# Signal extension modes

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:

> - `zero` - zero-padding - signal is extended by adding zero samples:

>       ... 0  0 | x1 x2 ... xn | 0  0 ...

> - `constant` - constant-padding - border values are replicated:

>       ... x1 x1 | x1 x2 ... xn | xn xn ...

> - `symmetric` - symmetric-padding - signal is extended by mirroring samples. This mode is also known as half-sample symmetric.:

>       ... x2 x1 | x1 x2 ... xn | xn xn-1 ...

> - `reflect` - reflect-padding - signal is extended by reflecting samples. This mode is also known as whole-sample symmetric.:

>       ... x3 x2 | x1 x2 ... xn | xn-1 xn-2 ...

> - `periodic` - periodic-padding - signal is treated as a periodic one:

>       ... xn-1 xn | x1 x2 ... xn | x1 x2 ...

> - `smooth` - smooth-padding - signal is extended according to the first derivatives calculated on the edges (straight line)

> - `antisymmetric` - anti-symmetric padding - signal is extended by mirroring and negating samples. This mode is also known as half-sample anti-symmetric:

>       ... -x2 -x1 | x1 x2 ... xn | -xn -xn-1 ...

> - `antireflect` - anti-symmetric-reflect padding - signal is extended by reflecting anti-symmetrically about the edge samples. This mode is also known as whole-sample anti-symmetric:

>       ... (2*x1 - x3) (2*x1 - x2) | x1 x2 ... xn | (2*xn - xn-1) (2*xn - xn-2) ...

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:

> - `periodization` - 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)
>     ['zero', 'constant', 'symmetric', 'periodic', 'smooth', 'periodization', 'reflect', 'antisymmetric', 'antireflect']

The following figure illustrates how a short signal (red) gets extended (black) outside of its original extent. Note that periodization first extends the signal to an even length prior to using periodic boundary conditions.

    """A visual illustration of the various signal extension modes supported in
    PyWavelets. For efficiency, in the C routines the array is not actually
    extended as is done here. This is just a demo for easier visual explanation of
    the behavior of the various boundary modes.

    In practice, which signal extension mode is beneficial will depend on the
    signal characteristics.  For this particular signal, some modes such as
    "periodic",  "antisymmetric" and "zero" result in large discontinuities that
    would lead to large amplitude boundary coefficients in the detail coefficients
    of a discrete wavelet transform.
    """
    import numpy as np
    from matplotlib import pyplot as plt
    from pywt._doc_utils import boundary_mode_subplot

    # synthetic test signal
    x = 5 - np.linspace(-1.9, 1.1, 9)**2

    # Create a figure with one subplots per boundary mode
    fig, axes = plt.subplots(3, 3, figsize=(10, 6))
    plt.subplots_adjust(hspace=0.5)
    axes = axes.ravel()
    boundary_mode_subplot(x, 'symmetric', axes[0], symw=False)
    boundary_mode_subplot(x, 'reflect', axes[1], symw=True)
    boundary_mode_subplot(x, 'periodic', axes[2], symw=False)
    boundary_mode_subplot(x, 'antisymmetric', axes[3], symw=False)
    boundary_mode_subplot(x, 'antireflect', axes[4], symw=True)
    boundary_mode_subplot(x, 'periodization', axes[5], symw=False)
    boundary_mode_subplot(x, 'smooth', axes[6], symw=False)
    boundary_mode_subplot(x, 'constant', axes[7], symw=False)
    boundary_mode_subplot(x, 'zero', axes[8], symw=False)
    plt.show()

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', 'smooth')
    >>> (a, d) = pywt.dwt([1,2,3,4,5,6], pywt.Wavelet('db2'), pywt.Modes.smooth)

Note

Extending data in context of PyWavelets does not mean reallocation of the data in the 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.

## Naming Conventions

The correspondence between PyWavelets edge modes and the extension modes available in Matlab’s dwtmode and numpy’s pad are tabulated here for reference.

PyWavelets | Matlab | numpy.pad
--- | --- | ---
symmetric | sym, symh | symmetric
reflect | symw | reflect
smooth | spd, sp1 | N/A
constant | sp0 | edge
zero | zpd | constant, cval=0
periodic | ppd | wrap
periodization | per | N/A
antisymmetric | asym, asymh | N/A
antireflect | asymw | reflect, reflect_type=’odd’

## Padding using PyWavelets Signal Extension Modes - `pad`

Pywavelets provides a function, `pad()`, that operate like `numpy.pad()`, but supporting the PyWavelets signal extension modes discussed above. For efficiency, the DWT routines in PyWavelets do not expclitly create padded signals using this function. It can be used to manually prepad signals to reduce boundary effects in functions such as `cwt()` and `swt()` that do not currently support all of these signal extension modes.