# WaveletExplorer

As of Version 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 "PrimalHighpass" 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 ScalingFunction 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 DWavelet 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 1. 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 BSpline 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 MRDecomposition is not directly supported with built-in functionality.

The functionality of WaveletTransform 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 MRDecomposition 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 CosPacketCoefficients, SinPacketCoefficients, CosPacketTransform, SinPacketTransform, InverseCosPacketTransform, InverseSinPacketTransform, MalvarTransform, and InverseMalvarTransform are not directly supported with built-in functionality.

To compute SinTransform, use the built-in function FourierDST.

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SinTransform with a specified second argument.

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

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Similarly, we can write CosPacketCoefficients 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 GrayCodeOrder and CumulativeEnergy 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 CumulativeEnergy can be written as follows.

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