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:
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:
.
.
where 
is the minimum dimension of data.
.
consists of coarse coefficients 
and detail coefficients 
with 
representing the input data. 
, 
, 
, and 
.
. 
are low-pass filter coefficients and 
are high-pass filter coefficients that are defined for each wavelet family. 
and 
are given by 
where 
is the input data dimension and fl is the filter length for the corresponding wspec.
to extract coefficient images:
:
:
:
:
:
:
padding: 
padding lessens boundary effects for non-periodic data: 
for exact computation:
, the computed coefficients are identical:
normalization at each refinement level: 
and 
wavelet coefficients:
coefficients of separately transformed image channels:
coefficient of DiscreteWaveletPacketTransform of original image:
EXAMPLES
Basic Examples 
Scope 
