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The Wavelet object#

Wavelet families and builtin Wavelets names#

pywt.Wavelet objects are really handy carriers of a bunch of DWT-specific data like quadrature mirror filters and some general properties associated with them.

At first let’s go through the methods of creating a Wavelet object. The easiest and the most convenient way is to use built-in named Wavelets.

These wavelets are organized into groups called wavelet families. The most commonly used families are:

import pywt
pywt.families()
['haar',
 'db',
 'sym',
 'coif',
 'bior',
 'rbio',
 'dmey',
 'gaus',
 'mexh',
 'morl',
 'cgau',
 'shan',
 'fbsp',
 'cmor']

The wavelist() function with family name passed as an argument is used to obtain the list of wavelet names in each family.

for family in pywt.families():
    print("%s family: " % family + ', '.join(pywt.wavelist(family)))
haar family: haar
db family: db1, db2, db3, db4, db5, db6, db7, db8, db9, db10, db11, db12, db13, db14, db15, db16, db17, db18, db19, db20, db21, db22, db23, db24, db25, db26, db27, db28, db29, db30, db31, db32, db33, db34, db35, db36, db37, db38
sym family: sym2, sym3, sym4, sym5, sym6, sym7, sym8, sym9, sym10, sym11, sym12, sym13, sym14, sym15, sym16, sym17, sym18, sym19, sym20
coif family: coif1, coif2, coif3, coif4, coif5, coif6, coif7, coif8, coif9, coif10, coif11, coif12, coif13, coif14, coif15, coif16, coif17
bior family: bior1.1, bior1.3, bior1.5, bior2.2, bior2.4, bior2.6, bior2.8, bior3.1, bior3.3, bior3.5, bior3.7, bior3.9, bior4.4, bior5.5, bior6.8
rbio family: rbio1.1, rbio1.3, rbio1.5, rbio2.2, rbio2.4, rbio2.6, rbio2.8, rbio3.1, rbio3.3, rbio3.5, rbio3.7, rbio3.9, rbio4.4, rbio5.5, rbio6.8
dmey family: dmey
gaus family: gaus1, gaus2, gaus3, gaus4, gaus5, gaus6, gaus7, gaus8
mexh family: mexh
morl family: morl
cgau family: cgau1, cgau2, cgau3, cgau4, cgau5, cgau6, cgau7, cgau8
shan family: shan
fbsp family: fbsp
cmor family: cmor

To get the full list of built-in wavelets’ names, just use the pywt.wavelist function without any arguments.

Creating Wavelet objects#

Now that we know all the names, let’s finally create a Wavelet object:

w = pywt.Wavelet('db3')

and, that’s it!

Wavelet properties#

But what can we do with Wavelet objects? Well, they carry some interesting pieces of information.

First, let’s try printing a Wavelet object that we used earlier. This shows a brief information about its name, its family name and some properties like orthogonality and symmetry.

print(w)
Wavelet db3
  Family name:    Daubechies
  Short name:     db
  Filters length: 6
  Orthogonal:     True
  Biorthogonal:   True
  Symmetry:       asymmetric
  DWT:            True
  CWT:            False

But the most important bits of information are the wavelet filters coefficients, which are used in Discrete Wavelet Transform. These coefficients can be obtained via the Wavelet.dec_lo, dec_hi, rec_lo, and the rec_hi attributes, which correspond to lowpass & highpass decomposition filters, and lowpass & highpass reconstruction filters respectively:

def print_array(arr):
    print("[%s]" % ", ".join(["%.14f" % x for x in arr]))

Another way to get the filters data is to use the filter_bank attribute, which returns all four filters in a tuple:

w.filter_bank == (w.dec_lo, w.dec_hi, w.rec_lo, w.rec_hi)
True

Other properties of a Wavelet object are:

name, short_family_name, and family_name:

print(w.name)
print(w.short_family_name)
print(w.family_name)
db3
db
Daubechies

Decomposition (dec_len) and reconstruction (rec_len) filter lengths:

w.dec_len
6
w.rec_len
6

Orthogonality (orthogonal) and biorthogonality (biorthogonal):

w.orthogonal
True
w.biorthogonal
True

Symmetry (symmetry):

print(w.symmetry)
asymmetric

Number of vanishing moments for the scaling function phi (vanishing_moments_phi) and the wavelet function psi (vanishing_moments_psi), associated with the filters:

 w.vanishing_moments_phi
0
w.vanishing_moments_psi
3

Now when we know a bit about the builtin Wavelets, let’s see how to create custom wavelets. These can be created in two ways:

  1. Passing the filter bank object that implements the filter_bank attribute. The attribute must return four filters coefficients.

class MyHaarFilterBank(object):
    @property
    def filter_bank(self):
        from math import sqrt
        return (
          [sqrt(2)/2, sqrt(2)/2], [-sqrt(2)/2, sqrt(2)/2],
          [sqrt(2)/2, sqrt(2)/2], [sqrt(2)/2, -sqrt(2)/2]
        )

my_wavelet = pywt.Wavelet('My Haar Wavelet', filter_bank=MyHaarFilterBank())
  1. Passing the filters coefficients directly as the filter_bank parameter.

from math import sqrt
my_filter_bank = (
  [sqrt(2)/2, sqrt(2)/2], [-sqrt(2)/2, sqrt(2)/2],
  [sqrt(2)/2, sqrt(2)/2], [sqrt(2)/2, -sqrt(2)/2]
)
my_wavelet = pywt.Wavelet('My Haar Wavelet', filter_bank=my_filter_bank)

Note that such custom wavelets will not have all the properties set to correct values and some of them could be missing:

print(my_wavelet)
Wavelet My Haar Wavelet
  Family name:
  Short name:
  Filters length: 2
  Orthogonal:     False
  Biorthogonal:   False
  Symmetry:       unknown
  DWT:            True
  CWT:            False

You can, however, set a couple of them on your own:

my_wavelet.orthogonal = True
my_wavelet.biorthogonal = True

Let’s view the values of the custom wavelet properties again:

print(my_wavelet)
Wavelet My Haar Wavelet
  Family name:
  Short name:
  Filters length: 2
  Orthogonal:     True
  Biorthogonal:   True
  Symmetry:       unknown
  DWT:            True
  CWT:            False

And now… the wavefun!#

We all know that the fun with wavelets is in wavelet functions. Now, what would this package be without a tool to compute wavelet and scaling functions approximations?

This is the purpose of the wavefun() method, which is used to approximate scaling function (phi) and wavelet function (psi) at the given level of refinement, based on the filters coefficients.

The number of returned values varies depending on the wavelet’s orthogonality property. For orthogonal wavelets, the result is a tuple with the scaling function, wavelet function, and the xgrid coordinates.

w = pywt.Wavelet('sym3')
w.orthogonal
True
(phi, psi, x) = w.wavefun(level=5)

For biorthogonal (non-orthogonal) wavelets, different scaling and wavelet functions are used for decomposition and reconstruction, and thus, five elements are returned: decomposition scaling and wavelet functions approximations, reconstruction scaling and wavelet functions approximations, and the xgrid.

w = pywt.Wavelet('bior1.3')
w.orthogonal
False
(phi_d, psi_d, phi_r, psi_r, x) = w.wavefun(level=5)

See also

You can find live examples of the usage of wavefun() 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.