This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# StationaryWaveletTransform

 StationaryWaveletTransform[data] gives the stationary wavelet transform (SWT) of an array of data. StationaryWaveletTransformgives the stationary wavelet transform using the wavelet wave. StationaryWaveletTransformgives the stationary wavelet transform using r levels of refinement. StationaryWaveletTransformgives the stationary wavelet transform of an image. StationaryWaveletTransformgives the stationary wavelet 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 and dimensions as the input data.
• The possible wavelets 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.  »
• 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 , where is the filter length for the corresponding wspec and is the length of input data.  »
• 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 the same as input data dimensions.
• The following options can be given:
 Method Automatic method to use WorkingPrecision MachinePrecision precision to use in internal computations
Compute a stationary wavelet transform using the HaarWavelet:
Use Normal to view all coefficients:
Transform an Image object:
Use to extract coefficient images:
Compute the inverse transform:
Transform a sampled Sound object:
Compute a stationary wavelet transform using the HaarWavelet:
 Out[1]=
Use Normal to view all coefficients:
 Out[2]=

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]=
 Scope   (34)
Compute a stationary 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 of the signal energy is left in :
With further refinement, is resolved into further components:
Compute the stationary wavelet 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:
Constant data:
All coefficients are small except coarse coefficients :
Data oscillating at the highest resolvable frequency (Nyquist frequency):
Only the first detail coefficient is nonzero:
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 stationary wavelet transform:
View the tree of wavelet coefficients:
Inverse transform to get back the original signal:
Use to visualize each coefficient as a MatrixPlot:
Visualize wavelet coefficients at higher refinement levels:
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 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:
Compute a three-dimensional stationary 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 arrays of data for each image channel:
Number of channels and dimensions of the original image are the same:
Get all coefficients as Image objects instead of arrays of data:
Get raw Image objects with no rescaling of color levels:
Get the inverse transform of the coefficient as an Image object:
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:
Number of channels and data length in the original sound are the same:
Get the coefficient as a Sound object:
Inverse transform of coefficient as a Sound object:
Browse all coefficients using a MenuView:
StationaryWaveletTransform 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:
Inverse transform recovers the input:
 Options   (3)
By default, WorkingPrecision is used:
Use higher-precision computation:
Use WorkingPrecision for exact computation:
 Applications   (3)
A simple wavelet-based inverse halftoning:
Apply GaussianFilter on the detail coefficients:
Differentiate noisy data using wavelet transform:
Translation-Rotation-Transform (TRT) is used to reduce boundary effects by subtracting a linear component from the input signal:
Since HaarWavelet has one vanishing moment, choose it to perform a wavelet transform on :
Detail coefficients give the differentiation of the data. Coefficients at refinement level 4 are chosen to minimize noise:
Rescale the differentiated values:
Compare wavelet-based numerical differentiation with exact differentiation:
Compare with standard Mathematica numerical differentiation:
Add texture to an existing image:
Perform wavelet transform on both images:
Combine detail coefficients of the two images by taking their mean:
Append the coarse coefficient of the first image:
Construct a new DiscreteWaveletData of the combined wavelet coefficients:
Reconstruct the combined image:
StationaryWaveletPacketTransform computes the full tree of wavelet coefficients:
StationaryWaveletTransform computes a subset of the full tree of coefficients:
DiscreteWaveletTransform coefficients halve in length with each level of refinement:
Rotated data gives different coefficients:
StationaryWaveletTransform coefficients have the same length as the original data:
Rotated data gives rotated coefficients:
The default refinement is given by :
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 stationary wavelet coefficients for periodic data:
Compute filter coefficients:
Coarse coefficients at level are given by :
Detail coefficients at level are given by :
Compute partial stationary inverse wavelet transform:
Compute filter coefficients:
Coarse coefficients at level are given:
Detail coefficients at level are given:
Inverse wavelet transform at level is given by :
Reconstruct coarse coefficients at refinement level :
Reconstruct coarse coefficients at refinement level :
Compute a Haar stationary wavelet transform in one dimension:
Compute and wavelet coefficients:
In two dimensions, a separate filter is applied in each dimension:
Low-pass and high-pass filters for a Haar wavelet:
Haar wavelet transform of matrix data:
Image channels are transformed individually:
Combine coefficients of separately transformed image channels:
Compare with coefficient of StationaryWaveletTransform of the original image:
The images are identical:
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