BUILT-IN MATHEMATICA SYMBOL

StationaryWaveletTransform

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

gives the stationary wavelet transform using the wavelet wave.

StationaryWaveletTransform

gives the stationary wavelet transform using r levels of refinement.

StationaryWaveletTransform

gives the stationary wavelet transform of an image.

StationaryWaveletTransform

gives the stationary wavelet transform of sampled sound.

MORE INFORMATION

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"].

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.

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

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 , 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

StationaryWaveletTransform uses periodic padding of data.

InverseWaveletTransform gives the inverse transform.

EXAMPLES

CLOSE ALL

Basic Examples (3)

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:

In[1]:=

Out[1]=

Use Normal to view all coefficients:

In[2]:=

Out[2]=

Transform an Image object:

In[1]:=

Out[1]=

Use

to extract coefficient images:

In[2]:=

Out[2]=

Compute the inverse transform:

In[3]:=

Out[3]=

Transform a sampled Sound object:

In[1]:=

Out[1]=

In[2]:=

Out[2]=

In[3]:=

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):

DaubechiesWavelet:

BattleLemarieWavelet:

BiorthogonalSplineWavelet:

CoifletWavelet:

MeyerWavelet:

ReverseBiorthogonalSplineWavelet:

ShannonWavelet:

SymletWavelet:

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:

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)

By default, WorkingPrecision->MachinePrecision 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:

Properties & Relations (12)

(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

:

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:

Compare with DiscreteWaveletPacketTransform:

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:

Compare with DiscreteWaveletPacketTransform using HaarWavelet:

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: