def convex_hull(self):
"""Return the smallest `IntervalProd` containing this grid.
The convex hull of a set is the union of all line segments
between points in the set. For a rectilinear grid, it is the
interval product given by the extremal coordinates.
Returns
-------
convex_hull : `IntervalProd`
Interval product defined by the minimum and maximum points
of the grid.
Examples
--------
>>> g = RectGrid([-1, 0, 3], [2, 4], [5], [2, 4, 7])
>>> g.convex_hull()
IntervalProd([-1., 2., 5., 2.], [ 3., 4., 5., 7.])
"""
return IntervalProd(self.min(), self.max())
def _resize_discr(discr, newshp, offset, discr_kwargs):
"""Return a space based on ``discr`` and ``newshp``.
Use the domain of ``discr`` and its partition to create a new
uniformly discretized space with ``newshp`` as shape. In axes where
``offset`` is given, it determines the number of added/removed cells to
the left. Where ``offset`` is ``None``, the points are distributed
evenly to left and right. The ``discr_kwargs`` parameter is passed
to `uniform_discr` for further specification of discretization
parameters.
"""
nodes_on_bdry = discr_kwargs.get('nodes_on_bdry', False)
if np.shape(nodes_on_bdry) == ():
nodes_on_bdry = ([(bool(nodes_on_bdry), bool(nodes_on_bdry))] *
discr.ndim)
elif discr.ndim == 1 and len(nodes_on_bdry) == 2:
nodes_on_bdry = [nodes_on_bdry]
elif len(nodes_on_bdry) != discr.ndim:
raise ValueError('`nodes_on_bdry` has length {}, expected {}'
''.format(len(nodes_on_bdry), discr.ndim))
dtype = discr_kwargs.pop('dtype', discr.dtype)
impl = discr_kwargs.pop('impl', discr.impl)
exponent = discr_kwargs.pop('exponent', discr.exponent)
interp = discr_kwargs.pop('interp', discr.interp)
weighting = discr_kwargs.pop('weighting', discr.weighting)
affected = np.not_equal(newshp, discr.shape)
ndim = discr.ndim
for i in range(ndim):
if affected[i] and not discr.is_uniform_byaxis[i]:
raise ValueError('cannot resize in non-uniformly discretized '
'axis {}'.format(i))
grid_min, grid_max = discr.grid.min(), discr.grid.max()
cell_size = discr.cell_sides
new_minpt, new_maxpt = [], []
for axis, (n_orig, n_new, off, on_bdry) in enumerate(
zip(discr.shape, newshp, offset, nodes_on_bdry)):
if not affected[axis]:
new_minpt.append(discr.min_pt[axis])
new_maxpt.append(discr.max_pt[axis])
continue
n_diff = n_new - n_orig
if off is None:
num_r = n_diff // 2
num_l = n_diff - num_r
else:
num_r = n_diff - off
num_l = off
try:
on_bdry_l, on_bdry_r = on_bdry
except TypeError:
on_bdry_l = on_bdry
on_bdry_r = on_bdry
if on_bdry_l:
new_minpt.append(grid_min[axis] - num_l * cell_size[axis])
else:
new_minpt.append(grid_min[axis] - (num_l + 0.5) * cell_size[axis])
if on_bdry_r:
new_maxpt.append(grid_max[axis] + num_r * cell_size[axis])
else:
new_maxpt.append(grid_max[axis] + (num_r + 0.5) * cell_size[axis])
fspace = FunctionSpace(IntervalProd(new_minpt, new_maxpt), out_dtype=dtype)
tspace = tensor_space(newshp,
dtype=dtype,
impl=impl,
exponent=exponent,
weighting=weighting)
# Stack together the (unchanged) nonuniform axes and the (new) uniform
# axes in the right order
part = uniform_partition([], [], ())
for i in range(ndim):
if discr.is_uniform_byaxis[i]:
part = part.append(
uniform_partition(new_minpt[i],
new_maxpt[i],
newshp[i],
nodes_on_bdry=nodes_on_bdry[i]))
else:
part = part.append(discr.partition.byaxis[i])
return DiscreteLp(fspace, part, tspace, interp=interp)
def uniform_partition_fromgrid(grid, min_pt=None, max_pt=None):
"""Return a partition of an interval product based on a given grid.
