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BiorthogonalSplineWavelet

BiorthogonalSplineWavelet[]

represents a biorthogonal spline wavelet of order 4 and dual order 2.

BiorthogonalSplineWavelet[n,m]

represents a biorthogonal spline wavelet of order n and dual order m.

Details

BiorthogonalSplineWavelet defines a family of biorthogonal wavelets.
BiorthogonalSplineWavelet[n,m] is defined for positive integers m and n where m+n is even.
The scaling function () and wavelet function () have compact support. The functions are symmetric.
BiorthogonalSplineWavelet can be used with such functions as DiscreteWaveletTransform, WaveletPhi, etc.

Examples

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Basic Examples (6)

Primal scaling function:

Primal wavelet function:

Dual scaling function:

Dual wavelet function:

Primal filter coefficients:

Dual filter coefficients:

Scope (17)

Basic Uses (10)

Compute primal lowpass filter coefficients:

Dual lowpass filter coefficients:

Primal highpass filter coefficients:

Dual highpass filter coefficients:

Lifting filter coefficients:

Generate a function to compute lifting wavelet transform:

Primal scaling function:

Dual scaling function:

Plot scaling function using different levels of recursion:

Primal wavelet function:

Dual wavelet function:

Plot scaling function using different levels of recursion:

Wavelet Transforms (5)

Compute a DiscreteWaveletTransform:

View the tree of wavelet coefficients:

Get the dimensions of wavelet coefficients:

Plot the wavelet coefficients:

Compute a DiscreteWaveletPacketTransform:

View the tree of wavelet coefficients:

Get the dimensions of wavelet coefficients:

Plot the wavelet coefficients:

Compute a StationaryWaveletTransform:

View the tree of wavelet coefficients:

Get the dimensions of wavelet coefficients:

Plot the wavelet coefficients:

Compute a StationaryWaveletPacketTransform:

View the tree of wavelet coefficients:

Get the dimensions of wavelet coefficients:

Plot the wavelet coefficients:

Compute a LiftingWaveletTransform:

View the tree of wavelet coefficients:

Get the dimensions of wavelet coefficients:

Plot the wavelet coefficients:

Higher Dimensions (2)

Multivariate scaling and wavelet functions are products of univariate ones:

Multivariate dual scaling and wavelet functions are products of univariate ones:

Properties & Relations (18)

BiorthogonalSplineWavelet[1,1] is equivalent to HaarWavelet:

Lowpass filter coefficients sum to unity; :

Highpass filter coefficients sum to zero; :

Dual lowpass filter coefficients sum to unity; :

Dual highpass filter coefficients sum to zero; :

Scaling function integrates to unity; :

Dual scaling function integrates to unity; :

Wavelet function integrates to zero; :

Dual wavelet function integrates to zero; :

Scaling function has compact support {n₁,n₂}:

Dual scaling function has compact support {nd₁,nd₂}:

Corresponding wavelet function has support ({n₁-nd₂+1)/2,(n₂-nd₁+1)/2}:

Dual wavelet function has support ({nd₁-n₂+1)/2,(nd₂-n₁+1)/2}:

satisfies the recursion equation :

Plot the components and the sum of the recursion:

satisfies the recursion equation :

Plot the components and the sum of the recursion:

satisfies the recursion equation :

Plot the components and the sum of the recursion:

satisfies the recursion equation :

Plot the components and the sum of the recursion:

Frequency response for is given by :

The filter is a lowpass filter:

Fourier transform of is given by :

Frequency response for is given by :

The filter is a dual lowpass filter:

Fourier transform of is given by :

Frequency response for is given by :

The filter is a lowpass filter:

Fourier transform of is given by :

Frequency response for is given by :

The filter is a lowpass filter:

Fourier transform of is given by :

Neat Examples (2)

Plot translates and dilations of scaling function:

Plot translates and dilations of wavelet function:

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