Compute a stationary wavelet transform:

The resulting DiscreteWaveletData represents a tree of transform 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 "IndexMap" 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 stationary wavelet 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} is nonzero:

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 stationary wavelet 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 wavelet coefficients at higher refinement levels:

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:

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

Data with horizontal discontinuity:

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

Compute a three-dimensional stationary 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 arrays of data for each image channel:

Number of channels and dimensions of the original image are the same:

Get all coefficients as Image objects instead of arrays of data:

Get raw Image objects with no rescaling of color levels:

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

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:

Number of channels and data length in the original sound are the same:

Get the {0,1} coefficient as a Sound object:

Inverse transform of {0,0,1} coefficient as a Sound object:

Browse all coefficients using a MenuView:

By default, WorkingPrecision->MachinePrecision is used:

Use higher-precision computation:

Use WorkingPrecision->∞ for exact computation:

A simple wavelet-based inverse halftoning:

Apply GaussianFilter on the detail coefficients:

Differentiate noisy data using wavelet transform:

Translation-Rotation-Transform (TRT) is used to reduce boundary effects by subtracting a linear component from the input signal:

Since HaarWavelet has one vanishing moment, choose it to perform a wavelet transform on :

Detail coefficients give the differentiation of the data. Coefficients at refinement level 4 are chosen to minimize noise:

Rescale the differentiated values:

Compare wavelet-based numerical differentiation with exact differentiation:

Compare with standard Wolfram Language numerical differentiation:

Add texture to an existing image:

Perform wavelet transform on both images:

Combine detail coefficients of the two images by taking their mean:

Append the coarse coefficient of the first image:

Construct a new DiscreteWaveletData of the combined wavelet coefficients:

Reconstruct the combined image: