BUILT-IN MATHEMATICA SYMBOL

InverseWaveletTransform

InverseWaveletTransform[dwd]
gives the inverse wavelet transform of a DiscreteWaveletData object dwd.

gives the inverse transform using the wavelet wave.

InverseWaveletTransform

gives the inverse transform from the wavelet coefficients specified by wind.

MORE INFORMATION

InverseWaveletTransform computes the inverse transform of discrete forward transforms such as DiscreteWaveletTransform etc.

The possible wavelets wave are the same as for the forward wavelet transforms.

The default wave is Automatic, which is taken to be dwd["Wavelet"].

The possible specifications for wind are the same as used by DiscreteWaveletData.

The default wind is Automatic, which is taken to be dwd["BasisIndex"].

InverseWaveletTransform computes the inverse transform using only the wavelet coefficients specified by wind; other coefficients are set to be zero.

The inverse transform works recursively by computing coefficients with wavelet index from coefficients with wavelet index .

An explicit wind specification needs to be consistent. A wind specification is consistent if for each that is included no for is included.

InverseWaveletTransform can be used to inverse transform the r lowest levels of the wavelet tree.

The default level r is given by the number of refinement levels n in dwd. With a new DiscreteWaveletData object is returned with refinement levels.

EXAMPLES

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Basic Examples (2)

Perform a discrete wavelet transform:

Inverse transform recovers the original data:

DiscreteWaveletData representing modified wavelet image coefficients:

Inverse transform gives filtered image:

Perform a discrete wavelet transform:

In[1]:=

Out[1]=

Inverse transform recovers the original data:

In[2]:=

Out[2]=

DiscreteWaveletData representing modified wavelet image coefficients:

In[1]:=

In[2]:=

Out[2]=

Inverse transform gives filtered image:

In[3]:=

Out[3]=

Scope (14)

Inverse wavelet transform DiscreteWaveletData from any discrete forward transform:

The inverse transform is exact for data directly from the forward transform:

Inverse wavelet transform modified DiscreteWaveletData:

The inverse is computed from the modified wavelet coefficients:

Inverse wavelet transform using specified wavelet coefficients only:

Inverse transform only the detail coefficient

:

Inverse transform all wavelet coefficients matching

, setting other coefficients to zero:

Inverse transform an explicitly constructed DiscreteWaveletData object:

The type of wavelet transform is inferred from the form of the specified coefficients:

Unspecified coefficients are taken to be zero:

Specify a different wavelet to use in the inverse transform:

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

For array data, the inverse transform is an array with the same dimensions:

For image data, the inverse transform is given as an Image object:

The image has the same dimensions and number of color channels as the original data:

For sound data, the inverse transform is given as a Sound object:

The sound has the same duration, sampling rate, and number of channels as the original data:

A subset of the available coefficients is used in the inverse transform:

dwd["BasisIndex"] gives the coefficients used by default in the inverse transform:

Use dwd["TreeView"] to get a tree plot of all coefficients with the default basis highlighted:

Compute the inverse transform using the default basis of coefficients:

Compute the inverse transform using a specific basis:

Use a specific subset of a basis, with the other coefficients taken to be zero:

Compute the inverse transform of a single coefficient:

A standard basis is chosen as the default for all non-packet DiscreteWaveletData:

The basis includes the detail coefficients

and the last coarse coefficient

:

The default basis for packet transform data uses coefficients at the highest refinement level:

For packet transform data, the wavelet basis can be changed using WaveletBestBasis:

Choose the basis that minimizes the information entropy of the coefficients:

Choose a specific basis:

Highlight each basis in a tree plot of all coefficients:

For data directly from packet transforms, the inverse transform is independent of the basis:

Compare the inverse transform for several different full wavelet basis specifications:

For modified wavelet data, the inverse transform can depend on the basis chosen:

Generalizations & Extensions (2)

Partial inverse wavelet transform:

Obtain a DiscreteWaveletData with 1 less level of refinement:

Visualize wavelet coefficients remaining after inverting the

lowest levels of refinement:

InverseWaveletTransform[dwd] is equivalent to inverting all levels of refinement:

Inverse transform of complex wavelet coefficients:

InverseWaveletTransform gives the exact inverse transform:

Applications (8)

Discrete wavelet data consisting of noise in detail coefficient

at refinement level

:

Use inverse wavelet transform to synthesize noise on a single specific scale:

Stationary wavelet matrix data consisting of noise in vertical detail coefficient

only:

Use inverse wavelet transform to synthesize noisy data:

Noise in different wavelet coefficients leads to different kinds of synthesized noisy data:

Discrete wavelet transform data with the first coarse coefficient equal to

:

Visualize the inverse transform for different coarse coefficients:

Discrete wavelet transform data with the first detail coefficient equal to

:

Visualize the inverse transform for different detail coefficients:

Discrete wavelet packet transform data with wavelet coefficients

:

Visualize the inverse transform of each coefficient separately:

The set of inverse transforms forms a basis for the signal space

:

Stationary wavelet transform data with the first coarse coefficient equal to

:

Visualize the inverse transform for different coarse coefficients:

Stationary wavelet transform data with the first detail coefficient equal to

:

Visualize the inverse transform for different detail coefficients:

Stationary wavelet packet transform data with wavelet coefficients

:

Visualize the inverse transform of each component

of each coefficient separately:

The set of 16 inverse transforms spans the signal space

:

Properties & Relations (7)

InverseWaveletTransform is the exact inverse of transforms using an orthogonal wavelet:

Use the orthogonal HaarWavelet:

The inverse is inexact for nonorthogonal wavelet families such as ShannonWavelet:

By default, coefficients in dwd["BasisIndex"] are used in the inverse:

Set all other coefficients to zero:

The inverse using the default basis is unaffected:

InverseWaveletTransform

effectively sets other wavelet coefficients to zero:

Explicitly set other wavelet coefficients to zero:

InverseWaveletTransform

computes the inverse from the coefficients wind:

computes the inverse of each coefficient separately:

The total gives the same result as InverseWaveletTransform:

Use ListLinePlot to plot inverse transforms of individual vector coefficients:

gives a simple list plot of each inverse transform:

WaveletListPlot[dwd, wind, Method->"Inverse"->True] plots inverses together:

Use MatrixPlot to plot inverse transforms of individual matrix coefficients:

gives a simple matrix plot of each inverse:

Inverse transforms of individual coefficients from image data are given as Image objects:

Possible Issues (1)

A wind specification is inconsistent if it includes both

and

for

:

The specification is inconsistent because the inverse transform computes

from

:

Use a consistent wind specification: