def show(self,
title=None,
method='scatter',
force_show=False,
fig=None,
**kwargs):
"""Display this vector graphically.
Parameters
----------
title : string, optional
Set the title of the figure
method : string, optional
The following plotting methods are available:
'scatter' : point plot
'plot' : graph plot
force_show : bool, optional
Whether the plot should be forced to be shown now or deferred until
later. Note that some backends always displays the plot, regardless
of this value.
fig : `matplotlib.figure.Figure`, optional
Figure to draw into. Expected to be of same "style" as
the figure given by this function. The most common use case
is that ``fig`` is the return value of an earlier call to
this function.
kwargs : {'figsize', 'saveto', ...}, optional
Extra keyword arguments passed on to the display method.
See the Matplotlib functions for documentation of extra
options.
Returns
-------
fig : `matplotlib.figure.Figure`
Resulting figure. If ``fig`` was given, the returned object
is a reference to it.
See Also
--------
odl.util.graphics.show_discrete_data : Underlying implementation
"""
from odl.util.graphics import show_discrete_data
from odl.discr import uniform_grid
grid = uniform_grid(0, self.size - 1, self.size)
return show_discrete_data(self.asarray(),
grid,
title=title,
method=method,
force_show=force_show,
fig=fig,
**kwargs)
def show(self, title=None, method='scatter', force_show=False, fig=None,
**kwargs):
"""Display this vector graphically.
Parameters
----------
title : string, optional
Set the title of the figure
method : string, optional
The following plotting methods are available:
'scatter' : point plot
'plot' : graph plot
force_show : bool, optional
Whether the plot should be forced to be shown now or deferred until
later. Note that some backends always displays the plot, regardless
of this value.
fig : `matplotlib.figure.Figure`, optional
Figure to draw into. Expected to be of same "style" as
the figure given by this function. The most common use case
is that ``fig`` is the return value of an earlier call to
this function.
kwargs : {'figsize', 'saveto', ...}, optional
Extra keyword arguments passed on to the display method.
See the Matplotlib functions for documentation of extra
options.
Returns
-------
fig : `matplotlib.figure.Figure`
Resulting figure. If ``fig`` was given, the returned object
is a reference to it.
See Also
--------
odl.util.graphics.show_discrete_data : Underlying implementation
"""
from odl.util.graphics import show_discrete_data
from odl.discr import uniform_grid
grid = uniform_grid(0, self.size - 1, self.size)
return show_discrete_data(self.asarray(), grid, title=title,
method=method, force_show=force_show,
fig=fig, **kwargs)
def show(self,
title=None,
method='',
indices=None,
force_show=False,
fig=None,
**kwargs):
"""Display the function graphically.
Parameters
----------
title : string, optional
Set the title of the figure
method : string, optional
1d methods:
``'plot'`` : graph plot
``'scatter'`` : scattered 2d points (2nd axis <-> value)
2d methods:
``'imshow'`` : image plot with coloring according to
value, including a colorbar.
``'scatter'`` : cloud of scattered 3d points
(3rd axis <-> value)
indices : index expression, optional
Display a slice of the array instead of the full array. The
index expression is most easily created with the `numpy.s_`
constructor, i.e. supply ``np.s_[:, 1, :]`` to display the
first slice along the second axis.
For data with 3 or more dimensions, the 2d slice in the first
two axes at the "middle" along the remaining axes is shown
(semantically ``[:, :, shape[2:] // 2]``).
This option is mutually exclusive to ``coords``.
force_show : bool, optional
Whether the plot should be forced to be shown now or deferred until
later. Note that some backends always displays the plot, regardless
of this value.
fig : `matplotlib.figure.Figure`, optional
The figure to show in. Expected to be of same "style", as
the figure given by this function. The most common use case
is that ``fig`` is the return value of an earlier call to
this function.
kwargs : {'figsize', 'saveto', 'clim', ...}, optional
Extra keyword arguments passed on to the display method.
See the Matplotlib functions for documentation of extra
options.
Returns
-------
fig : `matplotlib.figure.Figure`
The resulting figure. It is also shown to the user.
See Also
--------
odl.util.graphics.show_discrete_data : Underlying implementation
"""
from odl.discr import uniform_grid
from odl.util.graphics import show_discrete_data
# Default to showing x-y slice "in the middle"
if indices is None and self.ndim >= 3:
indices = tuple([slice(None)] * 2 +
[n // 2 for n in self.space.shape[2:]])
if isinstance(indices, (Integral, slice)):
indices = (indices, )
elif indices is None or indices == Ellipsis:
indices = (slice(None), ) * self.ndim
else:
indices = tuple(indices)
# Replace None by slice(None)
indices = tuple(slice(None) if idx is None else idx for idx in indices)
if Ellipsis in indices:
# Replace Ellipsis with the correct number of [:] expressions
pos = indices.index(Ellipsis)
indices = (indices[:pos] + (np.s_[:], ) *
(self.ndim - len(indices) + 1) + indices[pos + 1:])
if len(indices) < self.ndim:
raise ValueError('too few axes ({} < {})'.format(
len(indices), self.ndim))
if len(indices) > self.ndim:
raise ValueError('too many axes ({} > {})'.format(
len(indices), self.ndim))
# Squeeze grid and values according to the index expression
full_grid = uniform_grid([0] * self.ndim,
np.array(self.shape) - 1, self.shape)
grid = full_grid[indices].squeeze()
values = self.asarray()[indices].squeeze()
return show_discrete_data(values,
grid,
title=title,
method=method,
force_show=force_show,
fig=fig,
**kwargs)
def reciprocal_grid(grid, shift=True, axes=None, halfcomplex=False):
"""Return the reciprocal of the given regular grid.
This function calculates the reciprocal (Fourier/frequency space)
grid for a given regular grid defined by the nodes::
x[k] = x[0] + k * s,
where ``k = (k[0], ..., k[d-1])`` is a ``d``-dimensional index in
the range ``0 <= k < N`` (component-wise). The multi-index
``N`` is the shape of the input grid.
This grid's reciprocal is then given by the nodes::
xi[j] = xi[0] + j * sigma,
with the reciprocal grid stride ``sigma = 2*pi / (s * N)``.
The minimum frequency ``xi[0]`` can in principle be chosen
freely, but usually it is chosen in a such a way that the reciprocal
grid is centered around zero. For this, there are two possibilities:
1. Make the grid point-symmetric around 0.
2. Make the grid "almost" point-symmetric around zero by shifting
it to the left by half a reciprocal stride.
In the first case, the minimum frequency (per axis) is given as::
xi_1[0] = -pi/s + pi/(s*n) = -pi/s + sigma/2.
For the second case, it is::
xi_1[0] = -pi / s.
Note that the zero frequency is contained in case 1 for an odd
number of points, while for an even size, the second option
guarantees that 0 is contained.
If a real-to-complex (half-complex) transform is to be computed,
the reciprocal grid has the shape ``M[i] = floor(N[i]/2) + 1``
in the last transform axis ``i``.
Parameters
----------
grid : uniform `RectGrid`
Original sampling grid,.
shift : bool or sequence of bools, optional
If ``True``, the grid is shifted by half a stride in the negative
direction. With a sequence, this option is applied separately on
each axis.
axes : int or sequence of ints, optional
Dimensions in which to calculate the reciprocal. The sequence
must have the same length as ``shift`` if the latter is given
as a sequence. ``None`` means all axes in ``grid``.
halfcomplex : bool, optional
If ``True``, return the half of the grid with last coordinate
less than zero. This is related to the fact that for real-valued
functions, the other half is the mirrored complex conjugate of
the given half and therefore needs not be stored.
Returns
-------
reciprocal_grid : uniform `RectGrid`
The reciprocal grid.
"""
if axes is None:
axes = list(range(grid.ndim))
else:
try:
axes = [int(axes)]
except TypeError:
axes = list(axes)
# List indicating shift or not per "active" axis, same length as axes
shift_list = normalized_scalar_param_list(shift, length=len(axes),
param_conv=bool)
# Full-length vectors
stride = grid.stride
shape = np.array(grid.shape)
rmin = grid.min_pt.copy()
rmax = grid.max_pt.copy()
rshape = list(shape)
# Shifted axes (full length to avoid ugly double indexing)
shifted = np.zeros(grid.ndim, dtype=bool)
shifted[axes] = shift_list
rmin[shifted] = -np.pi / stride[shifted]
# Length min->max increases by double the shift, so we
# have to compensate by a full stride
rmax[shifted] = (-rmin[shifted] -
2 * np.pi / (stride[shifted] * shape[shifted]))
# Non-shifted axes
not_shifted = np.zeros(grid.ndim, dtype=bool)
not_shifted[axes] = np.logical_not(shift_list)
rmin[not_shifted] = ((-1.0 + 1.0 / shape[not_shifted]) *
np.pi / stride[not_shifted])
rmax[not_shifted] = -rmin[not_shifted]
# Change last axis shape and max if halfcomplex
if halfcomplex:
rshape[axes[-1]] = shape[axes[-1]] // 2 + 1
# - Odd and shifted: - stride / 2
# - Even and not shifted: + stride / 2
# - Otherwise: 0
last_odd = shape[axes[-1]] % 2 == 1
last_shifted = shift_list[-1]
half_rstride = np.pi / (shape[axes[-1]] * stride[axes[-1]])
if last_odd and last_shifted:
rmax[axes[-1]] = -half_rstride
elif not last_odd and not last_shifted:
rmax[axes[-1]] = half_rstride
else:
rmax[axes[-1]] = 0
return uniform_grid(rmin, rmax, rshape)
def realspace_grid(recip_grid, x0, axes=None, halfcomplex=False,
halfcx_parity='even'):
"""Return the real space grid from the given reciprocal grid.
Given a reciprocal grid::
xi[j] = xi[0] + j * sigma,
with a multi-index ``j = (j[0], ..., j[d-1])`` in the range
``0 <= j < M``, this function calculates the original grid::
x[k] = x[0] + k * s
by using a provided ``x[0]`` and calculating the stride ``s``.
If the reciprocal grid is interpreted as coming from a usual
complex-to-complex FFT, it is ``N == M``, and the stride is::
s = 2*pi / (sigma * N)
For a reciprocal grid from a real-to-complex (half-complex) FFT,
it is ``M[i] = floor(N[i]/2) + 1`` in the last transform axis ``i``.
To resolve the ambiguity regarding the parity of ``N[i]``, the
it must be specified if the output shape should be even or odd,
resulting in::
odd : N[i] = 2 * M[i] - 1
even: N[i] = 2 * M[i] - 2
The output stride is calculated with this ``N`` as above in this
case.
Parameters
----------
recip_grid : uniform `RectGrid`
Sampling grid in reciprocal space.
x0 : `array-like`
Desired minimum point of the real space grid.
axes : int or sequence of ints, optional
Dimensions in which to calculate the real space grid. The sequence
must have the same length as ``shift`` if the latter is given
as a sequence. ``None`` means "all axes".
halfcomplex : bool, optional
If ``True``, interpret the given grid as the reciprocal as used
in a half-complex FFT (see above). Otherwise, the grid is
regarded as being used in a complex-to-complex transform.
halfcx_parity : {'even', 'odd'}
Use this parity for the shape of the returned grid in the
last axis of ``axes`` in the case ``halfcomplex=True``
Returns
-------
irecip : uniform `RectGrid`
The inverse reciprocal grid.
"""
if axes is None:
axes = list(range(recip_grid.ndim))
else:
try:
axes = [int(axes)]
except TypeError:
axes = list(axes)
rstride = recip_grid.stride
rshape = recip_grid.shape
# Calculate shape of the output grid by adjusting in axes[-1]
irshape = list(rshape)
if halfcomplex:
if str(halfcx_parity).lower() == 'even':
irshape[axes[-1]] = 2 * rshape[axes[-1]] - 2
elif str(halfcx_parity).lower() == 'odd':
irshape[axes[-1]] = 2 * rshape[axes[-1]] - 1
else:
raise ValueError("`halfcomplex` parity '{}' not understood"
"".format(halfcx_parity))
irmin = np.asarray(x0)
irshape = np.asarray(irshape)
irstride = np.copy(rstride)
irstride[axes] = 2 * np.pi / (irshape[axes] * rstride[axes])
irmax = irmin + (irshape - 1) * irstride
return uniform_grid(irmin, irmax, irshape)
def reciprocal_grid(grid, shift=True, axes=None, halfcomplex=False):
"""Return the reciprocal of the given regular grid.
This function calculates the reciprocal (Fourier/frequency space)
grid for a given regular grid defined by the nodes::
x[k] = x[0] + k * s,
where ``k = (k[0], ..., k[d-1])`` is a ``d``-dimensional index in
the range ``0 <= k < N`` (component-wise). The multi-index
``N`` is the shape of the input grid.
This grid's reciprocal is then given by the nodes::
xi[j] = xi[0] + j * sigma,
with the reciprocal grid stride ``sigma = 2*pi / (s * N)``.
The minimum frequency ``xi[0]`` can in principle be chosen
freely, but usually it is chosen in a such a way that the reciprocal
grid is centered around zero. For this, there are two possibilities:
1. Make the grid point-symmetric around 0.
2. Make the grid "almost" point-symmetric around zero by shifting
it to the left by half a reciprocal stride.
In the first case, the minimum frequency (per axis) is given as::
xi_1[0] = -pi/s + pi/(s*n) = -pi/s + sigma/2.
For the second case, it is::
xi_1[0] = -pi / s.
Note that the zero frequency is contained in case 1 for an odd
number of points, while for an even size, the second option
guarantees that 0 is contained.
If a real-to-complex (half-complex) transform is to be computed,
the reciprocal grid has the shape ``M[i] = floor(N[i]/2) + 1``
in the last transform axis ``i``.
Parameters
----------
grid : uniform `RectGrid`
Original sampling grid,.
shift : bool or sequence of bools, optional
If ``True``, the grid is shifted by half a stride in the negative
direction. With a sequence, this option is applied separately on
each axis.
axes : int or sequence of ints, optional
Dimensions in which to calculate the reciprocal. The sequence
must have the same length as ``shift`` if the latter is given
as a sequence. ``None`` means all axes in ``grid``.
halfcomplex : bool, optional
If ``True``, return the half of the grid with last coordinate
less than zero. This is related to the fact that for real-valued
functions, the other half is the mirrored complex conjugate of
the given half and therefore needs not be stored.
Returns
-------
reciprocal_grid : uniform `RectGrid`
The reciprocal grid.
