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]
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.]
Violla! That’s it!
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 sym) for the border effect handling:
>>> w = pywt.Wavelet('sym3')
>>> cA, cD = pywt.dwt(x, wavelet=w, mode='cpd')
>>> 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 lenght depends not only on the input data length but also on the :class:Wavelet type (particularly on it’s filters lenght that are used in the transformation).
To find out what will be the output data size use the dwt_coeff_len() function:
>>> pywt.dwt_coeff_len(data_len=len(x), filter_len=w.dec_len, mode='sym')
6
>>> pywt.dwt_coeff_len(len(x), w, 'sym')
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 tradeoff 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
['zpd', 'cpd', 'sym', 'ppd', 'sp1', 'per']
>>> [pywt.dwt_coeff_len(len(x), w.dec_len, mode) for mode in pywt.MODES.modes]
[6, 6, 6, 6, 6, 4]
As you see in the above example, the per (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='per')
>>> print pywt.idwt(cA, cD, 'sym3', 'sym') # invalid mode
[ 1. 1. -2. 5.]
>>> print pywt.idwt(cA, cD, 'sym3', 'per')
[ 3. 7. 1. 1. -2. 5. 4. 6.]
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', 'sym')
[ 1.19006969 1.54362308 0.44828774 -0.25881905 0.48296291 0.8365163 ]
>>> print pywt.idwt([1, 2, 0, 1], [0, 0, 0, 0], 'db2', 'sym')
[ 1.19006969 1.54362308 0.44828774 -0.25881905 0.48296291 0.8365163 ]
>>> print pywt.idwt(None, [1, 2, 0, 1], 'db2', 'sym')
[ 0.57769726 -0.93125065 1.67303261 -0.96592583 -0.12940952 -0.22414387]
>>> print pywt.idwt([0, 0, 0, 0], [1, 2, 0, 1], 'db2', 'sym')
[ 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', 'sym')
Traceback (most recent call last):
...
ValueError: At least one coefficient parameter must be specified.
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', 'sym')
Traceback (most recent call last):
...
ValueError: Coefficients arrays must have the same size.
But for some applications like multilevel DWT and IDWT it is sometimes conveniet to allow for a small departure from this behaviour. When the correct_size flag is set, the approximation coefficients array can be larger from the details coefficient array by one element:
>>> print pywt.idwt([1, 2, 3, 4, 5], [1, 2, 3, 4], 'db2', 'sym', correct_size=True)
[ 1.76776695 0.61237244 3.18198052 0.61237244 4.59619408 0.61237244]
>>> print pywt.idwt([1, 2, 3, 4], [1, 2, 3, 4, 5], 'db2', 'sym', correct_size=True)
Traceback (most recent call last):
...
ValueError: Coefficients arrays must satisfy (0 <= len(cA) - len(cD) <= 1).
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, beacuse the minimal output length for dwt using db4 wavelet and the sym mode is 4, not 3:
>>> pywt.idwt([1,2,4], [4,1,3], 'db4', 'sym')
Traceback (most recent call last):
...
ValueError: Invalid coefficient arrays length for specified wavelet. Wavelet and mode must be the same as used for decomposition.
>>> pywt.dwt_coeff_len(1, pywt.Wavelet('db4').dec_len, 'sym')
4