Because the most common and practical way of representing digital signals in computer science is with finite arrays of values, some extrapolation of the input data has to be performed in order to extend the signal before computing the Discrete Wavelet Transform using the cascading filter banks algorithm.
Depending on the extrapolation method, significant artifacts at the signal’s borders can be introduced during that process, which in turn may lead to inaccurate computations of the DWT at the signal’s ends.
PyWavelets provides several methods of signal extrapolation that can be used to minimize this negative effect:
zpd - zero-padding - signal is extended by adding zero samples:
... 0 0 | x1 x2 ... xn | 0 0 ...
cpd - constant-padding - border values are replicated:
... x1 x1 | x1 x2 ... xn | xn xn ...
sym - symmetric-padding - signal is extended by mirroring samples:
... x2 x1 | x1 x2 ... xn | xn xn-1 ...
ppd - periodic-padding - signal is treated as a periodic one:
... xn-1 xn | x1 x2 ... xn | x1 x2 ...
- sp1 - smooth-padding - signal is extended according to the first derivatives calculated on the edges (straight line)
DWT performed for these extension modes is slightly redundant, but ensures perfect reconstruction. To receive the smallest possible number of coefficients, computations can be performed with the periodization mode:
- per - periodization - is like periodic-padding but gives the smallest possible number of decomposition coefficients. IDWT must be performed with the same mode.
Example:
>>> import pywt >>> print pywt.MODES.modes ['zpd', 'cpd', 'sym', 'ppd', 'sp1', 'per']
Notice that you can use any of the following ways of passing wavelet and mode parameters:
>>> import pywt
>>> (a, d) = pywt.dwt([1,2,3,4,5,6], 'db2', 'sp1')
>>> (a, d) = pywt.dwt([1,2,3,4,5,6], pywt.Wavelet('db2'), pywt.MODES.sp1)
Note
Extending data in context of PyWavelets does not mean reallocation of the data in computer’s physical memory and copying values, but rather computing the extra values only when they are needed. This feature saves extra memory and CPU resources and helps to avoid page swapping when handling relatively big data arrays on computers with low physical memory.