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

# DiscreteWaveletTransform

 DiscreteWaveletTransform[data] gives the discrete wavelet transform (DWT) of an array of data. DiscreteWaveletTransformgives the discrete wavelet transform using the wavelet wave. DiscreteWaveletTransformgives the discrete wavelet transform using r levels of refinement. DiscreteWaveletTransformgives the discrete wavelet transform of an image. DiscreteWaveletTransformgives the discrete 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 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 .  »
• 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:
 Method Automatic method to use Padding "Periodic" how to extend data beyond boundaries WorkingPrecision MachinePrecision precision to use in internal computations
Compute a discrete 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 discrete 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   (36)
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:
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 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:
DiscreteWaveletTransform works on arrays of symbolic quantities:
Inverse transform recovers the input exactly:
Specify any internal working precision:
Use complex-valued data:
The wavelet coefficients are complex:
Inverse transform recovers the input:
 Options   (5)
The settings for Padding are the same as the methods for ArrayPad, including :
:
:
:
:
:
:
Using padding has fewer boundary effects for nonperiodic data:
By default, WorkingPrecision 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:
 Applications   (11)
Compress data by finding a representation with few nonzero coefficients:
SymletWavelet[n] has n vanishing moments and represents polynomials of degree n:
Count counts the number of wavelet coefficients close to 0:
Visualize discontinuities in the wavelet domain:
Detail coefficients in the region of discontinuities have larger values:
Detect edges in an image:
Set coarse coefficients to 0 and reconstruct using detail coefficients only:
Compare the cumulative energy in a signal, its wavelet coefficients, and Fourier coefficients:
Compute the ordered cumulative energy in the signal:
Compute wavelet coefficients and Fourier coefficients:
The DWT captures more energy with fewer coefficients than the DFT:
Perform energy-dependent thresholding:
Computing the fraction of energy contained at each refinement level:
Set wavelet coefficients containing less than 1% energy to zero:
Perform an amplitude-dependent thresholding:
Use WaveletThreshold to perform thresholding:
Use Stein's unbiased risk estimator smoothing:
Denoise an Image:
Perform thresholding with threshold value computed adaptively at each level:
Invert thresholded coefficients:
Wavelet transforms can be used to filter frequencies:
To filter out the two signals, first perform a wavelet transform:
Use WaveletListPlot to visualize frequency distribution:
To filter low frequencies, keep only the coarse coefficients:
To filter high frequencies, keep only the detail coefficients:
Extract the stock price trend for IBM since January 1, 2000:
The trend of the series is captured in the low-pass filter coefficients:
Thresholding all detail coefficients and inverting the series gives the trend:
Detrend a financial series:
Detail coefficients captured the detrended series:
Remove the trend by removing the coarse coefficients and inverting:
Study variance of returns in a financial time series:
Perform a wavelet transform using HaarWavelet and SymletWavelet:
Since the GE return series does not exhibit low-frequency oscillations, higher-scale detail coefficients do not indicate large variations from zero:
Although both filters will capture the variance of the series, they distribute it differently because of their approximate band-pass properties:
SymletWavelet isolates features in a certain frequency interval better than HaarWavelet:
DiscreteWaveletPacketTransform computes the full tree of wavelet coefficients:
DiscreteWaveletTransform 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:
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 :
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:
Compute discrete wavelet coefficients for periodic data:
Define filter coefficients to have compact support:
Coarse coefficients at level are given by , with :
Detail coefficients at level are given by :
Compute a partial discrete inverse wavelet transform:
Define filter coefficients to have compact support:
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 the dimensions of wavelet coefficients:
At refinement level , the dimensions of wavelet coefficients are given by , where represents dimensions of input data:
Compare dimensions with coefficient dimensions in dwd:
Compute a Haar discrete 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 Haar wavelet:
Haar wavelet transform of matrix data:
Image channels are transformed individually:
Combine coefficients of separately transformed image channels:
Compare with coefficient of DiscreteWaveletTransform of the original image:
The images are identical:
DWT is similar to LiftingWaveletTransform with extra coefficients needed for padding:
Padding can affect the total energy of wavelet coefficients:
Energy is not conserved:
Pad with 0s to ensure energy conservation in the coefficients:
Create a padded matrix of data:
Create a 3D plot of the Haar DWT coefficients:
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