# ReverseBiorthogonalSplineWavelet

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

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

# Details • ReverseBiorthogonalSplineWavelet defines a family of biorthogonal wavelets.
• 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.
• ReverseBiorthogonalSplineWavelet can be used with such functions as DiscreteWaveletTransform and 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 wavelet function at different refinement scales:

### 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(19)

is equivalent to HaarWavelet:

ReverseBiorthogonalSplineWavelet is equivalent to BiorthogonalSplineWavelet :

ReverseBiorthogonalSplineWavelet is equivalent to BiorthogonalSplineWavelet :

ReverseBiorthogonalSplineWavelet is equivalent to BiorthogonalSplineWavelet :

ReverseBiorthogonalSplineWavelet is equivalent to BiorthogonalSplineWavelet :

Lowpass filter coefficients sum to unity; :

Highpass filter coefficients sum to zero; :

Dual 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: