StationaryWaveletTransform

StationaryWaveletTransform[data]

gives the stationary wavelet transform (SWT) of an array of data.

StationaryWaveletTransform[data,wave]

gives the stationary wavelet transform using the wavelet wave.

StationaryWaveletTransform[data,wave,r]

gives the stationary wavelet transform using r levels of refinement.

Details and Options

  • StationaryWaveletTransform is similar to DiscreteWaveletTransform except that no subsampling occurs at any refinement level and the resulting coefficient arrays all have the same dimensions as the original data.
  • StationaryWaveletTransform 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 any of the following:
  • listarbitrary-rank numerical array
    imagearbitrary Image object
    audioan Audio or sampled Sound object
  • The possible wavelets wave include:
  • BattleLemarieWavelet[]BattleLemarié 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
  • The default wave is HaarWavelet[].
  • With higher settings for the refinement level r, larger-scale features are resolved.
  • The default refinement level r is given by TemplateBox[{{{InterpretationBox[{log, _, DocumentationBuild`Utils`Private`Parenth[2]}, Log2, AutoDelete -> True], (, n, )}, +, {1, /, 2}}}, Floor], 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 lowpass filter coefficients and are highpass 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:
  • MethodAutomaticmethod to use
    WorkingPrecisionMachinePrecisionprecision to use in internal computations
  • StationaryWaveletTransform uses periodic padding of data.
  • InverseWaveletTransform gives the inverse transform.

Examples

open allclose all

Basic Examples  (3)

Compute a stationary wavelet transform using the HaarWavelet:

Use Normal to view all coefficients:

Transform an audio signal:

Use dwd[,"Audio"] to extract coefficient signals:

Verify lengths of all coefficient signals:

Compute the inverse transform:

Transform an Image object:

Use dwd[,"Image"] to extract coefficient images:

Compute the inverse transform:

Scope  (34)

Basic Uses  (6)

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 "TreeView" or "IndexMap" 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 {0,0,0}:

With further refinement, {0,0,0} is resolved into further components:

Wavelet Families  (10)

Compute the stationary wavelet transform using different wavelet families:

Compare the coefficients:

Use different families of wavelets to capture different features:

HaarWavelet (default):

DaubechiesWavelet:

BattleLemarieWavelet:

BiorthogonalSplineWavelet:

CoifletWavelet:

MeyerWavelet:

ReverseBiorthogonalSplineWavelet:

ShannonWavelet:

SymletWavelet:

Vector Data  (6)

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 {0,0,}:

Data oscillating at the highest resolvable frequency (Nyquist frequency):

Only the first detail coefficient {1} is nonzero:

Data with large discontinuities:

Coarse coefficients {0,} have the same large-scale structure as the data:

Detail coefficients are sensitive to discontinuities:

Data with both spatial and frequency structure:

Coarse coefficients {0,} track the local mean of the data:

The first detail coefficient identifies the oscillatory region:

All coefficients on a common vertical axis:

Matrix Data  (5)

Compute a two-dimensional stationary wavelet transform:

View the tree of wavelet coefficients:

Inverse transform to get back the original signal:

Use dwd[,"MatrixPlot"] 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 lowpass and highpass 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 {,1}, are nonzero:

Data with horizontal discontinuity:

Only the horizontal detail coefficients, wavelet index {,2}, are nonzero:

Array Data  (2)

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:

Image Data  (2)

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 {0,1} coefficient as an Image object:

Sound Data  (3)

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 {0,1} coefficient as a Sound object:

Inverse transform of {0,0,1} coefficient as a Sound object:

Browse all coefficients using a MenuView:

Generalizations & Extensions  (3)

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)

WorkingPrecision  (3)

By default, WorkingPrecision->MachinePrecision is used:

Use higher-precision computation:

Use WorkingPrecision-> for exact computation:

Applications  (3)

Inverse Halftoning  (1)

A simple wavelet-based inverse halftoning:

Apply GaussianFilter on the detail coefficients:

Numerical Differentiation  (1)

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 Wolfram Language numerical differentiation:

Image Fusion  (1)

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:

Properties & Relations  (12)

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 TemplateBox[{{{InterpretationBox[{log, _, DocumentationBuild`Utils`Private`Parenth[2]}, Log2, AutoDelete -> True], (, n, )}, +, {1, /, 2}}}, Floor]:

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 {0,0} at refinement level :

Reconstruct coarse coefficients {0} at refinement level :

Compute a Haar stationary wavelet transform in one dimension:

Compute {0} and {1} wavelet coefficients:

Compare with DiscreteWaveletPacketTransform:

In two dimensions, a separate filter is applied in each dimension:

Lowpass and highpass filters for a Haar wavelet:

Haar wavelet transform of matrix data:

Compare with DiscreteWaveletPacketTransform using HaarWavelet:

Image channels are transformed individually:

Combine {0} coefficients of separately transformed image channels:

Compare with {0} coefficient of StationaryWaveletTransform of the original image:

The images are identical:

Introduced in 2010
 (8.0)
 |
Updated in 2017
 (11.2)