Compute a wavelet transform:
The resulting
DiscreteWaveletData represents a tree of transform coefficients:
The inverse transform reconstructs the input:
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 the list of wavelet index specifications:
Extract all coefficients whose wavelet indexes match a pattern:
The
Automatic coefficients are used by default in functions like
WaveletListPlot:
Use a higher refinement level to increase the frequency resolution:
With a smaller refinement level, more signal energy is left in
:
With further refinement,
is resolved into further components:
Compute the discrete wavelet 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:
Constant data:
All coefficients are small except coarse coefficients
:
Data oscillating at the highest resolvable frequency (Nyquist frequency):
Only the first detail coefficient
is 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:
The first detail coefficient identifies the oscillatory region:
All coefficients on a common vertical axis:
Compute a two-dimensional discrete wavelet 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 wavelet transform at a higher refinement level:
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:
Only the vertical detail coefficients, wavelet index
, are nonzero:
Data with horizontal discontinuity:
Only the horizontal detail coefficients, wavelet index
, are nonzero:
Data with diagonal discontinuity:
Only the diagonal detail coefficients, wavelet index
, are nonzero:
Compute a three-dimensional discrete wavelet transform:
Tree view of all coefficients:
Inverse transform to get back the original signal:
Wavelet transform of a three-dimensional cross array:
Visualize 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:
Plot coefficients used in the inverse transform in a hierarchical grid using
WaveletImagePlot:
Image wavelet coefficients lie outside the valid range of
ImageType:
gives an unnormalized image wavelet coefficient:
The color channels lie outside its valid 0 to 1 range:
By default,
ImageAdjust is used to normalize coefficients:
The color channels are now within the valid 0 to 1 range:
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:
Browse all coefficients using a
MenuView:
.
is HaarWavelet
.
, 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 has fewer boundary effects for nonperiodic data: 
for exact computation: 
thresholding:
thresholding with threshold value 
computed adaptively at each level:
, the computed coefficients are identical:
: 
normalization at each refinement level: 
are given by 
, with 
:
are given by 
:
are given:
are given:
is given by 
: 
at refinement level 
:
at refinement level 
:
, the dimensions of wavelet coefficients are given by 
, where 
represents dimensions of input data:
dimensions with coefficient dimensions in dwd: 
and 
wavelet coefficients:
coefficients of separately transformed image channels:
coefficient of DiscreteWaveletTransform of the original image:
EXAMPLES
Basic Examples 
Scope 
