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DiscreteWaveletPacketTransform

DiscreteWaveletPacketTransform[data]
gives the discrete wavelet packet transform (DWPT) of an array of data.
DiscreteWaveletPacketTransform
gives the discrete wavelet packet transform using the wavelet wave.
DiscreteWaveletPacketTransform
gives the discrete wavelet packet transform using r levels of refinement.
DiscreteWaveletPacketTransform
gives the discrete wavelet packet transform of an image.
DiscreteWaveletPacketTransform
gives the discrete wavelet packet transform of sampled sound.
  • Properties of the DiscreteWaveletData dwd can be found using dwd["prop"], and a list of available properties can be found using dwd["Properties"].
  • 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
  • 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 low-pass filter coefficients and are high-pass 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:
MethodAutomaticmethod to use
Padding"Periodic"how to extend data beyond boundaries
WorkingPrecisionMachinePrecisionprecision to use in internal computations
Compute a wavelet packet transform:
The resulting DiscreteWaveletData represents a full tree of wavelet coefficients:
The inverse transform reconstructs the input:
Transform an Image object:
Use to extract coefficient images:
Compute the inverse transform:
Transform a sampled Sound object:
Compute a wavelet packet transform:
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The resulting DiscreteWaveletData represents a full tree of wavelet coefficients:
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The inverse transform reconstructs the input:
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Transform an Image object:
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Use to extract coefficient images:
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Compute the inverse transform:
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Transform a sampled Sound object:
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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 or 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 WaveletBestBasis to compute an optimal basis of wavelet packet coefficients:
Highlight the best basis in a block grid of all coefficients:
Extract the best basis using :
The computed best basis is used by default in functions like WaveletListPlot:
Use a higher refinement level to increase the frequency resolution:
With a smaller refinement level, more of the signal energy is left in :
With further refinement, is resolved into further components:
Compute the wavelet packet transform using different wavelet families:
Compare the coefficients:
Use different families of wavelets to capture different features:
HaarWavelet (default):
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:
WaveletScalogram of the best tree representation of the data:
Constant data:
All coefficients are small except coarse coefficients :
Data oscillating at the highest resolvable frequency (Nyquist frequency):
Only the first detail coefficient and its coarse child coefficients are not small:
Data with large discontinuities:
Coarse coefficients have the same large-scale structure as the data:
Detail coefficients are sensitive to discontinuities:
Data with both spatial and frequency structure:
Coarse coefficients track the local mean of the data:
First detail coefficient and its coarse child coefficients represent the oscillations:
All coefficients on a common vertical axis:
Compute a two-dimensional wavelet packet transform:
View the tree of wavelet coefficients:
Inverse transform to get back the original signal:
Use WaveletMatrixPlot to visualize the different wavelet coefficients:
WaveletMatrixPlot of best tree representation:
In two dimensions, the vector of filtering operations in each direction can be computed:
Interpreting these vectors as binary digit expansions results in wavelet index numbers:
Get the low-pass and high-pass 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 , are zero:
Data with horizontal discontinuity:
All vertical and diagonal detail coefficients, wavelet index , are zero:
Data with diagonal discontinuity:
All horizontal and vertical detail coefficients, wavelet index , are zero:
Compute a three-dimensional wavelet packet transform:
Block grid view of all coefficients:
Wavelet transform of a three-dimensional cross array:
Visualize low-pass wavelet coefficients :
Energy of the original data is conserved within the transformed coefficients:
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 coefficient as an Image object:
Compute a best tree of coefficients from a packet transform of image data:
Plot the best tree in a hierarchical grid using WaveletImagePlot:
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 coefficient as a Sound object:
Inverse transform of coefficient as a Sound object:
Compute a best tree of coefficients from a packet transform of sound data:
Browse the best tree coefficients using a MenuView:
DiscreteWaveletPacketTransform 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:
The settings for Padding are the same as the methods for ArrayPad, including :
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Padding can remove boundary effects:
Use the default padding:
padding lessens boundary effects for non-periodic data:
By default, WorkingPrecision->MachinePrecision is used:
Use higher-precision computation:
With numbers close to zero, accuracy is the better indicator of the number of correct digits:
Use WorkingPrecision for exact computation:
The default reconstruction tree contains the coefficients at the maximum refinement level:
Choose a reconstruction tree with energy concentrated in a small number of coefficients:
Plot the best tree coefficients against a common vertical axis:
Visualize default reconstruction tree coefficients for image data:
Compute the reconstruction tree whose coefficients have the smallest total log energy:
Lossless compression of matrix data:
In the best tree wavelet packet representation many coefficients are zero:
Count nonzero coefficients as a measure of compressed size:
Nonzero values in original data:
Length of data:
DiscreteWaveletPacketTransform computes the full tree of wavelet coefficients:
DiscreteWaveletTransform 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 original data:
Rotated data gives rotated coefficients:
Multidimensional discrete wavelet transform is related to one-dimensional packet transform:
For Haar wavelet (default) and data length , the computed coefficients are identical:
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:
Compensate for the normalization at each 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:
HaarWavelet corresponds to averaging (low-pass filter) and differencing (high-pass filter):
Compute and wavelet coefficients:
In two dimensions a separate filter is applied in each dimension:
Low-pass and high-pass filters for Haar wavelet:
Haar wavelet transform of matrix data:
Image channels are transformed individually:
Combine coefficients of separately transformed image channels:
Compare with coefficient of DiscreteWaveletPacketTransform of original image:
The images are identical:
Padding can affect the total energy of wavelet coefficients:
Energy is not conserved:
Pad with 0s to ensure energy conservation in the coefficients:
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