InverseContinuousWaveletTransform
✖
InverseContinuousWaveletTransform
gives the inverse transform from the wavelet coefficients specified by octvoc.
Details and Options

- InverseContinuousWaveletTransform computes the inverse transform of continuous forward transforms such as ContinuousWaveletTransform.
- The possible wavelets wave are the same as for ContinuousWaveletTransform.
- The default wave is Automatic, which is taken to be cwd["Wavelet"].
- The possible specifications for octvoc are the same as used by ContinuousWaveletData.
- The default octvoc is Automatic, which is taken to be cwd["WaveletIndex"].
- InverseContinuousWaveletTransform[cwd,wave,octvoc] computes the inverse transform using only the wavelet coefficients specified by octvoc; other coefficients are set to be zero.
Examples
open allclose allBasic Examples (1)Summary of the most common use cases
Perform a continuous wavelet transform:

https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-gn4cac

https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-k0y8ag

Inverse transform resynthesizes data from continuous wavelet coefficients:

https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-h2v3yy

Scope (5)Survey of the scope of standard use cases
Basic Uses (5)
Inverse transform ContinuousWaveletData from the forward transform:

https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-oqxid


https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-e8ehqp

The quality of the reconstruction depends on the number of octaves and voices:

https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-figdq9

Inverse transform modified ContinuousWaveletData:

https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-k7fyxt

https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-bo35zr

Plot the inverse transform of original and modified coefficients:

https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-hd71qp

Inverse transform selected octaves and voices only:

https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-so2xj

https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-jlrlq4

Inverse transform only the {2,5} coefficient:

https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-eex5p5

https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-nbxt85

Inverse transform the first octave {1,_}, setting other coefficients to zero:

https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-f9f1d5

Inverse transform an explicitly constructed ContinuousWaveletData object:

https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-chz5e7


https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-ch23tb

Unspecified coefficients are taken to be zero:

https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-sgqv3

Specify a different wavelet to use in the inverse transform:

https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-pplya

https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-fmsw2o

By default, the wavelet used in the forward transform is chosen:

https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-g02w3i

Options (4)Common values & functionality for each option
Method (4)
By default, Method option "LeastSquares" is used for data less than length 512:

https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-pu57jn


https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-8vd4z1

By default, Method option "DeltaFunction" is used for data greater than length 512:

https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-7ft4w7


https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-r6c6sf

Method option "LeastSquares" performs an exact inverse transform:

https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-9a1w82


https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-spr66n

Method option "DeltaFunction" performs an approximate inverse transform:

https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-pvv74q

Compare efficiency and accuracy of the two methods:

https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-zicz9u

https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-60tpcy

For large data, "LeastSquares" is slow:

https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-dvv0sq

Compare with the original data:

https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-mj49fg

Use "DeltaFunction" for large data:

https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-giyc1f

Compare with the original data:

https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-gxggsb

Applications (2)Sample problems that can be solved with this function
Scale & Time Filtering (2)
Filter one scale or frequency from a signal:

https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-bouvog

https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-cjsz8e

Identify separate signal components on a scalogram:

https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-twys2

https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-i8rnsd

Remove the feature at small scales {3 4 5,_}:

https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-hu4p7t


https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-jprgyp

Filter data with both time- and scale-dependent features:

https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-dfn2sj

https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-d322xr

Identify signal components as a function of scale and time:

https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-k3pzrz


https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-br432w

Excise transient feature using a step filter:

https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-prmem

https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-hz19u8

Show altered scalogram and synthesized filtered data:

https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-hjwcix

Properties & Relations (2)Properties of the function, and connections to other functions
InverseContinuousWaveletTransform synthesizes data from continuous wavelet coefficients:

https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-e7szgq

The synthesis operation is approximately the inverse of the forward continuous transform:

https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-dp53to

InverseWaveletTransform gives the inverse of discrete forward transforms:

https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-ci4xpq

The inverse is exact for all orthogonal wavelets including HaarWavelet[]:

https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-hj296

InverseContinuousWaveletTransform[…,…,octvoc] effectively zeros other coefficients:

https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-f0fcjb

https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-d8hwwt

Explicitly set other wavelet coefficients to zero:

https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-fic2og


https://wolfram.com/xid/0rb64d6k0j5l567nw3vde-elt7tg

Wolfram Research (2010), InverseContinuousWaveletTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/InverseContinuousWaveletTransform.html.
Text
Wolfram Research (2010), InverseContinuousWaveletTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/InverseContinuousWaveletTransform.html.
Wolfram Research (2010), InverseContinuousWaveletTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/InverseContinuousWaveletTransform.html.
CMS
Wolfram Language. 2010. "InverseContinuousWaveletTransform." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/InverseContinuousWaveletTransform.html.
Wolfram Language. 2010. "InverseContinuousWaveletTransform." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/InverseContinuousWaveletTransform.html.
APA
Wolfram Language. (2010). InverseContinuousWaveletTransform. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/InverseContinuousWaveletTransform.html
Wolfram Language. (2010). InverseContinuousWaveletTransform. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/InverseContinuousWaveletTransform.html
BibTeX
@misc{reference.wolfram_2025_inversecontinuouswavelettransform, author="Wolfram Research", title="{InverseContinuousWaveletTransform}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/InverseContinuousWaveletTransform.html}", note=[Accessed: 30-March-2025
]}
BibLaTeX
@online{reference.wolfram_2025_inversecontinuouswavelettransform, organization={Wolfram Research}, title={InverseContinuousWaveletTransform}, year={2010}, url={https://reference.wolfram.com/language/ref/InverseContinuousWaveletTransform.html}, note=[Accessed: 30-March-2025
]}