# Source code for pylops.basicoperators.laplacian

```__all__ = ["Laplacian"]

from typing import Tuple
from pylops.utils.typing import NDArray

from numpy.core.multiarray import normalize_axis_index

from pylops import LinearOperator
from pylops.basicoperators import SecondDerivative
from pylops.utils.typing import DTypeLike, InputDimsLike

[docs]class Laplacian(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

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.

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)

"""

def __init__(self, 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", name: str = "L"):
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")
self.axes = axes
self.weights = weights
self.sampling = sampling
self.edge = edge
self.kind = kind
Op = self._calc_l2op(dims=dims, axes=axes, sampling=sampling, edge=edge, kind=kind, dtype=dtype,
weights=weights)
super().__init__(Op=Op, name=name)

def _matvec(self, x: NDArray) -> NDArray:
return super()._matvec(x)

def _rmatvec(self, x: NDArray) -> NDArray:
return super()._rmatvec(x)

@staticmethod
def _calc_l2op(dims: InputDimsLike, axes: InputDimsLike, weights: Tuple[float, ...], sampling: Tuple[float, ...],
edge: bool, kind: str, dtype: DTypeLike):
l2op = SecondDerivative(
dims, axis=axes, sampling=sampling, edge=edge, kind=kind, dtype=dtype
)
dims = l2op.dims
l2op *= weights
for ax, samp, weight in zip(axes[1:], sampling[1:], weights[1:]):
l2op += weight * SecondDerivative(
dims, axis=ax, sampling=samp, edge=edge, dtype=dtype
)
return l2op
```