"""
if axes is None:
axes = list(range(grid.ndim))
else:
try:
axes = [int(axes)]
except TypeError:
axes = list(axes)
# List indicating shift or not per "active" axis, same length as axes
shift_list = normalized_scalar_param_list(shift, length=len(axes),
param_conv=bool)
# Full-length vectors
stride = grid.stride
shape = np.array(grid.shape)
rmin = grid.min_pt.copy()
rmax = grid.max_pt.copy()
rshape = list(shape)
# Shifted axes (full length to avoid ugly double indexing)
shifted = np.zeros(grid.ndim, dtype=bool)
shifted[axes] = shift_list
rmin[shifted] = -np.pi / stride[shifted]
# Length min->max increases by double the shift, so we
# have to compensate by a full stride
rmax[shifted] = (-rmin[shifted] -
2 * np.pi / (stride[shifted] * shape[shifted]))
# Non-shifted axes
not_shifted = np.zeros(grid.ndim, dtype=bool)
not_shifted[axes] = np.logical_not(shift_list)
rmin[not_shifted] = ((-1.0 + 1.0 / shape[not_shifted]) *
np.pi / stride[not_shifted])
rmax[not_shifted] = -rmin[not_shifted]
# Change last axis shape and max if halfcomplex
if halfcomplex:
rshape[axes[-1]] = shape[axes[-1]] // 2 + 1
# - Odd and shifted: - stride / 2
# - Even and not shifted: + stride / 2
# - Otherwise: 0
last_odd = shape[axes[-1]] % 2 == 1
last_shifted = shift_list[-1]
half_rstride = np.pi / (shape[axes[-1]] * stride[axes[-1]])
if last_odd and last_shifted:
rmax[axes[-1]] = -half_rstride
elif not last_odd and not last_shifted:
rmax[axes[-1]] = half_rstride
else:
rmax[axes[-1]] = 0
return uniform_grid(rmin, rmax, rshape)
def realspace_grid(recip_grid, x0, axes=None, halfcomplex=False,
halfcx_parity='even'):
"""Return the real space grid from the given reciprocal grid.
Given a reciprocal grid::
xi[j] = xi[0] + j * sigma,
with a multi-index ``j = (j[0], ..., j[d-1])`` in the range
``0 <= j < M``, this function calculates the original grid::
x[k] = x[0] + k * s
by using a provided ``x[0]`` and calculating the stride ``s``.
If the reciprocal grid is interpreted as coming from a usual
complex-to-complex FFT, it is ``N == M``, and the stride is::
s = 2*pi / (sigma * N)
For a reciprocal grid from a real-to-complex (half-complex) FFT,
it is ``M[i] = floor(N[i]/2) + 1`` in the last transform axis ``i``.
To resolve the ambiguity regarding the parity of ``N[i]``, the
it must be specified if the output shape should be even or odd,
resulting in::
odd : N[i] = 2 * M[i] - 1
even: N[i] = 2 * M[i] - 2
The output stride is calculated with this ``N`` as above in this
case.
Parameters
----------
recip_grid : uniform `RectGrid`
Sampling grid in reciprocal space.
x0 : `array-like`
Desired minimum point of the real space grid.
axes : int or sequence of ints, optional
Dimensions in which to calculate the real space grid. The sequence
must have the same length as ``shift`` if the latter is given
as a sequence. ``None`` means "all axes".
halfcomplex : bool, optional
If ``True``, interpret the given grid as the reciprocal as used
in a half-complex FFT (see above). Otherwise, the grid is
regarded as being used in a complex-to-complex transform.
halfcx_parity : {'even', 'odd'}
Use this parity for the shape of the returned grid in the
last axis of ``axes`` in the case ``halfcomplex=True``
Returns
-------
irecip : uniform `RectGrid`
The inverse reciprocal grid.
"""
if axes is None:
axes = list(range(recip_grid.ndim))
else:
try:
axes = [int(axes)]
except TypeError:
axes = list(axes)
rstride = recip_grid.stride
rshape = recip_grid.shape
# Calculate shape of the output grid by adjusting in axes[-1]
irshape = list(rshape)
if halfcomplex:
if str(halfcx_parity).lower() == 'even':
irshape[axes[-1]] = 2 * rshape[axes[-1]] - 2
elif str(halfcx_parity).lower() == 'odd':
irshape[axes[-1]] = 2 * rshape[axes[-1]] - 1
else:
raise ValueError("`halfcomplex` parity '{}' not understood"
"".format(halfcx_parity))
irmin = np.asarray(x0)
irshape = np.asarray(irshape)
irstride = np.copy(rstride)
irstride[axes] = 2 * np.pi / (irshape[axes] * rstride[axes])
irmax = irmin + (irshape - 1) * irstride
return uniform_grid(irmin, irmax, irshape)
