BUILT-IN MATHEMATICA SYMBOL

LiftingWaveletTransform

LiftingWaveletTransform[data]
gives the lifting wavelet transform (LWT) of an array of data.

LiftingWaveletTransform[data, wave]
gives the lifting wavelet transform using the wavelet wave.

LiftingWaveletTransform[data, wave, r]
gives the lifting wavelet transform using r levels of refinement.

LiftingWaveletTransform[image, ...]
gives the lifting wavelet transform of an image.

LiftingWaveletTransform[sound, ...]
gives the lifting wavelet transform of sampled sound.

Details and Options Details and Options

LiftingWaveletTransform gives a DiscreteWaveletData object.
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

The default is HaarWavelet[].
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 $TemplateBox[{{{InterpretationBox[{log, _, DocumentationBuild`Utils`Private`Parenth[2]}, Log2, AutoDelete -> True], (, n, )}, +, {1, /, 2}}}, Floor]$ .
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 $2^r TemplateBox[{{l, /, {2, ^, r}}}, Ceiling]$ , 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->"IntegerLifting", integer data will transform to integer coefficients, in which case input image data of type is converted to type .
InverseWaveletTransform gives the inverse transform.