from torch.nn.functional import conv2d, pad
from e2cnn.nn import init
from e2cnn.nn import FieldType
from e2cnn.nn import GeometricTensor
from e2cnn.gspaces import *
from ..equivariant_module import EquivariantModule
from .basisexpansion import BasisExpansion
from .basisexpansion_blocks import BlocksBasisExpansion
from typing import Callable, Union, Tuple, List
import torch
from torch.nn import Parameter
import numpy as np
import math
__all__ = ["R2Conv"]
[docs]class R2Conv(EquivariantModule):
def __init__(self,
in_type: FieldType,
out_type: FieldType,
kernel_size: int,
padding: int = 0,
stride: int = 1,
dilation: int = 1,
padding_mode: str = 'zeros',
groups: int = 1,
bias: bool = True,
basisexpansion: str = 'blocks',
sigma: Union[List[float], float] = None,
frequencies_cutoff: Union[float, Callable[[float], int]] = None,
rings: List[float] = None,
maximum_offset: int = None,
recompute: bool = False,
basis_filter: Callable[[dict], bool] = None,
initialize: bool = True,
):
r"""
G-steerable planar convolution mapping between the input and output :class:`~e2cnn.nn.FieldType` s specified by
the parameters ``in_type`` and ``out_type``.
This operation is equivariant under the action of :math:`\R^2\rtimes G` where :math:`G` is the
:attr:`e2cnn.nn.FieldType.fibergroup` of ``in_type`` and ``out_type``.
Specifically, let :math:`\rho_\text{in}: G \to \GL{\R^{c_\text{in}}}` and
:math:`\rho_\text{out}: G \to \GL{\R^{c_\text{out}}}` be the representations specified by the input and output
field types.
Then :class:`~e2cnn.nn.R2Conv` guarantees an equivariant mapping
.. math::
\kappa \star [\mathcal{T}^\text{in}_{g,u} . f] = \mathcal{T}^\text{out}_{g,u} . [\kappa \star f] \qquad\qquad \forall g \in G, u \in \R^2
where the transformation of the input and output fields are given by
.. math::
[\mathcal{T}^\text{in}_{g,u} . f](x) &= \rho_\text{in}(g)f(g^{-1} (x - u)) \\
[\mathcal{T}^\text{out}_{g,u} . f](x) &= \rho_\text{out}(g)f(g^{-1} (x - u)) \\
The equivariance of G-steerable convolutions is guaranteed by restricting the space of convolution kernels to an
equivariant subspace.
As proven in `3D Steerable CNNs <https://arxiv.org/abs/1807.02547>`_, this parametrizes the *most general
equivariant convolutional map* between the input and output fields.
For feature fields on :math:`\R^2` (e.g. images), the complete G-steerable kernel spaces for :math:`G \leq \O2`
is derived in `General E(2)-Equivariant Steerable CNNs <https://arxiv.org/abs/1911.08251>`_.
During training, in each forward pass the module expands the basis of G-steerable kernels with learned weights
before calling :func:`torch.nn.functional.conv2d`.
When :meth:`~torch.nn.Module.eval()` is called, the filter is built with the current trained weights and stored
for future reuse such that no overhead of expanding the kernel remains.
.. warning ::
When :meth:`~torch.nn.Module.train()` is called, the attributes :attr:`~e2cnn.nn.R2Conv.filter` and
:attr:`~e2cnn.nn.R2Conv.expanded_bias` are discarded to avoid situations of mismatch with the
learnable expansion coefficients.
See also :meth:`e2cnn.nn.R2Conv.train`.
This behaviour can cause problems when storing the :meth:`~torch.nn.Module.state_dict` of a model while in
a mode and lately loading it in a model with a different mode, as the attributes of the class change.
To avoid this issue, we recommend converting the model to eval mode before storing or loading the state
dictionary.
The learnable expansion coefficients of the this module can be initialized with the methods in
:mod:`e2cnn.nn.init`.
By default, the weights are initialized in the constructors using :func:`~e2cnn.nn.init.generalized_he_init`.
.. warning ::
This initialization procedure can be extremely slow for wide layers.
In case initializing the model is not required (e.g. before loading the state dict of a pre-trained model)
or another initialization method is preferred (e.g. :func:`~e2cnn.nn.init.deltaorthonormal_init`), the
parameter ``initialize`` can be set to ``False`` to avoid unnecessary overhead.
The parameters ``basisexpansion``, ``sigma``, ``frequencies_cutoff``, ``rings`` and ``maximum_offset`` are
optional parameters used to control how the basis for the filters is built, how it is sampled on the filter
grid and how it is expanded to build the filter. We suggest to keep these default values.
Args:
in_type (FieldType): the type of the input field, specifying its transformation law
out_type (FieldType): the type of the output field, specifying its transformation law
kernel_size (int): the size of the (square) filter
padding (int, optional): implicit zero paddings on both sides of the input. Default: ``0``
padding_mode(str, optional): ``zeros``, ``reflect``, ``replicate`` or ``circular``. Default: ``zeros``
stride (int, optional): the stride of the kernel. Default: ``1``
dilation (int, optional): the spacing between kernel elements. Default: ``1``
groups (int, optional): number of blocked connections from input channels to output channels.
It allows depthwise convolution. When used, the input and output types need to be
divisible in ``groups`` groups, all equal to each other.
Default: ``1``.
bias (bool, optional): Whether to add a bias to the output (only to fields which contain a
trivial irrep) or not. Default ``True``
basisexpansion (str, optional): the basis expansion algorithm to use
sigma (list or float, optional): width of each ring where the bases are sampled. If only one scalar
is passed, it is used for all rings.
frequencies_cutoff (callable or float, optional): function mapping the radii of the basis elements to the
maximum frequency accepted. If a float values is passed, the maximum frequency is equal to the
radius times this factor. By default (``None``), a more complex policy is used.
rings (list, optional): radii of the rings where to sample the bases
maximum_offset (int, optional): number of additional (aliased) frequencies in the intertwiners for finite
groups. By default (``None``), all additional frequencies allowed by the frequencies cut-off
are used.
recompute (bool, optional): if ``True``, recomputes a new basis for the equivariant kernels.
By Default (``False``), it caches the basis built or reuse a cached one, if it is found.
basis_filter (callable, optional): function which takes as input a descriptor of a basis element
(as a dictionary) and returns a boolean value: whether to preserve (``True``) or discard (``False``)
the basis element. By default (``None``), no filtering is applied.
initialize (bool, optional): initialize the weights of the model. Default: ``True``
Attributes:
~.weights (torch.Tensor): the learnable parameters which are used to expand the kernel
~.filter (torch.Tensor): the convolutional kernel obtained by expanding the parameters
in :attr:`~e2cnn.nn.R2Conv.weights`
~.bias (torch.Tensor): the learnable parameters which are used to expand the bias, if ``bias=True``
~.expanded_bias (torch.Tensor): the equivariant bias which is summed to the output, obtained by expanding
the parameters in :attr:`~e2cnn.nn.R2Conv.bias`
"""
assert in_type.gspace == out_type.gspace
assert isinstance(in_type.gspace, GeneralOnR2)
super(R2Conv, self).__init__()
self.space = in_type.gspace
self.in_type = in_type
self.out_type = out_type
self.kernel_size = kernel_size
self.stride = stride
self.dilation = dilation
self.padding = padding
self.padding_mode = padding_mode
self.groups = groups
if isinstance(padding, tuple) and len(padding) == 2:
_padding = padding
elif isinstance(padding, int):
_padding = (padding, padding)
else:
raise ValueError('padding needs to be either an integer or a tuple containing two integers but {} found'.format(padding))
padding_modes = {'zeros', 'reflect', 'replicate', 'circular'}
if padding_mode not in padding_modes:
raise ValueError("padding_mode must be one of [{}], but got padding_mode='{}'".format(padding_modes, padding_mode))
self._reversed_padding_repeated_twice = tuple(x for x in reversed(_padding) for _ in range(2))
if groups > 1:
# Check the input and output classes can be split in `groups` groups, all equal to each other
# first, check that the number of fields is divisible by `groups`
assert len(in_type) % groups == 0
assert len(out_type) % groups == 0
in_size = len(in_type) // groups
out_size = len(out_type) // groups
# then, check that all groups are equal to each other, i.e. have the same types in the same order
assert all(in_type.representations[i] == in_type.representations[i % in_size] for i in range(len(in_type)))
assert all(out_type.representations[i] == out_type.representations[i % out_size] for i in range(len(out_type)))
