# The Wavelet object

## Wavelet families and builtin Wavelets names

`Wavelet` objects are really a 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 builtin 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 builtin wavelets’ names just use the `wavelist()` with no argument.

## Creating Wavelet objects

Now when we know all the names let’s finally create a `Wavelet` object:

    >>> w = pywt.Wavelet('db3')

So.. that’s it.

## Wavelet properties

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

First, let’s try printing a `Wavelet` object. 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 information are the wavelet filters coefficients, which are used in Discrete Wavelet Transform. These coefficients can be obtained via the `dec_lo`, `Wavelet.dec_hi`, `rec_lo` and `rec_hi` attributes, which corresponds to lowpass and highpass decomposition filters and lowpass and 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 Wavelet’s properties are:

> Wavelet `name`, `short_family_name` and `family_name`:

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

> - Decomposition (`dec_len`) and reconstruction (`rec_len`) filter lengths:

>       >>> int(w.dec_len) # int() is for normalizing longs and ints for doctest
>       6
>       >>> int(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 objects. These can be done 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())

> 2. 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:

>     >>> 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

>     >>> 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 be this package 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 tuple with scaling function, wavelet function and 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 `wavefun()` usage and images of all the built-in wavelets on the Wavelet Properties Browser page.