from pylops.signalprocessing import ConvolveND
[docs]def Convolve2D(N, h, dims, offset=(0, 0), nodir=None, dtype="float64", method="fft"):
r"""2D convolution operator.
Apply two-dimensional convolution with a compact filter to model
(and data) along a pair of specific directions of a two or
three-dimensional array depending on the choice of ``nodir``.
Parameters
----------
N : :obj:`int`
Number of samples in model
h : :obj:`numpy.ndarray`
2d compact filter to be convolved to input signal
dims : :obj:`list`
Number of samples for each dimension
offset : :obj:`tuple`, optional
Indeces of the center of the compact filter
nodir : :obj:`int`, optional
Direction along which convolution is NOT applied
(set to ``None`` for 2d arrays)
dtype : :obj:`str`, optional
Type of elements in input array.
method : :obj:`str`, optional
Method used to calculate the convolution (``direct`` or ``fft``).
Returns
-------
cop : :obj:`pylops.LinearOperator`
Convolve2D linear operator
Notes
-----
The Convolve2D operator applies two-dimensional convolution
between the input signal :math:`d(t,x)` and a compact filter kernel
:math:`h(t,x)` in forward model:
.. math::
y(t,x) = \iint\limits_{-\infty}^{\infty}
h(t-\tau,x-\chi) d(\tau,\chi) \,\mathrm{d}\tau \,\mathrm{d}\chi
This operation can be discretized as follows
.. math::
y[i,n] = \sum_{j=-\infty}^{\infty} \sum_{m=-\infty}^{\infty} h[i-j,n-m] d[j,m]
as well as performed in the frequency domain.
.. math::
Y(f, k_x) = \mathscr{F} (h(t,x)) * \mathscr{F} (d(t,x))
Convolve2D operator uses :py:func:`scipy.signal.convolve2d`
that automatically chooses the best domain for the operation
to be carried out.
As the adjoint of convolution is correlation, Convolve2D operator
applies correlation in the adjoint mode.
In time domain:
.. math::
y(t,x) = \iint\limits_{-\infty}^{\infty}
h(t+\tau,x+\chi) d(\tau,\chi) \,\mathrm{d}\tau \,\mathrm{d}\chi
or in frequency domain:
.. math::
y(t, x) = \mathscr{F}^{-1} (H(f, k_x)^* * X(f, k_x))
"""
if h.ndim != 2:
raise ValueError("h must be 2-dimensional")
if nodir is None:
dirs = (0, 1)
elif nodir == 0:
dirs = (1, 2)
elif nodir == 1:
dirs = (0, 2)
else:
dirs = (0, 1)
cop = ConvolveND(N, h, dims, offset=offset, dirs=dirs, method=method, dtype=dtype)
return cop