# finally, retrieve the type associated to a single group in input.
# this type will be used to build a smaller kernel basis and a smaller filter
# as in PyTorch, to build a filter for grouped convolution, we build a filter which maps from one input
# group to all output groups. Then, PyTorch's standard convolution routine interpret this filter as `groups`
# different filters, each mapping an input group to an output group.
in_type = in_type.index_select(list(range(in_size)))
if bias:
# bias can be applied only to trivial irreps inside the representation
# to apply bias to a field we learn a bias for each trivial irreps it contains
# and, then, we transform it with the change of basis matrix to be able to apply it to the whole field
# this is equivalent to transform the field to its irreps through the inverse change of basis,
# sum the bias only to the trivial irrep and then map it back with the change of basis
# count the number of trivial irreps
trivials = 0
for r in self.out_type:
for irr in r.irreps:
if self.out_type.fibergroup.irreps[irr].is_trivial():
trivials += 1
# if there is at least 1 trivial irrep
if trivials > 0:
# matrix containing the columns of the change of basis which map from the trivial irreps to the
# field representations. This matrix allows us to map the bias defined only over the trivial irreps
# to a bias for the whole field more efficiently
bias_expansion = torch.zeros(self.out_type.size, trivials)
p, c = 0, 0
for r in self.out_type:
pi = 0
for irr in r.irreps:
irr = self.out_type.fibergroup.irreps[irr]
if irr.is_trivial():
bias_expansion[p:p+r.size, c] = torch.tensor(r.change_of_basis[:, pi])
c += 1
pi += irr.size
p += r.size
self.register_buffer("bias_expansion", bias_expansion)
self.bias = Parameter(torch.zeros(trivials), requires_grad=True)
self.register_buffer("expanded_bias", torch.zeros(out_type.size))
else:
self.bias = None
self.expanded_bias = None
else:
self.bias = None
self.expanded_bias = None
grid, basis_filter, rings, sigma, maximum_frequency = compute_basis_params(kernel_size,
frequencies_cutoff,
rings,
sigma,
dilation,
basis_filter)
# BasisExpansion: submodule which takes care of building the filter
self._basisexpansion = None
# notice that `in_type` is used instead of `self.in_type` such that it works also when `groups > 1`
if basisexpansion == 'blocks':
self._basisexpansion = BlocksBasisExpansion(in_type, out_type,
basis_generator=self.space.build_kernel_basis,
points=grid,
sigma=sigma,
rings=rings,
maximum_offset=maximum_offset,
maximum_frequency=maximum_frequency,
basis_filter=basis_filter,
recompute=recompute)
else:
raise ValueError('Basis Expansion algorithm "%s" not recognized' % basisexpansion)
if self.basisexpansion.dimension() == 0:
raise ValueError('''
The basis for the steerable filter is empty!
Tune the `frequencies_cutoff`, `kernel_size`, `rings`, `sigma` or `basis_filter` parameters to allow
for a larger basis.
''')
self.weights = Parameter(torch.zeros(self.basisexpansion.dimension()), requires_grad=True)
self.register_buffer("filter", torch.zeros(out_type.size, in_type.size, kernel_size, kernel_size))
if initialize:
# by default, the weights are initialized with a generalized form of He's weight initialization
init.generalized_he_init(self.weights.data, self.basisexpansion)
@property
def basisexpansion(self) -> BasisExpansion:
r"""
Submodule which takes care of building the filter.
It uses the learnt ``weights`` to expand a basis and returns a filter in the usual form used by conventional
convolutional modules.
It uses the learned ``weights`` to expand the kernel in the G-steerable basis and returns it in the shape
:math:`(c_\text{out}, c_\text{in}, s^2)`, where :math:`s` is the ``kernel_size``.
"""
return self._basisexpansion
[docs] def expand_parameters(self) -> Tuple[torch.Tensor, torch.Tensor]:
r"""
Expand the filter in terms of the :attr:`e2cnn.nn.R2Conv.weights` and the
expanded bias in terms of :class:`e2cnn.nn.R2Conv.bias`.
Returns:
the expanded filter and bias
"""
_filter = self.basisexpansion(self.weights)
_filter = _filter.reshape(_filter.shape[0], _filter.shape[1], self.kernel_size, self.kernel_size)
if self.bias is None:
_bias = None
else:
_bias = self.bias_expansion @ self.bias
return _filter, _bias
[docs] def forward(self, input: GeometricTensor):
r"""
Convolve the input with the expanded filter and bias.
Args:
input (GeometricTensor): input feature field transforming according to ``in_type``
Returns:
output feature field transforming according to ``out_type``
"""
assert input.type == self.in_type
if not self.training:
_filter = self.filter
_bias = self.expanded_bias
else:
# retrieve the filter and the bias
_filter, _bias = self.expand_parameters()
# use it for convolution and return the result
if self.padding_mode == 'zeros':
output = conv2d(input.tensor, _filter,
stride=self.stride,
padding=self.padding,
dilation=self.dilation,
groups=self.groups,
bias=_bias)
else:
output = conv2d(pad(input.tensor, self._reversed_padding_repeated_twice, self.padding_mode),
_filter,
stride=self.stride,
dilation=self.dilation,
padding=(0,0),
groups=self.groups,
bias=_bias)
return GeometricTensor(output, self.out_type)
[docs] def train(self, mode=True):
r"""
If ``mode=True``, the method sets the module in training mode and discards the :attr:`~e2cnn.nn.R2Conv.filter`
and :attr:`~e2cnn.nn.R2Conv.expanded_bias` attributes.
If ``mode=False``, it sets the module in evaluation mode. Moreover, the method builds the filter and the bias using
the current values of the trainable parameters and store them in :attr:`~e2cnn.nn.R2Conv.filter` and
:attr:`~e2cnn.nn.R2Conv.expanded_bias` such that they are not recomputed at each forward pass.
.. warning ::
This behaviour can cause problems when storing the :meth:`~torch.nn.Module.state_dict` of a model while in
a mode and lately loading it in a model with a different mode, as the attributes of this class change.
To avoid this issue, we recommend converting the model to eval mode before storing or loading the state
dictionary.
Args:
mode (bool, optional): whether to set training mode (``True``) or evaluation mode (``False``).
Default: ``True``.
