Wavelet Analysis¶

Woxi implements the Wolfram Language wavelet analysis functions: wavelet families, filter coefficients, discrete and continuous wavelet transforms, coefficient manipulation, and wavelet visualization.

Wavelet Families¶

Wavelet families are symbolic constructor objects:

$ wo 'HaarWavelet[]'
HaarWavelet[]

$ wo 'DaubechiesWavelet[4]'
DaubechiesWavelet[4]

Filter Coefficients¶

WaveletFilterCoefficients gives the filter as {index, coefficient} pairs. Lowpass coefficients sum to 1:

$ wo 'WaveletFilterCoefficients[HaarWavelet[]]'
{{0, 1/2}, {1, 1/2}}

$ wo 'WaveletFilterCoefficients[HaarWavelet[], "PrimalHighpass"]'
{{0, 1/2}, {1, -1/2}}

Exact coefficients with WorkingPrecision -> Infinity:

$ wo 'WaveletFilterCoefficients[DaubechiesWavelet[2], "PrimalLowpass", WorkingPrecision -> Infinity]'
{{0, (1 + Sqrt[3])/8}, {1, (3 + Sqrt[3])/8}, {2, (3 - Sqrt[3])/8}, {3, (1 - Sqrt[3])/8}}

Biorthogonal families have separate primal and dual filters:

$ wo 'WaveletFilterCoefficients[BiorthogonalSplineWavelet[2, 2], "DualLowpass"]'
{{-2, -1/8}, {-1, 1/4}, {0, 3/4}, {1, 1/4}, {2, -1/8}}

Discrete Wavelet Transform¶

DiscreteWaveletTransform gives a DiscreteWaveletData object; the coefficients are accessed with wavelet indices:

$ wo 'dwd = DiscreteWaveletTransform[{1, 2, 3, 4}, HaarWavelet[], 1]; dwd[All]'
{{0} -> {2.121320343559643, 4.949747468305833}, {1} -> {-0.7071067811865476, -0.7071067811865476}}

$ wo 'DiscreteWaveletTransform[{1, 2, 3, 4}]["BasisIndex"]'
{{0, 0}, {0, 1}, {1}}

InverseWaveletTransform reconstructs the data:

$ wo 'Round[InverseWaveletTransform[DiscreteWaveletTransform[{1, 2, 3, 4}]]]'
{1, 2, 3, 4}

Symbolic data transforms exactly:

$ wo 'Simplify[InverseWaveletTransform[DiscreteWaveletTransform[{a, b, c, d}, HaarWavelet[]]]]'
{a, b, c, d}

Lifting Wavelet Transform¶

The lifting transform computes exact results for exact input:

$ wo 'Normal[LiftingWaveletTransform[{1, 1, 3, 1}, HaarWavelet[], 1, WorkingPrecision -> Infinity]]'
{{0} -> {Sqrt[2], 2*Sqrt[2]}, {1} -> {0, Sqrt[2]}}

Wavelet and Scaling Functions¶

$ wo 'WaveletPhi[HaarWavelet[], x]'
Piecewise[{{1, Inequality[0, LessEqual, x, Less, 1]}}, 0]

$ wo 'WaveletPsi[MexicanHatWavelet[1], t]'
(2*E^(-1/2*t^2)*(1 - t^2))/(Sqrt[3]*Pi^(1/4))

Thresholding¶

WaveletThreshold shrinks detail coefficients (here soft thresholding with an explicit threshold value):

$ wo 'WaveletThreshold[DiscreteWaveletTransform[{1., 5., 2., 8.}], {"Soft", 1.}][{1}, "Values"]'
{-1.8284271247461903, -3.2426406871192857}

Continuous Wavelet Transform¶

$ wo 'cwd = ContinuousWaveletTransform[Range[32] // N]; {cwd["Octaves"], cwd["Voices"], cwd["Wavelet"]}'
{4, 4, MexicanHatWavelet[]}