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 "IndexMap" 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 WaveletBestBasis to compute an optimal basis of wavelet packet coefficients:

Highlight the best basis in a block grid of all coefficients:

Extract the best basis using "BasisIndex":

The computed best basis is used by default in functions like WaveletListPlot:

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:

WaveletScalogram of the best tree representation of the data:

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 wavelet packet transform:

View the tree of wavelet coefficients:

Inverse transform to get back the original signal:

Use WaveletMatrixPlot to visualize the different wavelet coefficients:

WaveletMatrixPlot of best tree representation:

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

Interpreting these vectors as binary digit expansions results in 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:

Data with diagonal discontinuity:

All horizontal and vertical detail coefficients, wavelet index {___,1|2,___}, are zero:

Compute a three-dimensional wavelet packet transform:

Block grid view of all 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 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:

Compute a best tree of coefficients from a packet transform of image data:

Plot the best tree in a hierarchical grid using WaveletImagePlot:

Transform a Sound object:

The inverse transform yields a reconstructed Sound object:

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

Get the {1,1} coefficient as a Sound object:

Inverse transform of {1,1} coefficient as a Sound object:

Compute a best tree of coefficients from a packet transform of sound data:

Browse the best tree coefficients using a MenuView:

The settings for Padding are the same as the methods for ArrayPad, including "Periodic":

"Reversed":

"ReversedNegation":

"Reflected":

"ReflectedDifferences":

"ReversedDifferences":

"Extrapolated":

Padding can remove boundary effects:

Use the default "Periodic" padding:

"Extrapolated" padding lessens boundary effects for nonperiodic data:

By default, WorkingPrecision->MachinePrecision is used:

Use higher-precision computation:

With numbers close to zero, accuracy is the better indicator of the number of correct digits:

Use WorkingPrecision->∞ for exact computation:

The default reconstruction tree contains the coefficients at the maximum refinement level:

Choose a reconstruction tree with energy concentrated in a small number of coefficients:

Plot the best tree coefficients against a common vertical axis:

Visualize default reconstruction tree coefficients for image data:

Compute the reconstruction tree whose coefficients have the smallest total log energy:

Lossless compression of matrix data:

In the best tree wavelet packet representation many coefficients are zero:

Count nonzero coefficients as a measure of compressed size:

Nonzero values in original data:

Length of data: