BUILT-IN MATHEMATICA SYMBOL

# DiscreteWaveletPacketTransform

gives the discrete wavelet packet transform (DWPT) of an array of data.

DiscreteWaveletPacketTransform[data, wave]
gives the discrete wavelet packet transform using the wavelet wave.

DiscreteWaveletPacketTransform[data, wave, r]
gives the discrete wavelet packet transform using r levels of refinement.

DiscreteWaveletPacketTransform[image, ...]
gives the discrete wavelet packet transform of an image.

DiscreteWaveletPacketTransform[sound, ...]
gives the discrete wavelet packet transform of sampled sound.

## Details and OptionsDetails and Options

• DiscreteWaveletPacketTransform gives a DiscreteWaveletData object.
• Properties of the DiscreteWaveletData dwd can be found using dwd["prop"], and a list of available properties can be found using dwd["Properties"].
• DiscreteWaveletPacketTransform is a generalization of DiscreteWaveletTransform where the full tree of wavelet coefficients is computed.
• The data can be a rectangular array of any depth.
• By default, input image is converted to an image of type .
• The resulting wavelet coefficients are arrays of the same depth as the input data.
• The possible wavelet wave include:
•  BattleLemarieWavelet[...] Battle-Lemarié wavelets based on B-spline BiorthogonalSplineWavelet[...] B-spline-based wavelet CoifletWavelet[...] symmetric variant of Daubechies wavelets DaubechiesWavelet[...] the Daubechies wavelets HaarWavelet[...] classic Haar wavelet MeyerWavelet[...] wavelet defined in the frequency domain ReverseBiorthogonalSplineWavelet[...] B-spline-based wavelet (reverse dual and primal) ShannonWavelet[...] sinc function-based wavelet SymletWavelet[...] least asymmetric orthogonal wavelet
• The default wave is .
• With higher settings for the refinement level r, larger scale features are resolved.
• The default refinement level r is given by where is the minimum dimension of data.
• With refinement level Full, r is given by .
• The tree of wavelet coefficients at level consists of coarse coefficients and detail coefficients with representing the input data.
• The forward transform is given by , , , and .
• The inverse transform is given by .
• The are lowpass filter coefficients and are highpass filter coefficients that are defined for each wavelet family.
• The dimensions of and are given by where is the input data dimension and fl is the filter length for the corresponding wspec.
• The following options can be given:
•  Method Automatic method to use Padding "Periodic" how to extend data beyond boundaries WorkingPrecision MachinePrecision precision to use in internal computations
• The settings for Padding are the same as available in ArrayPad.
• InverseWaveletTransform gives the inverse transform.
• By default, InverseWaveletTransform uses coefficients represented by dwd["BasisIndex"] for reconstruction. Use WaveletBestBasis to compute and set an optimal basis.

## ExamplesExamplesopen allclose all

### Basic Examples (3)Basic Examples (3)

Compute a wavelet packet transform:

 Out[1]=

The resulting DiscreteWaveletData represents a full tree of wavelet coefficients:

 Out[2]=

The inverse transform reconstructs the input:

 Out[3]=

Transform an Image object:

 Out[1]=

Use to extract coefficient images:

 Out[2]=

Compute the inverse transform:

 Out[3]=

Transform a sampled Sound object:

 Out[1]=
 Out[2]=
 Out[3]=