# DWT and IDWT¶

## Discrete Wavelet Transform¶

Let’s do a Discrete Wavelet Transform 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 - Inverse Discrete Wavelet Transform:

>>> 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 dwt() some more. For example let’s pass a Wavelet object instead of the wavelet name and specify signal extension mode (the default is 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 filters length that are used in the transformation).

To find out what will be the output data size use the 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 dwt_coeff_len() is the already mentioned signal extension mode (please refer to the PyWavelets’ documentation for the modes description). Currently there are six extension modes available:

>>> pywt.Modes.modes
['zero', 'constant', 'symmetric', 'periodic', 'smooth', 'periodization', 'reflect', 'antisymmetric', 'antireflect']


As you see in the above example, the periodization (periodization) mode is slightly different from the others. It’s aim when doing the 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 DWT and 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 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 idwt¶

When doing the 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 IDWT. In the following example the idwt() will fail because the input arrays are invalid - they couldn’t be created as a result of DWT, because the minimal output length for dwt using db4 wavelet and the 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