### Navigation

- index
- next |
- previous |
- Home »
- PyWavelets »
- Usage examples »

# 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 its 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 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='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 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]

### Navigation

- index
- next |
- previous |
- Home »
- PyWavelets »
- Usage examples »