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 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 length depends not only on the input
data length but also on the :class:Wavelet type (particularly on its
filters lenght
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='sym'))
6
>>> int(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 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
['zpd', 'cpd', 'sym', 'ppd', 'sp1', 'per']
>>> [int(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.]
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', '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.
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', '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 convenient 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
, because 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.
>>> int(pywt.dwt_coeff_len(1, pywt.Wavelet('db4').dec_len, 'sym'))
4