import numpy as np
from scipy.signal import convolve, fftconvolve
from pylops import LinearOperator
[docs]class Convolve1D(LinearOperator):
r"""1D convolution operator.
Apply one-dimensional convolution with a compact filter to model (and data)
along a specific direction of a multi-dimensional array depending on the
choice of ``dir``.
Parameters
----------
N : :obj:`int`
Number of samples in model.
h : :obj:`numpy.ndarray`
1d compact filter to be convolved to input signal
offset : :obj:`int`
Index of the center of the compact filter
dims : :obj:`tuple`
Number of samples for each dimension
(``None`` if only one dimension is available)
dir : :obj:`int`, optional
Direction along which convolution is applied
method : :obj:`str`, optional
Method used to calculate the convolution (``direct`` or ``fft``).
Note that ``fft`` approach is always used if ``dims=None``.
dtype : :obj:`str`, optional
Type of elements in input array.
Attributes
----------
shape : :obj:`tuple`
Operator shape
explicit : :obj:`bool`
Operator contains a matrix that can be solved
explicitly (``True``) or not (``False``)
Raises
------
ValueError
If ``offset`` is bigger than ``len(h) - 1``
Notes
-----
The Convolve1D operator applies convolution between the input signal
:math:`x(t)` and a compact filter kernel :math:`h(t)` in forward model:
.. math::
y(t) = \int_{-\inf}^{\inf} h(t-\tau) x(\tau) d\tau
This operation can be discretized as follows
.. math::
y[n] = \sum_{m=-\inf}^{\inf} h[n-m] x[m]
as well as performed in the frequency domain.
.. math::
Y(f) = \mathscr{F} (h(t)) * \mathscr{F} (x(t))
Convolve1D operator uses :py:func:`scipy.signal.convolve` that
automatically chooses the best domain for the operation to be carried out
for one dimensional inputs. The fft implementation
:py:func:`scipy.signal.fftconvolve` is however enforced for signals in
2 or more dimensions as this routine efficently operates on
multi-dimensional arrays.
As the adjoint of convolution is correlation, Convolve1D operator applies
correlation in the adjoint mode.
In time domain:
.. math::
x(t) = \int_{-\inf}^{\inf} h(t+\tau) x(\tau) d\tau
or in frequency domain:
.. math::
y(t) = \mathscr{F}^{-1} (H(f)^* * X(f))
"""
def __init__(self, N, h, offset=0, dims=None, dir=0, dtype='float64',
method='direct'):
if offset > len(h) - 1:
raise ValueError('offset must be smaller than len(h) - 1')
self.h = h
self.hstar = np.flip(self.h)
self.nh = len(h)
self.offset = 2*(self.nh // 2 - int(offset))
if self.nh % 2 == 0:
self.offset -= 1
if self.offset != 0:
self.h = \
np.pad(self.h, (self.offset if self.offset > 0 else 0,
-self.offset if self.offset < 0 else 0),
mode='constant')
self.hstar = np.flip(self.h)
if dims is not None:
# add dimensions to filter to match dimensions of model and data
hdims = [1] * len(dims)
hdims[dir] = len(self.h)
self.h = self.h.reshape(hdims)
self.hstar = self.hstar.reshape(hdims)
self.dir = dir
if dims is None:
self.dims = np.array([N, 1])
self.reshape = False
else:
if np.prod(dims) != N:
raise ValueError('product of dims must equal N!')
else:
self.dims = np.array(dims)
self.reshape = True
self.method = method
self.shape = (np.prod(self.dims), np.prod(self.dims))
self.dtype = np.dtype(dtype)
self.explicit = False
def _matvec(self, x):
if not self.reshape:
y = convolve(x.squeeze(), self.h, mode='same', method=self.method)
else:
x = np.reshape(x, self.dims)
y = fftconvolve(x, self.h, mode='same', axes=self.dir)
y = y.ravel()
return y
def _rmatvec(self, x):
if not self.reshape:
y = convolve(x.squeeze(), self.hstar, mode='same', method=self.method)
else:
x = np.reshape(x, self.dims)
y = fftconvolve(x, self.hstar, mode='same', axes=self.dir)
y = y.ravel()
return y