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DiscreteWaveletTransform
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DiscreteWaveletTransform

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gives the discrete wavelet transform (DWT) of an array of data.

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gives the discrete wavelet transform using the wavelet wave.

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gives the discrete wavelet transform using r levels of refinement.

Details and Options

  • DiscreteWaveletTransform gives a DiscreteWaveletData object representing a tree of wavelet coefficient arrays.
  • 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 resulting wavelet coefficients are arrays of the same depth 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 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 . »
  • 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 given by wd_(j+1)=TemplateBox[{{{1, /, 2}, , {(, {{wd, _, j}, +, fl, -, 2}, )}}}, Ceiling], where is the input data dimension and fl is the filter length for the corresponding wspec. »
  • The following options can be given:
  • MethodAutomaticmethod to use
    Padding "Periodic"how to extend data beyond boundaries
    WorkingPrecision MachinePrecisionprecision to use in internal computations
  • The settings for Padding are the same as those available in ArrayPad.
  • InverseWaveletTransform gives the inverse transform.

Examples

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

Compute a discrete wavelet transform using the HaarWavelet:

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In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-zjig9d
Direct link to example
Out[1]=1

Use Normal to view all coefficients:

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In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-s766la
Direct link to example
Out[2]=2

Transform an audio signal:

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In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-c4c4hc
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-i79gyn
Direct link to example
Out[2]=2

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

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In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-bxhoqo
Direct link to example
Out[3]=3

Compute the inverse transform:

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In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-ds1g1n
Direct link to example
Out[4]=4

Transform an Image object:

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In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-jza1au
Direct link to example
Out[1]=1

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

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In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-lg7vnk
Direct link to example
Out[2]=2

Compute the inverse transform:

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In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-xasr81
Direct link to example
Out[3]=3

Scope  (36)Survey of the scope of standard use cases

Basic Uses  (6)

Compute a wavelet transform:

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In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-h8yn3f
Direct link to example
Out[1]=1

The resulting DiscreteWaveletData represents a tree of transform coefficients:

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In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-hic6us
Direct link to example
Out[2]=2

The inverse transform reconstructs the input:

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In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-f3afir
Direct link to example
Out[3]=3

Useful properties can be extracted from the DiscreteWaveletData object:

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In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-f5g3t
Direct link to example
Out[1]=1

Get a full list of properties:

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In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-che078
Direct link to example
Out[2]=2

Get data and coefficient dimensions:

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In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-svmo5
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-dtl7ea
Direct link to example
Out[4]=4

Use Normal to get all wavelet coefficients explicitly:

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In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-4pudeo
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-pngz6t
Direct link to example
Out[2]=2

Also use All as an argument to get all coefficients:

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In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-fy98zc
Direct link to example
Out[3]=3

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

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In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-o4zzzw
Direct link to example
Out[4]=4

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

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In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-8z5pdm
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-7vy1zr
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-dbjopc
Direct link to example
Out[3]=3

Extract specific coefficient arrays:

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In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-limxd1
Direct link to example
Out[4]=4
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-ukcyo5
Direct link to example
Out[5]=5

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

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In[6]:=6
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-bw6a2j
Direct link to example
Out[6]=6

Extract all coefficients whose wavelet indexes match a pattern:

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In[7]:=7
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-57sk78
Direct link to example
Out[7]=7
Copy to clipboard.
In[8]:=8
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-u5etdi
Direct link to example
Out[8]=8

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

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In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-c115g
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-b2d2ce
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-lb6dy9
Direct link to example
Out[3]=3

Use a higher refinement level to increase the frequency resolution:

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In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-e0tfo1
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-d8rad6
Direct link to example
Out[2]=2

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

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In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-qhxfy
Direct link to example
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-b7oh62
Direct link to example
Out[4]=4

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

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In[5]:=5
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-pcyr03
Direct link to example
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-ox23n
Direct link to example
Out[6]=6

Wavelet Families  (10)

Compute the discrete wavelet transform using different wavelet families:

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In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-gavf2f
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-b3h8vd
Direct link to example

Compare the coefficients:

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In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-lu64r
Direct link to example
Out[3]=3

Use different families of wavelets to capture different features:

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In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-xux3yw
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-6jwo5u
Direct link to example
Out[2]=2

HaarWavelet (default):

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In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-nm4160
Direct link to example
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-7up01x
Direct link to example
Out[4]=4

DaubechiesWavelet:

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In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-022m3l
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-5guii
Direct link to example
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-588xab
Direct link to example
Out[3]=3

BattleLemarieWavelet:

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In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-64i349
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-8g8wur
Direct link to example
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-myg47y
Direct link to example
Out[3]=3

BiorthogonalSplineWavelet:

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In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-874w84
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-3itphu
Direct link to example
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-7q8w5p
Direct link to example
Out[3]=3

CoifletWavelet:

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In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-klv9q6
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-7eaj36
Direct link to example
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-kmt3r3
Direct link to example
Out[3]=3

MeyerWavelet:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-q6etlg
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-9nicex
Direct link to example
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-6c9di7
Direct link to example
Out[3]=3

ReverseBiorthogonalSplineWavelet:

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In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-fs79sv
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-gmiqe8
Direct link to example
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-hthxz8
Direct link to example
Out[3]=3

ShannonWavelet:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-ntfv84
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-2ydklr
Direct link to example
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-fe4q6k
Direct link to example
Out[3]=3

SymletWavelet:

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In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-l4xlsf
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-ovf86t
Direct link to example
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-qly5se
Direct link to example
Out[3]=3

Vector Data  (6)

Plot the coefficients over a common horizontal axis using WaveletListPlot:

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In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-gypdz4
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-xm5ld4
Direct link to example
Out[2]=2

Plot against a common vertical axis:

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In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-ez6l51
Direct link to example
Out[3]=3

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

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In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-c1j4n5
Direct link to example

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

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In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-di3ww
Direct link to example
Out[2]=2

Constant data:

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In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-i9a5e9
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-kwq8bn
Direct link to example
Out[2]=2

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

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In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-bfxyna
Direct link to example
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-lji86v
Direct link to example
Out[4]=4

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

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In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-c7n0wl
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-ne0a67
Direct link to example
Out[2]=2

Only the first detail coefficient {1} is not small:

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In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-cxc4yt
Direct link to example
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-nyp7xn
Direct link to example
Out[4]=4

Data with large discontinuities:

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In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-gl9hci
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-fp30
Direct link to example
Out[2]=2

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

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In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-fqn4ul
Direct link to example
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-g46hdy
Direct link to example
Out[4]=4

Detail coefficients are sensitive to discontinuities:

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In[5]:=5
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-bi93r6
Direct link to example
Out[5]=5

Data with both spatial and frequency structure:

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In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-qcsku6
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-c0drk1
Direct link to example
Out[2]=2

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

Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-dot1jc
Direct link to example
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-imnl1l
Direct link to example
Out[4]=4

The first detail coefficient identifies the oscillatory region:

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In[5]:=5
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-72xvt
Direct link to example
Out[5]=5

All coefficients on a common vertical axis:

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In[6]:=6
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-cu61t
Direct link to example
Out[6]=6

Matrix Data  (5)

Compute a two-dimensional discrete wavelet transform:

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In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-gflx0m
Direct link to example
Out[1]=1

View the tree of wavelet coefficients:

Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-q2twv6
Direct link to example
Out[2]=2

Inverse transform to get back the original signal:

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In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-xz2n41
Direct link to example

Use WaveletMatrixPlot to visualize the different wavelet coefficients:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-tj6pke
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-svtxpu
Direct link to example
Out[2]=2

WaveletMatrixPlot of wavelet transform at a higher refinement level:

Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-e9wf0j
Direct link to example
Out[3]=3

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

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-lhs0ib
Direct link to example
Out[1]=1

Interpreting these vectors as binary digit expansions results in wavelet index numbers:

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In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-e2sxk5
Direct link to example
Out[2]=2

Get the lowpass and highpass filters for a Haar wavelet:

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In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-lhi79n
Direct link to example
Out[1]=1

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

Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-dc3ief
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-ljd94e
Direct link to example
Out[3]=3

Wavelet transform of step data:

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In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-hfdje4
Direct link to example

Data with a vertical discontinuity:

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In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-n155zh
Direct link to example
Out[2]=2

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

Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-zxwe7i
Direct link to example
Out[3]=3

Data with horizontal discontinuity:

Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-yv487p
Direct link to example
Out[4]=4

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

Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-basqcr
Direct link to example
Out[5]=5

Data with diagonal discontinuity:

Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-mlwx9y
Direct link to example
Out[6]=6

Only the diagonal detail coefficients, wavelet index {…,3}, are nonzero:

Copy to clipboard.
In[7]:=7
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-xo9kch
Direct link to example
Out[7]=7

Array Data  (2)

Compute a three-dimensional discrete wavelet transform:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-wgoboy
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-f8249l
Direct link to example
Out[2]=2

Tree view of all coefficients:

Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-0ek2ai
Direct link to example
Out[3]=3

Inverse transform to get back the original signal:

Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-gsidhc
Direct link to example
Out[4]=4

Wavelet transform of a three-dimensional cross array:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-q65ymx
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-6gkgva
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-sxjayp
Direct link to example
Out[3]=3

Visualize wavelet coefficients:

Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-gzdpgu
Direct link to example
Out[4]=4

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

Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-57obwt
Direct link to example
Out[5]=5

Image Data  (4)

Transform an Image object:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-gmw99b
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-yzn8qe
Direct link to example
Out[2]=2

The inverse transform yields a reconstructed Image object:

Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-xlen9k
Direct link to example
Out[3]=3

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

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-79t4z
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-lmi7qq
Direct link to example
Out[2]=2

Get all coefficients as Image objects instead:

Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-9oty5
Direct link to example
Out[3]=3

Get raw Image objects with no rescaling of color levels:

Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-ctsnnz
Direct link to example
Out[4]=4

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

Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-hbdon
Direct link to example
Out[5]=5

Plot coefficients used in the inverse transform in a hierarchical grid using WaveletImagePlot:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-ec85oe
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-bxf3qx
Direct link to example
Out[2]=2

Image wavelet coefficients lie outside the valid range of ImageType:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-cwmmu5
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-vxwb7m
Direct link to example

"ImageFunction"->Identity gives an unnormalized image wavelet coefficient:

Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-ip2hqy
Direct link to example
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-gb5d3c
Direct link to example

The color channels lie outside its valid 0 to 1 range:

Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-hvl40e
Direct link to example
Out[5]=5

By default, "ImageFunction"->ImageAdjust is used to normalize coefficients:

Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-e6nuz0
Direct link to example
Copy to clipboard.
In[7]:=7
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-b96e38
Direct link to example

The color channels are now within the valid 0 to 1 range:

Copy to clipboard.
In[8]:=8
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-vns9un
Direct link to example
Out[8]=8

Sound Data  (3)

Transform a Sound object:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-mineo
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-jtcd5x
Direct link to example
Out[2]=2

The inverse transform yields a reconstructed Sound object:

Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-jx08w
Direct link to example
Out[3]=3

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

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-bj4f7m
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-kp8pm
Direct link to example
Out[2]=2

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

Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-hqd4o6
Direct link to example
Out[3]=3

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

Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-d4evuw
Direct link to example
Out[4]=4

Browse all coefficients using a MenuView:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-jhcy39
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-c9wmlc
Direct link to example
Out[2]=2

Generalizations & Extensions  (3)Generalized and extended use cases

DiscreteWaveletTransform works on arrays of symbolic quantities:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-iue08m
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-cqhoa
Direct link to example
Out[2]=2

Inverse transform recovers the input exactly:

Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-zu4y
Direct link to example
Out[3]=3

Specify any internal working precision:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-daoo2p
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-hlsinj
Direct link to example
Out[2]=2

Use complex-valued data:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-noliik
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-bmqscn
Direct link to example
Out[2]=2

The wavelet coefficients are complex:

Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-pma2rw
Direct link to example
Out[3]=3

Inverse transform recovers the input:

Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-l82pnk
Direct link to example
Out[4]=4

Options  (5)Common values & functionality for each option

Padding  (2)

The settings for Padding are the same as the methods for ArrayPad, including "Periodic":

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-dehf7e
Direct link to example
Out[1]=1

"Reversed":

Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-qpkfr
Direct link to example
Out[2]=2

"ReversedNegation":

Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-ldsb8a
Direct link to example
Out[3]=3

"Reflected":

Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-sme52
Direct link to example
Out[4]=4

"ReflectedDifferences":

Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-ntyqgp
Direct link to example
Out[5]=5

"ReversedDifferences":

Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-pojqqw
Direct link to example
Out[6]=6

"Extrapolated":

Copy to clipboard.
In[7]:=7
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-bsohej
Direct link to example
Out[7]=7

Padding can remove boundary effects:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-x52hhy
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-ung771
Direct link to example
Out[2]=2

Using the default "Periodic" padding:

Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-rtqg9j
Direct link to example
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-g93fui
Direct link to example
Out[4]=4

Using "Extrapolated" padding has fewer boundary effects for nonperiodic data:

Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-3alr6e
Direct link to example
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-5q71rx
Direct link to example
Out[6]=6

WorkingPrecision  (3)

By default, WorkingPrecision->MachinePrecision is used:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-btupk0
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-fvcmsi
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-iwm7rm
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-bomt1d
Direct link to example
Out[4]=4

Use higher-precision computation:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-e7oo7e
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-6lkrhl
Direct link to example
Out[2]=2

With numbers close to zero, accuracy is the better indicator of the number of correct digits:

Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-cqyqyd
Direct link to example
Out[3]=3

Use WorkingPrecision->∞ for exact computation:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-m5fvtb
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-5ais8y
Direct link to example
Out[2]=2

Applications  (11)Sample problems that can be solved with this function

Wavelet Compression  (1)

Compress data by finding a representation with few nonzero coefficients:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-6esnso
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-j1b7ln
Direct link to example
Out[2]=2

SymletWavelet[n] has n vanishing moments and represents polynomials of degree n:

Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-1870cg
Direct link to example

Count counts the number of wavelet coefficients close to 0:

Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-v3m8sz
Direct link to example
Out[4]=4

Detect Discontinuities and Edges  (2)

Visualize discontinuities in the wavelet domain:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-i2926z
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-xndaw6
Direct link to example
Out[2]=2

Detail coefficients in the region of discontinuities have larger values:

Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-vobk7q
Direct link to example
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-mtt4yw
Direct link to example
Out[4]=4

Detect edges in an image:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-glhsr6
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-oyzynz
Direct link to example
Out[2]=2

Set coarse coefficients to 0 and reconstruct using detail coefficients only:

Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-tz24k7
Direct link to example
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-f6jjqj
Direct link to example
Out[4]=4

Energy Comparison  (1)

Compare the cumulative energy in a signal, its wavelet coefficients, and Fourier coefficients:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-ghuz1d
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-vrds9l
Direct link to example
Out[2]=2

Compute the ordered cumulative energy in the signal:

Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-d0i4sz
Direct link to example
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-q8ovws
Direct link to example
Out[4]=4

Compute wavelet coefficients and Fourier coefficients:

Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-1n8h94
Direct link to example
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-cdyfw6
Direct link to example

The DWT captures more energy with fewer coefficients than the DFT:

Copy to clipboard.
In[7]:=7
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-dpf8jz
Direct link to example
Out[7]=7

Denoising  (3)

Perform energy-dependent thresholding:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-y01hyh
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-dd9xsz
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-s2redz
Direct link to example

Computing the fraction of energy contained at each refinement level:

Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-efq61x
Direct link to example
Out[4]=4

Set wavelet coefficients containing less than 1% energy to zero:

Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-r4nuw6
Direct link to example
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-fvrhqf
Direct link to example
Copy to clipboard.
In[7]:=7
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-386uh2
Direct link to example
Out[7]=7

