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StationaryWaveletTransform
✖
StationaryWaveletTransform

✖

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

✖

gives the stationary wavelet transform using the wavelet wave.

✖

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[…]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 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
    WorkingPrecision MachinePrecisionprecision to use in internal computations
  • StationaryWaveletTransform uses periodic padding of data.
  • InverseWaveletTransform gives the inverse transform.

Examples

open allclose all

Basic Examples  (3)Summary of the most common use cases

Compute a stationary wavelet transform using the HaarWavelet:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-zjig9d
Out[1]=1

Use Normal to view all coefficients:

In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-s766la
Out[2]=2

Transform an audio signal:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-c4c4hc
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-i79gyn
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-bxhoqo
Out[3]=3

Verify lengths of all coefficient signals:

In[4]:=4
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-qrq5z9
Out[4]=4

Compute the inverse transform:

In[5]:=5
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-ds1g1n
Out[5]=5

Transform an Image object:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-jza1au
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-lg7vnk
Out[2]=2

Compute the inverse transform:

In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-xasr81
Out[3]=3

Scope  (34)Survey of the scope of standard use cases

Basic Uses  (6)

Compute a stationary wavelet transform:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-h8yn3f
Out[1]=1

The resulting DiscreteWaveletData represents a tree of transform coefficients:

In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-hic6us
Out[2]=2

The inverse transform reconstructs the input:

In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-f3afir
Out[3]=3

Useful properties can be extracted from the DiscreteWaveletData object:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-f5g3t
Out[1]=1

Get a full list of properties:

In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-che078
Out[2]=2

Get data and coefficient dimensions:

In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-svmo5
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-dtl7ea
Out[4]=4

Use Normal to get all wavelet coefficients explicitly:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-4pudeo
In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-pngz6t
Out[2]=2

Also use All as an argument to get all coefficients:

In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-fy98zc
Out[3]=3

Use Automatic to get only the coefficients used in the inverse transform:

In[4]:=4
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-o4zzzw
Out[4]=4

Use the "TreeView" or "IndexMap" to find out what wavelet coefficients are available:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-8z5pdm
In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-7vy1zr
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-dbjopc
Out[3]=3

Extract specific coefficient arrays:

In[4]:=4
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-limxd1
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-ukcyo5
Out[5]=5

Extract several wavelet coefficients corresponding to the list of wavelet index specifications:

In[6]:=6
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-bw6a2j
Out[6]=6

Extract all coefficients whose wavelet indexes match a pattern:

In[7]:=7
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-57sk78
Out[7]=7
In[8]:=8
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-u5etdi
Out[8]=8

The Automatic coefficients are used by default in functions like WaveletListPlot:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-c115g
In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-b2d2ce
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-lb6dy9
Out[3]=3

Use a higher refinement level to increase the frequency resolution:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-e0tfo1
In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-d8rad6
Out[2]=2

With a smaller refinement level, more of the signal energy is left in {0,0,0}:

In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-qhxfy
In[4]:=4
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-b7oh62
Out[4]=4

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

In[5]:=5
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-pcyr03
In[6]:=6
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-ox23n
Out[6]=6

Wavelet Families  (10)

Compute the stationary wavelet transform using different wavelet families:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-gavf2f
In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-b3h8vd

Compare the coefficients:

In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-lu64r
Out[3]=3

Use different families of wavelets to capture different features:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-iax26u
In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-6jwo5u
Out[2]=2

HaarWavelet (default):

In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-nm4160
In[4]:=4
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-7up01x
Out[4]=4

DaubechiesWavelet:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-022m3l
In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-5guii
In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-588xab
Out[3]=3

BattleLemarieWavelet:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-64i349
In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-8g8wur
In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-myg47y
Out[3]=3

BiorthogonalSplineWavelet:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-874w84
In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-3itphu
In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-7q8w5p
Out[3]=3

CoifletWavelet:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-klv9q6
In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-7eaj36
In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-kmt3r3
Out[3]=3

MeyerWavelet:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-q6etlg
In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-9nicex
In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-6c9di7
Out[3]=3

ReverseBiorthogonalSplineWavelet:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-fs79sv
In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-gmiqe8
In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-hthxz8
Out[3]=3

ShannonWavelet:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-ntfv84
In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-2ydklr
In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-fe4q6k
Out[3]=3

SymletWavelet:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-l4xlsf
In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-ovf86t
In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-qly5se
Out[3]=3

Vector Data  (6)

Plot the coefficients over a common horizontal axis using WaveletListPlot:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-gypdz4
In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-xm5ld4
Out[2]=2

Plot against a common vertical axis:

In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-ez6l51
Out[3]=3

Visualize coefficients as a function of time and refinement level using WaveletScalogram:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-c1j4n5

The coefficient indexes appear as tooltips when the mouse pointer is moved over a coefficient:

In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-di3ww
Out[2]=2

Constant data:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-i9a5e9
In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-kwq8bn
Out[2]=2

All coefficients are small except coarse coefficients {0,0,…}:

In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-bfxyna
In[4]:=4
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-lji86v
Out[4]=4

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

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-c7n0wl
In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-ne0a67
Out[2]=2

Only the first detail coefficient {1} is nonzero:

In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-cxc4yt
In[4]:=4
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-nyp7xn
Out[4]=4

Data with large discontinuities:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-gl9hci
In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-fp30
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-fqn4ul
In[4]:=4
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-g46hdy
Out[4]=4

Detail coefficients are sensitive to discontinuities:

In[5]:=5
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-bi93r6
Out[5]=5

Data with both spatial and frequency structure:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-qcsku6
In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-c0drk1
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-dot1jc
In[4]:=4
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-imnl1l
Out[4]=4

The first detail coefficient identifies the oscillatory region:

In[5]:=5
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-72xvt
Out[5]=5

All coefficients on a common vertical axis:

In[6]:=6
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-gxq0kb
Out[6]=6

Matrix Data  (5)

Compute a two-dimensional stationary wavelet transform:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-gflx0m
Out[1]=1

View the tree of wavelet coefficients:

In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-q2twv6
Out[2]=2

Inverse transform to get back the original signal:

In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-xz2n41

Use dwd[…,"MatrixPlot"] to visualize each coefficient as a MatrixPlot:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-tj6pke
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-svtxpu
Out[2]=2

Visualize wavelet coefficients at higher refinement levels:

In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-c7lg5h
In[4]:=4
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-ldtwbr
Out[4]=4

In two dimensions, the vector of filtering operations in each direction can be computed:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-lhs0ib
Out[1]=1

Interpreting these vectors as binary digit expansions, you get wavelet index numbers:

In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-e2sxk5
Out[2]=2

Get the lowpass and highpass filters for a Haar wavelet:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-lhi79n
Out[1]=1

The resulting 2D filters are outer products of filters in the two directions:

In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-dc3ief
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-ljd94e
Out[3]=3

Wavelet transform of step data:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-hfdje4

Data with a vertical discontinuity:

In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-n155zh
Out[2]=2

Only the vertical detail coefficients, wavelet index {…,1}, are nonzero:

In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-fcnbpi
Out[3]=3

Data with horizontal discontinuity:

In[4]:=4
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-yv487p
Out[4]=4

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

In[5]:=5
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-bpz7u8
Out[5]=5

Array Data  (2)

Compute a three-dimensional stationary wavelet transform:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-wgoboy
In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-f8249l
Out[2]=2

Tree view of all coefficients:

In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-0ek2ai
Out[3]=3

Inverse transform to get back the original signal:

In[4]:=4
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-gsidhc
Out[4]=4

Wavelet transform of a three-dimensional cross array:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-q65ymx
In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-6gkgva
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-sxjayp
Out[3]=3

Visualize wavelet coefficients:

In[4]:=4
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-gzdpgu
Out[4]=4

Energy of the original data is conserved within the transformed coefficients:

In[5]:=5
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-57obwt
Out[5]=5

Image Data  (2)

Transform an Image object:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-gmw99b
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-yzn8qe
Out[2]=2

The inverse transform yields a reconstructed Image object:

In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-xlen9k
Out[3]=3

Wavelet coefficients are normally given as arrays of data for each image channel:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-79t4z
In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-lmi7qq
Out[2]=2

Number of channels and dimensions of the original image are the same:

In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-kqf5in
Out[3]=3

Get all coefficients as Image objects instead of arrays of data:

In[4]:=4
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-9oty5
Out[4]=4

Get raw Image objects with no rescaling of color levels:

In[5]:=5
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-ctsnnz
Out[5]=5

Get the inverse transform of the {0,1} coefficient as an Image object:

In[6]:=6
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-hbdon
Out[6]=6

Sound Data  (3)

Transform a Sound object:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-mineo
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-jtcd5x
Out[2]=2

The inverse transform yields a reconstructed Sound object:

In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-jx08w
Out[3]=3

By default, coefficients are given as lists of data for each sound channel:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-bj4f7m
In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-kp8pm
Out[2]=2

Number of channels and data length in the original sound are the same:

In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-chexw1
Out[3]=3

Get the {0,1} coefficient as a Sound object:

In[4]:=4
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-hqd4o6
Out[4]=4

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

In[5]:=5
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-d4evuw
Out[5]=5

Browse all coefficients using a MenuView:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-jhcy39
In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-c9wmlc
Out[2]=2

Generalizations & Extensions  (3)Generalized and extended use cases

StationaryWaveletTransform works on arrays of symbolic quantities:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-iue08m
In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-cqhoa
Out[2]=2

Inverse transform recovers the input exactly:

In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-zu4y
Out[3]=3

Specify any internal working precision:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-daoo2p
In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-hlsinj
Out[2]=2

Use complex-valued data:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-noliik
In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-bmqscn
Out[2]=2

The wavelets coefficients are complex:

In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-pma2rw
Out[3]=3

Inverse transform recovers the input:

In[4]:=4
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-l82pnk
Out[4]=4

Options  (3)Common values & functionality for each option

WorkingPrecision  (3)

By default, WorkingPrecision->MachinePrecision is used:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-btupk0
In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-fvcmsi
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-iwm7rm
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-bomt1d
Out[4]=4

Use higher-precision computation:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-e7oo7e
In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-6lkrhl
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-cqyqyd
Out[3]=3

Use WorkingPrecision->∞ for exact computation:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-m5fvtb
In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-5ais8y
Out[2]=2

Applications  (3)Sample problems that can be solved with this function

Inverse Halftoning  (1)

A simple wavelet-based inverse halftoning:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-1xzz51
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-6cn82n

Apply GaussianFilter on the detail coefficients:

In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-1hv5q1
Out[3]=3

Numerical Differentiation  (1)

Differentiate noisy data using wavelet transform:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-08loxn
In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-lg248n
In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-7wollf
Out[3]=3

Translation-Rotation-Transform (TRT) is used to reduce boundary effects by subtracting a linear component from the input signal:

In[4]:=4
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-9i8iun
In[5]:=5
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-0e9yv5
In[6]:=6
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-w2krwf

Since HaarWavelet has one vanishing moment, choose it to perform a wavelet transform on :

In[7]:=7
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-5v5pvi
In[8]:=8
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-yhyx6q

Detail coefficients give the differentiation of the data. Coefficients at refinement level 4 are chosen to minimize noise:

In[9]:=9
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-le9d3k

Rescale the differentiated values:

In[10]:=10
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-8yk1yj
In[11]:=11
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-hyr1f

Compare wavelet-based numerical differentiation with exact differentiation:

In[12]:=12
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-hm2ezh
Out[12]=12

Compare with standard Wolfram Language numerical differentiation:

In[13]:=13
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-842g6d
In[14]:=14
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-orceoc
Out[14]=14

Image Fusion  (1)

Add texture to an existing image:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-smhiig

Perform wavelet transform on both images:

In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-tzko9p
In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-b10r6p

Combine detail coefficients of the two images by taking their mean:

In[4]:=4
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-3bn18q
In[5]:=5
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-4yojav
In[6]:=6
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-vw7m1m
In[7]:=7
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-mzgj7z
In[8]:=8
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-wrzk24

Append the coarse coefficient of the first image:

In[9]:=9
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-gvwllb

Construct a new DiscreteWaveletData of the combined wavelet coefficients:

In[10]:=10
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-lq3bzd
Out[10]=10

Reconstruct the combined image:

In[11]:=11
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-fhr7kh
Out[11]=11

Properties & Relations  (12)Properties of the function, and connections to other functions

StationaryWaveletPacketTransform computes the full tree of wavelet coefficients:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-cgjhd3
In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-e4t1li
Out[2]=2

StationaryWaveletTransform computes a subset of the full tree of coefficients:

In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-p3dq
In[4]:=4
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-wochi
Out[4]=4

DiscreteWaveletTransform coefficients halve in length with each level of refinement:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-geoy4j
Out[1]=1

Rotated data gives different coefficients:

In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-fspmhk
Out[2]=2

StationaryWaveletTransform coefficients have the same length as the original data:

In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-ce5j56
Out[3]=3

Rotated data gives rotated coefficients:

In[4]:=4
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-f9p8kx
Out[4]=4

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

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-nvnob0
In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-k74t7s
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-ccu2mo
Out[3]=3

In higher dimensions:

In[4]:=4
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-dzom29
In[5]:=5
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-nqp5e2
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-lsny6n
Out[6]=6

The energy norm is conserved for orthogonal wavelet families:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-40jjuu
In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-fgbdya
In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-ks8c24
Out[3]=3

The energy norm is approximately conserved for biorthogonal wavelet families:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-ihj98y
In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-hz2t90
In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-etc633
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-bm042g
Out[4]=4

The mean of the data is captured at the maximum refinement level of the transform:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-zu45nt
In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-7tclnf

Extract the coefficient for the maximum refinement level:

In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-hqx9zn
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-t3qkc
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-5slunz
Out[5]=5

