Wavelets

Wavelet families()

pywt.families(short=True)

Returns a list of available built-in wavelet families.

Currently the built-in families are:

  • Haar (haar)

  • Daubechies (db)

  • Symlets (sym)

  • Coiflets (coif)

  • Biorthogonal (bior)

  • Reverse biorthogonal (rbio)

  • “Discrete” FIR approximation of Meyer wavelet (dmey)

  • Gaussian wavelets (gaus)

  • Mexican hat wavelet (mexh)

  • Morlet wavelet (morl)

  • Complex Gaussian wavelets (cgau)

  • Shannon wavelets (shan)

  • Frequency B-Spline wavelets (fbsp)

  • Complex Morlet wavelets (cmor)

Parameters
shortbool, optional

Use short names (default: True).

Returns
familieslist

List of available wavelet families.

Examples

>>> import pywt
>>> pywt.families()
['haar', 'db', 'sym', 'coif', 'bior', 'rbio', 'dmey', 'gaus', 'mexh', 'morl', 'cgau', 'shan', 'fbsp', 'cmor']
>>> pywt.families(short=False)
['Haar', 'Daubechies', 'Symlets', 'Coiflets', 'Biorthogonal', 'Reverse biorthogonal', 'Discrete Meyer (FIR Approximation)', 'Gaussian', 'Mexican hat wavelet', 'Morlet wavelet', 'Complex Gaussian wavelets', 'Shannon wavelets', 'Frequency B-Spline wavelets', 'Complex Morlet wavelets']

Built-in wavelets - wavelist()

pywt.wavelist(family=None, kind='all')

Returns list of available wavelet names for the given family name.

Parameters
familystr, optional

Short family name. If the family name is None (default) then names of all the built-in wavelets are returned. Otherwise the function returns names of wavelets that belong to the given family. Valid names are:

'haar', 'db', 'sym', 'coif', 'bior', 'rbio', 'dmey', 'gaus',
'mexh', 'morl', 'cgau', 'shan', 'fbsp', 'cmor'
kind{‘all’, ‘continuous’, ‘discrete’}, optional

Whether to return only wavelet names of discrete or continuous wavelets, or all wavelets. Default is 'all'. Ignored if family is specified.

Returns
wavelistlist of str

List of available wavelet names.

Examples

>>> import pywt
>>> pywt.wavelist('coif')
['coif1', 'coif2', 'coif3', 'coif4', 'coif5', 'coif6', 'coif7', ...
>>> pywt.wavelist(kind='continuous')
['cgau1', 'cgau2', 'cgau3', 'cgau4', 'cgau5', 'cgau6', 'cgau7', ...

Custom discrete wavelets are also supported through the Wavelet object constructor as described below.

Wavelet object

class pywt.Wavelet(name[, filter_bank=None])

Describes properties of a discrete wavelet identified by the specified wavelet name. For continuous wavelets see pywt.ContinuousWavelet instead. In order to use a built-in wavelet the name parameter must be a valid wavelet name from the pywt.wavelist() list.

Custom Wavelet objects can be created by passing a user-defined filters set with the filter_bank parameter.

Parameters

  • name – Wavelet name

  • filter_bank – Use a user supplied filter bank instead of a built-in Wavelet.

The filter bank object can be a list of four filters coefficients or an object with filter_bank attribute, which returns a list of such filters in the following order:

[dec_lo, dec_hi, rec_lo, rec_hi]

Wavelet objects can also be used as a base filter banks. See section on using custom wavelets for more information.

Example:

>>> import pywt
>>> wavelet = pywt.Wavelet('db1')
name

Wavelet name.

short_name

Short wavelet name.

dec_lo

Decomposition filter values.

dec_hi

Decomposition filter values.

rec_lo

Reconstruction filter values.

rec_hi

Reconstruction filter values.

dec_len

Decomposition filter length.

rec_len

Reconstruction filter length.

filter_bank

Returns filters list for the current wavelet in the following order:

[dec_lo, dec_hi, rec_lo, rec_hi]
inverse_filter_bank

Returns list of reverse wavelet filters coefficients. The mapping from the filter_coeffs list is as follows:

[rec_lo[::-1], rec_hi[::-1], dec_lo[::-1], dec_hi[::-1]]
short_family_name

Wavelet short family name

family_name

Wavelet family name

orthogonal

Set if wavelet is orthogonal

biorthogonal

Set if wavelet is biorthogonal

symmetry

asymmetric, near symmetric, symmetric

vanishing_moments_psi

Number of vanishing moments for the wavelet function

vanishing_moments_phi

Number of vanishing moments for the scaling function

Example:

>>> def format_array(arr):
...     return "[%s]" % ", ".join(["%.14f" % x for x in arr])

>>> import pywt
>>> wavelet = pywt.Wavelet('db1')
>>> print(wavelet)
Wavelet db1
  Family name:    Daubechies
  Short name:     db
  Filters length: 2
  Orthogonal:     True
  Biorthogonal:   True
  Symmetry:       asymmetric
  DWT:            True
  CWT:            False
>>> print(format_array(wavelet.dec_lo), format_array(wavelet.dec_hi))
[0.70710678118655, 0.70710678118655] [-0.70710678118655, 0.70710678118655]
>>> print(format_array(wavelet.rec_lo), format_array(wavelet.rec_hi))
[0.70710678118655, 0.70710678118655] [0.70710678118655, -0.70710678118655]

Approximating wavelet and scaling functions - Wavelet.wavefun()

Wavelet.wavefun(level)

Changed in version 0.2: The time (space) localisation of approximation function points was added.

The wavefun() method can be used to calculate approximations of scaling function (phi) and wavelet function (psi) at the given level of refinement.

