# Source code for pylops.basicoperators.FirstDerivative

import numpy as np
from pylops import LinearOperator
from pylops.utils.backend import get_array_module

[docs]class FirstDerivative(LinearOperator):
r"""First derivative.

Apply first derivative.

Parameters
----------
N : :obj:int
Number of samples in model.
dims : :obj:tuple, optional
Number of samples for each dimension
(None if only one dimension is available)
dir : :obj:int, optional
Direction along which the derivative is applied.
sampling : :obj:float, optional
Sampling step dx.
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.
kind : :obj:str, optional
Derivative kind (forward, centered, or backward).

Attributes
----------
shape : :obj:tuple
Operator shape
explicit : :obj:bool
Operator contains a matrix that can be solved explicitly (True) or
not (False)

Notes
-----
The FirstDerivative operator applies a first derivative to any chosen
direction of a multi-dimensional array using either a second-order
centered stencil or first-order forward/backward stencils.

For simplicity, given a one dimensional array, the second-order centered
first derivative is:

.. math::
y[i] = (0.5x[i+1] - 0.5x[i-1]) / dx

while the first-order forward stencil is:

.. math::
y[i] = (x[i+1] - x[i]) / dx

and the first-order backward stencil is:

.. math::
y[i] = (x[i] - x[i-1]) / dx

"""
def __init__(self, N, dims=None, dir=0, sampling=1.,
edge=False, dtype='float64', kind='centered'):
self.N = N
self.sampling = sampling
self.edge = edge
if dims is None:
self.dims = (self.N,)
self.reshape = False
else:
if np.prod(dims) != self.N:
raise ValueError('product of dims must equal N')
else:
self.dims = dims
self.reshape = True
self.dir = dir if dir >= 0 else len(self.dims) + dir
self.kind = kind
self.shape = (self.N, self.N)
self.dtype = np.dtype(dtype)
self.explicit = False

# choose _matvec and _rmatvec kind
if self.kind == 'forward':
self._matvec = self._matvec_forward
self._rmatvec = self._rmatvec_forward
elif self.kind == 'centered':
self._matvec = self._matvec_centered
self._rmatvec = self._rmatvec_centered
elif self.kind == 'backward':
self._matvec = self._matvec_backward
self._rmatvec = self._rmatvec_backward
else:
raise NotImplementedError('kind must be forward, centered, '
'or backward')

def _matvec_forward(self, x):
ncp = get_array_module(x)
if not self.reshape:
x = x.squeeze()
y = ncp.zeros(self.N, self.dtype)
y[:-1] = (x[1:] - x[:-1]) / self.sampling
else:
x = ncp.reshape(x, self.dims)
if self.dir > 0:  # need to bring the dim. to derive to first dim.
x = ncp.swapaxes(x, self.dir, 0)
y = ncp.zeros(x.shape, self.dtype)
y[:-1] = (x[1:] - x[:-1]) / self.sampling
if self.dir > 0:
y = ncp.swapaxes(y, 0, self.dir)
y = y.ravel()
return y

def _rmatvec_forward(self, x):
ncp = get_array_module(x)
if not self.reshape:
x = x.squeeze()
y = ncp.zeros(self.N, self.dtype)
y[:-1] -= x[:-1] / self.sampling
y[1:] += x[:-1] / self.sampling
else:
x = ncp.reshape(x, self.dims)
if self.dir > 0:  # need to bring the dim. to derive to first dim.
x = ncp.swapaxes(x, self.dir, 0)
y = ncp.zeros(x.shape, self.dtype)
y[:-1] -= x[:-1] / self.sampling
y[1:] += x[:-1] / self.sampling
if self.dir > 0:
y = ncp.swapaxes(y, 0, self.dir)
y = y.ravel()
return y

def _matvec_centered(self, x):
ncp = get_array_module(x)
if not self.reshape:
x = x.squeeze()
y = ncp.zeros(self.N, self.dtype)
y[1:-1] = (0.5 * x[2:] - 0.5 * x[0:-2]) / self.sampling
if self.edge:
y[0] = (x[1] - x[0]) / self.sampling
y[-1] = (x[-1] - x[-2]) / self.sampling
else:
x = ncp.reshape(x, self.dims)
if self.dir > 0:  # need to bring the dim. to derive to first dim.
x = ncp.swapaxes(x, self.dir, 0)
y = ncp.zeros(x.shape, self.dtype)
y[1:-1] = (0.5 * x[2:] - 0.5 * x[0:-2]) / self.sampling
if self.edge:
y[0] = (x[1] - x[0]) / self.sampling
y[-1] = (x[-1] - x[-2]) / self.sampling
if self.dir > 0:
y = ncp.swapaxes(y, 0, self.dir)
y = y.ravel()
return y

def _rmatvec_centered(self, x):
ncp = get_array_module(x)
if not self.reshape:
x = x.squeeze()
y = ncp.zeros(self.N, self.dtype)
y[0:-2] -= (0.5 * x[1:-1]) / self.sampling
y[2:] += (0.5 * x[1:-1]) / self.sampling
if self.edge:
y[0] -= x[0] / self.sampling
y[1] += x[0] / self.sampling
y[-2] -= x[-1] / self.sampling
y[-1] += x[-1] / self.sampling
else:
x = ncp.reshape(x, self.dims)
if self.dir > 0:  # need to bring the dim. to derive to first dim.
x = ncp.swapaxes(x, self.dir, 0)
y = ncp.zeros(x.shape, self.dtype)
y[0:-2] -= (0.5 * x[1:-1]) / self.sampling
y[2:] += (0.5 * x[1:-1]) / self.sampling
if self.edge:
y[0] -= x[0] / self.sampling
y[1] += x[0] / self.sampling
y[-2] -= x[-1] / self.sampling
y[-1] += x[-1] / self.sampling
if self.dir > 0:
y = ncp.swapaxes(y, 0, self.dir)
y = y.ravel()
return y

def _matvec_backward(self, x):
ncp = get_array_module(x)
if not self.reshape:
x = x.squeeze()
y = ncp.zeros(self.N, self.dtype)
y[1:] = (x[1:] - x[:-1]) / self.sampling
else:
x = ncp.reshape(x, self.dims)
if self.dir > 0:  # need to bring the dim. to derive to first dim.
x = ncp.swapaxes(x, self.dir, 0)
y = ncp.zeros(x.shape, self.dtype)
y[1:] = (x[1:] - x[:-1]) / self.sampling
if self.dir > 0:
y = ncp.swapaxes(y, 0, self.dir)
y = y.ravel()
return y

def _rmatvec_backward(self, x):
ncp = get_array_module(x)
if not self.reshape:
x = x.squeeze()
y = ncp.zeros(self.N, self.dtype)
y[:-1] -= x[1:] / self.sampling
y[1:] += x[1:] / self.sampling
else:
x = ncp.reshape(x, self.dims)
if self.dir > 0:  # need to bring the dim. to derive to first dim.
x = ncp.swapaxes(x, self.dir, 0)
y = ncp.zeros(x.shape, self.dtype)
y[:-1] -= x[1:] / self.sampling
y[1:] += x[1:] / self.sampling
if self.dir > 0:
y = ncp.swapaxes(y, 0, self.dir)
y = y.ravel()
return y