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StationaryWaveletPacketTransform
✖
StationaryWaveletPacketTransform

✖

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

✖

gives the stationary wavelet packet transform using the wavelet wave.

✖

gives the stationary wavelet packet transform using r levels of refinement.

Details and Options

Examples

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

Compute a stationary wavelet packet transform:

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-5h58m
Out[1]=1

The resulting DiscreteWaveletData represents a full tree of wavelet coefficients:

In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-u3lci
Out[2]=2

The inverse transform reconstructs the input:

In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-f3afir
Out[3]=3

Transform an audio signal:

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-c4c4hc
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-i79gyn
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-bxhoqo
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-bdb5ev
Out[4]=4

Compute the inverse transform:

In[5]:=5
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-ds1g1n
Out[5]=5

Transform an Image object:

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-jza1au
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-lg7vnk
Out[2]=2

Compute the inverse transform:

In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-xasr81
Out[3]=3

Scope  (33)Survey of the scope of standard use cases

Basic Uses  (4)

Useful properties can be extracted from the DiscreteWaveletData object:

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-imvzqi
Out[1]=1

Get a full list of properties:

In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-1bqqmu
Out[2]=2

Get data and coefficient dimensions:

In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-svmo5
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-m50o40
Out[4]=4

Use Normal to get all wavelet coefficients explicitly:

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-4pudeo
In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-pngz6t
Out[2]=2

Also use All as an argument to get all coefficients:

In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-fy98zc
Out[3]=3

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

In[4]:=4
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-o4zzzw
Out[4]=4

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

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-8z5pdm
In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-7vy1zr
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-dbjopc
Out[3]=3

Extract specific coefficient arrays:

In[4]:=4
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-limxd1
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-ukcyo5
Out[5]=5

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

In[6]:=6
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-bw6a2j
Out[6]=6

Extract all coefficients whose wavelet indexes match a pattern:

In[7]:=7
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-57sk78
Out[7]=7
In[8]:=8
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-u5etdi
Out[8]=8

Use a higher refinement level to increase the frequency resolution:

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-e0tfo1
In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-d8rad6
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-qhxfy
In[4]:=4
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-b7oh62
Out[4]=4

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

In[5]:=5
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-pcyr03
In[6]:=6
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-ox23n
Out[6]=6

Wavelet Families  (10)

Compute the wavelet packet transform using different wavelet families:

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-gavf2f
In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-b3h8vd

Compare the coefficients:

In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-lu64r
Out[3]=3

Use different families of wavelets to capture different features:

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-iax26u
In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-6jwo5u
Out[2]=2

HaarWavelet (default):

In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-nm4160
In[4]:=4
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-7up01x
Out[4]=4

DaubechiesWavelet:

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-022m3l
In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-5guii
In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-588xab
Out[3]=3

BattleLemarieWavelet:

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-64i349
In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-8g8wur
In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-myg47y
Out[3]=3

BiorthogonalSplineWavelet:

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-874w84
In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-3itphu
In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-7q8w5p
Out[3]=3

CoifletWavelet:

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-klv9q6
In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-7eaj36
In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-kmt3r3
Out[3]=3

MeyerWavelet:

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-q6etlg
In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-9nicex
In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-6c9di7
Out[3]=3

ReverseBiorthogonalSplineWavelet:

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-fs79sv
In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-gmiqe8
In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-hthxz8
Out[3]=3

ShannonWavelet:

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-ntfv84
In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-2ydklr
In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-fe4q6k
Out[3]=3

SymletWavelet:

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-l4xlsf
In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-ovf86t
In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-qly5se
Out[3]=3

1-Dimensional Data  (6)

Plot the coefficients over a common horizontal axis using WaveletListPlot:

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-gypdz4
In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-xm5ld4
Out[2]=2

Plot against a common vertical axis:

In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-ez6l51
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-c1j4n5

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

In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-di3ww
Out[2]=2

Constant data:

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-i9a5e9
In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-kwq8bn
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-bfxyna
In[4]:=4
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-lji86v
Out[4]=4

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

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-c7n0wl
In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-ne0a67
Out[2]=2

Only the first detail coefficient {1} and its coarse child coefficients {1,0,0,…} are not small:

In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-cxc4yt
In[4]:=4
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-nyp7xn
Out[4]=4

Data with large discontinuities:

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-gl9hci
In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-fp30
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-fqn4ul
In[4]:=4
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-g46hdy
Out[4]=4

Detail coefficients are sensitive to discontinuities:

In[5]:=5
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-bi93r6
Out[5]=5

Data with both spatial and frequency structure:

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-qcsku6
In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-c0drk1
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-dot1jc
In[4]:=4
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-imnl1l
Out[4]=4

First detail coefficient {1} and its coarse child coefficients {1,0,…} represent the oscillations:

In[5]:=5
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-72xvt
Out[5]=5

All coefficients on a common vertical axis:

In[6]:=6
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-gxq0kb
Out[6]=6

2-Dimensional Data  (5)

Compute a two-dimensional stationary wavelet packet transform:

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-gflx0m
Out[1]=1

View the tree of wavelet coefficients:

In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-q2twv6
Out[2]=2

Inverse transform to get back the original signal:

In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-xz2n41

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

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-tj6pke
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-svtxpu
Out[2]=2

Visualize diagonal detail coefficient {3} and its child coefficients {3,__}:

In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-c7lg5h
In[4]:=4
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-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/0ejoqibx4bzks4kh6tbpu-lhs0ib
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-e2sxk5
Out[2]=2

Get the lowpass and highpass filters for a Haar wavelet:

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-lhi79n
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-dc3ief
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-ljd94e
Out[3]=3

Wavelet transform of step data:

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-hfdje4

Data with a vertical discontinuity:

In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-n155zh
Out[2]=2

All horizontal and diagonal detail coefficients, wavelet index {___,2|3,___}, are zero:

In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-zxwe7i
Out[3]=3

Data with horizontal discontinuity:

In[4]:=4
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-yv487p
Out[4]=4

All vertical and diagonal detail coefficients, wavelet index {___,1|3,___}, are zero:

In[5]:=5
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-lx6rc
Out[5]=5

Higher-Dimensional Data  (2)

Compute a three-dimensional wavelet packet transform:

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-wgoboy
In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-f8249l
Out[2]=2

List all computed wavelet coefficients:

In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-c7b4j4
Out[3]=3

Inverse transform to get back the original signal:

In[4]:=4
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-gsidhc
Out[4]=4

Wavelet transform of a three-dimensional cross array:

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-q65ymx
In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-6gkgva
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-sxjayp
Out[3]=3

Visualize lowpass wavelet coefficients {___,0}:

In[4]:=4
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-dlfuxh
Out[4]=4

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

In[5]:=5
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-57obwt
Out[5]=5

Audio Data  (2)

Transform an Audio object:

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-mineo
In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-jtcd5x
Out[2]=2

The inverse transform yields a reconstructed audio:

In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-jx08w
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-bj4f7m
In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-kp8pm
Out[2]=2

Get the {1,1} coefficient as an Audio object:

In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-hqd4o6
Out[3]=3

Inverse transform of {1,1} coefficient as an Audio object:

In[4]:=4
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-d4evuw
Out[4]=4

Sound Data  (2)

Transform a Sound object:

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-slfk3h
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-yw8jwa
Out[2]=2

The inverse transform yields a reconstructed audio object:

In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-rcs2ag
Out[3]=3

Browse all coefficients using a MenuView:

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-jhcy39
In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-c9wmlc
Out[2]=2

Image Data  (2)

Transform an Image object:

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-gmw99b
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-yzn8qe
Out[2]=2

The inverse transform yields a reconstructed Image object:

In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-xlen9k
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-79t4z
In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-lmi7qq
Out[2]=2

Get all coefficients as Image objects instead:

In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-9oty5

Get raw Image objects with no rescaling of color levels:

In[4]:=4
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-ctsnnz

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

In[5]:=5
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-hbdon
Out[5]=5

Generalizations & Extensions  (3)Generalized and extended use cases

StationaryWaveletPacketTransform works on arrays of symbolic quantities:

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-iue08m
In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-cqhoa
Out[2]=2

Inverse transform recovers the input exactly:

In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-zu4y
Out[3]=3

Specify any internal working precision:

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-daoo2p
In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-hlsinj
Out[2]=2

Use complex-valued data:

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-noliik
In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-bmqscn
Out[2]=2

The wavelets coefficients are complex:

In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-pma2rw
Out[3]=3

Options  (3)Common values & functionality for each option

WorkingPrecision  (3)

By default, WorkingPrecision->MachinePrecision is used:

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-btupk0
In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-fvcmsi
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-iwm7rm
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-bomt1d
Out[4]=4

Use higher-precision computation:

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-e7oo7e
In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-6lkrhl
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-cqyqyd
Out[3]=3

Use WorkingPrecision->∞ for exact computation:

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-m5fvtb
In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-5ais8y
Out[2]=2

Properties & Relations  (10)Properties of the function, and connections to other functions

