Details on the supported Wavelet families in the DiscreteTransforms package
Orthogonal Wavelet Families
All wavelets have been normalized to have (L2) norm 1. This means that values given here may be different (usually by a factor of 2 or 12) from values listed in references.
The Daubechies Wavelets are a family of orthogonal wavelets with vanishing moments, and were developed by Ingrid Daubechies.
In WaveletCoefficients(Daubechies,n), n can be any positive even number.
n is the size of the resulting filters.
The Daubechies wavelet of size n has n2 vanishing moments.
The values given by WaveletCoefficients for the Daubechies Coefficients, when multiplied by 2, agree with those in "Ten Lectures on Wavelets" by Ingrid Daubechies.
Symlets are also know as the Daubechies least asymmetric wavelets. Their construction is very similar to the Daubechies wavelets.
Whereas the Daubechies wavelets have maximal phase, the Symlets have minimal phase.
In WaveletCoefficients(Symlet,n), n can be any positive even number.
The Symlet wavelet of size n has n2 vanishing moments.
The values given by WaveletCoefficients, when normalized, agree with those listed in "Ten Lectures on Wavelets" by Ingrid Daubechies.
Coiflets are a family of orthogonal wavelets designed by Ingrid Daubechies to have better symmetry than the Daubechies wavelets.
Note: Currently, only Coiflets 1-7 are supported.
In WaveletCoefficients(Coiflet,n), n can be 1,2,3,4,5,6, or 7.
The nth Coiflet has size 6⁢n. Coiflet scaling functions have 2⁢n−1 vanishing moments, and their wavelet functions have 2⁢n vanishing moments.
The algorithm used to generated Coiflets is a modification of the one given in "Orthonormal Bases of Compactly Supported Wavelets II," by Ingrid Daubechies.
The values generated agree with those in "Ten Lectures on Wavelets" by Ingrid Daubechies, when normalized.
Battle-Lemarie wavelets, also know as orthogonal spline wavelets, are a family of wavelets developed from a multi-resolutional analysis of spaces of piecewise polynomial, continuously differentiable functions. Unlike many other wavelets, they have closed form representations in the frequency domain.
Battle-Lemarie wavelets use guarddigits=5 by default. This greatly speeds up WaveletCoefficients by allowing it to do hardware float integration.
WARNING: Because of the low default setting of guarddigits, a call to WaveletCoefficients for BattleLemarie with Digits=10 will result in an answer that is not necessarily accurate to full hardware float precision.
Battle-Lemarie wavelets do not have compact support. That is, the associated filters do not have finite length.
WaveletCoefficients(BattleLemarie, 4, 5) will give the 4th Battle-Lemarie wavelet with 11 coefficients. In general, WaveletCoefficients(BattleLemarie, n, m) will give the nth Battle-Lemarie wavelet with 2⁢m+1 coefficients.
The coefficients in the Battle-Lemarie wavelets converge very quickly to zero, so although WaveletCoefficients(BattleLemarie,n,m) will give filters that are not quite orthogonal, they are usually almost orthogonal.
Increasing m will improve the orthogonality of the resulting wavelet.
WaveletCoefficients(BattleLemarie, n, m) gives the middle 2⁢m+1 coefficients of WaveletCoefficients(BattleLemarie, n, m+1).
Because WaveletCoefficients(BattleLemarie,n,m) uses numerical integration, increasing the Digits setting will significantly affect performance.
The Cohen-Daubechies-Feauveau 9 tap 7 tap wavelet, or CDF wavelet, is used in the JPEG 2000 image compression standard.
WaveletCoefficients(CDF) gives the CDF wavelet. It in fact returns four length 10 Vectors. This is to allow for offsets.
Biorthogonal spline wavelets are a family of biorthogonal wavelets.
In WaveletCoefficients(BiorthogonalSpline, b, c), b and c can be any positive integers whose sum is even.
b and c are the number of vanishing moments of the analysis and synthesis filters, respectively.
Daubechies, Ingrid. "Orthonormal Bases of Compactly Supported Wavelets II: Variations on a Theme." SIAM J MATH ANAL. (March 1993).
Daubechies, Ingrid. "Ten Lectures on Wavelets." SIAM. 1992.
Wavelet Examples and Applications
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