Perform an amplitude-dependent thresholding:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-4w1av0
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-83jmoj
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-o7a6zp
Direct link to example

Use WaveletThreshold to perform "Universal" thresholding:

Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-fkwcst
Direct link to example
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-s72xb9
Direct link to example
Out[5]=5

Use Stein's unbiased risk estimator smoothing:

Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-5jmg82
Direct link to example
Copy to clipboard.
In[7]:=7
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-tqi507
Direct link to example
Out[7]=7

Denoise an Image:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-deapzr
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-tol1xp
Direct link to example

Perform "Soft" thresholding with threshold value "SURE" computed adaptively at each level:

Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-faki6e
Direct link to example

Invert thresholded coefficients:

Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-k43fr0
Direct link to example
Out[4]=4

Frequency Filtering  (1)

Wavelet transforms can be used to filter frequencies:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-i7w7q9
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-p6wez
Direct link to example
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-dzkhmb
Direct link to example
Out[3]=3

To filter out the two signals, first perform a wavelet transform:

Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-5bmd19
Direct link to example

Use WaveletListPlot to visualize frequency distribution:

Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-ye4b6c
Direct link to example
Out[5]=5

To filter low frequencies, keep only the coarse coefficients:

Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-cvprwr
Direct link to example
Copy to clipboard.
In[7]:=7
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-f9lost
Direct link to example
Out[7]=7

To filter high frequencies, keep only the detail coefficients:

Copy to clipboard.
In[8]:=8
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-bc096r
Direct link to example
Copy to clipboard.
In[9]:=9
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-nq3kza
Direct link to example
Out[9]=9

Finance  (3)

Extract the stock price trend for IBM since January 1, 2000:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-drjdck
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-c8bnyn
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-ixvjvh
Direct link to example

The trend of the series is captured in the lowpass filter coefficients:

Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-exk77h
Direct link to example
Out[4]=4

Thresholding all detail coefficients and inverting the series gives the trend:

Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-e6tyv4
Direct link to example
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-8n8fif
Direct link to example
Out[6]=6

Detrend a financial series:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-mpggpu
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-4c7u5b
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-yijg5u
Direct link to example
Out[3]=3

Detail coefficients captured the detrended series:

Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-qntazs
Direct link to example
Out[4]=4

Remove the trend by removing the coarse coefficients and inverting:

Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-090if2
Direct link to example
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-gse9p1
Direct link to example
Out[6]=6

Study variance of returns in a financial time series:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-lkmihg
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-5zqr0v
Direct link to example
Out[2]=2

Perform a wavelet transform using HaarWavelet and SymletWavelet:

Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-q5bl6e
Direct link to example

Since the GE return series does not exhibit low-frequency oscillations, higher-scale detail coefficients do not indicate large variations from zero:

Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-x8spio
Direct link to example
Out[4]=4

Although both filters will capture the variance of the series, they distribute it differently because of their approximate bandpass properties:

Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-v0bxk
Direct link to example
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-ede51
Direct link to example

SymletWavelet isolates features in a certain frequency interval better than HaarWavelet:

Copy to clipboard.
In[7]:=7
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-0m1lgq
Direct link to example
Out[7]=7

Properties & Relations  (15)Properties of the function, and connections to other functions

DiscreteWaveletPacketTransform computes the full tree of wavelet coefficients:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-om6men
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-1zexfr
Direct link to example
Out[2]=2

DiscreteWaveletTransform computes a subset of the full tree of coefficients:

Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-r42jth
Direct link to example
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-vlpb84
Direct link to example
Out[4]=4

DiscreteWaveletTransform coefficients halve in length with each level of refinement:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-72op15
Direct link to example
Out[1]=1

Rotated data gives different coefficients:

Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-nswego
Direct link to example
Out[2]=2

StationaryWaveletTransform coefficients have the same length as the original data:

Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-zer7un
Direct link to example
Out[3]=3

Rotated data gives rotated coefficients:

Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-uovhur
Direct link to example
Out[4]=4

Multidimensional discrete wavelet transform is related to one-dimensional packet transform:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-ybflpq
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-wddau6
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-9afbpd
Direct link to example
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-6n5odr
Direct link to example
Out[4]=4

