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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='symmetric')

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 (the Node2D class) 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

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 (Node2D) are arranged in a tree. Each node in a WP tree is uniquely identified and addressed by a Node2D.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 2D (WaveletPacket2D) (as well as 1D WaveletPacket) using compound paths is just the same as indexing the 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.

Oops, we have reached the maximum level of decomposition for the 'aaaa' path, which, by the way, was:

print(wp.maxlevel)

3

Now, try an 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 the Node2D class (which in turn inherits from a BaseNode class 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)

symmetric

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):
    if (i+1) % 4 == 0:
        print(path)
    else:
        print(path, end=' ')

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):
    if (i+1) % 8 == 0:
        print(path)
    else:
        print(path, end=' ')

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='symmetric')

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 the 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):
    if (i+1) % 8 == 0:
        print(path)
    else:
        try:
            print(path, end=' ')
        except:
            print(path, end=' ')

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 the demonstration of PyWavelets’ 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='symmetric')

At first, the wp’s attribute a is None

print(wp.a)

None

Remember that you should not rely on the attribute access.

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.]]

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.]]