"""
if mode:
# TODO thoroughly check this is not causing problems
if hasattr(self, "filter"):
del self.filter
if hasattr(self, "expanded_bias"):
del self.expanded_bias
elif self.training:
# avoid re-computation of the filter and the bias on multiple consecutive calls of `.eval()`
_filter, _bias = self.expand_parameters()
self.register_buffer("filter", _filter)
if _bias is not None:
self.register_buffer("expanded_bias", _bias)
else:
self.expanded_bias = None
return super(R2Conv, self).train(mode)
def evaluate_output_shape(self, input_shape: Tuple[int, int, int, int]) -> Tuple[int, int, int, int]:
assert len(input_shape) == 4
assert input_shape[1] == self.in_type.size
b, c, hi, wi = input_shape
ho = math.floor((hi + 2 * self.padding - self.dilation * (self.kernel_size - 1) - 1) / self.stride + 1)
wo = math.floor((wi + 2 * self.padding - self.dilation * (self.kernel_size - 1) - 1) / self.stride + 1)
return b, self.out_type.size, ho, wo
def check_equivariance(self, atol: float = 0.1, rtol: float = 0.1, assertion: bool = True, verbose: bool = True):
# np.set_printoptions(precision=5, threshold=30 *self.in_type.size**2, suppress=False, linewidth=30 *self.in_type.size**2)
feature_map_size = 33
last_downsampling = 5
first_downsampling = 5
initial_size = (feature_map_size * last_downsampling - 1 + self.kernel_size) * first_downsampling
c = self.in_type.size
import matplotlib.image as mpimg
from skimage.measure import block_reduce
from skimage.transform import resize
x = mpimg.imread('../group/testimage.jpeg').transpose((2, 0, 1))[np.newaxis, 0:c, :, :]
x = resize(
x,
(x.shape[0], x.shape[1], initial_size, initial_size),
anti_aliasing=True
)
x = x / 255.0 - 0.5
if x.shape[1] < c:
to_stack = [x for i in range(c // x.shape[1])]
if c % x.shape[1] > 0:
to_stack += [x[:, :(c % x.shape[1]), ...]]
x = np.concatenate(to_stack, axis=1)
x = GeometricTensor(torch.FloatTensor(x), self.in_type)
def shrink(t: GeometricTensor, s) -> GeometricTensor:
return GeometricTensor(torch.FloatTensor(block_reduce(t.tensor.detach().numpy(), s, func=np.mean)), t.type)
errors = []
for el in self.space.testing_elements:
out1 = self(shrink(x, (1, 1, 5, 5))).transform(el).tensor.detach().numpy()
out2 = self(shrink(x.transform(el), (1, 1, 5, 5))).tensor.detach().numpy()
out1 = block_reduce(out1, (1, 1, 5, 5), func=np.mean)
out2 = block_reduce(out2, (1, 1, 5, 5), func=np.mean)
b, c, h, w = out2.shape
center_mask = np.zeros((2, h, w))
center_mask[1, :, :] = np.arange(0, w) - w / 2
center_mask[0, :, :] = np.arange(0, h) - h / 2
center_mask[0, :, :] = center_mask[0, :, :].T
center_mask = center_mask[0, :, :] ** 2 + center_mask[1, :, :] ** 2 < (h / 4) ** 2
out1 = out1[..., center_mask]
out2 = out2[..., center_mask]
out1 = out1.reshape(-1)
out2 = out2.reshape(-1)
errs = np.abs(out1 - out2)
esum = np.maximum(np.abs(out1), np.abs(out2))
esum[esum == 0.0] = 1
relerr = errs / esum
if verbose:
print(el, relerr.max(), relerr.mean(), relerr.var(), errs.max(), errs.mean(), errs.var())
tol = rtol * esum + atol
if np.any(errs > tol) and verbose:
print(out1[errs > tol])
print(out2[errs > tol])
print(tol[errs > tol])
if assertion:
assert np.all(errs < tol), 'The error found during equivariance check with element "{}" is too high: max = {}, mean = {} var ={}'.format(el, errs.max(), errs.mean(), errs.var())
errors.append((el, errs.mean()))
return errors
# init.deltaorthonormal_init(self.weights.data, self.basisexpansion)
# filter = self.basisexpansion()
# center = self.s // 2
# filter = filter[..., center, center]
# assert torch.allclose(torch.eye(filter.shape[1]), filter.t() @ filter, atol=3e-7)
[docs] def export(self):
r"""
Export this module to a normal PyTorch :class:`torch.nn.Conv2d` module and set to "eval" mode.
