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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

open allclose all

Basic Examples  (2)Summary of the most common use cases

Get a ContinuousWaveletData object:

In[1]:=1
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-bggie2
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-clgxrw
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-j4ukwq
Out[3]=3

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

In[1]:=1
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-i3o0fe
Out[1]=1

Scope  (12)Survey of the scope of standard use cases

Basic Uses  (10)

Get a ContinuousWaveletData object from ContinuousWaveletTransform:

In[1]:=1
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-dua9ap
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-fho3xu
Out[2]=2

Get the coefficient arrays as a list of rules:

In[3]:=3
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-dwznbn
Out[3]=3

InverseContinuousWaveletTransform operates on ContinuousWaveletData:

In[1]:=1
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-cgp6wm
In[2]:=2
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-3exuz
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-kbjry7
Out[3]=3

Extract coefficients corresponding to specific octaves and voices:

In[1]:=1
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-q1m6j
Out[1]=1

Extract a single coefficient array:

In[2]:=2
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-ewddiq
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-kdvwmd
Out[3]=3

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

In[4]:=4
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-bas555
Out[4]=4

Get coefficient arrays in different forms:

In[1]:=1
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-igjnm6
Out[1]=1

Get as a list of rules (default):

In[2]:=2
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-xstdo
Out[2]=2

Get values only:

In[3]:=3
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-fi53sf
Out[3]=3

Get coefficients as small list plots:

In[4]:=4
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-bbagie
Out[4]=4

Get an inverse transform of each coefficient array:

In[5]:=5
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-1ojkx
Out[5]=5

Combine forms:

In[6]:=6
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-bfa3z1
Out[6]=6

Get sound wavelet coefficients as Sound objects:

In[1]:=1
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-c5i7eb
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-efuxli
Out[2]=2

Extract properties of the wavelet transform data:

In[1]:=1
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-ca22gi
Out[1]=1

Number of octaves and voices, and wavelet used:

In[2]:=2
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-bhkppp
Out[2]=2

All available properties:

In[3]:=3
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-fcmoa
Out[3]=3

Use ContinuousWaveletData in other wavelet functions:

In[1]:=1
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-5ve1p
In[2]:=2
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-x0a1p
Out[2]=2

Inverse transform:

In[3]:=3
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-zw6ko
Out[3]=3

Scalogram:

In[4]:=4
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-f7srhe
Out[4]=4

Transform ContinuousWaveletData:

In[1]:=1
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-ccpsk
In[2]:=2
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-in57yu
Out[2]=2

Apply an arbitrary function to each coefficient:

In[3]:=3
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-c0frxo
Out[3]=3

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

In[4]:=4
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-dtle8m
Out[4]=4

Compare scalograms:

In[5]:=5
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-f31e44
Out[5]=5

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

In[1]:=1
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-d7j35v
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-ei3p5y
Out[2]=2

The other coefficients are assumed to be zero:

In[3]:=3
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-h2s7um
Out[3]=3

Construct a ContinuousWaveletData object using a specified wavelet:

In[1]:=1
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-islkuu
Out[1]=1

The specified wavelet is used in the inverse transform:

In[2]:=2
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-dgjcuc
Out[2]=2

Properties  (2)

Get properties of the continuous wavelet transform:

In[1]:=1
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-cwo7hg
Out[1]=1

Wavelet and wavelet scale used:

In[2]:=2
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-dvucjy
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-irom8w
Out[3]=3

List of all available {oct,voc}:

In[4]:=4
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-i4upjp
Out[4]=4

Numerical scales corresponding to each {oct,voc}:

In[5]:=5
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-duczw1
Out[5]=5

Properties related to input data:

In[1]:=1
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-g77h01
In[2]:=2
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-btkqm6
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-2b963
Out[3]=3

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

In[4]:=4
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-e509ba
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-ko26av
Out[5]=5

Options  (5)Common values & functionality for each option

SampleRate  (1)

For Sound input, SampleRate is automatically computed:

