Compute a lifting wavelet transform:

The resulting DiscreteWaveletData represents a tree of transform coefficients:

The inverse transform reconstructs the input:

Compute an integer lifting wavelet transform:

Use Normal to view all integer coefficients:

The inverse transform reconstructs the input:

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 the list of wavelet index specifications:

Extract all coefficients whose wavelet indexes match a pattern:

The Automatic coefficients are 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 {0,0,0}:

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

Compute the discrete wavelet transform using different wavelet families:

Compare the coefficients:

Use different families of wavelets to capture different features:

HaarWavelet (default):

DaubechiesWavelet:

BiorthogonalSplineWavelet:

CoifletWavelet:

ReverseBiorthogonalSplineWavelet:

SymletWavelet:

Use LiftingFilterData as an input to LiftingWaveletTransform:

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} is 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:

The first detail coefficient identifies the oscillatory region:

All coefficients on a common vertical axis:

Compute a two-dimensional lifting wavelet transform:

View the tree of wavelet coefficients:

Inverse transform to get back the original signal:

Use WaveletMatrixPlot to visualize the different wavelet coefficients:

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

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

Wavelet transform of step data:

Data with a vertical discontinuity:

Only the vertical detail coefficients, wavelet index {…,1}, are nonzero:

Data with horizontal discontinuity:

Only the horizontal detail coefficients, wavelet index {…,2}, are nonzero:

Data with diagonal discontinuity:

Only the diagonal detail coefficients, wavelet index {…,3}, are nonzero:

Compute a three-dimensional lifting wavelet transform:

Tree view of all coefficients:

Inverse transform to get back the original signal:

Wavelet transform of a three-dimensional cross array:

Visualize wavelet coefficients:

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:

Plot coefficients used in inverse transform 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} coefficient as a Sound object:

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

Browse all coefficients using a MenuView:

By default, lifting wavelet transform maps integers to real values:

Use Normal to view all coefficients:

Perform an inverse:

With Method->"IntegerLifting", integers are mapped to integers:

Display all coefficients:

Use inverse wavelet transform to invert integer lifting:

Padding can remove boundary effects:

Treat data as cyclic using "Periodic" padding:

Pad data with 0 to reduce boundary effects on the left:

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:

Preprocess images to improve performance of image processing functions:

Plot transformed image coefficients:

Extract vertical and diagonal components from the wavelet-transformed image and reconstruct:

Threshold image pixel values below 0.5:

With vertical components intensified, apply ImageLines: