import numpy as np
from pylops.basicoperators import FirstDerivative, VStack
[docs]def Gradient(dims, sampling=1, edge=False, dtype='float64'):
r"""Gradient.
Apply gradient operator to a multi-dimensional
array (at least 2 dimensions are required).
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``).
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 is:
.. 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.
"""
ndims = len(dims)
if isinstance(sampling, (int, float)):
sampling = [sampling] * ndims
gop = VStack([FirstDerivative(np.prod(dims), dims=dims, dir=idir,
sampling=sampling[idir],
edge=edge, dtype=dtype)
for idir in range(ndims)])
return gop