ContinuousWaveletData

ContinuousWaveletData[{{oct1,voc1}->coef1,…},wave]

yields a continuous wavelet data object with wavelet coefficients coefi corresponding to octave and voice {octi,voci} and wavelet wave.

Details and Options

  • ContinuousWaveletData[{{oct1,voc1}->coef1,…},…] is always converted to an optimized standard form with structure ContinuousWaveletData[coefs,octvocs,…].
  • The coefficients coefi can be vectors, Sound[…], or SampledSoundList[…] objects.
  • The options used by ContinuousWaveletTransform can also be used as options to ContinuousWaveletData.
  • In standard output format, only the number of octaves, voices, and dimensions of the original data are printed.
  • Normal[ContinuousWaveletData[…]] gives a list of rules {{oct1,voc1}->coef1,{oct2,voc2}->coef2,…} which gives the correspondence between each octave and voice {octi,voci} and the corresponding coefficient array coefi.
  • ContinuousWaveletData represents a continuous wavelet transform w(u,s)=1/(sqrt(s))sum_(k=1)^nx_k TemplateBox[{psi}, Conjugate]((Delta (k-u))/s) at multiple scales .
  • Each scale is specified by an octave number and voice number and is given by .
  • The scale {oct,voc} can be used to extract wavelet coefficients from a ContinuousWaveletData object cwd. The following specifications can be given:
  • cwd[{oct,voc}]extract coefficients corresponding to {oct,voc}
    cwd[{{oct1,voc1},{oct2,voc2},…}]extract several wavelet coefficient arrays
    cwd[ovpatt]extract all coefficients whose scale matches ovpatt
    cwd[All]extract all coefficients
  • By default, coefficients are returned as a list of rules {{oct1,voc1}->coef1,{oct2,voc2}->coef2,…}.
  • cwd[…,{form1,form2,…}] can be used to control the output form. Possible formi include:
  • "Rules"rules {{oct1,voc1}->…}
    "Values"coefficients only
    "Inverse"inverse transform individual coefficients
    "ListPlot"simple list plots for 1D coefficients
    "Sound"sound objects for sound coefficients
    "SampledSoundList"sampled sound objects for sound coefficients
  • Properties of a wavelet representation are obtained from ContinuousWaveletData[…]["prop"].
  • ContinuousWaveletData[…]["Properties"] gives a list of properties available for the ContinuousWaveletData object.
  • Properties related to transform coefficients include:
  • "Octaves"the number of octaves used
    "Voices"the number of voices per octave used
    "Scales"wavelet scales used
    "Wavelet"wavelet family used
    "WaveletScale"smallest resolvable scale
    "WaveletIndex"list of all wavelet indices {octi,voci}
    "LogScalogramFunction"gives the function
    "LinearScalogramFunction"gives the function
  • Properties related to input data include:
  • "DataDimensions"dimensions of original data
    "DataChannels"the number of channels of data
    "DataMean"the mean of original data
    "DataWrapper"wrapper function applied to data after reconstruction
    "SampleRate"sample rate used for input data

Examples

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Basic Examples  (2)

Get a ContinuousWaveletData object:

The ContinuousWaveletData represents arrays of coefficients at different scales {oct,voc}:

Extract properties including numerical scale corresponding to each {oct,voc}:

Compute the inverse wavelet transform of arbitrary continuous wavelet transform coefficients:

Scope  (12)

Basic Uses  (10)

Get a ContinuousWaveletData object from ContinuousWaveletTransform:

Show the list of computed scales {oct,voc}:

Get the coefficient arrays as a list of rules:

InverseContinuousWaveletTransform operates on ContinuousWaveletData:

With enough octaves and voices, the inverse transform gives an accurate reconstruction:

Extract coefficients corresponding to specific octaves and voices:

Extract a single coefficient array:

Extract all coefficients in the first octave using the pattern {1,_}:

Specify a list including both specific {oct,voc} and patterns:

Get coefficient arrays in different forms:

Get as a list of rules (default):

Get values only:

Get coefficients as small list plots:

Get an inverse transform of each coefficient array:

Combine forms:

Get sound wavelet coefficients as Sound objects:

Extract properties of the wavelet transform data:

Number of octaves and voices, and wavelet used:

All available properties:

Use ContinuousWaveletData in other wavelet functions:

Inverse transform:

Scalogram:

Transform ContinuousWaveletData:

Apply an arbitrary function to each coefficient:

Set all coefficients matching a specified {oct,voc} to zero:

Compare scalograms:

Construct a ContinuousWaveletData object from a list of rules giving coefficient arrays:

The result represents a range of octaves and voices including the specified coefficients:

The other coefficients are assumed to be zero:

Construct a ContinuousWaveletData object using a specified wavelet:

The specified wavelet is used in the inverse transform:

Properties  (2)

Get properties of the continuous wavelet transform:

Wavelet and wavelet scale used:

Properties of transform coefficients, including number of octaves and voices:

List of all available {oct,voc}:

Numerical scales corresponding to each {oct,voc}:

Properties related to input data:

Data dimensions, number of audio channels, and sample rate:

Wrapper function that is automatically applied to the result of an inverse transform:

Options  (5)

SampleRate  (1)

For Sound input, SampleRate is automatically computed:

By default, SampleRate is extracted from the first coefficient rule:

Specify SampleRate explicitly:

WaveletScale  (2)

By default, WaveletScale is automatically computed:

Set ContinuousWaveletData:

Explicitly set WaveletScale:

WorkingPrecision  (2)

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:

Properties & Relations  (5)

The length of each wavelet coefficient array equals the length of the data:

ContinuousWaveletData represents continuous transform coefficients at a set of scales:

DiscreteWaveletData represents a tree of discrete transform coefficients:

Reconstruct a ContinuousWaveletData from its coefficients and properties:

Specify the coefficients and the wavelet used:

The data dimensions are determined from the coefficients:

Compare inverse transforms:

Equivalent ways to get all coefficients as a list of rules:

Use Normal:

Explicitly extract All coefficients:

Specify the pattern Blank[] (_), which matches any octave and voice:

Equivalent ways to get only coefficient lists:

Apply Last to each rule returned by cwd[{oct,voc}]:

Use Part:

Explicitly get only coefficient values:

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

Text

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

CMS

Wolfram Language. 2010. "ContinuousWaveletData." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ContinuousWaveletData.html.

APA

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

BibTeX

@misc{reference.wolfram_2024_continuouswaveletdata, author="Wolfram Research", title="{ContinuousWaveletData}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/ContinuousWaveletData.html}", note=[Accessed: 22-November-2024 ]}

BibLaTeX

@online{reference.wolfram_2024_continuouswaveletdata, organization={Wolfram Research}, title={ContinuousWaveletData}, year={2010}, url={https://reference.wolfram.com/language/ref/ContinuousWaveletData.html}, note=[Accessed: 22-November-2024 ]}