"""
Defines a scalar field over a grid
.. codeauthor:: David Zwicker <david.zwicker@ds.mpg.de>
"""
from __future__ import annotations
import numbers
from collections.abc import Sequence
from pathlib import Path
from typing import TYPE_CHECKING, Callable, Literal
import numpy as np
from numpy.typing import DTypeLike
from ..grids import CartesianGrid, UnitGrid
from ..grids.base import DimensionError, DomainError, GridBase
from ..grids.boundaries.axes import BoundariesData
from ..tools.docstrings import fill_in_docstring
from ..tools.misc import Number
from ..tools.typing import NumberOrArray
from .base import DataFieldBase
if TYPE_CHECKING:
from .vectorial import VectorField
[docs]
class ScalarField(DataFieldBase):
"""Scalar field discretized on a grid"""
rank = 0
[docs]
@classmethod
@fill_in_docstring
def from_expression(
cls,
grid: GridBase,
expression: str,
*,
user_funcs: dict[str, Callable] | None = None,
consts: dict[str, NumberOrArray] | None = None,
label: str | None = None,
dtype: DTypeLike | None = None,
) -> ScalarField:
"""create a scalar field on a grid from a given expression
Warning:
{WARNING_EXEC}
Args:
grid (:class:`~pde.grids.base.GridBase`):
Grid defining the space on which this field is defined
expression (str):
Mathematical expression for the scalar value as a function of the
position on the grid. The expression may contain standard mathematical
functions and it may depend on the axes labels of the grid.
More information can be found in the
:ref:`expression documentation <documentation-expressions>`.
user_funcs (dict, optional):
A dictionary with user defined functions that can be used in the
expression
consts (dict, optional):
A dictionary with user defined constants that can be used in the
expression. The values of these constants should either be numbers or
:class:`~numpy.ndarray`.
label (str, optional):
Name of the field
dtype (numpy dtype):
The data type of the field. If omitted, it will be determined from
`data` automatically.
"""
from ..tools.expressions import ScalarExpression
# parse the expression
expr = ScalarExpression(
expression=expression,
signature=grid.axes,
user_funcs=user_funcs,
consts=consts,
repl=grid.c._axes_alt_repl,
)
# obtain the coordinates of the grid points
points = [grid.cell_coords[..., i] for i in range(grid.num_axes)]
try:
# try evaluating the expression using a vectorized call
data = expr(*points)
except ValueError:
# if this fails, evaluate expression point-wise
data = np.empty(grid.shape)
for cells in np.ndindex(*grid.shape):
data[cells] = expr(*grid.cell_coords[cells])
return cls(grid=grid, data=data, label=label, dtype=dtype)
[docs]
@classmethod
def from_image(
cls,
path: Path | str,
bounds=None,
periodic=False,
*,
label: str | None = None,
) -> ScalarField:
"""create a scalar field from an image
Args:
path (:class:`Path` or str):
The path to the image file
bounds (tuple, optional):
Gives the coordinate range for each axis. This should be two tuples of
two numbers each, which mark the lower and upper bound for each axis.
periodic (bool or list):
Specifies which axes possess periodic boundary conditions. This is
either a list of booleans defining periodicity for each individual axis
or a single boolean value specifying the same periodicity for all axes.
label (str, optional):
Name of the field
"""
from matplotlib.pyplot import imread
# read image and convert to grayscale
data = imread(str(path))
if data.ndim == 2:
pass # is already gray scale
elif data.ndim == 3:
# convert to gray scale using ITU-R 601-2 luma transform:
weights = np.array([0.299, 0.587, 0.114])
data = data[..., :3] @ weights
else:
raise RuntimeError(f"Image data has wrong shape: {data.shape}")
# transpose data to use mathematical conventions for axes
data = data.T[:, ::-1]
# determine the associated grid
if bounds is None:
grid: GridBase = UnitGrid(data.shape, periodic=periodic)
else:
grid = CartesianGrid(bounds, data.shape, periodic=periodic)
return cls(grid, data, label=label)
@DataFieldBase._data_flat.setter # type: ignore
def _data_flat(self, value):
"""set the data from a value from a collection"""
self._data_full = value[0]
def __array_ufunc__(self, ufunc, method, *inputs, **kwargs):
"""support unary numpy ufuncs, like np.sin, but also np.multiply"""
if method == "__call__":
# only support unary functions in simple calls
# check the input
arrs = []
for arg in inputs:
if isinstance(arg, numbers.Number):
arrs.append(arg)
elif isinstance(arg, np.ndarray):
if arg.shape != self.data.shape:
raise RuntimeError(
f"Data shapes incompatible ({arg.shape} != {self.data.shape}"
)
arrs.append(arg)
elif isinstance(arg, self.__class__):
self.assert_field_compatible(arg)
arrs.append(arg.data)
else:
# unsupported type
return NotImplemented
if "out" in kwargs:
# write to given field
out = kwargs.pop("out")[0]
self.assert_field_compatible(out)
kwargs["out"] = (out.data,)
ufunc(*arrs, **kwargs)
return out
else:
# return new field
return self.__class__(self.grid, data=ufunc(*arrs, **kwargs))
else:
return NotImplemented
[docs]
@fill_in_docstring
def laplace(
self,
bc: BoundariesData | None,
out: ScalarField | None = None,
**kwargs,
) -> ScalarField:
"""apply Laplace operator and return result as a field
Args:
bc:
The boundary conditions applied to the field.
{ARG_BOUNDARIES_OPTIONAL}
out (ScalarField, optional):
Optional scalar field to which the result is written.
label (str, optional):
Name of the returned field
backend (str):
The backend (e.g., 'numba' or 'scipy') used for this operator.
Returns:
:class:`~pde.fields.scalar.ScalarField`: the Laplacian of the field
"""
return self.apply_operator("laplace", bc=bc, out=out, **kwargs) # type: ignore
[docs]
@fill_in_docstring
def gradient_squared(
self,
bc: BoundariesData | None,
out: ScalarField | None = None,
**kwargs,
) -> ScalarField:
r"""apply squared gradient operator and return result as a field
This evaluates :math:`|\nabla \phi|^2` for the scalar field :math:`\phi`
Args:
bc:
The boundary conditions applied to the field.
{ARG_BOUNDARIES_OPTIONAL}
out (ScalarField, optional):
Optional vector field to which the result is written.
label (str, optional):
Name of the returned field
central (bool):
Determines whether a central difference approximation is used for the
gradient operator or not. If not, the squared gradient is calculated as
the mean of the squared values of the forward and backward derivatives,
which thus includes the value at a support point in the result at the
same point.
Returns:
:class:`~pde.fields.scalar.ScalarField`: the squared gradient of the field
"""
return self.apply_operator("gradient_squared", bc=bc, out=out, **kwargs) # type: ignore
[docs]
@fill_in_docstring
def gradient(
self,
bc: BoundariesData | None,
out: VectorField | None = None,
**kwargs,
) -> VectorField:
"""apply gradient operator and return result as a field
Args:
bc:
The boundary conditions applied to the field.
{ARG_BOUNDARIES_OPTIONAL}
out (VectorField, optional):
Optional vector field to which the result is written.
label (str, optional):
Name of the returned field
Returns:
:class:`~pde.fields.vectorial.VectorField`: result of applying the operator
"""
return self.apply_operator("gradient", bc=bc, out=out, **kwargs) # type: ignore
@property
def integral(self) -> Number:
"""Number: integral of the scalar field over space"""
return self.grid.integrate(self.data) # type: ignore
[docs]
def project(
self,
axes: str | Sequence[str],
method: Literal["integral", "average", "mean"] = "integral",
label: str | None = None,
) -> ScalarField:
"""project scalar field along given axes
Args:
axes (list of str):
The names of the axes that are removed by the projection
operation. The valid names for a given grid are the ones in
the :attr:`GridBase.axes` attribute.
method (str):
The projection method. This can be either 'integral' to
integrate over the removed axes or 'average' to perform an
average instead.
label (str, optional):
The label of the returned field
Returns:
:class:`~pde.fields.scalar.ScalarField`: The projected data in a scalar
field with a subgrid of the original grid.
"""
if isinstance(axes, str):
axes = [axes]
if any(ax not in self.grid.axes for ax in axes):
raise ValueError(
f"The axes {axes} are not all contained in {self.grid} with axes "
f"{self.grid.axes}"
)
# determine the axes after projection
ax_all = range(self.grid.num_axes)
ax_remove = tuple(self.grid.axes.index(ax) for ax in axes)
ax_retain = tuple(sorted(set(ax_all) - set(ax_remove)))
# determine the new grid
sliced_grid = self.grid.slice(ax_retain)
# calculate the new data
if method == "integral":
subdata = self.grid.integrate(self.data, axes=ax_remove)
elif method == "average" or method == "mean":
integrals = self.grid.integrate(self.data, axes=ax_remove)
volumes = self.grid.integrate(1, axes=ax_remove)
subdata = integrals / volumes
else:
raise ValueError(f"Unknown projection method `{method}`")
# create the new field instance
return self.__class__(grid=sliced_grid, data=subdata, label=label)
[docs]
def slice(
self,
position: dict[str, float],
*,
method: Literal["nearest"] = "nearest",
label: str | None = None,
) -> ScalarField:
"""slice data at a given position
Note:
This method should not be used to evaluate fields right at the boundary
since it does not respect boundary conditions. Use
:meth:`~ScalarField.get_boundary_field` to obtain the values directly on the
boundary.
Args:
position (dict):
Determines the location of the slice using a dictionary supplying
coordinate values for a subset of axes. Axes not mentioned in the
dictionary are retained and form the slice. For instance, in a 2d
Cartesian grid, `position = {'x': 1}` slices along the y-direction at
x=1. Additionally, the special positions 'low', 'mid', and 'high' are
supported to reference relative positions along the axis.
method (str):
The method used for slicing. Currently, we only support `nearest`, which
takes data from cells defined on the grid.
label (str, optional):
The label of the returned field
Returns:
:class:`~pde.fields.scalar.ScalarField`: The sliced data in a scalar field
with a subgrid of the original grid.
"""
grid = self.grid
# parse the positions and determine the axes to remove
ax_remove, pos_values = [], np.zeros(grid.num_axes)
for ax, pos in position.items():
# check the axis
try:
i = grid.axes.index(ax)
except ValueError:
raise ValueError(
f"The axes {ax} is not contained in "
f"{self.grid} with axes {self.grid.axes}"
)
ax_remove.append(i)
# check the position
if isinstance(pos, str):
if pos in {"min", "low", "lower"}:
pos_values[i] = grid.axes_coords[i][0]
elif pos in {"max", "high", "upper"}:
pos_values[i] = grid.axes_coords[i][-1]
elif pos in {"mid", "middle", "center"}:
pos_values[i] = np.mean(grid.axes_bounds[i])
else:
raise ValueError(f"Unknown position `{pos}`")
else:
pos_values[i] = float(pos)
# determine the axes left after slicing and the new grid
ax_all = range(grid.num_axes)
ax_retain = tuple(sorted(set(ax_all) - set(ax_remove)))
sliced_grid = grid.slice(ax_retain)
# obtain the sliced data
if method == "nearest":
idx: list[int | slice] = []
for i in range(grid.num_axes):
if i in ax_remove:
pos = pos_values[i]
axis_bounds = grid.axes_bounds[i]
if pos < axis_bounds[0] or pos > axis_bounds[1]:
raise DomainError(
f"Position {grid.axes[i]} = {pos} is outside the domain"
)
# add slice that is closest to pos
idx.append(int(np.argmin((grid.axes_coords[i] - pos) ** 2)))
else:
idx.append(slice(None))
subdata = self.data[tuple(idx)]
else:
raise ValueError(f"Unknown slicing method `{method}`")
# create the new field instance
return self.__class__(grid=sliced_grid, data=subdata, label=label)
[docs]
def to_scalar(
self, scalar: str | Callable = "auto", *, label: str | None = None
) -> ScalarField:
"""return a modified scalar field by applying method `scalar`
Args:
scalar (str or callable):
Determines the method used for obtaining the scalar. If this is a
callable, it is simply applied to self.data and a new scalar field with
this data is returned. Alternatively, pre-defined methods can be
selected using strings. Here, `abs` and `norm` denote the norm of each
entry of the field, while `norm_squared` returns the squared norm. The
default `auto` is to return a (unchanged) copy of a real field and the
norm of a complex field.
label (str, optional):
Name of the returned field
Returns:
:class:`~pde.fields.scalar.ScalarField`: Scalar field after applying the
operation
"""
if callable(scalar):
data = scalar(self.data)
elif scalar == "auto":
if np.iscomplexobj(self.data):
data = np.abs(self.data)
else:
data = self.data
elif scalar == "abs" or scalar == "norm":
data = np.abs(self.data)
elif scalar == "norm_squared":
data = self.data * self.data.conjugate()
else:
raise ValueError(f"Unknown method `{scalar}` for `to_scalar`")
return ScalarField(grid=self.grid, data=data, label=label)
[docs]
@fill_in_docstring
def interpolate_to_grid(
self: ScalarField,
grid: GridBase,
*,
bc: BoundariesData | None = None,
fill: Number | None = None,
label: str | None = None,
) -> ScalarField:
"""interpolate the data of this scalar field to another grid.
Args:
grid (:class:`~pde.grids.base.GridBase`):
The grid of the new field onto which the current field is interpolated.
bc:
The boundary conditions applied to the field, which affects values close
to the boundary. If omitted, the argument `fill` is used to determine
values outside the domain.
{ARG_BOUNDARIES_OPTIONAL}
fill (Number, optional):
Determines how values out of bounds are handled. If `None`, a
`ValueError` is raised when out-of-bounds points are requested.
Otherwise, the given value is returned.
label (str, optional):
Name of the returned field
Returns:
Field of the same rank as the current one.
"""
if self.grid.dim != grid.dim:
raise DimensionError(
f"Incompatible grid dimensions ({self.grid.dim:d} != {grid.dim:d})"
)
# determine the points at which data needs to be calculated
if isinstance(grid, CartesianGrid):
# convert Cartesian coordinates to coordinates in current grid
points = self.grid.transform(grid.cell_coords, "cartesian", "grid")
elif (
self.grid.__class__ is grid.__class__
and self.grid.num_axes == grid.num_axes
):
# convert within the same grid class
points = grid.cell_coords
else:
# this type of interpolation is not supported
grid_in = self.grid.__class__.__name__
grid_out = grid.__class__.__name__
raise NotImplementedError(f"Can't interpolate from {grid_in} to {grid_out}")
# interpolate the data to the grid
data = self.interpolate(points, bc=bc, fill=fill)
return self.__class__(grid, data, label=label)
[docs]
@fill_in_docstring
def get_boundary_field(
self,
index: str | tuple[int, bool],
bc: BoundariesData | None = None,
*,
label: str | None = None,
) -> ScalarField:
"""get the field on the specified boundary
Args:
index (str or tuple):
Index specifying the boundary. Can be either a string given in
:attr:`~pde.grids.base.GridBase.boundary_names`, like :code:`"left"`, or
a tuple of the axis index perpendicular to the boundary and a boolean
specifying whether the boundary is at the upper side of the axis or not,
e.g., :code:`(1, True)`.
bc:
The boundary conditions applied to the field.
{ARG_BOUNDARIES_OPTIONAL}
label (str):
Label of the returned field
Returns:
:class:`~pde.fields.scalar.ScalarField`: The field on the boundary
"""
axis, upper = self.grid._get_boundary_index(index)
data = self.get_boundary_values(axis, upper, bc)
boundary_axes = tuple(i for i in range(self.grid.num_axes) if i != axis)
if boundary_axes:
# the boundary is an actual field (the original grid had more than 2 axes)
grid = self.grid.slice(boundary_axes)
else:
# the boundary is a singular point => return a UnitGrid
grid = UnitGrid([1])
return self.__class__(grid=grid, data=data, label=label, dtype=self.dtype)