This method is complementary to `uniform_partition_fromintv` in that
it infers the set to be partitioned from a given grid and optional
parameters for ``min_pt`` and ``max_pt`` of the set.
Parameters
----------
grid : `RectGrid`
Grid on which the partition is based
min_pt, max_pt : float, sequence of floats, or dict, optional
Spatial points defining the lower/upper limits of the intervals
to be partitioned. The points can be specified in two ways:
float or sequence: The values are used directly as ``min_pt``
and/or ``max_pt``.
dict: Index-value pairs specifying an axis and a spatial
coordinate to be used in that axis. In axes which are not a key
in the dictionary, the coordinate for the vector is calculated
as::
min_pt = x[0] - (x[1] - x[0]) / 2
max_pt = x[-1] + (x[-1] - x[-2]) / 2
See ``Examples`` below.
In general, ``min_pt`` may not be larger than ``grid.min_pt``,
and ``max_pt`` not smaller than ``grid.max_pt`` in any component.
``None`` is equivalent to an empty dictionary, i.e. the values
are calculated in each dimension.
See Also
--------
uniform_partition_fromintv
Examples
--------
Have ``min_pt`` and ``max_pt`` of the bounding box automatically
calculated:
>>> grid = odl.uniform_grid(0, 1, 3)
>>> grid.coord_vectors
(array([ 0. , 0.5, 1. ]),)
>>> part = odl.uniform_partition_fromgrid(grid)
>>> part.cell_boundary_vecs
(array([-0.25, 0.25, 0.75, 1.25]),)
``min_pt`` and ``max_pt`` can be given explicitly:
>>> part = odl.uniform_partition_fromgrid(grid, min_pt=0, max_pt=1)
>>> part.cell_boundary_vecs
(array([ 0. , 0.25, 0.75, 1. ]),)
Using dictionaries, selective axes can be explicitly set. The
keys refer to axes, the values to the coordinates to use:
>>> grid = odl.uniform_grid([0, 0], [1, 1], (3, 3))
>>> part = odl.uniform_partition_fromgrid(grid,
... min_pt={0: -1}, max_pt={-1: 3})
>>> part.cell_boundary_vecs[0]
array([-1. , 0.25, 0.75, 1.25])
>>> part.cell_boundary_vecs[1]
array([-0.25, 0.25, 0.75, 3. ])
"""
# Make dictionaries from `min_pt` and `max_pt` and fill with `None` where
# no value is given (taking negative indices into account)
if min_pt is None:
min_pt = {i: None for i in range(grid.ndim)}
elif not hasattr(min_pt, 'items'): # array-like
min_pt = np.atleast_1d(min_pt)
min_pt = {i: float(v) for i, v in enumerate(min_pt)}
else:
min_pt.update({
i: None
for i in range(grid.ndim)
if i not in min_pt and i - grid.ndim not in min_pt
})
if max_pt is None:
max_pt = {i: None for i in range(grid.ndim)}
elif not hasattr(max_pt, 'items'):
max_pt = np.atleast_1d(max_pt)
max_pt = {i: float(v) for i, v in enumerate(max_pt)}
else:
max_pt.update({
i: None
for i in range(grid.ndim)
if i not in max_pt and i - grid.ndim not in max_pt
})
# Set the values in the vectors by computing (None) or directly from the
# given vectors (otherwise).
min_pt_vec = np.empty(grid.ndim)
for ax, xmin in min_pt.items():
if xmin is None:
cvec = grid.coord_vectors[ax]
if len(cvec) == 1:
raise ValueError('in axis {}: cannot calculate `min_pt` with '
'only 1 grid point'.format(ax))
min_pt_vec[ax] = cvec[0] - (cvec[1] - cvec[0]) / 2
else:
min_pt_vec[ax] = xmin
max_pt_vec = np.empty(grid.ndim)
for ax, xmax in max_pt.items():
if xmax is None:
cvec = grid.coord_vectors[ax]
if len(cvec) == 1:
raise ValueError('in axis {}: cannot calculate `max_pt` with '
'only 1 grid point'.format(ax))
max_pt_vec[ax] = cvec[-1] + (cvec[-1] - cvec[-2]) / 2
else:
max_pt_vec[ax] = xmax
return RectPartition(IntervalProd(min_pt_vec, max_pt_vec), grid)
def __getitem__(self, indices):
"""Return ``self[indices]``.