"""
# set to eval mode so the filter and the bias are updated with the current
# values of the weights
self.eval()
_filter = self.filter
_bias = self.expanded_bias
if self.padding_mode not in ['zeros']:
x, y = torch.__version__.split('.')[:2]
if int(x) < 1 or int(y) < 5:
if self.padding_mode == 'circular':
raise ImportError(
"'{}' padding mode had some issues in old `torch` versions. Therefore, we only support conversion from version 1.5 but only version {} is installed.".format(
self.padding_mode, torch.__version__
)
)
else:
raise ImportError(
"`torch` supports '{}' padding mode only from version 1.5 but only version {} is installed.".format(
self.padding_mode, torch.__version__
)
)
# build the PyTorch Conv2d module
has_bias = self.bias is not None
conv = torch.nn.Conv2d(self.in_type.size,
self.out_type.size,
self.kernel_size,
padding=self.padding,
padding_mode=self.padding_mode,
stride=self.stride,
dilation=self.dilation,
groups=self.groups,
bias=has_bias)
# set the filter and the bias
conv.weight.data = _filter.data
if has_bias:
conv.bias.data = _bias.data
return conv
def __repr__(self):
extra_lines = []
extra_repr = self.extra_repr()
if extra_repr:
extra_lines = extra_repr.split('\n')
main_str = self._get_name() + '('
if len(extra_lines) == 1:
main_str += extra_lines[0]
else:
main_str += '\n ' + '\n '.join(extra_lines) + '\n'
main_str += ')'
return main_str
def extra_repr(self):
s = ('{in_type}, {out_type}, kernel_size={kernel_size}, stride={stride}')
if self.padding != 0 and self.padding != (0, 0):
s += ', padding={padding}'
if self.dilation != 1 and self.dilation != (1, 1):
s += ', dilation={dilation}'
if self.groups != 1:
s += ', groups={groups}'
if self.bias is None:
s += ', bias=False'
return s.format(**self.__dict__)
def bandlimiting_filter(frequency_cutoff: Union[float, Callable[[float], float]]) -> Callable[[dict], bool]:
r"""
Returns a method which takes as input the attributes (as a dictionary) of a basis element and returns a boolean
value: whether to preserve that element (True) or not (False)
If the parameter ``frequency_cutoff`` is a scalar value, the maximum frequency allowed at a certain radius is
proportional to the radius itself. In thi case, the parameter ``frequency_cutoff`` is the factor controlling this
proportionality relation.
If the parameter ``frequency_cutoff`` is a callable, it needs to take as input a radius (a scalar value) and return
the maximum frequency which can be sampled at that radius.
Args:
frequency_cutoff (float): factor controlling the bandlimiting
Returns:
a function which checks the attributes of individual basis elements and chooses whether to discard them or not
"""
if isinstance(frequency_cutoff, float):
frequency_cutoff = lambda r, fco=frequency_cutoff: r * frequency_cutoff
def bl_filter(attributes: dict) -> bool:
return math.fabs(attributes["frequency"]) <= frequency_cutoff(attributes["radius"])
return bl_filter
def get_grid_coords(kernel_size: int, dilation: int = 1):
actual_size = dilation * (kernel_size -1) + 1
origin = actual_size / 2 - 0.5
points = []
for y in range(kernel_size):
y *= dilation
for x in range(kernel_size):
x *= dilation
p = (x - origin, -y + origin)
points.append(p)
points = np.array(points)
assert points.shape == (kernel_size ** 2, 2), points.shape
return points.T
def compute_basis_params(kernel_size: int,
frequencies_cutoff: Union[float, Callable[[float], float]] = None,
rings: List[float] = None,
sigma: List[float] = None,
dilation: int = 1,
custom_basis_filter: Callable[[dict], bool] = None,
):
# compute the coordinates of the centers of the cells in the grid where the filter is sampled
grid = get_grid_coords(kernel_size, dilation)
max_radius = np.sqrt((grid **2).sum(0)).max()
# max_radius = kernel_size // 2
# by default, the number of rings equals half of the filter size
if rings is None:
n_rings = math.ceil(kernel_size / 2)
# if self.group.order() > 0:
# # compute the number of edges of the polygon inscribed in the filter (which is a square)
# # whose points stay inside the filter under the action of the group
# # the number of edges is lcm(group's order, 4)
# n_edges = self.group.order()
# while n_edges % 4 > 0:
# n_edges *= 2
# # the largest ring we can sample has radius equal to the circumradius of the polygon described above
# n_rings /= math.cos(math.pi/n_edges)
# n_rings = s // 2 + 1
# rings = torch.linspace(1 - s % 2, s // 2, n_rings)
rings = torch.linspace(0, (kernel_size - 1) // 2, n_rings) * dilation
rings = rings.tolist()
assert all([max_radius >= r >= 0 for r in rings])
if sigma is None:
sigma = [0.6] * (len(rings) - 1) + [0.4]
for i, r in enumerate(rings):
if r == 0.:
sigma[i] = 0.005
elif isinstance(sigma, float):
sigma = [sigma] * len(rings)
# TODO - use a string name for this setting
if frequencies_cutoff is None:
frequencies_cutoff = -1.
if isinstance(frequencies_cutoff, float):
if frequencies_cutoff == -3:
frequencies_cutoff = _manual_fco3(kernel_size // 2)
elif frequencies_cutoff == -2:
frequencies_cutoff = _manual_fco2(kernel_size // 2)
elif frequencies_cutoff == -1:
frequencies_cutoff = _manual_fco1(kernel_size // 2)
else:
frequencies_cutoff = lambda r, fco=frequencies_cutoff: fco * r
# check if the object is a callable function
assert callable(frequencies_cutoff)
maximum_frequency = int(max(frequencies_cutoff(r) for r in rings))
fco_filter = bandlimiting_filter(frequencies_cutoff)
if custom_basis_filter is not None:
basis_filter = lambda d, custom_basis_filter=custom_basis_filter, fco_filter=fco_filter: (custom_basis_filter(d) and fco_filter(d))
else:
basis_filter = fco_filter
return grid, basis_filter, rings, sigma, maximum_frequency
def _manual_fco3(max_radius: float) -> Callable[[float], float]:
r"""
Returns a method which takes as input the radius of a ring and returns the maximum frequency which can be sampled
on that ring.
Args:
max_radius (float): radius of the last ring touching the border of the grid
Returns:
a function which checks the attributes of individual basis elements and chooses whether to discard them or not
"""
def bl_filter(r: float) -> float:
max_freq = 0 if r == 0. else 1 if r == max_radius else 2
return max_freq
return bl_filter
def _manual_fco2(max_radius: float) -> Callable[[float], float]:
r"""
Returns a method which takes as input the radius of a ring and returns the maximum frequency which can be sampled
on that ring.
Args:
max_radius (float): radius of the last ring touching the border of the grid
Returns:
a function which checks the attributes of individual basis elements and chooses whether to discard them or not
"""
def bl_filter(r: float) -> float:
max_freq = 0 if r == 0. else min(2 * r, 1 if r == max_radius else 2 * r - (r + 1) % 2)
return max_freq
return bl_filter
def _manual_fco1(max_radius: float) -> Callable[[float], float]:
r"""
Returns a method which takes as input the radius of a ring and returns the maximum frequency which can be sampled
on that ring.
Args:
max_radius (float): radius of the last ring touching the border of the grid
Returns:
a function which checks the attributes of individual basis elements and chooses whether to discard them or not
"""
def bl_filter(r: float) -> float:
max_freq = 0 if r == 0. else min(2 * r, 2 if r == max_radius else 2 * r - (r + 1) % 2)
return max_freq
return bl_filter