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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:
-
list arbitrary-rank numerical array image arbitrary Image object audio an 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 , 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:
-
Method Automatic method to use WorkingPrecision MachinePrecision precision to use in internal computations - StationaryWaveletTransform uses periodic padding of data.
- InverseWaveletTransform gives the inverse transform.
Examples
open allclose allBasic Examples (3)Summary of the most common use cases
Compute a stationary wavelet transform using the HaarWavelet:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-zjig9d
Use Normal to view all coefficients:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-s766la
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-c4c4hc
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-i79gyn
Use dwd[…,"Audio"] to extract coefficient signals:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-bxhoqo
Verify lengths of all coefficient signals:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-qrq5z9
Compute the inverse transform:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-ds1g1n
Transform an Image object:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-jza1au
Use dwd[…,"Image"] to extract coefficient images:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-lg7vnk
Compute the inverse transform:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-xasr81
Scope (34)Survey of the scope of standard use cases
Basic Uses (6)
Compute a stationary wavelet transform:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-h8yn3f
The resulting DiscreteWaveletData represents a tree of transform coefficients:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-hic6us
The inverse transform reconstructs the input:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-f3afir
Useful properties can be extracted from the DiscreteWaveletData object:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-f5g3t
Get a full list of properties:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-che078
Get data and coefficient dimensions:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-svmo5
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-dtl7ea
Use Normal to get all wavelet coefficients explicitly:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-4pudeo
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-pngz6t
Also use All as an argument to get all coefficients:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-fy98zc
Use Automatic to get only the coefficients used in the inverse transform:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-o4zzzw
Use the "TreeView" or "IndexMap" to find out what wavelet coefficients are available:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-8z5pdm
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-7vy1zr
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-dbjopc
Extract specific coefficient arrays:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-limxd1
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-ukcyo5
Extract several wavelet coefficients corresponding to the list of wavelet index specifications:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-bw6a2j
Extract all coefficients whose wavelet indexes match a pattern:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-57sk78
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-u5etdi
The Automatic coefficients are used by default in functions like WaveletListPlot:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-c115g
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-b2d2ce
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-lb6dy9
Use a higher refinement level to increase the frequency resolution:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-e0tfo1
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-d8rad6
With a smaller refinement level, more of the signal energy is left in {0,0,0}:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-qhxfy
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-b7oh62
With further refinement, {0,0,0} is resolved into further components:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-pcyr03
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-ox23n
Wavelet Families (10)
Compute the stationary wavelet transform using different wavelet families:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-gavf2f
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-b3h8vd
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-lu64r
Use different families of wavelets to capture different features:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-iax26u
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-6jwo5u
HaarWavelet (default):
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-nm4160
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-7up01x
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-022m3l
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-5guii
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-588xab
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-64i349
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-8g8wur
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-myg47y
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-874w84
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-3itphu
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-7q8w5p
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-klv9q6
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-7eaj36
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-kmt3r3
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-q6etlg
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-9nicex
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-6c9di7
ReverseBiorthogonalSplineWavelet:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-fs79sv
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-gmiqe8
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-hthxz8
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-ntfv84
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-2ydklr
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-fe4q6k
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-l4xlsf
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-ovf86t
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-qly5se
Vector Data (6)
Plot the coefficients over a common horizontal axis using WaveletListPlot:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-gypdz4
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-xm5ld4
Plot against a common vertical axis:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-ez6l51
Visualize coefficients as a function of time and refinement level using WaveletScalogram:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-c1j4n5
The coefficient indexes appear as tooltips when the mouse pointer is moved over a coefficient:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-di3ww
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-i9a5e9
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-kwq8bn
All coefficients are small except coarse coefficients {0,0,…}:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-bfxyna
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-lji86v
Data oscillating at the highest resolvable frequency (Nyquist frequency):
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-c7n0wl
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-ne0a67
Only the first detail coefficient {1} is nonzero:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-cxc4yt
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-nyp7xn
Data with large discontinuities:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-gl9hci
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-fp30
Coarse coefficients {0,…} have the same large-scale structure as the data:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-fqn4ul
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-g46hdy
Detail coefficients are sensitive to discontinuities:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-bi93r6
Data with both spatial and frequency structure:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-qcsku6
