>>> from __future__ import print_function

Wavelet Packets

Import pywt

>>> import pywt

>>> def format_array(a):
...     """Consistent array representation across different systems"""
...     import numpy
...     a = numpy.where(numpy.abs(a) < 1e-5, 0, a)
...     return numpy.array2string(a, precision=5, separator=' ', suppress_small=True)

Create Wavelet Packet structure

Ok, let’s create a sample WaveletPacket:

>>> x = [1, 2, 3, 4, 5, 6, 7, 8]
>>> wp = pywt.WaveletPacket(data=x, wavelet='db1', mode='symmetric')

The input data and decomposition coefficients are stored in the WaveletPacket.data attribute:

>>> print(wp.data)
[1, 2, 3, 4, 5, 6, 7, 8]

Nodes are identified by paths. For the root node the path is '' and the decomposition level is 0.

>>> print(repr(wp.path))
''
>>> print(wp.level)
0

The maxlevel, if not given as param in the constructor, is automatically computed:

>>> print(wp['ad'].maxlevel)
3

Traversing WP tree:

Accessing subnodes:

>>> x = [1, 2, 3, 4, 5, 6, 7, 8]
>>> wp = pywt.WaveletPacket(data=x, wavelet='db1', mode='symmetric')

First check what is the maximum level of decomposition:

>>> print(wp.maxlevel)
3

and try accessing subnodes of the WP tree:

  • 1st level:

    >>> print(wp['a'].data)
[  2.12132034   4.94974747   7.77817459  10.60660172]
>>> print(wp['a'].path)
a

  • 2nd level:

    >>> print(wp['aa'].data)
[  5.  13.]
>>> print(wp['aa'].path)
aa

  • 3rd level:

    >>> print(wp['aaa'].data)
[ 12.72792206]
>>> print(wp['aaa'].path)
aaa

    Ups, we have reached the maximum level of decomposition and got an IndexError:

    >>> print(wp['aaaa'].data)
Traceback (most recent call last):
...
IndexError: Path length is out of range.

Now try some invalid path:

>>> print(wp['ac'])
Traceback (most recent call last):
...
ValueError: Subnode name must be in ['a', 'd'], not 'c'.

which just yielded a ValueError.

Accessing Node’s attributes:

WaveletPacket object is a tree data structure, which evaluates to a set of Node objects. WaveletPacket is just a special subclass of the Node class (which in turn inherits from the BaseNode).

Tree nodes can be accessed using the obj[x] (Node.__getitem__()) operator. Each tree node has a set of attributes: data, path, node_name, parent, level, maxlevel and mode.

>>> x = [1, 2, 3, 4, 5, 6, 7, 8]
>>> wp = pywt.WaveletPacket(data=x, wavelet='db1', mode='symmetric')

>>> print(wp['ad'].data)
[-2. -2.]

>>> print(wp['ad'].path)
ad

>>> print(wp['ad'].node_name)
d

>>> print(wp['ad'].parent.path)
a

>>> print(wp['ad'].level)
2

>>> print(wp['ad'].maxlevel)
3

>>> print(wp['ad'].mode)
symmetric

Collecting nodes

>>> x = [1, 2, 3, 4, 5, 6, 7, 8]
>>> wp = pywt.WaveletPacket(data=x, wavelet='db1', mode='symmetric')

We can get all nodes on the particular level either in natural order:

>>> print([node.path for node in wp.get_level(3, 'natural')])
['aaa', 'aad', 'ada', 'add', 'daa', 'dad', 'dda', 'ddd']

or sorted based on the band frequency (freq):

>>> print([node.path for node in wp.get_level(3, 'freq')])
['aaa', 'aad', 'add', 'ada', 'dda', 'ddd', 'dad', 'daa']

Note that WaveletPacket.get_level() also performs automatic decomposition until it reaches the specified level.

Reconstructing data from Wavelet Packets:

>>> x = [1, 2, 3, 4, 5, 6, 7, 8]
>>> wp = pywt.WaveletPacket(data=x, wavelet='db1', mode='symmetric')

Now create a new Wavelet Packet and set its nodes with some data.

>>> new_wp = pywt.WaveletPacket(data=None, wavelet='db1', mode='symmetric')

>>> new_wp['aa'] = wp['aa'].data
>>> new_wp['ad'] = [-2., -2.]

For convenience, Node.data gets automatically extracted from the Node object:

>>> new_wp['d'] = wp['d']

And reconstruct the data from the aa, ad and d packets.

>>> print(new_wp.reconstruct(update=False))
[ 1.  2.  3.  4.  5.  6.  7.  8.]

If the update param in the reconstruct method is set to False, the node’s data will not be updated.

>>> print(new_wp.data)
None

Otherwise, the data attribute will be set to the reconstructed value.

>>> print(new_wp.reconstruct(update=True))
[ 1.  2.  3.  4.  5.  6.  7.  8.]
>>> print(new_wp.data)
[ 1.  2.  3.  4.  5.  6.  7.  8.]

>>> print([n.path for n in new_wp.get_leaf_nodes(False)])
['aa', 'ad', 'd']

>>> print([n.path for n in new_wp.get_leaf_nodes(True)])
['aaa', 'aad', 'ada', 'add', 'daa', 'dad', 'dda', 'ddd']

Removing nodes from Wavelet Packet tree:

Let’s create a sample data:

>>> x = [1, 2, 3, 4, 5, 6, 7, 8]
>>> wp = pywt.WaveletPacket(data=x, wavelet='db1', mode='symmetric')

First, start with a tree decomposition at level 2. Leaf nodes in the tree are:

>>> dummy = wp.get_level(2)
>>> for n in wp.get_leaf_nodes(False):
...     print(n.path, format_array(n.data))
aa [  5.  13.]
ad [-2. -2.]
da [-1. -1.]
dd [ 0.  0.]

>>> node = wp['ad']
>>> print(node)
ad: [-2. -2.]

To remove a node from the WP tree, use Python’s del obj[x] (Node.__delitem__):

>>> del wp['ad']

The leaf nodes that left in the tree are:

>>> for n in wp.get_leaf_nodes():
...     print(n.path, format_array(n.data))
aa [  5.  13.]
da [-1. -1.]
dd [ 0.  0.]

And the reconstruction is:

>>> print(wp.reconstruct())
[ 2.  3.  2.  3.  6.  7.  6.  7.]

Now restore the deleted node value.

>>> wp['ad'].data = node.data

Printing leaf nodes and tree reconstruction confirms the original state of the tree:

>>> for n in wp.get_leaf_nodes(False):
...     print(n.path, format_array(n.data))
aa [  5.  13.]
ad [-2. -2.]
da [-1. -1.]
dd [ 0.  0.]

>>> print(wp.reconstruct())
[ 1.  2.  3.  4.  5.  6.  7.  8.]

Lazy evaluation:

Note

This section is for demonstration of pywt internals purposes only. Do not rely on the attribute access to nodes as presented in this example.

>>> x = [1, 2, 3, 4, 5, 6, 7, 8]
>>> wp = pywt.WaveletPacket(data=x, wavelet='db1', mode='symmetric')

  1. At first the wp’s attribute a is None

    >>> print(wp.a)
None

    Remember that you should not rely on the attribute access.

  2. At first attempt to access the node it is computed via decomposition of its parent node (the wp object itself).

    >>> print(wp['a'])
a: [  2.12132034   4.94974747   7.77817459  10.60660172]

  3. Now the wp.a is set to the newly created node:

    >>> print(wp.a)
a: [  2.12132034   4.94974747   7.77817459  10.60660172]

    And so is wp.d:

    >>> print(wp.d)
d: [-0.70710678 -0.70710678 -0.70710678 -0.70710678]