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# 2D Wavelet Packets

## Import pywt

    >>> import pywt
    >>> import numpy

## Create 2D Wavelet Packet structure

Start with preparing test data:

    >>> x = numpy.array([[1, 2, 3, 4, 5, 6, 7, 8]] * 8, 'd')
    >>> print x
    [[ 1.  2.  3.  4.  5.  6.  7.  8.]
     [ 1.  2.  3.  4.  5.  6.  7.  8.]
     [ 1.  2.  3.  4.  5.  6.  7.  8.]
     [ 1.  2.  3.  4.  5.  6.  7.  8.]
     [ 1.  2.  3.  4.  5.  6.  7.  8.]
     [ 1.  2.  3.  4.  5.  6.  7.  8.]
     [ 1.  2.  3.  4.  5.  6.  7.  8.]
     [ 1.  2.  3.  4.  5.  6.  7.  8.]]

Now create a `2D Wavelet Packet` object:

    >>> wp = pywt.WaveletPacket2D(data=x, wavelet='db1', mode='sym')

The input data and decomposition coefficients are stored in the `WaveletPacket2D.data` attribute:

    >>> print wp.data
    [[ 1.  2.  3.  4.  5.  6.  7.  8.]
     [ 1.  2.  3.  4.  5.  6.  7.  8.]
     [ 1.  2.  3.  4.  5.  6.  7.  8.]
     [ 1.  2.  3.  4.  5.  6.  7.  8.]
     [ 1.  2.  3.  4.  5.  6.  7.  8.]
     [ 1.  2.  3.  4.  5.  6.  7.  8.]
     [ 1.  2.  3.  4.  5.  6.  7.  8.]
     [ 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 `WaveletPacket2D.maxlevel`, if not given in the constructor, is automatically computed based on the data size:

    >>> print wp.maxlevel
    3

## Traversing WP tree:

Wavelet Packet `nodes` are arranged in a tree. Each node in a WP tree is uniquely identified and addressed by a `path` string.

In the 1D `WaveletPacket` case nodes were accessed using `'a'` (approximation) and `'d'` (details) path names (each node has two 1D children).

Because now we deal with a bit more complex structure (each node has four children), we have four basic path names based on the dwt 2D output convention to address the WP2D structure:

> - `a` - LL, low-low coefficients
> - `h` - LH, low-high coefficients
> - `v` - HL, high-low coefficients
> - `d` - HH, high-high coefficients

In other words, subnode naming corresponds to the `dwt2()` function output naming convention (as wavelet packet transform is based on the dwt2 transform):

                                -------------------
                                |        |        |
                                | cA(LL) | cH(LH) |
                                |        |        |
    (cA, (cH, cV, cD))  <--->   -------------------
                                |        |        |
                                | cV(HL) | cD(HH) |
                                |        |        |
                                -------------------

       (fig.1: DWT 2D output and interpretation)

Knowing what the nodes names are, we can now access them using the indexing operator obj[x] (`WaveletPacket2D.__getitem__()`):

    >>> print wp['a'].data
    [[  3.   7.  11.  15.]
     [  3.   7.  11.  15.]
     [  3.   7.  11.  15.]
     [  3.   7.  11.  15.]]
    >>> print wp['h'].data
    [[ 0.  0.  0.  0.]
     [ 0.  0.  0.  0.]
     [ 0.  0.  0.  0.]
     [ 0.  0.  0.  0.]]
    >>> print wp['v'].data
    [[-1. -1. -1. -1.]
     [-1. -1. -1. -1.]
     [-1. -1. -1. -1.]
     [-1. -1. -1. -1.]]
    >>> print wp['d'].data
    [[ 0.  0.  0.  0.]
     [ 0.  0.  0.  0.]
     [ 0.  0.  0.  0.]
     [ 0.  0.  0.  0.]]

Similarly, a subnode of a subnode can be accessed by:

    >>> print wp['aa'].data
    [[ 10.  26.]
     [ 10.  26.]]

Indexing base `WaveletPacket2D` (as well as 1D `WaveletPacket`) using compound path is just the same as indexing WP subnode:

    >>> node = wp['a']
    >>> print node['a'].data
    [[ 10.  26.]
     [ 10.  26.]]
    >>> print wp['a']['a'].data is wp['aa'].data
    True

Following down the decomposition path:

    >>> print wp['aaa'].data
    [[ 36.]]
    >>> print wp['aaaa'].data
    Traceback (most recent call last):
    ...
    IndexError: Path length is out of range.