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-c0drk1
Coarse coefficients {0,…} track the local mean of the data:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-dot1jc
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-imnl1l
The first detail coefficient identifies the oscillatory region:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-72xvt
All coefficients on a common vertical axis:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-gxq0kb
Matrix Data (5)
Compute a two-dimensional stationary wavelet transform:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-gflx0m
View the tree of wavelet coefficients:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-q2twv6
Inverse transform to get back the original signal:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-xz2n41
Use dwd[…,"MatrixPlot"] to visualize each coefficient as a MatrixPlot:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-tj6pke
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-svtxpu
Visualize wavelet coefficients at higher refinement levels:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-c7lg5h
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-ldtwbr
In two dimensions, the vector of filtering operations in each direction can be computed:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-lhs0ib
Interpreting these vectors as binary digit expansions, you get wavelet index numbers:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-e2sxk5
Get the lowpass and highpass filters for a Haar wavelet:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-lhi79n
The resulting 2D filters are outer products of filters in the two directions:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-dc3ief
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-ljd94e
Wavelet transform of step data:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-hfdje4
Data with a vertical discontinuity:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-n155zh
Only the vertical detail coefficients, wavelet index {…,1}, are nonzero:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-fcnbpi
Data with horizontal discontinuity:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-yv487p
Only the horizontal detail coefficients, wavelet index {…,2}, are nonzero:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-bpz7u8
Array Data (2)
Compute a three-dimensional stationary wavelet transform:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-wgoboy
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-f8249l
Tree view of all coefficients:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-0ek2ai
Inverse transform to get back the original signal:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-gsidhc
Wavelet transform of a three-dimensional cross array:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-q65ymx
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-6gkgva
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-sxjayp
Visualize wavelet coefficients:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-gzdpgu
Energy of the original data is conserved within the transformed coefficients:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-57obwt
Image Data (2)
Transform an Image object:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-gmw99b
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-yzn8qe
The inverse transform yields a reconstructed Image object:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-xlen9k
Wavelet coefficients are normally given as arrays of data for each image channel:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-79t4z
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-lmi7qq
Number of channels and dimensions of the original image are the same:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-kqf5in
Get all coefficients as Image objects instead of arrays of data:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-9oty5
Get raw Image objects with no rescaling of color levels:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-ctsnnz
Get the inverse transform of the {0,1} coefficient as an Image object:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-hbdon
Sound Data (3)
Transform a Sound object:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-mineo
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-jtcd5x
The inverse transform yields a reconstructed Sound object:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-jx08w
By default, coefficients are given as lists of data for each sound channel:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-bj4f7m
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-kp8pm
Number of channels and data length in the original sound are the same:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-chexw1
Get the {0,1} coefficient as a Sound object:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-hqd4o6
Inverse transform of {0,0,1} coefficient as a Sound object:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-d4evuw
Browse all coefficients using a MenuView:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-jhcy39
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-c9wmlc
Generalizations & Extensions (3)Generalized and extended use cases
StationaryWaveletTransform works on arrays of symbolic quantities:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-iue08m
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-cqhoa
Inverse transform recovers the input exactly:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-zu4y
Specify any internal working precision:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-daoo2p
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-hlsinj
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-noliik
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-bmqscn
The wavelets coefficients are complex:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-pma2rw
Inverse transform recovers the input:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-l82pnk
Options (3)Common values & functionality for each option
WorkingPrecision (3)
By default, WorkingPrecision->MachinePrecision is used:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-btupk0
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-fvcmsi
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-iwm7rm
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-bomt1d
Use higher-precision computation:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-e7oo7e
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-6lkrhl
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-cqyqyd
Use WorkingPrecision->∞ for exact computation:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-m5fvtb
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-5ais8y
Applications (3)Sample problems that can be solved with this function
Inverse Halftoning (1)
A simple wavelet-based inverse halftoning:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-1xzz51
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-6cn82n
Apply GaussianFilter on the detail coefficients:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-1hv5q1
Numerical Differentiation (1)
Differentiate noisy data using wavelet transform:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-08loxn
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-lg248n
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-7wollf
Translation-Rotation-Transform (TRT) is used to reduce boundary effects by subtracting a linear component from the input signal:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-9i8iun
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-0e9yv5
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-w2krwf
Since HaarWavelet has one vanishing moment, choose it to perform a wavelet transform on :
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-5v5pvi
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-yhyx6q
Detail coefficients give the differentiation of the data. Coefficients at refinement level 4 are chosen to minimize noise:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-le9d3k
Rescale the differentiated values:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-8yk1yj
