__all__ = ["Convolve2D"]
from typing import Union
from pylops import LinearOperator
from pylops.signalprocessing import ConvolveND
from pylops.utils.typing import DTypeLike, InputDimsLike, NDArray
[docs]def Convolve2D(
dims: Union[int, InputDimsLike],
h: NDArray,
offset: InputDimsLike = (0, 0),
axes: InputDimsLike = (-2, -1),
method: str = "fft",
dtype: DTypeLike = "float64",
name: str = "C",
) -> LinearOperator:
r"""2D convolution operator.
Apply two-dimensional convolution with a compact filter to model
(and data) along a pair of ``axes`` of a two or
three-dimensional array.
Parameters
----------
dims : :obj:`list` or :obj:`int`
Number of samples for each dimension
h : :obj:`numpy.ndarray`
2d compact filter to be convolved to input signal
offset : :obj:`tuple`, optional
Indices of the center of the compact filter
axes : :obj:`int`, optional
.. versionadded:: 2.0.0
Axes along which convolution is applied
method : :obj:`str`, optional
Method used to calculate the convolution (``direct`` or ``fft``).
dtype : :obj:`str`, optional
Type of elements in input array.
name : :obj:`str`, optional
.. versionadded:: 2.0.0
Name of operator (to be used by :func:`pylops.utils.describe.describe`)
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")
cop = ConvolveND(dims, h, offset=offset, axes=axes, method=method, dtype=dtype)
cop.name = name
return cop