MATHEMATICA COMPATIBILITY INFORMATION

Upgrading from:

WaveletExplorer

As of Mathematica 8, the functionality of the Wavelet Explorer add-on has been integrated into the Mathematica kernel.

Wavelet Filters

The following is a list of filters available in Wavelet Explorer, along with the equivalent form in Mathematica 8.

HaarFilter[]WaveletFilterCoefficients[HaarWavelet[]]
DaubechiesFilter[n]WaveletFilterCoefficients[DaubechiesWavelet[n]]
LeastAsymmetricFilter[n]WaveletFilterCoefficients[SymletWavelet[n]]
CoifletFilter[n]WaveletFilterCoefficients[CoifletWavelet[n]]
ShannonFilter[lim]WaveletFilterCoefficients[ShannonWavelet[lim]]
MeyerFilter[n,lim]WaveletFilterCoefficients[MeyerWavelet[n,lim]]
SplineFilter[n,lim]WaveletFilterCoefficients[BattleLemarieWavelet[n,lim]]
BiorthogonalSplineFilter[n,m]WaveletFilterCoefficients[BiorthogonalSplineWavelet[n,m]]
HighpassFilter[h]WaveletFilterCoefficients[wave,"PrimalHighpass"]
Built-in function equivalents.

To compute wavelet coefficients, use the built-in function WaveletFilterCoefficients.

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Note that all wavelet coefficients are scaled by relative to the results from Wavelet Explorer, so to get the equivalent values, you must multiply the result by .

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To compute high-pass filter coefficients, use the argument to WaveletFilterCoefficients.

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Scaling and Wavelet Functions

The following is a list of functions available in Wavelet Explorer, along with the equivalent form in Mathematica 8.

ScalingFunction[filt,j]WaveletPhi[wave]
Wavelet[wave,j]WaveletPsi[wave]
ShannonPhi[t]WaveletPhi[ShannonWavelet[lim],t]
ShannonPsi[t]WaveletPsi[ShannonWavelet[lim],t]
MeyerPhi[n,t,lim]WaveletPhi[MeyerWavelet[n,lim],t]
MeyerPsi[n,t,lim]WaveletPsi[MeyerWavelet[n,lim],t]
SplinePhi[n,t,lim]WaveletPhi[BattleLemarieWavelet[n,lim],t]
SplinePsi[n,t,lim]WaveletPsi[BattleLemarieWavelet[n,lim],t]
BSpline[n,t]BSplineBasis[{n,{u1,u2,...}},0,t]
DScalingFunction[filt,jmax,m]Dt[WaveletPhi[wave,t],{t,m}]
DWavelet[filt,jmax,m]Dt[WaveletPsi[wave,t],{t,m}]
Built-in function equivalents.

The functionality of is now available by using WaveletPhi.

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To find the derivative of the scaling function, use Dt and WaveletPhi.

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The functionality of is now available by using Dt and WaveletPsi.

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Compute higher-order derivatives of the scaling and wavelet function.

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The InterpolatingFunction outputted from WaveletPhi and WaveletPsi has InterpolationOrder set to . Hence the second derivative comes out to be 0.

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Resampling and interpolating with a higher InterpolationOrder resolves the issue.

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The functionality of is now available by using the built-in function BSplineBasis.

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Wavelet Transforms

The following is a list of wavelet transforms available in Wavelet Explorer, along with the equivalent form in Mathematica 8.

WaveletTransform[data,filt,j]DiscreteWaveletTransform[data,wave,j]
InverseWaveletTransform[wd,filt]InverseWaveletTransform[dwd]
WaveletPacketCoefficients[data,filt,j]DiscreteWaveletPacketTransform[data,filt,j]
WaveletPacketTransform[data,filt,l]WaveletBestBasis[DiscreteWaveletPacketTransform[...]]
InverseWaveletPacketTransform[wpdata,filt]InverseWaveletTransform[dwd]
Built-in function equivalents. The function is not directly supported with built-in functionality.

The functionality of is now available by using DiscreteWaveletTransform.

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To compute a packet transform, use DiscreteWaveletPacketTransform.

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Use InverseWaveletTransform to compute the inverse:

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The functionality of can be replicated as follows.

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Sine & Cosine Transforms

The following is a list of functions available in Wavelet Explorer, along with the equivalent form in Mathematica 8.

CosTransform[data,n, BasisType->m]FourierDCT[data,m]
SinTransform[data,n,BasisType->m]FourierDST[data,m]
InverseCosTransform[cdata]FourierDCT[cdata,m]
InverseSinTransform[sdata]FourierDST[sdata,m]
Built-in function equivalents. The functions , , , , , , , and are not directly supported with built-in functionality.

To compute , use the built-in function FourierDST.

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with a specified second argument.

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In one dimension, the functionality of can be replicated as follows.

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Similarly, we can write using FourierDCT.

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Other Utilities

The following is a list of functions available in Wavelet Explorer, along with the equivalent form in Mathematica 8.

PlotCoefficients[wd]WaveletListPlot[dwd]
PhaseSpacePlot[wd]WaveletScalogram[dwd]
ShowBasisPosition[wd]DiscreteWaveletData[...]["BestBasisBlockView"]
PlotCoefficients2D[wd]WaveletMatrixPlot[dwd]
ShowBasisPosition2D[wd]DiscreteWaveletData[...]["BestBasisBlockView"]
WaveletCompress[wd,...]WaveletThreshold[dwd,tspec]
Built-in function equivalents. The functions and are not directly supported with built-in functionality.

To plot wavelet coefficients, use WaveletScalogram.

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Use WaveletThreshold for data compression.

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The function can be written as follows.

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