__all__ = ["Gradient"]
from typing import Union
from pylops import LinearOperator
from pylops.basicoperators import FirstDerivative, VStack
from pylops.utils._internal import _value_or_sized_to_tuple
from pylops.utils.typing import DTypeLike, InputDimsLike
[docs]def Gradient(
dims: Union[int, InputDimsLike],
sampling: int = 1,
edge: bool = False,
kind: str = "centered",
dtype: DTypeLike = "float64",
) -> LinearOperator:
r"""Gradient.
Apply gradient operator to a multi-dimensional array.
.. note:: At least 2 dimensions are required, use
:py:func:`pylops.FirstDerivative` for 1d arrays.
Parameters
----------
dims : :obj:`tuple`
Number of samples for each dimension.
sampling : :obj:`tuple`, optional
Sampling steps for each direction.
edge : :obj:`bool`, optional
Use reduced order derivative at edges (``True``) or
ignore them (``False``).
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`
Gradient linear operator
Notes
-----
The Gradient operator applies a first-order derivative to each dimension of
a multi-dimensional array in forward mode.
For simplicity, given a three dimensional array, the Gradient in forward
mode using a centered stencil can be expressed as:
.. math::
\mathbf{g}_{i, j, k} =
(f_{i+1, j, k} - f_{i-1, j, k}) / d_1 \mathbf{i_1} +
(f_{i, j+1, k} - f_{i, j-1, k}) / d_2 \mathbf{i_2} +
(f_{i, j, k+1} - f_{i, j, k-1}) / d_3 \mathbf{i_3}
which is discretized as follows:
.. math::
\mathbf{g} =
\begin{bmatrix}
\mathbf{df_1} \\
\mathbf{df_2} \\
\mathbf{df_3}
\end{bmatrix}
In adjoint mode, the adjoints of the first derivatives along different
axes are instead summed together.
"""
dims = _value_or_sized_to_tuple(dims)
ndims = len(dims)
sampling = _value_or_sized_to_tuple(sampling, repeat=ndims)
gop = VStack(
[
FirstDerivative(
dims,
axis=iax,
sampling=sampling[iax],
edge=edge,
kind=kind,
dtype=dtype,
)
for iax in range(ndims)
]
)
gop.dims = dims
gop.dimsd = (ndims, *gop.dims)
gop.sampling = sampling
gop.edge = edge
gop.kind = kind
return gop