Details on the supported Wavelet families in the DiscreteTransforms package - Maple Programming Help

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Details on the supported Wavelet families in the DiscreteTransforms package

 Note: All wavelets have been normalized to have (${L}_{2}$) norm 1. This means that values given here may be different (usually by a factor of $\sqrt{2}$ or $\frac{1}{\sqrt{2}}$) from values listed in references.

Orthogonal Wavelet Families

Daubechies

 • 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 $\frac{n}{2}$ vanishing moments.
 • The values given by WaveletCoefficients for the Daubechies Coefficients, when multiplied by $\sqrt{2}$, agree with those in "Ten Lectures on Wavelets" by Ingrid Daubechies.

Symlet

 • 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.
 • n is the size of the resulting filters.
 • The Symlet wavelet of size n has $\frac{n}{2}$ vanishing moments.
 • The values given by WaveletCoefficients, when normalized, agree with those listed in "Ten Lectures on Wavelets" by Ingrid Daubechies.

Coiflet

 • 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 $6n$. Coiflet scaling functions have $2n-1$ vanishing moments, and their wavelet functions have $2n$ 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

 • 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 $2m+1$ 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 $2m+1$ coefficients of WaveletCoefficients(BattleLemarie, n, m+1).
 • Because WaveletCoefficients(BattleLemarie,n,m) uses numerical integration, increasing the Digits setting will significantly affect performance.

Biorthogonal Wavelets

CDF

 • 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

 • 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.

References

 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.