import logging
import numpy as np
from pylops import LinearOperator
from pylops.basicoperators import Diagonal, MatrixMult, Restriction, Transpose
from pylops.utils.backend import get_array_module
logging.basicConfig(format="%(levelname)s: %(message)s", level=logging.WARNING)
def _checkunique(iava):
_, count = np.unique(iava, return_counts=True)
if np.any(count > 1):
raise ValueError("Repeated values in iava array")
def _nearestinterp(M, iava, dims=None, dir=0, dtype="float64"):
"""Nearest neighbour interpolation."""
iava = np.round(iava).astype(int)
_checkunique(iava)
return Restriction(M, iava, dims=dims, dir=dir, dtype=dtype), iava
def _linearinterp(M, iava, dims=None, dir=0, dtype="float64"):
"""Linear interpolation."""
ncp = get_array_module(iava)
if np.issubdtype(iava.dtype, np.integer):
iava = iava.astype(np.float64)
if dims is None:
lastsample = M
dimsd = None
else:
lastsample = dims[dir]
dimsd = list(dims)
dimsd[dir] = len(iava)
dimsd = tuple(dimsd)
# ensure that samples are not beyond the last sample, in that case set to
# penultimate sample and raise a warning
outside = iava >= lastsample - 1
if sum(outside) > 0:
logging.warning(
"at least one value is beyond penultimate sample, "
"forced to be at penultimate sample"
)
iava[outside] = lastsample - 1 - 1e-10
_checkunique(iava)
# find indices and weights
iva_l = ncp.floor(iava).astype(int)
iva_r = iva_l + 1
weights = iava - iva_l
# create operators
Op = Diagonal(1 - weights, dims=dimsd, dir=dir, dtype=dtype) * Restriction(
M, iva_l, dims=dims, dir=dir, dtype=dtype
) + Diagonal(weights, dims=dimsd, dir=dir, dtype=dtype) * Restriction(
M, iva_r, dims=dims, dir=dir, dtype=dtype
)
return Op, iava
def _sincinterp(M, iava, dims=None, dir=0, dtype="float64"):
"""Sinc interpolation."""
ncp = get_array_module(iava)
_checkunique(iava)
# create sinc interpolation matrix
nreg = M if dims is None else dims[dir]
ireg = ncp.arange(nreg)
sinc = ncp.tile(iava[:, np.newaxis], (1, nreg)) - ncp.tile(ireg, (len(iava), 1))
sinc = ncp.sinc(sinc)
# identify additional dimensions and create MatrixMult operator
otherdims = None
if dims is not None:
otherdims = ncp.array(dims)
otherdims = ncp.roll(otherdims, -dir)
otherdims = otherdims[1:]
Op = MatrixMult(sinc, dims=otherdims, dtype=dtype)
# create Transpose operator that brings dir to first dimension
if dir > 0:
axes = np.arange(len(dims), dtype=int)
axes = np.roll(axes, -dir)
dimsd = list(dims)
dimsd[dir] = len(iava)
Top = Transpose(dims, axes=axes, dtype=dtype)
T1op = Transpose(dimsd, axes=axes, dtype=dtype)
Op = T1op.H * Op * Top
return Op
[docs]def Interp(M, iava, dims=None, dir=0, kind="linear", dtype="float64"):
r"""Interpolation operator.
Apply interpolation along direction ``dir``
from regularly sampled input vector into fractionary positions ``iava``
using one of the following algorithms:
- *Nearest neighbour* interpolation
is a thin wrapper around :obj:`pylops.Restriction` at ``np.round(iava)``
locations.
- *Linear interpolation* extracts values from input vector
at locations ``np.floor(iava)`` and ``np.floor(iava)+1`` and linearly
combines them in forward mode, places weighted versions of the
interpolated values at locations ``np.floor(iava)`` and
``np.floor(iava)+1`` in an otherwise zero vector in adjoint mode.
- *Sinc interpolation* performs sinc interpolation at locations ``iava``.
Note that this is the most accurate method but it has higher computational
cost as it involves multiplying the input data by a matrix of size
:math:`N \times M`.
.. note:: The vector ``iava`` should contain unique values. If the same
index is repeated twice an error will be raised. This also applies when
values beyond the last element of the input array for
*linear interpolation* as those values are forced to be just before this
element.
Parameters
----------
M : :obj:`int`
Number of samples in model.
iava : :obj:`list` or :obj:`numpy.ndarray`
Floating indices of locations of available samples for interpolation.
dims : :obj:`list`, optional
Number of samples for each dimension
(``None`` if only one dimension is available)
dir : :obj:`int`, optional
Direction along which restriction is applied.
kind : :obj:`str`, optional
Kind of interpolation (``nearest``, ``linear``, and ``sinc`` are
currently supported)
dtype : :obj:`str`, optional
Type of elements in input array.
Returns
-------
op : :obj:`pylops.LinearOperator`
Linear intepolation operator
iava : :obj:`list` or :obj:`numpy.ndarray`
Corrected indices of locations of available samples
(samples at ``M-1`` or beyond are forced to be at ``M-1-eps``)
Raises
------
ValueError
If the vector ``iava`` contains repeated values.
NotImplementedError
If ``kind`` is not ``nearest``, ``linear`` or ``sinc``
See Also
--------
pylops.Restriction : Restriction operator
Notes
-----
*Linear interpolation* of a subset of :math:`N` values at locations
``iava`` from an input (or model) vector :math:`\mathbf{x}` of size
:math:`M` can be expressed as:
.. math::
y_i = (1-w_i) x_{l^{l}_i} + w_i x_{l^{r}_i}
\quad \forall i=1,2,\ldots,N
where :math:`\mathbf{l^l}=[\lfloor l_1 \rfloor, \lfloor l_2 \rfloor,\ldots,
\lfloor l_N \rfloor]` and :math:`\mathbf{l^r}=[\lfloor l_1 \rfloor +1,
\lfloor l_2 \rfloor +1,\ldots,
\lfloor l_N \rfloor +1]` are vectors containing the indeces
of the original array at which samples are taken, and
:math:`\mathbf{w}=[l_1 - \lfloor l_1 \rfloor, l_2 - \lfloor l_2 \rfloor,
..., l_N - \lfloor l_N \rfloor]` are the linear interpolation weights.
This operator can be implemented by simply summing two
:class:`pylops.Restriction` operators which are weighted
using :class:`pylops.basicoperators.Diagonal` operators.
*Sinc interpolation* of a subset of :math:`N` values at locations
``iava`` from an input (or model) vector :math:`\mathbf{x}` of size
:math:`M` can be expressed as:
.. math::
\DeclareMathOperator{\sinc}{sinc}
y_i = \sum_{j=0}^{M} x_j \sinc(i-j) \quad \forall i=1,2,\ldots,N
This operator can be implemented using the :class:`pylops.MatrixMult`
operator with a matrix containing the values of the sinc function at all
:math:`i,j` possible combinations.
"""
if kind == "nearest":
interpop, iava = _nearestinterp(M, iava, dims=dims, dir=dir, dtype=dtype)
elif kind == "linear":
interpop, iava = _linearinterp(M, iava, dims=dims, dir=dir, dtype=dtype)
elif kind == "sinc":
interpop = _sincinterp(M, iava, dims=dims, dir=dir, dtype=dtype)
else:
raise NotImplementedError("kind is not correct...")
return LinearOperator(interpop), iava