In[1]:=1
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-xc74fv
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-u0mal1
Out[2]=2

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

In[3]:=3
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-vh5x4x
In[4]:=4
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-45juhj
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-8fcs5p
Out[5]=5

Specify SampleRate explicitly:

In[6]:=6
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-5n0c5k
Out[6]=6

WaveletScale  (2)

By default, WaveletScale is automatically computed:

In[1]:=1
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-27hc1i
In[2]:=2
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-3zaus0
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-6qn914
Out[3]=3

Set ContinuousWaveletData:

In[4]:=4
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-i05g7p
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-kj84om
Out[5]=5

Explicitly set WaveletScale:

In[1]:=1
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-oxidu2
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-dp4fu9
Out[2]=2

WorkingPrecision  (2)

By default, WorkingPrecision->MachinePrecision is used:

In[1]:=1
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-fvcmsi
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-qlff3u
Out[2]=2
In[3]:=3
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-7qfsoe
Out[3]=3

Use higher-precision computation:

In[1]:=1
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-6lkrhl
Out[1]=1

With numbers close to zero, accuracy is the better indicator of the number of correct digits:

In[2]:=2
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-cqyqyd
Out[2]=2

Properties & Relations  (5)Properties of the function, and connections to other functions

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

In[1]:=1
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-fm6yrb
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-fujd56
Out[2]=2

ContinuousWaveletData represents continuous transform coefficients at a set of scales:

In[1]:=1
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-fpkibb
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-ohmn9
Out[2]=2

DiscreteWaveletData represents a tree of discrete transform coefficients:

In[3]:=3
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-zzvt6
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-jalso3
Out[4]=4

Reconstruct a ContinuousWaveletData from its coefficients and properties:

In[1]:=1
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-bfsdx
Out[1]=1

Specify the coefficients and the wavelet used:

In[2]:=2
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-gd9q04
Out[2]=2

The data dimensions are determined from the coefficients:

In[3]:=3
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-u5chd
Out[3]=3

Compare inverse transforms:

In[4]:=4
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-9hi4n7
Out[4]=4

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

In[1]:=1
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-gja6x7
Out[1]=1

Use Normal:

In[2]:=2
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-kygfk
Out[2]=2

Explicitly extract All coefficients:

In[3]:=3
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-e6kezw
Out[3]=3

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

In[4]:=4
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-bhh5s
Out[4]=4

Equivalent ways to get only coefficient lists:

In[1]:=1
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-i1xfk7
Out[1]=1

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

In[2]:=2
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-by6nzu
Out[2]=2

Use Part:

In[3]:=3
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-fbkorx
Out[3]=3

Explicitly get only coefficient values:

In[4]:=4
✖
https://wolfram.com/xid/0bswe5jb6i7vyzk8wy-fgn9vv
Out[4]=4
Wolfram Research (2010), ContinuousWaveletData, Wolfram Language function, https://reference.wolfram.com/language/ref/ContinuousWaveletData.html.
✖
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.

✖
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.

✖
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

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

BibTeX

@misc{reference.wolfram_2025_continuouswaveletdata, author="Wolfram Research", title="{ContinuousWaveletData}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/ContinuousWaveletData.html}", note=[Accessed: 25-March-2025 ]}

✖
@misc{reference.wolfram_2025_continuouswaveletdata, author="Wolfram Research", title="{ContinuousWaveletData}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/ContinuousWaveletData.html}", note=[Accessed: 25-March-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_continuouswaveletdata, organization={Wolfram Research}, title={ContinuousWaveletData}, year={2010}, url={https://reference.wolfram.com/language/ref/ContinuousWaveletData.html}, note=[Accessed: 25-March-2025 ]}

✖
@online{reference.wolfram_2025_continuouswaveletdata, organization={Wolfram Research}, title={ContinuousWaveletData}, year={2010}, url={https://reference.wolfram.com/language/ref/ContinuousWaveletData.html}, note=[Accessed: 25-March-2025 ]}