Wavelet Toolbox

Wavelet Families: Additional Discussion

There are different types of wavelet families whose qualities vary according to several criteria. The main criteria are

• The support of , (and , ): the speed of convergence to 0 of these functions ( or ) when the time t or the frequency goes to infinity, which quantifies both time and frequency localizations
• The symmetry, which is useful in avoiding dephasing in image processing
• The number of vanishing moments for or for (if it exists), which is useful for compression purposes
• The regularity, which is useful for getting nice features, like smoothness of the reconstructed signal or image, and for the estimated function in nonlinear regression analysis

These are associated with two properties that allow fast algorithm and space-saving coding:

• The existence of a scaling function
• The orthogonality or the biorthogonality of the resulting analysis

They may also be associated with these less important properties:

• The existence of an explicit expression
• The ease of tabulating
• The familiarity with use

Typing waveinfo in command-line mode displays a survey of the main properties of all wavelet families available in the toolbox.

Note that the and functions can be computed using wavefun; the filters are generated using wfilters. We provide definition equations for several wavelets. Some are given explicitly by their time definitions, others by their frequency definitions, and still others by their filters.

The table below outlines the wavelet families included in the toolbox.

Wavelet Family Short Name	Wavelet Family Name
'haar'	Haar wavelet
'db'	Daubechies wavelets
'sym'	Symlets
'coif'	Coiflets
'bior'	Biorthogonal wavelets
'rbio'	Reverse biorthogonal wavelets
'meyr'	Meyer wavelet
'dmey'	Discrete approximation of Meyer wavelet
'gaus'	Gaussian wavelets
'mexh'	Mexican hat wavelet
'morl'	Morlet wavelet
'cgau'	Complex Gaussian wavelets
'shan'	Shannon wavelets
'fbsp'	Frequency B-Spline wavelets
'cmor'	Complex Morlet wavelets

Frequently Asked Questions

Daubechies Wavelets: dbN

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