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 {0,1}:

Inverse transform all wavelet coefficients matching {0,_}, 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 {…,1} and the last coarse coefficient {…,0}:

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:

Discrete wavelet data consisting of noise in detail coefficient {0,…,1} 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 {2,2} 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 :