__all__ = ["Laplacian"]
from typing import Tuple
from numpy.core.multiarray import normalize_axis_index
from pylops import LinearOperator, aslinearoperator
from pylops.basicoperators import SecondDerivative
from pylops.utils.typing import DTypeLike, InputDimsLike
[docs]def Laplacian(
dims: InputDimsLike,
axes: InputDimsLike = (-2, -1),
weights: Tuple[float, ...] = (1, 1),
sampling: Tuple[float, ...] = (1, 1),
edge: bool = False,
kind: str = "centered",
dtype: DTypeLike = "float64",
) -> LinearOperator:
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.
axes : :obj:`int`, optional
.. versionadded:: 2.0.0
Axes along which the 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
kind : :obj:`str`, optional
Derivative kind (``forward``, ``centered``, or ``backward``)
dtype : :obj:`str`, optional
Type of elements in input array.
Returns
-------
l2op : :obj:`pylops.LinearOperator`
Laplacian linear operator
Raises
------
ValueError
If ``axes``. ``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)
"""
axes = tuple(normalize_axis_index(ax, len(dims)) for ax in axes)
if not (len(axes) == len(weights) == len(sampling)):
raise ValueError("axes, weights, and sampling have different size")
l2op = SecondDerivative(
dims, axis=axes[0], sampling=sampling[0], edge=edge, kind=kind, dtype=dtype
)
dims, dimsd = l2op.dims, l2op.dimsd
l2op *= weights[0]
for ax, samp, weight in zip(axes[1:], sampling[1:], weights[1:]):
l2op += weight * SecondDerivative(
dims, axis=ax, sampling=samp, edge=edge, dtype=dtype
)
l2op = aslinearoperator(l2op)
l2op.dims = dims
l2op.dimsd = dimsd
l2op.axes = axes
l2op.weights = weights
l2op.sampling = sampling
l2op.edge = edge
l2op.kind = kind
return l2op