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.. _reg-dwt-idwt:
.. currentmodule:: pywt
DWT and IDWT
============
Discrete Wavelet Transform
--------------------------
Let's do a :func:`Discrete Wavelet Transform <dwt>` of a sample data ``x``
using the ``db2`` wavelet. It's simple..
>>> import pywt
>>> x = [3, 7, 1, 1, -2, 5, 4, 6]
>>> cA, cD = pywt.dwt(x, 'db2')
And the approximation and details coefficients are in ``cA`` and ``cD``
respectively:
>>> print(cA)
[ 5.65685425 7.39923721 0.22414387 3.33677403 7.77817459]
>>> print(cD)
[-2.44948974 -1.60368225 -4.44140056 -0.41361256 1.22474487]
Inverse Discrete Wavelet Transform
----------------------------------
Now let's do an opposite operation
- :func:`Inverse Discrete Wavelet Transform <idwt>`:
>>> print(pywt.idwt(cA, cD, 'db2'))
[ 3. 7. 1. 1. -2. 5. 4. 6.]
Voilà! That's it!
More Examples
-------------
Now let's experiment with the :func:`dwt` some more. For example let's pass a
:class:`Wavelet` object instead of the wavelet name and specify signal
extension mode (the default is :ref:`symmetric <Modes.symmetric>`) for the
border effect handling:
>>> w = pywt.Wavelet('sym3')
>>> cA, cD = pywt.dwt(x, wavelet=w, mode='constant')
>>> print(cA)
[ 4.38354585 3.80302657 7.31813271 -0.58565539 4.09727044 7.81994027]
>>> print(cD)
[-1.33068221 -2.78795192 -3.16825651 -0.67715519 -0.09722957 -0.07045258]
Note that the output coefficients arrays length depends not only on the input
data length but also on the :class:Wavelet type (particularly on its
:attr:`filters length <~Wavelet.dec_len>` that are used in the transformation).
To find out what will be the output data size use the :func:`dwt_coeff_len`
function:
>>> # int() is for normalizing Python integers and long integers for documentation tests
>>> int(pywt.dwt_coeff_len(data_len=len(x), filter_len=w.dec_len, mode='symmetric'))
6
>>> int(pywt.dwt_coeff_len(len(x), w, 'symmetric'))
6
>>> len(cA)
6
Looks fine. (And if you expected that the output length would be a half of the
input data length, well, that's the trade-off that allows for the perfect
reconstruction...).
The third argument of the :func:`dwt_coeff_len` is the already mentioned signal
extension mode (please refer to the PyWavelets' documentation for the
:ref:`modes <modes>` description). Currently there are six
:ref:`extension modes <Modes>` available:
>>> pywt.Modes.modes
['zero', 'constant', 'symmetric', 'periodic', 'smooth', 'periodization', 'reflect', 'antisymmetric', 'antireflect']
As you see in the above example, the :ref:`periodization <Modes.periodization>`
(periodization) mode is slightly different from the others. It's aim when
doing the :func:`DWT <dwt>` transform is to output coefficients arrays that
are half of the length of the input data.
Knowing that, you should never mix the periodization mode with other modes when
doing :func:`DWT <dwt>` and :func:`IDWT <idwt>`. Otherwise, it will produce
**invalid results**:
>>> x
[3, 7, 1, 1, -2, 5, 4, 6]
>>> cA, cD = pywt.dwt(x, wavelet=w, mode='periodization')
>>> print(pywt.idwt(cA, cD, 'sym3', 'symmetric')) # invalid mode
[ 1. 1. -2. 5.]
>>> print(pywt.idwt(cA, cD, 'sym3', 'periodization'))
[ 3. 7. 1. 1. -2. 5. 4. 6.]
Tips & tricks
-------------
Passing ``None`` instead of coefficients data to :func:`idwt`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Now some tips & tricks. Passing ``None`` as one of the coefficient arrays
parameters is similar to passing a *zero-filled* array. The results are simply
the same:
>>> print(pywt.idwt([1,2,0,1], None, 'db2', 'symmetric'))
[ 1.19006969 1.54362308 0.44828774 -0.25881905 0.48296291 0.8365163 ]
>>> print(pywt.idwt([1, 2, 0, 1], [0, 0, 0, 0], 'db2', 'symmetric'))
[ 1.19006969 1.54362308 0.44828774 -0.25881905 0.48296291 0.8365163 ]
>>> print(pywt.idwt(None, [1, 2, 0, 1], 'db2', 'symmetric'))
[ 0.57769726 -0.93125065 1.67303261 -0.96592583 -0.12940952 -0.22414387]
>>> print(pywt.idwt([0, 0, 0, 0], [1, 2, 0, 1], 'db2', 'symmetric'))
[ 0.57769726 -0.93125065 1.67303261 -0.96592583 -0.12940952 -0.22414387]
Remember that only one argument at a time can be ``None``:
>>> print(pywt.idwt(None, None, 'db2', 'symmetric'))
Traceback (most recent call last):
...
ValueError: At least one coefficient parameter must be specified.
Coefficients data size in :attr:`idwt`
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
When doing the :func:`IDWT <idwt>` transform, usually the coefficient arrays
must have the same size.
>>> print(pywt.idwt([1, 2, 3, 4, 5], [1, 2, 3, 4], 'db2', 'symmetric'))
Traceback (most recent call last):
...
ValueError: Coefficients arrays must have the same size.
Not every coefficient array can be used in :func:`IDWT <idwt>`. In the
following example the :func:`idwt` will fail because the input arrays are
invalid - they couldn't be created as a result of :func:`DWT <dwt>`, because
the minimal output length for dwt using ``db4`` wavelet and the :ref:`symmetric
<Modes.symmetric>` mode is ``4``, not ``3``:
>>> pywt.idwt([1,2,4], [4,1,3], 'db4', 'symmetric')
Traceback (most recent call last):
...
ValueError: Invalid coefficient arrays length for specified wavelet. Wavelet and mode must be the same as used for decomposition.
>>> int(pywt.dwt_coeff_len(1, pywt.Wavelet('db4').dec_len, 'symmetric'))
4
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