Wavelet

The term wavelet which a continuous or discrete wavelet transform underlying functions are referred. The word is a neologism from the French " ondelette " or "small wave " means and partly literal ( " onde " → "wave" ), partly phonetically ( " palette " → " let" ) was translated into English. The term wavelet was in the 1980s in geophysics (Jean Morlet, Alex Grossmann ) coined for functions which generalize the Short -Time Fourier Transform, but is used exclusively in the present-day importance since the late 1980s. In the 1990s, a real wavelet boom, triggered by the discovery of compact, continuous ( up to any order of differentiability ) and orthogonal wavelets by Ingrid Daubechies (1988 ) and the development of the algorithm of the fast wavelet transform ( FWT ) was using the multi-scale analysis ( multiresolution analysis - MRA) by Stéphane Mallat and Yves Meyer ( 1989).

Wavelets and transformations

In contrast to the sine and cosine functions of the Fourier transform of the wavelets used usually have not only location in the frequency spectrum, but also in the time domain. Where " locality" to be understood in the sense of small scattering. The probability density is the normalized absolute square of the function under consideration and of their Fourier transforms. The product of the two variances is always greater than a constant, analogous to the heisenberg 's uncertainty relation, see also the WKS sampling theorem. From this restriction out originated in the functional analysis, the Paley -Wiener theory ( Raymond Paley, Norbert Wiener), a forerunner of the discrete wavelet transform and the Calderón - Zygmund theory (Alberto Calderon, Antoni Zygmund ), the continuous wavelet transformation corresponds.

The integral of a wavelet function is always 0, therefore, typically takes the form of the wavelet outwardly expire ( become smaller ) waves (ie " ripples " = Ondelettes = wavelets) to.

Important examples of the wavelets are the Haar wavelet ( Alfréd Hair 1909), named after Ingrid Daubechies Daubechies wavelets ( around 1990 ), who also designed her Coiflet wavelets and the more theoretically significant Meyer wavelet ( Yves Meyer, around 1988 ).

Wavelets are available for spaces of arbitrary dimension, usually a tensor product of one-dimensional wavelet basis is used. Due to the fractal nature of the two- scale equation in the most MRA wavelets have a complicated shape, most have no closed form.

Application

Find use in methods of the wavelet signal processing, especially the signal compression, which include as a first step, a Discrete Wavelet Transform. These were touted as a milestone of image compression and audio compression since the early 1990s. Nevertheless, outside of specialized applications, such as in geophysics or computer tomography, such wavelet compression methods implemented only in the JPEG2000 standard and its direct predecessors such as DjVu and the LuraWave format. So far JPEG2000 not widespread. In a broad sense, the integrated JPEG is based on a wavelet transform, the discrete cosine transform used may be interpreted as a Haar wavelet. In methods of signal analysis rather the continuous wavelet transform is used in discretised form.

Wavelets of the discrete wavelet transform

A wavelet is here the generating function of an affine system of functions which form a Hilbert basis, ie a complete orthonormal system in function space. The representation of a function by means of these functions is called wavelet transform:

And inverse wavelet transform

The most basic example is the Haar wavelet. It is helpful if the wavelet function is associated to a multiscale analysis since then in practical calculation, the evaluation of many of the integrals that are behind the dot products can be replaced by repeated folding of even coefficients obtained sequences with finite filter effects. This accelerated procedure is called accordingly fast wavelet transform.

Signal processing

The relationship between wavelets and filters for signal processing is now quite clearly: The Waveletmaske corresponds to the impulse response of a bandpass filter with a certain sharpness in time ( filter length ) and in frequency ( bandwidth). Filter length and bandwidth are inversely proportional, as a " stretching " of the filter by a factor of 2 the bandwidth is halved.

Extensions

It is possible and useful to consider other scale factors. So corresponds to the DCT variant in the JPEG algorithm a Haar wavelet for block size 8 Under further weakening of the analytical requirements, wavelet frames found ( see inset ) and framelets, these generate a redundant signal transformation that is preferable under certain circumstances, for for example in the noise reduction.

A arisen lately variant, the so-called multiwavelets that not one, but have a vector of scaling functions in the MRA and accordingly matrix-valued scaling consequences.

The new standard JPEG2000 image compression can use ( biorthogonal, 5/3 and 9 /7) wavelets.