StationaryWaveletPacketTransform

StationaryWaveletPacketTransform[data]

gives the stationary wavelet packet transform (SWPT) of an array of data.

StationaryWaveletPacketTransform[data,wave]

gives the stationary wavelet packet transform using the wavelet wave.

StationaryWaveletPacketTransform[data,wave,r]

gives the stationary wavelet packet transform using r levels of refinement.

Details and Options

Examples

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Basic Examples  (3)

Compute a stationary wavelet packet transform:

The resulting DiscreteWaveletData represents a full tree of wavelet coefficients:

The inverse transform reconstructs the input:

Transform an audio signal:

Use dwd[,"Audio"] to extract coefficient signals:

Compute the inverse transform:

Transform an Image object:

Use dwd[,"Image"] to extract coefficient images:

Compute the inverse transform:

Scope  (32)

Basic Uses  (4)

Useful properties can be extracted from the DiscreteWaveletData object:

Get a full list of properties:

Get data and coefficient dimensions:

Use Normal to get all wavelet coefficients explicitly:

Also use All as an argument to get all coefficients:

Use Automatic to get only the coefficients used in the inverse transform:

Use the "TreeView" or "WaveletIndex" to find out what wavelet coefficients are available:

Extract specific coefficient arrays:

Extract several wavelet coefficients corresponding to a list of wavelet index specifications:

Extract all coefficients whose wavelet indexes match a pattern:

Use a higher refinement level to increase the frequency resolution:

With a smaller refinement level, more of the signal energy is left in {0,0}:

With further refinement, {0,0} is resolved into further components:

Wavelet Families  (10)

Compute the wavelet packet transform using different wavelet families:

Compare the coefficients:

Use different families of wavelets to capture different features:

HaarWavelet (default):

DaubechiesWavelet:

BattleLemarieWavelet:

BiorthogonalSplineWavelet:

CoifletWavelet:

MeyerWavelet:

ReverseBiorthogonalSplineWavelet:

ShannonWavelet:

SymletWavelet:

Vector Data  (6)

Plot the coefficients over a common horizontal axis using WaveletListPlot:

Plot against a common vertical axis:

Visualize coefficients as a function of time and refinement level using WaveletScalogram:

The coefficient indexes appear as tooltips when the mouse pointer is moved over a coefficient:

Constant data:

All coefficients are small except coarse coefficients {0,0,}:

Data oscillating at the highest resolvable frequency (Nyquist frequency):

Only the first detail coefficient {1} and its coarse child coefficients {1,0,0,} are not small:

Data with large discontinuities:

Coarse coefficients {0,} have the same large-scale structure as the data:

Detail coefficients are sensitive to discontinuities:

Data with both spatial and frequency structure:

Coarse coefficients {0,} track the local mean of the data:

First detail coefficient {1} and its coarse child coefficients {1,0,} represent the oscillations:

All coefficients on a common vertical axis:

Matrix Data  (5)

Compute a two-dimensional stationary wavelet packet transform:

View the tree of wavelet coefficients:

Inverse transform to get back the original signal:

Use dwd[,"MatrixPlot"] to visualize each coefficient as a MatrixPlot:

Visualize diagonal detail coefficient {3} and its child coefficients {3,__}:

In two dimensions, the vector of filtering operations in each direction can be computed:

Interpreting these vectors as binary digit expansions, you get wavelet index numbers:

Get the lowpass and highpass filters for a Haar wavelet:

The resulting 2D filters are outer products of filters in the two directions:

Wavelet transform of step data:

Data with a vertical discontinuity:

All horizontal and diagonal detail coefficients, wavelet index {___,2|3,___}, are zero:

Data with horizontal discontinuity:

All vertical and diagonal detail coefficients, wavelet index {___,1|3,___}, are zero:

Array Data  (2)

Compute a three-dimensional wavelet packet transform:

List all computed wavelet coefficients:

Inverse transform to get back the original signal:

Wavelet transform of a three-dimensional cross array:

Visualize lowpass wavelet coefficients {___,0}:

Energy of the original data is conserved within the transformed coefficients:

Image Data  (2)

Transform an Image object:

The inverse transform yields a reconstructed Image object:

Wavelet coefficients are normally given as lists of data for each image channel:

Get all coefficients as Image objects instead:

Get raw Image objects with no rescaling of color levels:

Get the inverse transform of the {0,1} coefficient as an Image object:

Sound Data  (3)

Transform a Sound object:

The inverse transform yields a reconstructed Sound object:

By default, coefficients are given as lists of data for each sound channel:

Get the {1,1} coefficient as a Sound object:

Inverse transform of {1,1} coefficient as a Sound object:

Browse all coefficients using a MenuView:

Generalizations & Extensions  (3)

StationaryWaveletPacketTransform works on arrays of symbolic quantities:

Inverse transform recovers the input exactly:

Specify any internal working precision:

Use complex-valued data:

The wavelets coefficients are complex:

Options  (3)

WorkingPrecision  (3)

By default, WorkingPrecision->MachinePrecision is used:

Use higher-precision computation:

Use WorkingPrecision-> for exact computation:

Properties & Relations  (10)

StationaryWaveletPacketTransform computes the full tree of wavelet coefficients:

StationaryWaveletTransform computes a subset of the full tree of coefficients:

DiscreteWaveletPacketTransform coefficients halve in length with each level of refinement:

Rotated data gives different coefficients:

StationaryWaveletPacketTransform coefficients have the same length as the data:

Rotated data gives rotated coefficients:

The default refinement is given by Min[Round[Log2[Min[Dimensions[data]]]],4]:

In higher dimensions:

The energy norm is conserved for orthogonal wavelet families:

The energy norm is approximately conserved for biorthogonal wavelet families:

The mean of the data is captured at the maximum refinement level of the transform:

Extract the coefficient for the maximum refinement level:

The sum of inverse transforms from individual coefficient arrays gives the original data:

Individually inverse transform each wavelet coefficient array:

The sum gives the original data:

Compute a Haar stationary wavelet packet transform in one dimension:

Compute {0} and {1} wavelet coefficients:

Compare with StationaryWaveletPacketTransform:

In two dimensions, a separate filter is applied in each dimension:

Lowpass and highpass filters for Haar wavelet:

Haar wavelet transform of matrix data:

Compare with StationaryWaveletPacketTransform using HaarWavelet:

Image channels are transformed individually:

Combine {0} coefficients of separately transformed image channels:

Compare with {0} coefficient of StationaryWaveletPacketTransform of original image:

The images are identical:

Introduced in 2010
 (8.0)
 |
Updated in 2017
 (11.2)