For Haar wavelet (default) and data length , the computed coefficients are identical:

Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-l7z28z
Direct link to example
Out[5]=5

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

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-nvnob0
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-k74t7s
Direct link to example
Out[2]=2
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-ccu2mo
Direct link to example
Out[3]=3

In higher dimensions:

Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-dzom29
Direct link to example
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-nqp5e2
Direct link to example
Out[5]=5
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-lsny6n
Direct link to example
Out[6]=6

The energy norm is conserved for orthogonal wavelet families:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-hh4od4
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-z58fpr
Direct link to example
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-0j3fcn
Direct link to example
Out[3]=3

The energy norm is approximately conserved for biorthogonal wavelet families:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-ftqzwu
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-uqfzo
Direct link to example
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-2wyhvw
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-cvwt6v
Direct link to example
Out[4]=4

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

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-jsrwt6
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-esuqos
Direct link to example

Extract the coefficient for the maximum refinement level:

Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-rn4mma
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-2firkb
Direct link to example
Out[4]=4

Compensate for the normalization at each refinement level:

Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-tl1xr4
Direct link to example
Out[5]=5
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-yoic8t
Direct link to example
Out[6]=6

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

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-71wxdw
Direct link to example
Out[1]=1
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-qm9cuf
Direct link to example
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-islxwj
Direct link to example
Out[3]=3

Individually inverse transform each wavelet coefficient array:

Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-qadvnh
Direct link to example
Out[4]=4
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-jvt70z
Direct link to example
Out[5]=5
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-ov8xp2
Direct link to example
Out[6]=6

The sum gives the original data:

Copy to clipboard.
In[7]:=7
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-fswsdm
Direct link to example
Out[7]=7

Compute discrete wavelet coefficients for periodic data:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-81usyq
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-c6pxyh
Direct link to example
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-8059v1
Direct link to example

Define filter coefficients to have compact support:

Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-8xpc58
Direct link to example
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-yrprzl
Direct link to example

Coarse coefficients at level are given by , with :

Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-9iulp7
Direct link to example
Copy to clipboard.
In[7]:=7
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-0usjr
Direct link to example
Copy to clipboard.
In[8]:=8
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-8kw0h5
Direct link to example
Out[8]=8

Detail coefficients at level are given by :

Copy to clipboard.
In[9]:=9
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-l1w7i4
Direct link to example
Copy to clipboard.
In[10]:=10
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-c7zdu6
Direct link to example
Copy to clipboard.
In[11]:=11
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-wl5c72
Direct link to example
Out[11]=11

Compute a partial discrete inverse wavelet transform:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-knxsvp
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-115wba
Direct link to example
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-ut3bvb
Direct link to example

Define filter coefficients to have compact support:

Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-upz69t
Direct link to example
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-jcggoi
Direct link to example

Coarse coefficients at level are given:

Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-0y4m9q
Direct link to example
Copy to clipboard.
In[7]:=7
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-kubtpu
Direct link to example

Detail coefficients at level are given:

Copy to clipboard.
In[8]:=8
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-78x6bs
Direct link to example
Copy to clipboard.
In[9]:=9
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-oo7c0k
Direct link to example

Inverse wavelet transform at level is given by :

Copy to clipboard.
In[10]:=10
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-66znis
Direct link to example
Copy to clipboard.
In[11]:=11
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-bnpxaq
Direct link to example

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

Copy to clipboard.
In[12]:=12
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-t9kl1n
Direct link to example
Out[12]=12

Reconstruct coarse coefficients {0} at refinement level :

Copy to clipboard.
In[13]:=13
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-vxh7ou
Direct link to example
Out[13]=13

Compute the dimensions of wavelet coefficients:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-59looe
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-5vlm1c
Direct link to example

At refinement level , the dimensions of wavelet coefficients are given by wd_(j+1)=TemplateBox[{{{1, /, 2}, , {(, {{wd, _, j}, +, fl, -, 2}, )}}}, Ceiling], where represents dimensions of input data:

Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-8cklxm
Direct link to example
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-6tsyjd
Direct link to example

Compare dimensions with coefficient dimensions in dwd:

Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-6aog5d
Direct link to example
Out[5]=5
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-xlpkkm
Direct link to example
Out[6]=6

Compute a Haar discrete wavelet transform in one dimension:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-kywxlf
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-f0nc2a
Direct link to example

Compute {0} and {1} wavelet coefficients:

Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-bzen0a
Direct link to example
Out[3]=3

Compare with DiscreteWaveletTransform:

Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-zcrryy
Direct link to example
Out[4]=4

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

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-cq5vc4
Direct link to example

Lowpass and highpass filters for Haar wavelet:

Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-bzl44r
Direct link to example

Haar wavelet transform of matrix data:

Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-5n59bt
Direct link to example
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-6zl4c
Direct link to example
Out[4]=4

Compare with DiscreteWaveletTransform using HaarWavelet:

Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-maqks4
Direct link to example
Copy to clipboard.
In[6]:=6
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-s166s
Direct link to example
Out[6]=6

Image channels are transformed individually:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-gfhws8
Direct link to example
Out[1]=1

Combine {0} coefficients of separately transformed image channels:

Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-kh8lp
Direct link to example
Out[2]=2

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

Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-t91ou
Direct link to example
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-byduzg
Direct link to example
Out[4]=4

The images are identical:

Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-6hevx
Direct link to example
Out[5]=5

DWT is similar to LiftingWaveletTransform with extra coefficients needed for padding:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-vk5frs
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-x4f7ks
Direct link to example
Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-3m2nag
Direct link to example
Out[3]=3
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-15s3zl
Direct link to example
Out[4]=4

Possible Issues  (1)Common pitfalls and unexpected behavior

Padding can affect the total energy of wavelet coefficients:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-hkzuox
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-bgqoqz
Direct link to example

Energy is not conserved:

Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-ir1xix
Direct link to example
Out[3]=3

Pad with 0s to ensure energy conservation in the coefficients:

Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-qvgvx8
Direct link to example
Copy to clipboard.
In[5]:=5
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-0zly8
Direct link to example
Out[5]=5

Neat Examples  (1)Surprising or curious use cases

Create a padded matrix of data:

Copy to clipboard.
In[1]:=1
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-qpxv4y
Direct link to example
Copy to clipboard.
In[2]:=2
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-rnzvdf
Direct link to example
Out[2]=2

Create a 3D plot of the Haar DWT coefficients:

Copy to clipboard.
In[3]:=3
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-oan0i9
Direct link to example
Copy to clipboard.
In[4]:=4
✖
https://wolfram.com/xid/0d4ktomcm5gltdhu-qbabha
Direct link to example
Out[4]=4
Wolfram Research (2010), DiscreteWaveletTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/DiscreteWaveletTransform.html (updated 2017).
Copy to clipboard.
✖
Wolfram Research (2010), DiscreteWaveletTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/DiscreteWaveletTransform.html (updated 2017).

Text

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

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

CMS

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

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

APA

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

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

BibTeX

@misc{reference.wolfram_2025_discretewavelettransform, author="Wolfram Research", title="{DiscreteWaveletTransform}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/DiscreteWaveletTransform.html}", note=[Accessed: 05-April-2025 ]}

Copy to clipboard.
✖
@misc{reference.wolfram_2025_discretewavelettransform, author="Wolfram Research", title="{DiscreteWaveletTransform}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/DiscreteWaveletTransform.html}", note=[Accessed: 05-April-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_discretewavelettransform, organization={Wolfram Research}, title={DiscreteWaveletTransform}, year={2017}, url={https://reference.wolfram.com/language/ref/DiscreteWaveletTransform.html}, note=[Accessed: 05-April-2025 ]}

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@online{reference.wolfram_2025_discretewavelettransform, organization={Wolfram Research}, title={DiscreteWaveletTransform}, year={2017}, url={https://reference.wolfram.com/language/ref/DiscreteWaveletTransform.html}, note=[Accessed: 05-April-2025 ]}