The sum of inverse transforms from individual coefficient arrays gives the original data:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-8csip2
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-uwqql4
In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-sacrp
Out[3]=3

Individually inverse transform each wavelet coefficient array:

In[4]:=4
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-f3ljg7
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-bh52zs
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-u8p9i
Out[6]=6

The sum gives the original data:

In[7]:=7
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-fuhvs4
Out[7]=7

Compute stationary wavelet coefficients for periodic data:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-81usyq
In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-c6pxyh
In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-8059v1

Compute filter coefficients:

In[4]:=4
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-8xpc58
In[5]:=5
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-nbr339

Coarse coefficients at level are given by :

In[6]:=6
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-nmsfh7
In[7]:=7
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-iqmt4l
In[8]:=8
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-8kw0h5
Out[8]=8

Detail coefficients at level are given by :

In[9]:=9
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-ofmhs6
In[10]:=10
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-ccnq2x
In[11]:=11
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-wl5c72
Out[11]=11

Compute partial stationary inverse wavelet transform:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-knxsvp
In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-115wba
In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-u99tzo

Compute filter coefficients:

In[4]:=4
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-hive0q
In[5]:=5
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-rsdr3f

Coarse coefficients at level are given:

In[6]:=6
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-24erqj
In[7]:=7
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-moxip

Detail coefficients at level are given:

In[8]:=8
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-qejkrz
In[9]:=9
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-kpgqsd

Inverse wavelet transform at level is given by :

In[10]:=10
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-xu196g
In[11]:=11
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-j987r4

Reconstruct coarse coefficients {0,0} at refinement level :

In[12]:=12
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-t9kl1n
Out[12]=12

Reconstruct coarse coefficients {0} at refinement level :

In[13]:=13
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-vxh7ou
Out[13]=13

Compute a Haar stationary wavelet transform in one dimension:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-kywxlf
In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-f0nc2a

Compute {0} and {1} wavelet coefficients:

In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-bzen0a
Out[3]=3

Compare with DiscreteWaveletPacketTransform:

In[4]:=4
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-zcrryy
Out[4]=4

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

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-cq5vc4

Lowpass and highpass filters for a Haar wavelet:

In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-bzl44r

Haar wavelet transform of matrix data:

In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-5n59bt
In[4]:=4
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-6zl4c
Out[4]=4

Compare with DiscreteWaveletPacketTransform using HaarWavelet:

In[5]:=5
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-maqks4
In[6]:=6
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-s166s
Out[6]=6

Image channels are transformed individually:

In[1]:=1
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-gfhws8

Combine {0} coefficients of separately transformed image channels:

In[2]:=2
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-kh8lp
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-t91ou
In[4]:=4
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-byduzg
Out[4]=4

The images are identical:

In[5]:=5
✖
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-kbuh9g
Out[5]=5
Wolfram Research (2010), StationaryWaveletTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/StationaryWaveletTransform.html (updated 2017).
✖
Wolfram Research (2010), StationaryWaveletTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/StationaryWaveletTransform.html (updated 2017).

Text

Wolfram Research (2010), StationaryWaveletTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/StationaryWaveletTransform.html (updated 2017).

✖
Wolfram Research (2010), StationaryWaveletTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/StationaryWaveletTransform.html (updated 2017).

CMS

Wolfram Language. 2010. "StationaryWaveletTransform." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2017. https://reference.wolfram.com/language/ref/StationaryWaveletTransform.html.

✖
Wolfram Language. 2010. "StationaryWaveletTransform." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2017. https://reference.wolfram.com/language/ref/StationaryWaveletTransform.html.

APA

Wolfram Language. (2010). StationaryWaveletTransform. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/StationaryWaveletTransform.html

✖
Wolfram Language. (2010). StationaryWaveletTransform. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/StationaryWaveletTransform.html

BibTeX

@misc{reference.wolfram_2025_stationarywavelettransform, author="Wolfram Research", title="{StationaryWaveletTransform}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/StationaryWaveletTransform.html}", note=[Accessed: 29-March-2025 ]}

✖
@misc{reference.wolfram_2025_stationarywavelettransform, author="Wolfram Research", title="{StationaryWaveletTransform}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/StationaryWaveletTransform.html}", note=[Accessed: 29-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_stationarywavelettransform, organization={Wolfram Research}, title={StationaryWaveletTransform}, year={2017}, url={https://reference.wolfram.com/language/ref/StationaryWaveletTransform.html}, note=[Accessed: 29-March-2025 ]}

✖
@online{reference.wolfram_2025_stationarywavelettransform, organization={Wolfram Research}, title={StationaryWaveletTransform}, year={2017}, url={https://reference.wolfram.com/language/ref/StationaryWaveletTransform.html}, note=[Accessed: 29-March-2025 ]}