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-hyr1f
Compare wavelet-based numerical differentiation with exact differentiation:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-hm2ezh
Compare with standard Wolfram Language numerical differentiation:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-842g6d
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-orceoc
Image Fusion (1)
Add texture to an existing image:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-smhiig
Perform wavelet transform on both images:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-tzko9p
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-b10r6p
Combine detail coefficients of the two images by taking their mean:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-3bn18q
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-4yojav
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-vw7m1m
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-mzgj7z
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-wrzk24
Append the coarse coefficient of the first image:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-gvwllb
Construct a new DiscreteWaveletData of the combined wavelet coefficients:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-lq3bzd
Reconstruct the combined image:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-fhr7kh
Properties & Relations (12)Properties of the function, and connections to other functions
StationaryWaveletPacketTransform computes the full tree of wavelet coefficients:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-cgjhd3
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-e4t1li
StationaryWaveletTransform computes a subset of the full tree of coefficients:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-p3dq
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-wochi
DiscreteWaveletTransform coefficients halve in length with each level of refinement:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-geoy4j
Rotated data gives different coefficients:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-fspmhk
StationaryWaveletTransform coefficients have the same length as the original data:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-ce5j56
Rotated data gives rotated coefficients:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-f9p8kx
The default refinement is given by :
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-nvnob0
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-k74t7s
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-ccu2mo
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-dzom29
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-nqp5e2
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-lsny6n
The energy norm is conserved for orthogonal wavelet families:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-40jjuu
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-fgbdya
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-ks8c24
The energy norm is approximately conserved for biorthogonal wavelet families:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-ihj98y
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-hz2t90
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-etc633
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-bm042g
The mean of the data is captured at the maximum refinement level of the transform:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-zu45nt
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-7tclnf
Extract the coefficient for the maximum refinement level:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-hqx9zn
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-t3qkc
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-5slunz
The sum of inverse transforms from individual coefficient arrays gives the original data:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-8csip2
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-uwqql4
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-sacrp
Individually inverse transform each wavelet coefficient array:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-f3ljg7
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-bh52zs
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-u8p9i
The sum gives the original data:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-fuhvs4
Compute stationary wavelet coefficients for periodic data:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-81usyq
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-c6pxyh
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-8059v1
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-8xpc58
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-nbr339
Coarse coefficients at level are given by :
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-nmsfh7
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-iqmt4l
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-8kw0h5
Detail coefficients at level are given by :
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-ofmhs6
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-ccnq2x
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-wl5c72
Compute partial stationary inverse wavelet transform:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-knxsvp
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-115wba
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-u99tzo
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-hive0q
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-rsdr3f
Coarse coefficients at level are given:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-24erqj
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-moxip
Detail coefficients at level are given:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-qejkrz
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-kpgqsd
Inverse wavelet transform at level is given by :
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-xu196g
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-j987r4
Reconstruct coarse coefficients {0,0} at refinement level :
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-t9kl1n
Reconstruct coarse coefficients {0} at refinement level :
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-vxh7ou
Compute a Haar stationary wavelet transform in one dimension:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-kywxlf
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-f0nc2a
Compute {0} and {1} wavelet coefficients:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-bzen0a
Compare with DiscreteWaveletPacketTransform:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-zcrryy
In two dimensions, a separate filter is applied in each dimension:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-cq5vc4
Lowpass and highpass filters for a Haar wavelet:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-bzl44r
Haar wavelet transform of matrix data:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-5n59bt
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-6zl4c
Compare with DiscreteWaveletPacketTransform using HaarWavelet:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-maqks4
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-s166s
Image channels are transformed individually:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-gfhws8
Combine {0} coefficients of separately transformed image channels:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-kh8lp
Compare with {0} coefficient of StationaryWaveletTransform of the original image:
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-t91ou
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-byduzg
https://wolfram.com/xid/0bsv1gx2v9710xb0ia-kbuh9g
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_2024_stationarywavelettransform, author="Wolfram Research", title="{StationaryWaveletTransform}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/StationaryWaveletTransform.html}", note=[Accessed: 08-January-2025
]}
BibLaTeX
@online{reference.wolfram_2024_stationarywavelettransform, organization={Wolfram Research}, title={StationaryWaveletTransform}, year={2017}, url={https://reference.wolfram.com/language/ref/StationaryWaveletTransform.html}, note=[Accessed: 08-January-2025
]}