Useful properties can be extracted from the DiscreteWaveletData object:

Get a full list of properties:

Get data and coefficient dimensions:

Use Normal to get all wavelet coefficients explicitly:

Also use All as an argument to get all coefficients:

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

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

Extract specific coefficient arrays:

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

Extract all coefficients whose wavelet indexes match a pattern:

Use a higher refinement level to increase the frequency resolution:

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

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

Compute the wavelet packet transform using different wavelet families:

Compare the coefficients:

Use different families of wavelets to capture different features:

HaarWavelet (default):

DaubechiesWavelet:

BattleLemarieWavelet:

BiorthogonalSplineWavelet:

CoifletWavelet:

MeyerWavelet:

ReverseBiorthogonalSplineWavelet:

ShannonWavelet:

SymletWavelet:

Plot the coefficients over a common horizontal axis using WaveletListPlot:

Plot against a common vertical axis:

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

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

Constant data:

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

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

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

Data with large discontinuities:

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

Detail coefficients are sensitive to discontinuities:

Data with both spatial and frequency structure:

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

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

All coefficients on a common vertical axis:

Compute a two-dimensional stationary wavelet packet transform:

View the tree of wavelet coefficients:

Inverse transform to get back the original signal:

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

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

In two dimensions, the vector of filtering operations in each direction can be computed:

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

Get the lowpass and highpass filters for a Haar wavelet:

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

Wavelet transform of step data:

Data with a vertical discontinuity:

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

Data with horizontal discontinuity:

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

Compute a three-dimensional wavelet packet transform:

List all computed wavelet coefficients:

Inverse transform to get back the original signal:

Wavelet transform of a three-dimensional cross array:

Visualize lowpass wavelet coefficients {___,0}:

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

Transform an Audio object:

The inverse transform yields a reconstructed audio:

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

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

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

Transform a Sound object:

The inverse transform yields a reconstructed audio object:

Browse all coefficients using a MenuView:

Transform an Image object:

The inverse transform yields a reconstructed Image object:

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

Get all coefficients as Image objects instead:

Get raw Image objects with no rescaling of color levels:

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

By default, WorkingPrecision->MachinePrecision is used:

Use higher-precision computation:

Use WorkingPrecision->∞ for exact computation: