Upgrading from:

WaveletExplorer

As of Version 8, the functionality of the Wavelet Explorer add-on has been integrated into the Wolfram System.

Wavelet Filters

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

Wavelet Explorer Needs["Wavelets`Wavelets`"] DaubechiesFilter[2]

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 .

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

Wavelet Explorer Needs["Wavelets`Wavelets`"] HighpassFilter[DaubechiesFilter[2]]

Scaling and Wavelet Functions

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

Wavelet Explorer Needs["Wavelets`Wavelets`"] ListLinePlot[ScalingFunction[DaubechiesFilter[2], 8]]

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

Wavelet Explorer Needs["Wavelets`Wavelets`"] ListLinePlot[DScalingFunction[DaubechiesFilter[6], 8, 1]]

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

Wavelet Explorer Needs["Wavelets`Wavelets`"] ListLinePlot[DWavelet[LeastAsymmetricFilter[10], 8, 1]]

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

The InterpolatingFunction outputted from WaveletPhi and WaveletPsi has InterpolationOrder set to 1. Hence the second derivative comes out to be 0.

Resampling and interpolating with a higher InterpolationOrder resolves the issue.

Wavelet Explorer Needs["Wavelets`Wavelets`"] ListLinePlot[DWavelet[LeastAsymmetricFilter[10], 7, 2]]

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

Wavelet Explorer Needs["Wavelets`Wavelets`"] Plot[Evaluate[BSpline[6, t]], {t, -3, 4}]

Wavelet Transforms

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

Wavelet Explorer Needs["Wavelets`Wavelets`"] Simplify[WaveletTransform[Range[4], HaarFilter[]]]

To compute a packet transform, use DiscreteWaveletPacketTransform.

Wavelet Explorer Needs["Wavelets`Wavelets`"] wd = WaveletPacketTransform[Range[8], DaubechiesFilter[2], 2]

Use InverseWaveletTransform to compute the inverse:

Wavelet Explorer InverseWaveletPacketTransform[wd, DaubechiesFilter[2]]

The functionality of MRDecomposition can be replicated as follows.

Wavelet Explorer Needs["Wavelets`Wavelets`"] wd = WaveletTransform[N@Range[8], HaarFilter[]]; MRDecomposition[wd, HaarFilter[]]

Sine & Cosine Transforms

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

Wavelet Explorer Needs["Wavelets`Wavelets`"] SinTransform[Range[8], Taper -> False]

SinTransform with a specified second argument.

Wavelet Explorer Needs["Wavelets`Wavelets`"] SinTransform[Range[8], 2, Taper -> False]

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

Wavelet Explorer Needs["Wavelets`Wavelets`"] SinPacketCoefficients[Range[8], 8, 3, Taper -> False]

Similarly, we can write CosPacketCoefficients using FourierDCT.

Wavelet Explorer Needs["Wavelets`Wavelets`"] CosPacketCoefficients[Range[8], 8, 3, Taper -> False]

Other Utilities

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

Wavelet Explorer Needs["Wavelets`Wavelets`"] data = Table[Sin[x^3], {x, -6, 6, 12/1023}]; wd = WaveletPacketTransform[data, N@HaarFilter[], 2] PhaseSpacePlot[wd, Frame -> True, FrameTicks -> None, AspectRatio -> 1/2]

Use WaveletThreshold for data compression.

Wavelet Explorer Needs["Wavelets`Wavelets`"] wd = WaveletTransform[Range[8], N@HaarFilter[], 3] Reverse@WaveletCompress[wd, 4]

The function CumulativeEnergy can be written as follows.

Wavelet Explorer Needs["Wavelets`Wavelets`"] wd = WaveletTransform[N@Range[8], HaarFilter[]]; CumulativeEnergy[wd]