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based on an earlier version of the Wolfram Language.
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Wavelet Analysis
Wavelets are short wavelike functions that can be scaled and translated. Wavelet transforms take any signal and express it in terms of scaled and translated wavelets. The resulting wavelet transform is a representation of the signal at different scales. The transform allows you to manipulate features at different scales independently, such as suppressing or strengthening some particular feature.
Mathematica provides a full-featured implementation of wavelet analysis, supporting many wavelet families, continuous wavelet transform, and several discrete wavelet transforms (standard, stationary, lifting, packet, ...). Discrete wavelet transforms directly work with arrays of any dimension as well as sounds and images, giving a symbolic representation of the transform, which can be directly processed or visualized, etc.
Scaling and Wavelet Functions
WaveletPhi scaling function ("father wavelet") for any wavelet family
WaveletPsi wavelet function ("mother wavelet") for any wavelength family
WaveletFilterCoefficients wavelet filter coefficients
Discrete Wavelet Families
HaarWavelet representation of the Haar wavelet
DaubechiesWavelet Daubechies wavelet family
Continuous Wavelet Families
MexicanHatWavelet derivative of Gaussian wavelet family
GaborWavelet Gabor wavelet family
Discrete Wavelet Transforms
DiscreteWaveletTransform discrete wavelet transform (DWT)
StationaryWaveletTransform stationary wavelet transform (SWT)
LiftingWaveletTransform lifting wavelet transform (LWT)
DiscreteWaveletPacketTransform discrete wavelet packet transform (DWPT)
StationaryWaveletPacketTransform stationary wavelet packet transform (SWPT)
InverseWaveletTransform inverse discrete wavelet transform (IDWT, ISWT, IDWPT, ISWPT, ILWT)
Padding  ▪ WorkingPrecision  ▪ ArrayPad  ▪
Continuous Wavelet Transforms
ContinuousWaveletTransform continuous wavelet transform (CWT)
InverseContinuousWaveletTransform inverse wavelet transform (ICWT)
Padding  ▪ WorkingPrecision  ▪ ArrayPad  ▪
Transform Manipulation
ContinuousWaveletData symbolic representation of continuous wavelet decomposition
"Wavelet"  ▪ "Scales"  ▪ "Octaves"  ▪
DiscreteWaveletData symbolic representation of a discrete wavelet decomposition
"Transform"  ▪ "TreeView"  ▪ "BestBasisBlockView"  ▪
LiftingFilterData symbolic representation of wavelet lifting filters
"ForwardLiftingFunction"  ▪ "ForwardIntegerLiftingFunction"  ▪
WaveletMapIndexed apply a function to the tree of transformed data
WaveletThreshold threshold filtering of wavelet representations
WaveletBestBasis compute the best tree for reconstruction
Wavelet Visualization
WaveletScalogram plot 1D wavelet scalogram
WaveletListPlot plot 1D wavelet transforms
WaveletMatrixPlot plot 2D wavelet transforms
WaveletImagePlot plot image wavelet transforms