Parameters
----------
indices : index expression
Object determining which parts of the partition to extract.
``None`` (new axis) and empty axes are not supported.
Examples
--------
>>> intvp = odl.IntervalProd([-1, 1, 4, 2], [3, 6, 5, 7])
>>> grid = odl.RectGrid([-1, 0, 3], [2, 4], [5], [2, 4, 7])
>>> part = odl.RectPartition(intvp, grid)
>>> part
nonuniform_partition(
[-1.0, 0.0, 3.0],
[2.0, 4.0],
[5.0],
[2.0, 4.0, 7.0],
min_pt=[-1.0, 1.0, 4.0, 2.0], max_pt=[3.0, 6.0, 5.0, 7.0]
)
Take an advanced slice (every second along the first axis,
the last in the last axis and everything in between):
>>> part[::2, ..., -1]
nonuniform_partition(
[-1.0, 3.0],
[2.0, 4.0],
[5.0],
[7.0],
min_pt=[-1.0, 1.0, 4.0, 5.5], max_pt=[3.0, 6.0, 5.0, 7.0]
)
Too few indices are filled up with an ellipsis from the right:
>>> part[1]
nonuniform_partition(
[0.0],
[2.0, 4.0],
[5.0],
[2.0, 4.0, 7.0],
min_pt=[-0.5, 1.0, 4.0, 2.0], max_pt=[1.5, 6.0, 5.0, 7.0]
)
Colons etc work as expected:
>>> part[:] == part
True
>>> part[:, :, :] == part
True
>>> part[...] == part
True
"""
# Special case of index list: slice along first axis
if isinstance(indices, list):
if indices == []:
new_min_pt = new_max_pt = []
else:
new_min_pt = [self.cell_boundary_vecs[0][:-1][indices][0]]
new_max_pt = [self.cell_boundary_vecs[0][1:][indices][-1]]
for cvec in self.cell_boundary_vecs[1:]:
new_min_pt.append(cvec[0])
new_max_pt.append(cvec[-1])
new_intvp = IntervalProd(new_min_pt, new_max_pt)
new_grid = self.grid[indices]
return RectPartition(new_intvp, new_grid)
indices = normalized_index_expression(indices,
self.shape,
int_to_slice=True)
# Build the new partition
new_min_pt, new_max_pt = [], []
for cvec, idx in zip(self.cell_boundary_vecs, indices):
# Determine the subinterval min_pt and max_pt vectors. Take the
# first min_pt as new min_pt and the last max_pt as new max_pt.
sub_min_pt = cvec[:-1][idx]
sub_max_pt = cvec[1:][idx]
new_min_pt.append(sub_min_pt[0])
new_max_pt.append(sub_max_pt[-1])
new_intvp = IntervalProd(new_min_pt, new_max_pt)
new_grid = self.grid[indices]
return RectPartition(new_intvp, new_grid)
def nonuniform_partition(*coord_vecs, **kwargs):
"""Return a partition with un-equally sized cells.
Parameters
----------
coord_vecs1, ... coord_vecsN : `array-like`
Arrays of coordinates of the mid-points of the partition cells.
min_pt, max_pt : float or sequence of floats, optional
Vectors defining the lower/upper limits of the intervals in an
`IntervalProd` (a rectangular box). ``None`` entries mean
"compute the value".
nodes_on_bdry : bool or sequence, optional
If a sequence is provided, it determines per axis whether to
place the last grid point on the boundary (``True``) or shift it
by half a cell size into the interior (``False``). In each axis,
an entry may consist in a single bool or a 2-tuple of
bool. In the latter case, the first tuple entry decides for
the left, the second for the right boundary. The length of the
sequence must be ``array.ndim``.
A single boolean is interpreted as a global choice for all
boundaries.
Cannot be given with both min_pt and max_pt since they determine the
same thing.
Default: ``False``
See Also
--------
uniform_partition : uniformly spaced points
uniform_partition_fromintv : partition an existing set
uniform_partition_fromgrid : use an existing grid as basis
Examples
--------
With uniformly spaced points the result is the same as a
uniform partition:
>>> odl.nonuniform_partition([0, 1, 2, 3])
uniform_partition(-0.5, 3.5, 4)
>>> odl.nonuniform_partition([0, 1, 2, 3], [1, 2])
uniform_partition([-0.5, 0.5], [3.5, 2.5], (4, 2))
If the points are not uniformly spaced, a nonuniform partition is
created. Note that the containing interval is calculated by assuming
that the points are in the middle of the sub-intervals:
>>> odl.nonuniform_partition([0, 1, 3])
nonuniform_partition(
[0.0, 1.0, 3.0]
)
Higher dimensional partitions are created by specifying the gridpoints
along each dimension:
>>> odl.nonuniform_partition([0, 1, 3], [1, 2])
nonuniform_partition(
[0.0, 1.0, 3.0],
[1.0, 2.0]
)
If the endpoints should be on the boundary, the ``nodes_on_bdry`` parameter
can be used:
>>> odl.nonuniform_partition([0, 1, 3], nodes_on_bdry=True)
nonuniform_partition(
[0.0, 1.0, 3.0],
nodes_on_bdry=True
)
Users can also manually specify the containing intervals dimensions by
using the ``min_pt`` and ``max_pt`` arguments:
>>> odl.nonuniform_partition([0, 1, 3], min_pt=-2, max_pt=3)
nonuniform_partition(
[0.0, 1.0, 3.0],
min_pt=-2.0, max_pt=3.0
)
"""
# Get parameters from kwargs
min_pt = kwargs.pop('min_pt', None)
max_pt = kwargs.pop('max_pt', None)
nodes_on_bdry = kwargs.pop('nodes_on_bdry', False)
# np.size(None) == 1
sizes = [len(coord_vecs)] + [np.size(p) for p in (min_pt, max_pt)]
ndim = int(np.max(sizes))
min_pt = normalized_scalar_param_list(min_pt,
ndim,
param_conv=float,
keep_none=True)
max_pt = normalized_scalar_param_list(max_pt,
ndim,
param_conv=float,
keep_none=True)
nodes_on_bdry = normalized_nodes_on_bdry(nodes_on_bdry, ndim)
# Calculate the missing parameters in min_pt, max_pt
for i, (xmin, xmax, (bdry_l, bdry_r),
coords) in enumerate(zip(min_pt, max_pt, nodes_on_bdry,
coord_vecs)):
# Check input for redundancy
if xmin is not None and bdry_l:
raise ValueError('in axis {}: got both `min_pt` and '
'`nodes_on_bdry=True`'.format(i))
if xmax is not None and bdry_r:
raise ValueError('in axis {}: got both `max_pt` and '
'`nodes_on_bdry=True`'.format(i))
# Compute boundary position if not given by user
if xmin is None:
if bdry_l:
min_pt[i] = coords[0]
else:
min_pt[i] = coords[0] - (coords[1] - coords[0]) / 2.0
if xmax is None:
if bdry_r:
max_pt[i] = coords[-1]
else:
max_pt[i] = coords[-1] + (coords[-1] - coords[-2]) / 2.0
interval = IntervalProd(min_pt, max_pt)
grid = RectGrid(*coord_vecs)
return RectPartition(interval, grid)
def uniform_partition(min_pt=None,
max_pt=None,
shape=None,
cell_sides=None,
nodes_on_bdry=False):
"""Return a partition with equally sized cells.
Parameters
----------
min_pt, max_pt : float or sequence of float, optional
Vectors defining the lower/upper limits of the intervals in an
`IntervalProd` (a rectangular box). ``None`` entries mean
"compute the value".
shape : int or sequence of ints, optional
Number of nodes per axis. ``None`` entries mean
"compute the value".
cell_sides : float or sequence of floats, optional
Side length of the partition cells per axis. ``None`` entries mean
"compute the value".
nodes_on_bdry : bool or sequence, optional
If a sequence is provided, it determines per axis whether to
place the last grid point on the boundary (``True``) or shift it
by half a cell size into the interior (``False``). In each axis,
an entry may consist in a single bool or a 2-tuple of
bool. In the latter case, the first tuple entry decides for
the left, the second for the right boundary. The length of the
sequence must be ``array.ndim``.
A single boolean is interpreted as a global choice for all
boundaries.
Notes
-----
In each axis, 3 of the 4 possible parameters ``min_pt``, ``max_pt``,
``shape`` and ``cell_sides`` must be given. If all four are
provided, they are checked for consistency.
See Also
--------
uniform_partition_fromintv : partition an existing set
uniform_partition_fromgrid : use an existing grid as basis
Examples
--------
Any combination of three of the four parameters can be used for
creation of a partition:
>>> part = odl.uniform_partition(min_pt=0, max_pt=2, shape=4)
>>> part.cell_boundary_vecs
(array([ 0. , 0.5, 1. , 1.5, 2. ]),)
>>> part = odl.uniform_partition(min_pt=0, shape=4, cell_sides=0.5)
>>> part.cell_boundary_vecs
(array([ 0. , 0.5, 1. , 1.5, 2. ]),)
>>> part = odl.uniform_partition(max_pt=2, shape=4, cell_sides=0.5)
>>> part.cell_boundary_vecs
(array([ 0. , 0.5, 1. , 1.5, 2. ]),)
>>> part = odl.uniform_partition(min_pt=0, max_pt=2, cell_sides=0.5)
>>> part.cell_boundary_vecs
(array([ 0. , 0.5, 1. , 1.5, 2. ]),)
In higher dimensions, the parameters can be given differently in
each axis. Where ``None`` is given, the value will be computed:
>>> part = odl.uniform_partition(min_pt=[0, 0], max_pt=[1, 2],
... shape=[4, 2])
>>> part.cell_boundary_vecs
(array([ 0. , 0.25, 0.5 , 0.75, 1. ]), array([ 0., 1., 2.]))
>>> part = odl.uniform_partition(min_pt=[0, 0], max_pt=[1, 2],
... shape=[None, 2], cell_sides=[0.25, None])
>>> part.cell_boundary_vecs
(array([ 0. , 0.25, 0.5 , 0.75, 1. ]), array([ 0., 1., 2.]))
>>> part = odl.uniform_partition(min_pt=[0, None], max_pt=[None, 2],
... shape=[4, 2], cell_sides=[0.25, 1])
>>> part.cell_boundary_vecs
(array([ 0. , 0.25, 0.5 , 0.75, 1. ]), array([ 0., 1., 2.]))
By default, no grid points are placed on the boundary:
>>> part = odl.uniform_partition(0, 1, 4)
>>> part.nodes_on_bdry
False
>>> part.cell_boundary_vecs
(array([ 0. , 0.25, 0.5 , 0.75, 1. ]),)
>>> part.grid.coord_vectors
(array([ 0.125, 0.375, 0.625, 0.875]),)
This can be changed with the nodes_on_bdry parameter:
>>> part = odl.uniform_partition(0, 1, 3, nodes_on_bdry=True)
>>> part.nodes_on_bdry
True
>>> part.cell_boundary_vecs
(array([ 0. , 0.25, 0.75, 1. ]),)
>>> part.grid.coord_vectors
(array([ 0. , 0.5, 1. ]),)
We can specify this per axis, too. In this case we choose both
in the first axis and only the rightmost in the second:
>>> part = odl.uniform_partition([0, 0], [1, 1], (3, 3),
... nodes_on_bdry=(True, (False, True)))
...
>>> part.cell_boundary_vecs[0] # first axis, as above
array([ 0. , 0.25, 0.75, 1. ])
>>> part.grid.coord_vectors[0]
array([ 0. , 0.5, 1. ])
>>> part.cell_boundary_vecs[1] # second, asymmetric axis
array([ 0. , 0.4, 0.8, 1. ])
>>> part.grid.coord_vectors[1]
array([ 0.2, 0.6, 1. ])
"""
# Normalize partition parameters
# np.size(None) == 1, so that would screw it for sizes 0 of the rest
sizes = [
np.size(p) for p in (min_pt, max_pt, shape, cell_sides)
if p is not None
]
ndim = int(np.max(sizes))
min_pt = normalized_scalar_param_list(min_pt,
ndim,
param_conv=float,
keep_none=True)
max_pt = normalized_scalar_param_list(max_pt,
ndim,
param_conv=float,
keep_none=True)
shape = normalized_scalar_param_list(shape,
ndim,
param_conv=safe_int_conv,
keep_none=True)
cell_sides = normalized_scalar_param_list(cell_sides,
ndim,
param_conv=float,
keep_none=True)
nodes_on_bdry = normalized_nodes_on_bdry(nodes_on_bdry, ndim)
# Calculate the missing parameters in min_pt, max_pt, shape
for i, (xmin, xmax, n, dx, on_bdry) in enumerate(
zip(min_pt, max_pt, shape, cell_sides, nodes_on_bdry)):
num_params = sum(p is not None for p in (xmin, xmax, n, dx))
if num_params < 3:
raise ValueError('in axis {}: expected at least 3 of the '
'parameters `min_pt`, `max_pt`, `shape`, '
'`cell_sides`, got {}'
''.format(i, num_params))
# Unpack the tuple if possible, else use bool globally for this axis
try:
bdry_l, bdry_r = on_bdry
except TypeError:
bdry_l = bdry_r = on_bdry
# For each node on the boundary, we subtract 1/2 from the number of
# full cells between min_pt and max_pt.
if xmin is None:
min_pt[i] = xmax - (n - sum([bdry_l, bdry_r]) / 2.0) * dx
elif xmax is None:
max_pt[i] = xmin + (n - sum([bdry_l, bdry_r]) / 2.0) * dx
elif n is None:
# Here we add to n since (e-b)/s gives the reduced number of cells.
n_calc = (xmax - xmin) / dx + sum([bdry_l, bdry_r]) / 2.0
n_round = int(round(n_calc))
if abs(n_calc - n_round) > 1e-5:
raise ValueError('in axis {}: calculated number of nodes '
'{} = ({} - {}) / {} too far from integer'
''.format(i, n_calc, xmax, xmin, dx))
shape[i] = n_round
elif dx is None:
pass
else:
xmax_calc = xmin + (n - sum([bdry_l, bdry_r]) / 2.0) * dx
if not np.isclose(xmax, xmax_calc):
raise ValueError('in axis {}: calculated endpoint '
'{} = {} + {} * {} too far from given '
'endpoint {}.'
''.format(i, xmax_calc, xmin, n, dx, xmax))
return uniform_partition_fromintv(IntervalProd(min_pt, max_pt), shape,
nodes_on_bdry)
def uniform_grid(min_pt, max_pt, shape, nodes_on_bdry=True):
"""Return a grid from sampling an implicit interval product uniformly.
Parameters
----------
min_pt : float or sequence of float
Vectors of lower ends of the intervals in the product.
max_pt : float or sequence of float
Vectors of upper ends of the intervals in the product.
shape : int or sequence of ints
Number of nodes per axis. Entries corresponding to degenerate axes
must be equal to 1.
nodes_on_bdry : bool or sequence, optional
If a sequence is provided, it determines per axis whether to
place the last grid point on the boundary (``True``) or shift it
by half a cell size into the interior (``False``). In each axis,
an entry may consist in a single bool or a 2-tuple of
bool. In the latter case, the first tuple entry decides for
the left, the second for the right boundary. The length of the
sequence must be ``array.ndim``.
A single boolean is interpreted as a global choice for all
boundaries.
Returns
-------
uniform_grid : `RectGrid`
The resulting uniform grid.
See Also
--------
uniform_grid_fromintv :
sample a given interval product
odl.discr.partition.uniform_partition :
divide implicitly defined interval product into equally
sized subsets
Examples
--------
By default, the min/max points are included in the grid:
>>> grid = odl.uniform_grid([-1.5, 2], [-0.5, 3], (3, 3))
>>> grid.coord_vectors
(array([-1.5, -1. , -0.5]), array([ 2. , 2.5, 3. ]))
If ``shape`` is supposed to refer to small subvolumes, and the grid
should be their centers, use the option ``nodes_on_bdry=False``:
>>> grid = odl.uniform_grid([-1.5, 2], [-0.5, 3], (2, 2),
... nodes_on_bdry=False)
>>> grid.coord_vectors
(array([-1.25, -0.75]), array([ 2.25, 2.75]))
In 1D, we don't need sequences:
>>> grid = odl.uniform_grid(0, 1, 3)
>>> grid.coord_vectors
(array([ 0. , 0.5, 1. ]),)
"""
return uniform_grid_fromintv(IntervalProd(min_pt, max_pt), shape,
nodes_on_bdry=nodes_on_bdry)
