavelet transform, the wavelet compression methods are adequate for representing transients, such as percussion sounds in audio, or highfrequency components in twodimensional 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.
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).
See Diary Of An x264 Developer: The problems with wavelets (2010) for discussion of practical issues of current methods using wavelets for video compression.
Method
First a wavelet transform is applied. This produces as many coefficients as there are pixels in the image (i.e., there is no compression yet since it is only a transform). These coefficients 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 coefficients are quantized and the quantized values are entropy encoded and/or run length encoded.
A few 1D and 2D applications of wavelet compression use a technique called "wavelet footprints".^{[2]}^{[3]}
Comparison with wavelet transformation, Fourier transformation and timefrequency analysis
Transformation 
Representation 
Output

Fourier transform 
$f(\backslash xi)\; =\; \backslash int\_\{\backslash infty\}^\{\backslash infty\}\; f(x)e^\{2\; \backslash pi\; ix\; \backslash xi\}\backslash ,\; dx$ 
ξ, frequency

Timefrequency analysis 
$X(t,\; f)$ 
t, time; f, frequency

Wavelet transform 
$X(a,b)\; =\; \backslash frac\{1\}\{\backslash sqrt\{a\}\}\backslash int\_\{\backslash infty\}^\{\backslash infty\}\backslash overline\{\backslash Psi\backslash left(\backslash frac\{t\; \; b\}\{a\}\backslash right)\}\; x(t)\backslash ,\; dt$ 
a, scaling; b, time

Other practical applications
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,^{[4]} and for fault detection.^{[5]}
(1) Discretizing of the cτaxis
Applied the following discretization of frequency and time:
 $\backslash begin\{align\}$
c_n &= c_0^n \\
\tau_m &= m \cdot T \cdot c_0^n
\end{align}
Leading to wavelets of the form, the discrete formula for the basis wavelet:
 $\backslash Psi(k,\; n,\; m)\; =\; \backslash frac\{1\}\{\backslash sqrt\{c\_0^n\}\}\backslash cdot\backslash Psi\backslash left[\backslash frac\{k\; \; m\; c\_0^n\}\{c\_0^n\}T\backslash right]\; =\; \backslash frac\{1\}\{\backslash sqrt\{c\_0^n\}\}\backslash cdot\backslash Psi\backslash left[\backslash left(\backslash frac\{k\}\{c\_0^n\}\; \; m\backslash right)T\backslash right]$
Such discrete wavelets can be used for the transformation:
 $Y\_\{DW\}(n,\; m)\; =\; \backslash frac\{1\}\{\backslash sqrt\{c\_0^n\}\}\backslash cdot\backslash sum\_\{k=0\}^\{K\; \; 1\}\; y(k)\backslash cdot\backslash Psi\backslash left[\backslash left(\backslash frac\{k\}\{c\_0^n\}\; \; m\backslash right)T\backslash right]$
(2) Implementation via the FFT (fast Fourier transform)
As apparent from wavelettransformation representation (shown below)
 $Y\_W(c,\; \backslash tau)\; =\; \backslash frac\{1\}\{\backslash sqrt\{c\}\}\backslash cdot\backslash int\_\{\backslash infty\}^\{\backslash infty\}\; y(k)\; \backslash cdot\; \backslash Psi\backslash left(\backslash frac\{t\; \; \backslash tau\}\{c\}\backslash right)\backslash ,\; dt$
where c is scaling factor, τ represents time shift factor
and as already mentioned in this context, the wavelettransformation corresponds to a convolution of a function y(t) and a waveletfunction. A convolution can be implemented as a multiplication in the frequency domain. With this the following approach of implementation results into:
 Fouriertransformation of signal y(k) with the FFT
 Selection of a discrete scaling factor $c\_n$
 Scaling of the waveletbasisfunction by this factor $c\_n$ and subsequent FFT of this function
 Multiplication with the transformed signal YFFT of the first step
 Inverse transformation of the product into the time domain results in Y_{W}$(c,\; \backslash tau)$ for different discrete values of τ and a discrete value of $c\_n$
 Back to the second step, until all discrete scaling values for $c\_n$are processed
There are large different types of wavelet transforms for specific purposes. See also a full list of waveletrelated transforms but the common ones are listed below: Mexican hat wavelet, Haar Wavelet, Daubechies wavelet, triangular wavelet.
See also
References
External links
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