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='sym')

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='sym')

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='sym')

>>> 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
sym

Collecting nodes

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

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='sym')

Now create a new Wavelet Packet and set it’s nodes with some data.

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

>>> 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='sym')

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 eveluation:

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='sym')

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 it’s 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]