Compute a 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 "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 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:

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 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 discrete wavelet 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 wavelet transform at a higher refinement level:

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:

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 discrete 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 the inverse transform in a hierarchical grid using WaveletImagePlot:

Image wavelet coefficients lie outside the valid range of ImageType:

"ImageFunction"->Identity gives an unnormalized image wavelet coefficient:

The color channels lie outside its valid 0 to 1 range:

By default, "ImageFunction"->ImageAdjust is used to normalize coefficients:

The color channels are now within the valid 0 to 1 range:

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 {0,1} coefficient as a Sound object:

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

Browse all 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:

Using the default "Periodic" padding:

Using "Extrapolated" padding has fewer 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:

Compress data by finding a representation with few nonzero coefficients:

SymletWavelet[n] has n vanishing moments and represents polynomials of degree n:

Count counts the number of wavelet coefficients close to 0:

Visualize discontinuities in the wavelet domain:

Detail coefficients in the region of discontinuities have larger values:

Detect edges in an image:

Set coarse coefficients to 0 and reconstruct using detail coefficients only:

Compare the cumulative energy in a signal, its wavelet coefficients, and Fourier coefficients:

Compute the ordered cumulative energy in the signal:

Compute wavelet coefficients and Fourier coefficients:

The DWT captures more energy with fewer coefficients than the DFT:

Perform energy-dependent thresholding:

Computing the fraction of energy contained at each refinement level:

Set wavelet coefficients containing less than 1% energy to zero:

Perform an amplitude-dependent thresholding:

Use WaveletThreshold to perform "Universal" thresholding:

Use Stein's unbiased risk estimator smoothing:

Denoise an Image:

Perform "Soft" thresholding with threshold value "SURE" computed adaptively at each level:

Invert thresholded coefficients:

Wavelet transforms can be used to filter frequencies:

To filter out the two signals, first perform a wavelet transform:

Use WaveletListPlot to visualize frequency distribution:

To filter low frequencies, keep only the coarse coefficients:

To filter high frequencies, keep only the detail coefficients:

Extract the stock price trend for IBM since January 1, 2000:

The trend of the series is captured in the lowpass filter coefficients:

Thresholding all detail coefficients and inverting the series gives the trend:

Detrend a financial series:

Detail coefficients captured the detrended series:

Remove the trend by removing the coarse coefficients and inverting:

Study variance of returns in a financial time series:

Perform a wavelet transform using HaarWavelet and SymletWavelet:

Since the GE return series does not exhibit low-frequency oscillations, higher-scale detail coefficients do not indicate large variations from zero:

Although both filters will capture the variance of the series, they distribute it differently because of their approximate bandpass properties:

SymletWavelet isolates features in a certain frequency interval better than HaarWavelet: