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


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The Daubechies Wavelets are a family of orthogonal wavelets with vanishing moments, and were developed by Ingrid Daubechies.

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In WaveletCoefficients(Daubechies,n), n can be any positive even number.

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n is the size of the resulting filters.

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The Daubechies wavelet of size n has $\frac{1}{2}n$ vanishing moments.

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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


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Symlets are also know as the Daubechies least asymmetric wavelets. Their construction is very similar to the Daubechies wavelets.

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Whereas the Daubechies wavelets have maximal phase, the Symlets have minimal phase.

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In WaveletCoefficients(Symlet,n), n can be any positive even number.

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n is the size of the resulting filters.

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The Symlet wavelet of size n has $\frac{1}{2}n$ vanishing moments.

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The values given by WaveletCoefficients, when normalized, agree with those listed in "Ten Lectures on Wavelets" by Ingrid Daubechies.



Coiflet


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Coiflets are a family of orthogonal wavelets designed by Ingrid Daubechies to have better symmetry than the Daubechies wavelets.

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Note: Currently, only Coiflets 17 are supported.

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In WaveletCoefficients(Coiflet,n), n can be 1,2,3,4,5,6, or 7.

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The nth Coiflet has size $6n$. Coiflet scaling functions have $2n1$ vanishing moments, and their wavelet functions have $2n$ vanishing moments.

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The algorithm used to generated Coiflets is a modification of the one given in "Orthonormal Bases of Compactly Supported Wavelets II," by Ingrid Daubechies.

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The values generated agree with those in "Ten Lectures on Wavelets" by Ingrid Daubechies, when normalized.



BattleLemarie


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BattleLemarie wavelets, also know as orthogonal spline wavelets, are a family of wavelets developed from a multiresolutional analysis of spaces of piecewise polynomial, continuously differentiable functions. Unlike many other wavelets, they have closed form representations in the frequency domain.

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BattleLemarie 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.
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BattleLemarie wavelets do not have compact support. That is, the associated filters do not have finite length.

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WaveletCoefficients(BattleLemarie, 4, 5) will give the 4th BattleLemarie wavelet with 11 coefficients. In general, WaveletCoefficients(BattleLemarie, n, m) will give the nth BattleLemarie wavelet with $2m\+1$ coefficients.

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The coefficients in the BattleLemarie 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.

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Increasing m will improve the orthogonality of the resulting wavelet.

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WaveletCoefficients(BattleLemarie, n, m) gives the middle $2m\+1$ coefficients of WaveletCoefficients(BattleLemarie, n, m+1).

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Because WaveletCoefficients(BattleLemarie,n,m) uses numerical integration, increasing the Digits setting will significantly affect performance.




Biorthogonal Wavelets



CDF


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The CohenDaubechiesFeauveau 9 tap 7 tap wavelet, or CDF wavelet, is used in the JPEG 2000 image compression standard.

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WaveletCoefficients(CDF) gives the CDF wavelet. It in fact returns four length 10 Vectors. This is to allow for offsets.



Biorthogonal Spline


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Biorthogonal spline wavelets are a family of biorthogonal wavelets.

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In WaveletCoefficients(BiorthogonalSpline, b, c), b and c can be any positive integers whose sum is even.

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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.


