DiscreteWaveletTransform

DiscreteWaveletTransform[data]

gives the discrete wavelet transform (DWT) of an array of data.

DiscreteWaveletTransform[data,wave]

gives the discrete wavelet transform using the wavelet wave.

DiscreteWaveletTransform[data,wave,r]

gives the discrete wavelet transform using r levels of refinement.

Details and Options

  • DiscreteWaveletTransform gives a DiscreteWaveletData object representing a tree of wavelet coefficient arrays.
  • Properties of the DiscreteWaveletData dwd can be found using dwd["prop"], and a list of available properties can be found using dwd["Properties"].
  • The data can be any of the following:
  • listarbitrary-rank numerical array
    imagearbitrary Image object
    audioan Audio or sampled Sound object
  • The resulting wavelet coefficients are arrays of the same depth as the input data.
  • The possible wavelets wave include:
  • BattleLemarieWavelet[]BattleLemarié wavelets based on B-spline
    BiorthogonalSplineWavelet[]B-spline-based wavelet
    CoifletWavelet[]symmetric variant of Daubechies wavelets
    DaubechiesWavelet[]the Daubechies wavelets
    HaarWavelet[]classic Haar wavelet
    MeyerWavelet[]wavelet defined in the frequency domain
    ReverseBiorthogonalSplineWavelet[]B-spline-based wavelet (reverse dual and primal)
    ShannonWavelet[]sinc function-based wavelet
    SymletWavelet[]least asymmetric orthogonal wavelet
  • The default wave is HaarWavelet[].
  • With higher settings for the refinement level r, larger-scale features are resolved.
  • The default refinement level r is given by TemplateBox[{{{InterpretationBox[{log, _, DocumentationBuild`Utils`Private`Parenth[2]}, Log2, AutoDelete -> True], (, n, )}, +, {1, /, 2}}}, Floor], where is the minimum dimension of data. »
  • The tree of wavelet coefficients at level consists of coarse coefficients and detail coefficients , with representing the input data.
  • The forward transform is given by and . »
  • The inverse transform is given by . »
  • The are lowpass filter coefficients and are highpass filter coefficients that are defined for each wavelet family.
  • The dimensions of and are given by wd_(j+1)=TemplateBox[{{{1, /, 2},  , {(, {{wd, _, j}, +, fl, -, 2}, )}}}, Ceiling], where is the input data dimension and fl is the filter length for the corresponding wspec. »
  • The following options can be given:
  • MethodAutomaticmethod to use
    Padding"Periodic"how to extend data beyond boundaries
    WorkingPrecisionMachinePrecisionprecision to use in internal computations
  • The settings for Padding are the same as those available in ArrayPad.
  • InverseWaveletTransform gives the inverse transform.

Examples

open allclose all

Basic Examples  (3)

Compute a discrete wavelet transform using the HaarWavelet:

Use Normal to view all coefficients:

Transform an audio signal:

Use dwd[,"Audio"] to extract coefficient signals:

Compute the inverse transform:

Transform an Image object:

Use dwd[,"Image"] to extract coefficient images:

Compute the inverse transform:

Scope  (36)

Basic Uses  (6)

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:

Wavelet Families  (10)

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:

Vector Data  (6)

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:

Matrix Data  (5)

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:

Array Data  (2)

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:

Image Data  (4)

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:

Sound Data  (3)

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:

Generalizations & Extensions  (3)

DiscreteWaveletTransform works on arrays of symbolic quantities:

Inverse transform recovers the input exactly:

Specify any internal working precision:

Use complex-valued data:

The wavelet coefficients are complex:

Inverse transform recovers the input:

Options  (5)

Padding  (2)

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:

WorkingPrecision  (3)

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:

Applications  (11)

Wavelet Compression  (1)

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:

Detect Discontinuities and Edges  (2)

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:

Energy Comparison  (1)

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:

Denoising  (3)

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:

Frequency Filtering  (1)

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:

Finance  (3)

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:

Properties & Relations  (15)

DiscreteWaveletPacketTransform computes the full tree of wavelet coefficients:

DiscreteWaveletTransform computes a subset of the full tree of coefficients:

DiscreteWaveletTransform coefficients halve in length with each level of refinement:

Rotated data gives different coefficients:

StationaryWaveletTransform coefficients have the same length as the original data:

Rotated data gives rotated coefficients:

Multidimensional discrete wavelet transform is related to one-dimensional packet transform:

For Haar wavelet (default) and data length , the computed coefficients are identical:

The default refinement is given by TemplateBox[{{{InterpretationBox[{log, _, DocumentationBuild`Utils`Private`Parenth[2]}, Log2, AutoDelete -> True], (, n, )}, +, {1, /, 2}}}, Floor]:

In higher dimensions:

The energy norm is conserved for orthogonal wavelet families:

The energy norm is approximately conserved for biorthogonal wavelet families:

The mean of the data is captured at the maximum refinement level of the transform:

Extract the coefficient for the maximum refinement level:

Compensate for the normalization at each refinement level:

The sum of inverse transforms from individual coefficient arrays gives the original data:

Individually inverse transform each wavelet coefficient array:

The sum gives the original data:

Compute discrete wavelet coefficients for periodic data:

Define filter coefficients to have compact support:

Coarse coefficients at level are given by , with :

Detail coefficients at level are given by :

Compute a partial discrete inverse wavelet transform:

Define filter coefficients to have compact support:

Coarse coefficients at level are given:

Detail coefficients at level are given:

Inverse wavelet transform at level is given by :

Reconstruct coarse coefficients {0,0} at refinement level :

Reconstruct coarse coefficients {0} at refinement level :

Compute the dimensions of wavelet coefficients:

At refinement level , the dimensions of wavelet coefficients are given by wd_(j+1)=TemplateBox[{{{1, /, 2},  , {(, {{wd, _, j}, +, fl, -, 2}, )}}}, Ceiling], where represents dimensions of input data:

Compare dimensions with coefficient dimensions in dwd:

Compute a Haar discrete wavelet transform in one dimension:

Compute {0} and {1} wavelet coefficients:

Compare with DiscreteWaveletTransform:

In two dimensions, a separate filter is applied in each dimension:

Lowpass and highpass filters for Haar wavelet:

Haar wavelet transform of matrix data:

Compare with DiscreteWaveletTransform using HaarWavelet:

Image channels are transformed individually:

Combine {0} coefficients of separately transformed image channels:

Compare with {0} coefficient of DiscreteWaveletTransform of the original image:

The images are identical:

DWT is similar to LiftingWaveletTransform with extra coefficients needed for padding:

Possible Issues  (1)

Padding can affect the total energy of wavelet coefficients:

Energy is not conserved:

Pad with 0s to ensure energy conservation in the coefficients:

Neat Examples  (1)

Create a padded matrix of data:

Create a 3D plot of the Haar DWT coefficients:

Wolfram Research (2010), DiscreteWaveletTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/DiscreteWaveletTransform.html (updated 2017).

Text

Wolfram Research (2010), DiscreteWaveletTransform, Wolfram Language function, https://reference.wolfram.com/language/ref/DiscreteWaveletTransform.html (updated 2017).

BibTeX

@misc{reference.wolfram_2020_discretewavelettransform, author="Wolfram Research", title="{DiscreteWaveletTransform}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/DiscreteWaveletTransform.html}", note=[Accessed: 01-December-2020 ]}

BibLaTeX

@online{reference.wolfram_2020_discretewavelettransform, organization={Wolfram Research}, title={DiscreteWaveletTransform}, year={2017}, url={https://reference.wolfram.com/language/ref/DiscreteWaveletTransform.html}, note=[Accessed: 01-December-2020 ]}

CMS

Wolfram Language. 2010. "DiscreteWaveletTransform." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2017. https://reference.wolfram.com/language/ref/DiscreteWaveletTransform.html.

APA

Wolfram Language. (2010). DiscreteWaveletTransform. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/DiscreteWaveletTransform.html