Ups, we have reached the maximum level of decomposition for the `'aaaa'` path, which btw. was:

    >>> print wp.maxlevel
    3

Now try some invalid path:

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

### Accessing Node2D’s attributes:

`WaveletPacket2D` is a tree data structure, which evaluates to a set of `Node2D` objects. `WaveletPacket2D` is just a special subclass of the `Node2D` class (which in turn inherits from a `BaseNode`, just like with `Node` and `WaveletPacket` for the 1D case.).

    >>> print wp['av'].data
    [[-4. -4.]
     [-4. -4.]]

    >>> print wp['av'].path
    av

    >>> print wp['av'].node_name
    v

    >>> print wp['av'].parent.path
    a

    >>> print wp['av'].parent.data
    [[  3.   7.  11.  15.]
     [  3.   7.  11.  15.]
     [  3.   7.  11.  15.]
     [  3.   7.  11.  15.]]

    >>> print wp['av'].level
    2

    >>> print wp['av'].maxlevel
    3

    >>> print wp['av'].mode
    sym

### Collecting nodes

We can get all nodes on the particular level using the `WaveletPacket2D.get_level()` method:

> - 0 level - the root wp node:

>       >>> len(wp.get_level(0))
>       1
>       >>> print [node.path for node in wp.get_level(0)]
>       ['']

> - 1st level of decomposition:

>       >>> len(wp.get_level(1))
>       4
>       >>> print [node.path for node in wp.get_level(1)]
>       ['a', 'h', 'v', 'd']

> - 2nd level of decomposition:

>       >>> len(wp.get_level(2))
>       16
>       >>> paths = [node.path for node in wp.get_level(2)]
>       >>> for i, path in enumerate(paths):
>       ...     print path,
>       ...     if (i+1) % 4 == 0: print
>       aa ah av ad
>       ha hh hv hd
>       va vh vv vd
>       da dh dv dd

> - 3rd level of decomposition:

>       >>> print len(wp.get_level(3))
>       64
>       >>> paths = [node.path for node in wp.get_level(3)]
>       >>> for i, path in enumerate(paths):
>       ...     print path,
>       ...     if (i+1) % 8 == 0: print
>       aaa aah aav aad aha ahh ahv ahd
>       ava avh avv avd ada adh adv add
>       haa hah hav had hha hhh hhv hhd
>       hva hvh hvv hvd hda hdh hdv hdd
>       vaa vah vav vad vha vhh vhv vhd
>       vva vvh vvv vvd vda vdh vdv vdd
>       daa dah dav dad dha dhh dhv dhd
>       dva dvh dvv dvd dda ddh ddv ddd

Note that `WaveletPacket2D.get_level()` performs automatic decomposition until it reaches the given level.

## Reconstructing data from Wavelet Packets:

Let’s create a new empty 2D Wavelet Packet structure and set its nodes values with known data from the previous examples:

>     >>> new_wp = pywt.WaveletPacket2D(data=None, wavelet='db1', mode='sym')

>     >>> new_wp['vh'] = wp['vh'].data # [[0.0, 0.0], [0.0, 0.0]]
>     >>> new_wp['vv'] = wp['vh'].data # [[0.0, 0.0], [0.0, 0.0]]
>     >>> new_wp['vd'] = [[0.0, 0.0], [0.0, 0.0]]

>     >>> new_wp['a'] = [[3.0, 7.0, 11.0, 15.0], [3.0, 7.0, 11.0, 15.0],
>     ...                [3.0, 7.0, 11.0, 15.0], [3.0, 7.0, 11.0, 15.0]]
>     >>> new_wp['d'] = [[0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0],
>     ...                [0.0, 0.0, 0.0, 0.0], [0.0, 0.0, 0.0, 0.0]]

> For convenience, `Node2D.data` gets automatically extracted from the base `Node2D` object:

>     >>> new_wp['h'] = wp['h'] # all zeros

> Note: just remember to not assign to the node.data parameter directly (todo).

And reconstruct the data from the `a`, `d`, `vh`, `vv`, `vd` and `h` packets (Note that `va` node was not set and the WP tree is “not complete” - the `va` branch will be treated as zero-array):

    >>> print new_wp.reconstruct(update=False)
    [[ 1.5  1.5  3.5  3.5  5.5  5.5  7.5  7.5]
     [ 1.5  1.5  3.5  3.5  5.5  5.5  7.5  7.5]
     [ 1.5  1.5  3.5  3.5  5.5  5.5  7.5  7.5]
     [ 1.5  1.5  3.5  3.5  5.5  5.5  7.5  7.5]
     [ 1.5  1.5  3.5  3.5  5.5  5.5  7.5  7.5]
     [ 1.5  1.5  3.5  3.5  5.5  5.5  7.5  7.5]
     [ 1.5  1.5  3.5  3.5  5.5  5.5  7.5  7.5]
     [ 1.5  1.5  3.5  3.5  5.5  5.5  7.5  7.5]]

Now set the `va` node with the known values and do the reconstruction again:

    >>> new_wp['va'] = wp['va'].data # [[-2.0, -2.0], [-2.0, -2.0]]
    >>> print new_wp.reconstruct(update=False)
    [[ 1.  2.  3.  4.  5.  6.  7.  8.]
     [ 1.  2.  3.  4.  5.  6.  7.  8.]
     [ 1.  2.  3.  4.  5.  6.  7.  8.]
     [ 1.  2.  3.  4.  5.  6.  7.  8.]
     [ 1.  2.  3.  4.  5.  6.  7.  8.]
     [ 1.  2.  3.  4.  5.  6.  7.  8.]
     [ 1.  2.  3.  4.  5.  6.  7.  8.]
     [ 1.  2.  3.  4.  5.  6.  7.  8.]]

which is just the same as the base sample data x.

Of course we can go the other way and remove nodes from the tree. If we delete the `va` node, again, we get the “not complete” tree from one of the previous examples:

    >>> del new_wp['va']
    >>> print new_wp.reconstruct(update=False)
    [[ 1.5  1.5  3.5  3.5  5.5  5.5  7.5  7.5]
     [ 1.5  1.5  3.5  3.5  5.5  5.5  7.5  7.5]
     [ 1.5  1.5  3.5  3.5  5.5  5.5  7.5  7.5]
     [ 1.5  1.5  3.5  3.5  5.5  5.5  7.5  7.5]
     [ 1.5  1.5  3.5  3.5  5.5  5.5  7.5  7.5]
     [ 1.5  1.5  3.5  3.5  5.5  5.5  7.5  7.5]
     [ 1.5  1.5  3.5  3.5  5.5  5.5  7.5  7.5]
     [ 1.5  1.5  3.5  3.5  5.5  5.5  7.5  7.5]]

Just restore the node before next examples.

    >>> new_wp['va'] = wp['va'].data

If the update param in the `WaveletPacket2D.reconstruct()` method is set to `False`, the node’s `Node2D.data` attribute will not be updated.

    >>> print new_wp.data
    None

Otherwise, the `WaveletPacket2D.data` attribute will be set to the reconstructed value.

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

Since we have an interesting WP structure built, it is a good occasion to present the `WaveletPacket2D.get_leaf_nodes()` method, which collects non-zero leaf nodes from the WP tree:

    >>> print [n.path for n in new_wp.get_leaf_nodes()]
    ['a', 'h', 'va', 'vh', 'vv', 'vd', 'd']

Passing the decompose=True parameter to the method will force the WP object to do a full decomposition up to the maximum level of decomposition:

    >>> paths = [n.path for n in new_wp.get_leaf_nodes(decompose=True)]
    >>> len(paths)
    64
    >>> for i, path in enumerate(paths):
    ...     print path,
    ...     if (i+1) % 8 == 0: print
    aaa aah aav aad aha ahh ahv ahd
    ava avh avv avd ada adh adv add
    haa hah hav had hha hhh hhv hhd
    hva hvh hvv hvd hda hdh hdv hdd
    vaa vah vav vad vha vhh vhv vhd
    vva vvh vvv vvd vda vdh vdv vdd
    daa dah dav dad dha dhh dhv dhd
    dva dvh dvv dvd dda ddh ddv ddd

## 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 = numpy.array([[1, 2, 3, 4, 5, 6, 7, 8]] * 8)
    >>> wp = pywt.WaveletPacket2D(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. During the first attempt to access the node it is computed via decomposition of its parent node (the wp object itself).

       >>> print wp['a']
       a: [[  3.   7.  11.  15.]
        [  3.   7.  11.  15.]
        [  3.   7.  11.  15.]
        [  3.   7.  11.  15.]]

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

       >>> print wp.a
       a: [[  3.   7.  11.  15.]
        [  3.   7.  11.  15.]
        [  3.   7.  11.  15.]
        [  3.   7.  11.  15.]]

   And so is wp.d:

       >>> print wp.d
       d: [[ 0.  0.  0.  0.]
        [ 0.  0.  0.  0.]
        [ 0.  0.  0.  0.]
        [ 0.  0.  0.  0.]]

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