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.

ReferenceReference

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

BattleLemarieWavelet ▪ BiorthogonalSplineWavelet ▪ CDFWavelet ▪ CoifletWavelet ▪ MeyerWavelet ▪ ReverseBiorthogonalSplineWavelet ▪ ShannonWavelet ▪ SymletWavelet

Continuous Wavelet Families

MexicanHatWavelet derivative of Gaussian wavelet family

GaborWavelet Gabor wavelet family

DGaussianWavelet ▪ MorletWavelet ▪ PaulWavelet

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

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