import numpy as np
from pylops.basicoperators import SecondDerivative
from pylops.LinearOperator import aslinearoperator
[docs]def Laplacian(
dims,
dirs=(0, 1),
weights=(1, 1),
sampling=(1, 1),
edge=False,
dtype="float64",
kind="centered",
):
r"""Laplacian.
Apply second-order centered Laplacian operator to a multi-dimensional array.
.. note:: At least 2 dimensions are required, use
:py:func:`pylops.SecondDerivative` for 1d arrays.
Parameters
----------
dims : :obj:`tuple`
Number of samples for each dimension.
dirs : :obj:`tuple`, optional
Directions along which laplacian is applied.
weights : :obj:`tuple`, optional
Weight to apply to each direction (real laplacian operator if
``weights=[1,1]``)
sampling : :obj:`tuple`, optional
Sampling steps for each direction
edge : :obj:`bool`, optional
Use reduced order derivative at edges (``True``) or
ignore them (``False``) for centered derivative
dtype : :obj:`str`, optional
Type of elements in input array.
kind : :obj:`str`, optional
Derivative kind (``forward``, ``centered``, or ``backward``)
Returns
-------
l2op : :obj:`pylops.LinearOperator`
Laplacian linear operator
Raises
------
ValueError
If ``dirs``. ``weights``, and ``sampling`` do not have the same size
Notes
-----
The Laplacian operator applies a second derivative along two directions of
a multi-dimensional array.
For simplicity, given a two dimensional array, the Laplacian is:
.. math::
y[i, j] = (x[i+1, j] + x[i-1, j] + x[i, j-1] +x[i, j+1] - 4x[i, j])
/ (\Delta x \Delta y)
"""
if not (len(dirs) == len(weights) == len(sampling)):
raise ValueError("dirs, weights, and sampling have different size")
l2op = weights[0] * SecondDerivative(
np.prod(dims),
dims=dims,
dir=dirs[0],
sampling=sampling[0],
edge=edge,
kind=kind,
dtype=dtype,
)
for dir, samp, weight in zip(dirs[1:], sampling[1:], weights[1:]):
l2op += weight * SecondDerivative(
np.prod(dims),
dims=dims,
dir=dir,
sampling=samp,
edge=edge,
dtype=dtype,
)
return aslinearoperator(l2op)