import numpy as np
from pylops import LinearOperator
from pylops.utils.backend import get_array_module
[docs]class Symmetrize(LinearOperator):
r"""Symmetrize along an axis.
Symmetrize a multi-dimensional array along a specified direction ``dir``.
Parameters
----------
N : :obj:`int`
Number of samples in model. Symmetric data has :math:`2N-1` samples
dims : :obj:`list`, optional
Number of samples for each dimension
(``None`` if only one dimension is available)
dir : :obj:`int`, optional
Direction along which symmetrization is applied
dtype : :obj:`str`, optional
Type of elements in input array
Attributes
----------
shape : :obj:`tuple`
Operator shape
explicit : :obj:`bool`
Operator contains a matrix that can be solved explicitly
(``True``) or not (``False``)
Notes
-----
The Symmetrize operator constructs a symmetric array given an input model
in forward mode, by pre-pending the input model in reversed order.
For simplicity, given a one dimensional array, the forward operation can
be expressed as:
.. math::
y[i] = \begin{cases}
x[i-N+1],& i\geq N\\
x[N-1-i],& \text{otherwise}
\end{cases}
for :math:`i=0,1,2,\ldots,2N-2`, where :math:`N` is the lenght of
the input model.
In adjoint mode, the Symmetrize operator assigns the sums of the elements
in position :math:`N-1-i` and :math:`N-1+i` to position :math:`i` as follows:
.. math::
\begin{multline}
x[i] = y[N-1-i]+y[N-1+i] \quad \forall i=0,2,\ldots,N-1
\end{multline}
apart from the central sample where :math:`x[0] = y[N-1]`.
"""
def __init__(self, N, dims=None, dir=0, dtype="float64"):
self.N = N
self.dir = dir
if dims is None:
self.dims = (self.N,)
self.dimsd = (self.N * 2 - 1,)
self.reshape = False
else:
if np.prod(dims) != self.N:
raise ValueError("product of dims must equal N")
else:
self.dims = dims
self.dimsd = list(dims)
self.dimsd[self.dir] = dims[self.dir] * 2 - 1
self.reshape = True
self.nsym = self.dims[self.dir]
self.shape = (np.prod(self.dimsd), np.prod(self.dims))
self.dtype = np.dtype(dtype)
self.explicit = False
def _matvec(self, x):
ncp = get_array_module(x)
y = ncp.zeros(self.dimsd, dtype=self.dtype)
if self.reshape:
x = ncp.reshape(x, self.dims)
if self.dir > 0: # bring the dimension to symmetrize to first
x = ncp.swapaxes(x, self.dir, 0)
y = ncp.swapaxes(y, self.dir, 0)
y[self.nsym - 1 :] = x
y[: self.nsym - 1] = x[-1:0:-1]
if self.dir > 0:
y = ncp.swapaxes(y, 0, self.dir)
if self.reshape:
y = y.ravel()
return y
def _rmatvec(self, x):
ncp = get_array_module(x)
if self.reshape:
x = ncp.reshape(x, self.dimsd)
if self.dir > 0: # bring the dimension to symmetrize to first
x = ncp.swapaxes(x, self.dir, 0)
y = x[self.nsym - 1 :].copy()
y[1:] += x[self.nsym - 2 :: -1]
if self.dir > 0:
y = ncp.swapaxes(y, 0, self.dir)
if self.reshape:
y = y.ravel()
return y