StationaryWaveletPacketTransform computes the full tree of wavelet coefficients:

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-cgjhd3
In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-e4t1li
Out[2]=2

StationaryWaveletTransform computes a subset of the full tree of coefficients:

In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-p3dq
In[4]:=4
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-wochi
Out[4]=4

DiscreteWaveletPacketTransform coefficients halve in length with each level of refinement:

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-geoy4j
Out[1]=1

Rotated data gives different coefficients:

In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-fspmhk
Out[2]=2

StationaryWaveletPacketTransform coefficients have the same length as the data:

In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-ce5j56
Out[3]=3

Rotated data gives rotated coefficients:

In[4]:=4
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-f9p8kx
Out[4]=4

The default refinement is given by Min[Round[Log2[Min[Dimensions[data]]]],4]:

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-nvnob0
In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-dfpfdn
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-ccu2mo
Out[3]=3

In higher dimensions:

In[4]:=4
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-dzom29
In[5]:=5
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-nqp5e2
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-lsny6n
Out[6]=6

The energy norm is conserved for orthogonal wavelet families:

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-40jjuu
In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-fgbdya
In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-ks8c24
Out[3]=3

The energy norm is approximately conserved for biorthogonal wavelet families:

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-ihj98y
In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-hz2t90
In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-etc633
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-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/0ejoqibx4bzks4kh6tbpu-zu45nt
In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-7tclnf

Extract the coefficient for the maximum refinement level:

In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-hqx9zn
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-t3qkc
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-5slunz
Out[5]=5

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

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-8csip2
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-uwqql4
In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-sacrp
Out[3]=3

Individually inverse transform each wavelet coefficient array:

In[4]:=4
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-f3ljg7
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-bh52zs
Out[5]=5
In[6]:=6
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-u8p9i
Out[6]=6
In[7]:=7
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-bevtad
Out[7]=7

The sum gives the original data:

In[8]:=8
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-fuhvs4
Out[8]=8

Compute a Haar stationary wavelet packet transform in one dimension:

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-kywxlf
In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-f0nc2a

Compute {0} and {1} wavelet coefficients:

In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-bzen0a
Out[3]=3

Compare with StationaryWaveletPacketTransform:

In[4]:=4
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-zcrryy
Out[4]=4

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

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-cq5vc4

Lowpass and highpass filters for Haar wavelet:

In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-bzl44r

Haar wavelet transform of matrix data:

In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-5n59bt
In[4]:=4
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-6zl4c
Out[4]=4

Compare with StationaryWaveletPacketTransform using HaarWavelet:

In[5]:=5
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-maqks4
In[6]:=6
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-s166s
Out[6]=6

Image channels are transformed individually:

In[1]:=1
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-gfhws8
Out[1]=1

Combine {0} coefficients of separately transformed image channels:

In[2]:=2
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-kh8lp
Out[2]=2

Compare with {0} coefficient of StationaryWaveletPacketTransform of original image:

In[3]:=3
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-t91ou
In[4]:=4
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-byduzg
Out[4]=4

The images are identical:

In[5]:=5
✖
https://wolfram.com/xid/0ejoqibx4bzks4kh6tbpu-kbuh9g
Out[5]=5
Wolfram Research (2010), StationaryWaveletPacketTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/StationaryWaveletPacketTransform.html (updated 2017).
✖
Wolfram Research (2010), StationaryWaveletPacketTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/StationaryWaveletPacketTransform.html (updated 2017).

Text

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

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

CMS

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

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

APA

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

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

BibTeX

@misc{reference.wolfram_2025_stationarywaveletpackettransform, author="Wolfram Research", title="{StationaryWaveletPacketTransform}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/StationaryWaveletPacketTransform.html}", note=[Accessed: 20-June-2025 ]}

✖
@misc{reference.wolfram_2025_stationarywaveletpackettransform, author="Wolfram Research", title="{StationaryWaveletPacketTransform}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/StationaryWaveletPacketTransform.html}", note=[Accessed: 20-June-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_stationarywaveletpackettransform, organization={Wolfram Research}, title={StationaryWaveletPacketTransform}, year={2017}, url={https://reference.wolfram.com/language/ref/StationaryWaveletPacketTransform.html}, note=[Accessed: 20-June-2025 ]}

✖
@online{reference.wolfram_2025_stationarywaveletpackettransform, organization={Wolfram Research}, title={StationaryWaveletPacketTransform}, year={2017}, url={https://reference.wolfram.com/language/ref/StationaryWaveletPacketTransform.html}, note=[Accessed: 20-June-2025 ]}