__all__ = ["Radon2D"]
import logging
from typing import Callable, Optional, Tuple
import numpy as np
from pylops.basicoperators import Spread
from pylops.utils import deps
from pylops.utils.typing import DTypeLike, NDArray
jit_message = deps.numba_import("the radon2d module")
if jit_message is None:
from numba import jit
from ._radon2d_numba import (
_create_table_numba,
_hyperbolic_numba,
_indices_2d_onthefly_numba,
_linear_numba,
_parabolic_numba,
)
logging.basicConfig(format="%(levelname)s: %(message)s", level=logging.WARNING)
def _linear(
x: NDArray,
t: int,
px: float,
) -> NDArray:
return t + px * x
def _parabolic(
x: NDArray,
t: int,
px: float,
) -> NDArray:
return t + px * x**2
def _hyperbolic(
x: NDArray,
t: int,
px: float,
) -> NDArray:
return np.sqrt(t**2 + (x / px) ** 2)
def _indices_2d(
f: Callable,
x: NDArray,
px: float,
t: int,
nt: int,
interp: bool = True,
) -> Tuple[NDArray, NDArray, Optional[NDArray]]:
"""Compute time and space indices of parametric line in ``f`` function
Parameters
----------
f : :obj:`func`
Function computing values of parametric line for stacking
x : :obj:`np.ndarray`
Spatial axis (must be symmetrical around 0 and with sampling 1)
px : :obj:`float`
Slowness/curvature
t : :obj:`int`
Time sample (time axis is assumed to have sampling 1)
nt : :obj:`int`
Size of time axis
interp : :obj:`bool`, optional
Apply linear interpolation (``True``) or nearest interpolation
(``False``) during stacking/spreading along parametric curve
Returns
-------
xscan : :obj:`np.ndarray`
Spatial indices
tscan : :obj:`np.ndarray`
Time indices
dtscan : :obj:`np.ndarray`
Decimal time variations for interpolation
"""
tdecscan = f(x, t, px)
if not interp:
xscan = (tdecscan >= 0) & (tdecscan < nt)
else:
xscan = (tdecscan >= 0) & (tdecscan < nt - 1)
tscan = tdecscan[xscan].astype(int)
if interp:
dtscan = tdecscan[xscan] - tscan
return xscan, tscan, (dtscan if interp else None)
def _indices_2d_onthefly(
f: Callable,
x: NDArray,
px: NDArray,
ip: int,
t: int,
nt: int,
interp: bool = True,
) -> Tuple[NDArray, NDArray, Optional[NDArray]]:
"""Wrapper around _indices_2d to allow on-the-fly computation of
parametric curves"""
tscan = np.full(len(x), np.nan, dtype=np.float32)
if interp:
dtscan = np.full(len(x), np.nan)
xscan, tscan1, dtscan1 = _indices_2d(f, x, px[ip], t, nt, interp=interp)
tscan[xscan] = tscan1
if interp:
dtscan[xscan] = dtscan1
return xscan, tscan, (dtscan if interp else None)
def _create_table(
f: Callable,
x: NDArray,
pxaxis: NDArray,
nt: int,
npx: int,
nx: int,
interp: bool,
) -> Tuple[NDArray, Optional[NDArray]]:
"""Create look up table"""
table = np.full((npx, nt, nx), np.nan, dtype=np.float32)
if interp:
dtable = np.full((npx, nt, nx), np.nan)
else:
dtable = None
for ipx, px in enumerate(pxaxis):
for it in range(nt):
xscan, tscan, dtscan = _indices_2d(f, x, px, it, nt, interp=interp)
table[ipx, it, xscan] = tscan
if interp:
dtable[ipx, it, xscan] = dtscan
return table, dtable
[docs]def Radon2D(
taxis: NDArray,
haxis: NDArray,
pxaxis: NDArray,
kind: str = "linear",
centeredh: bool = True,
interp: bool = True,
onthefly: bool = False,
engine: str = "numpy",
dtype: DTypeLike = "float64",
name: str = "R",
):
r"""Two dimensional Radon transform.
Apply two dimensional Radon forward (and adjoint) transform to a
2-dimensional array of size :math:`[n_{p_x} \times n_t]`
(and :math:`[n_x \times n_t]`).
In forward mode this entails to spreading the model vector
along parametric curves (lines, parabolas, or hyperbolas depending on the
choice of ``kind``), while stacking values in the data vector
along the same parametric curves is performed in adjoint mode.
Parameters
----------
taxis : :obj:`np.ndarray`
Time axis
haxis : :obj:`np.ndarray`
Spatial axis
pxaxis : :obj:`np.ndarray`
Axis of scanning variable :math:`p_x` of parametric curve
kind : :obj:`str`, optional
Curve to be used for stacking/spreading (``linear``, ``parabolic``,
and ``hyperbolic`` are currently supported) or a function that takes
:math:`(x, t_0, p_x)` as input and returns :math:`t` as output
centeredh : :obj:`bool`, optional
Assume centered spatial axis (``True``) or not (``False``). If ``True``
the original ``haxis`` is ignored and a new axis is created.
interp : :obj:`bool`, optional
Apply linear interpolation (``True``) or nearest interpolation
(``False``) during stacking/spreading along parametric curve
onthefly : :obj:`bool`, optional
Compute stacking parametric curves on-the-fly as part of forward
and adjoint modelling (``True``) or at initialization and store them
in look-up table (``False``). Using a look-up table is computationally
more efficient but increases the memory burden
engine : :obj:`str`, optional
Engine used for computation (``numpy`` or ``numba``)
dtype : :obj:`str`, optional
Type of elements in input array.
name : :obj:`str`, optional
.. versionadded:: 2.0.0
Name of operator (to be used by :func:`pylops.utils.describe.describe`)
Returns
-------
r2op : :obj:`pylops.LinearOperator`
Radon operator
Raises
------
KeyError
If ``engine`` is neither ``numpy`` nor ``numba``
NotImplementedError
If ``kind`` is not ``linear``, ``parabolic``, or ``hyperbolic``
See Also
--------
pylops.signalprocessing.Radon3D: Three dimensional Radon transform
pylops.Spread: Spread operator
Notes
-----
The Radon2D operator applies the following linear transform in adjoint mode
to the data after reshaping it into a 2-dimensional array of
size :math:`[n_x \times n_t]`:
.. math::
m(p_x, t_0) = \int{d(x, t = f(p_x, x, t))} \,\mathrm{d}x
where :math:`f(p_x, x, t) = t_0 + p_x x` where
:math:`p_x = \sin(\theta)/v` in linear mode,
:math:`f(p_x, x, t) = t_0 + p_x x^2` in parabolic mode, and
:math:`f(p_x, x, t) = \sqrt{t_0^2 + x^2 / p_x^2}` in hyperbolic mode. Note
that internally the :math:`p_x` axis will be normalized by the ratio of the
spatial and time axes and used alongside unitless axes. Whilst this makes
the linear mode fully unitless, users are required to apply additional
scalings to the :math:`p_x` axis for other relationships:
- :math:`p_x` should be pre-multipled by :math:`d_x` for the parabolic
relationship;
- :math:`p_x` should be pre-multipled by :math:`(d_t/d_x)^2` for the
hyperbolic relationship.
As the adjoint operator can be interpreted as a repeated summation of sets
of elements of the model vector along chosen parametric curves, the
forward is implemented as spreading of values in the data vector along the
same parametric curves. This operator is actually a thin wrapper around
the :class:`pylops.Spread` operator.
"""
# engine
if engine not in ["numpy", "numba"]:
raise KeyError("engine must be numpy or numba")
if engine == "numba" and jit_message is not None:
engine = "numpy"
# axes
nt, nh, npx = taxis.size, haxis.size, pxaxis.size
if kind == "linear":
f = _linear if engine == "numpy" else _linear_numba
elif kind == "parabolic":
f = _parabolic if engine == "numpy" else _parabolic_numba
elif kind == "hyperbolic":
f = _hyperbolic if engine == "numpy" else _hyperbolic_numba
elif callable(kind):
f = kind
else:
raise NotImplementedError("kind must be linear, " "parabolic, or hyperbolic...")
# make axes unitless
dh, dt = np.abs(haxis[1] - haxis[0]), np.abs(taxis[1] - taxis[0])
dpx = dh / dt
pxaxis = pxaxis * dpx
if not centeredh:
haxisunitless = haxis / dh
else:
haxisunitless = np.arange(nh) - nh // 2 + ((nh + 1) % 2) / 2
dims = (npx, nt)
dimsd = (nh, nt)
if onthefly:
if engine == "numba":
@jit(nopython=True, nogil=True)
def ontheflyfunc(x, y):
return _indices_2d_onthefly_numba(
f, haxisunitless, pxaxis, x, y, nt, interp=interp
)[1:]
else:
if interp:
def ontheflyfunc(x, y):
return _indices_2d_onthefly(
f, haxisunitless, pxaxis, x, y, nt, interp=interp
)[1:]
else:
def ontheflyfunc(x, y):
return _indices_2d_onthefly(
f, haxisunitless, pxaxis, x, y, nt, interp=interp
)[1]
r2op = Spread(
dims, dimsd, fh=ontheflyfunc, interp=interp, engine=engine, dtype=dtype
)
else:
if engine == "numba":
tablefunc = _create_table_numba
else:
tablefunc = _create_table
table, dtable = tablefunc(f, haxisunitless, pxaxis, nt, npx, nh, interp)
if not interp:
dtable = None
r2op = Spread(
dims,
dimsd,
table=table,
dtable=dtable,
interp=interp,
engine=engine,
dtype=dtype,
)
r2op.name = name
return r2op
