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

# LiftingWaveletTransform

 LiftingWaveletTransform[data] gives the lifting wavelet transform (LWT) of an array of data. LiftingWaveletTransformgives the lifting wavelet transform using the wavelet wave. LiftingWaveletTransformgives the lifting wavelet transform using r levels of refinement. LiftingWaveletTransformgives the lifting wavelet transform of an image. LiftingWaveletTransformgives the lifting 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:
 BiorthogonalSplineWavelet[...] B-spline-based wavelet CDFWavelet[...] Cohen-Daubechies-Feauveau wavelet CoifletWavelet[...] symmetric variant of Daubechies wavelets DaubechiesWavelet[...] the Daubechies wavelets HaarWavelet[...] classic Haar wavelet ReverseBiorthogonalSplineWavelet[...] B-spline-based wavelet (reverse dual and primal) SymletWavelet[...] least asymmetric orthogonal wavelet
• With higher settings for the refinement level r, larger scale features are resolved.
• With refinement level r, LiftingWaveletTransform internally pre-pads data so that each dimension is a multiple of . The padding values used for pre-padding are given by the setting of the Padding option.  »
• With refinement level Full, r is given by .
• The default levels of refinement r are given by , where is the integer factorization of the length of data. For multi-dimensional data, the same computation is done for each dimension and the resulting minimum refinement level is used.  »
• The tree of wavelet coefficients at level consists of coarse coefficients and detail coefficients , with representing the input data.
• The dimensions of and are given by , where is given by , where is the input data dimension.  »
• 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
• The settings for Padding include for periodic repetition of the dataset in each dimension and for constant padding.
• With the setting Method, integer data will transform to integer coefficients, in which case input image data of type is converted to type .
Compute a lifting 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 lifting 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   (33)
Compute a lifting wavelet transform:
The resulting DiscreteWaveletData represents a tree of transform coefficients:
The inverse transform reconstructs the input:
Compute an integer lifting wavelet transform:
Use Normal to view all integer 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 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 lifting wavelet transform:
View the tree of wavelet coefficients:
Inverse transform to get back the original signal:
Use WaveletMatrixPlot to visualize the different wavelet coefficients:
In two dimensions, the vector of filtering operations in each direction can be computed:
Interpreting these vectors as binary digit expansions, we get wavelet index numbers:
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 lifting 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 inverse transform in a heirarchical 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:
Browse all coefficients using a MenuView:
LiftingWaveletTransform 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   (6)
By default, lifting wavelet transform maps integers to real values:
Use Normal to view all coefficients:
Perform an inverse:
With Method, integers are mapped to integers:
Display all coefficients:
Use inverse wavelet transform to invert integer lifting:
Treat data as cyclic using padding:
Pad data with 0 to reduce boundary effects on the left:
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   (1)
Preprocess images to improve performance of image processing functions:
Plot transformed image coefficients:
Extract vertical and diagonal components from the wavelet-transformed image and reconstruct:
Threshold image pixel values below 0.5:
With vertical components intensified, apply ImageLines:
DWT is similar to LiftingWaveletTransform with extra coefficients needed for padding:
LiftingWaveletTransform coefficients halve in length with each level of refinement:
Rotated data give different coefficients:
StationaryWaveletTransform coefficients have the same length as the original data:
Rotated data give rotated coefficients:
The default refinement level is given by integer factorization of the length of data:
Odd length data is padded with zeros to make it even:
In higher dimensions:
For a specified refinement level r, input data of length l is padded as Ceiling[l, 2r]-l:
For input data of length 10 with 4 refinement levels, padding lengths are given:
In higher dimensions:
For data with 10 rows and 15 columns with 3 refinement levels, padding in each dimension is given:
Compute dimensions of wavelet coefficients:
The dimensions of wavelet coefficients are given by , where is the input data dimension:
Compare dimensions with coefficient dimensions in dwd:
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:
Image channels are transformed individually:
Combine coefficients of separately transformed image channels:
Compare with coefficient of LiftingWaveletTransform of the original image:
The images are identical:
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