For orthogonal wavelets returns approximations of scaling function and wavelet function with corresponding x-grid coordinates:

[phi, psi, x] = wavelet.wavefun(level)

Example:

>>> import pywt
>>> wavelet = pywt.Wavelet('db2')
>>> phi, psi, x = wavelet.wavefun(level=5)

For other (biorthogonal but not orthogonal) wavelets returns approximations of scaling and wavelet function both for decomposition and reconstruction and corresponding x-grid coordinates:

[phi_d, psi_d, phi_r, psi_r, x] = wavelet.wavefun(level)

Example:

>>> import pywt
>>> wavelet = pywt.Wavelet('bior3.5')
>>> phi_d, psi_d, phi_r, psi_r, x = wavelet.wavefun(level=5)

See also

You can find live examples of wavefun() usage and images of all the built-in wavelets on the Wavelet Properties Browser page. However, this website is no longer actively maintained and does not include every wavelet present in PyWavelets. The precision of the wavelet coefficients at that site is also lower than those included in PyWavelets.

Using custom wavelets

PyWavelets comes with a long list of the most popular wavelets built-in and ready to use. If you need to use a specific wavelet which is not included in the list it is very easy to do so. Just pass a list of four filters or an object with a filter_bank attribute as a filter_bank argument to the Wavelet constructor.

The filters list, either in a form of a simple Python list or returned via the filter_bank attribute, must be in the following order:

  • lowpass decomposition filter

  • highpass decomposition filter

  • lowpass reconstruction filter

  • highpass reconstruction filter

just as for the filter_bank attribute of the Wavelet class.

The Wavelet object created in this way is a standard Wavelet instance.

The following example illustrates the way of creating custom Wavelet objects from plain Python lists of filter coefficients and a filter bank-like object.

Example:

>>> import pywt, math
>>> c = math.sqrt(2)/2
>>> dec_lo, dec_hi, rec_lo, rec_hi = [c, c], [-c, c], [c, c], [c, -c]
>>> filter_bank = [dec_lo, dec_hi, rec_lo, rec_hi]
>>> myWavelet = pywt.Wavelet(name="myHaarWavelet", filter_bank=filter_bank)
>>>
>>> class HaarFilterBank(object):
...     @property
...     def filter_bank(self):
...         c = math.sqrt(2)/2
...         dec_lo, dec_hi, rec_lo, rec_hi = [c, c], [-c, c], [c, c], [c, -c]
...         return [dec_lo, dec_hi, rec_lo, rec_hi]
>>> filter_bank = HaarFilterBank()
>>> myOtherWavelet = pywt.Wavelet(name="myHaarWavelet", filter_bank=filter_bank)

ContinuousWavelet object

class pywt.ContinuousWavelet(name, dtype=np.float64)

Describes properties of a continuous wavelet identified by the specified wavelet name. In order to use a built-in wavelet the name parameter must be a valid wavelet name from the pywt.wavelist() list.

Parameters

  • name – Wavelet name

  • dtype – numpy.dtype to use for the wavelet. Can be numpy.float64 or numpy.float32.

Example:

>>> import pywt
>>> wavelet = pywt.ContinuousWavelet('gaus1')
name

Continuous Wavelet name.

short_family_name

Wavelet short family name

family_name

Wavelet family name

orthogonal

Set if wavelet is orthogonal

biorthogonal

Set if wavelet is biorthogonal

complex_cwt

Returns if wavelet is complex

lower_bound

Set the lower bound of the effective support

upper_bound

Set the upper bound of the effective support

center_frequency

Set the center frequency for the shan, fbsp and cmor wavelets

bandwidth_frequency

Set the bandwidth frequency for the shan, fbsp and cmor wavelets

fbsp_order

Set the order for the fbsp wavelet

symmetry

asymmetric, near symmetric, symmetric, anti-symmetric

Example:

>>> import pywt
>>> wavelet = pywt.ContinuousWavelet('gaus1')
>>> print(wavelet)
ContinuousWavelet gaus1
  Family name:    Gaussian
  Short name:     gaus
  Symmetry:       anti-symmetric
  DWT:            False
  CWT:            True
  Complex CWT:    False

Approximating wavelet functions - ContinuousWavelet.wavefun()

ContinuousWavelet.wavefun(level, length=None)

The wavefun() method can be used to calculate approximations of scaling function (psi) with grid (x). The vector length is set by length. The vector length can also be defined by 2**level if length is not set.

For complex_cwt wavelets returns a complex approximations of wavelet function with corresponding x-grid coordinates:

[psi, x] = wavelet.wavefun(level)

Example:

>>> import pywt
>>> wavelet = pywt.ContinuousWavelet('gaus1')
>>> psi, x = wavelet.wavefun(level=5)

Approximating wavelet functions - ContinuousWavelet.wavefun()

pywt.DiscreteContinuousWavelet(name[, filter_bank = None])
The DiscreteContinuousWavelet() returns a

Wavelet or a ContinuousWavelet object depending on the given name.

Example:

>>> import pywt
>>> wavelet = pywt.DiscreteContinuousWavelet('db1')
>>> print(wavelet)
Wavelet db1
  Family name:    Daubechies
  Short name:     db
  Filters length: 2
  Orthogonal:     True
  Biorthogonal:   True
  Symmetry:       asymmetric
  DWT:            True
  CWT:            False
>>> wavelet = pywt.DiscreteContinuousWavelet('gaus1')
>>> print(wavelet)
ContinuousWavelet gaus1
  Family name:    Gaussian
  Short name:     gaus
  Symmetry:       anti-symmetric
  DWT:            False
  CWT:            True
  Complex CWT:    False