# 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) 6Orthogonality (

`orthogonal`

) and biorthogonality (`biorthogonal`

):>>> w.orthogonal True >>> w.biorthogonal TrueSymmetry (

`symmetry`

):>>> print(w.symmetry) asymmetricNumber 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:

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())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: FalseYou 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.