venv/lib/python3.11/site-packages/torchmetrics/functional/segmentation/utils.py

# Copyright The Lightning team.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
#     http://www.apache.org/licenses/LICENSE-2.0
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.
import functools
import math
from typing import Optional, Union

import torch
from torch import Tensor
from torch.nn.functional import conv2d, conv3d, pad, unfold
from typing_extensions import Literal

from torchmetrics.utilities.checks import _check_same_shape
from torchmetrics.utilities.imports import _SCIPY_AVAILABLE


def _ignore_background(preds: Tensor, target: Tensor) -> tuple[Tensor, Tensor]:
    """Ignore the background class in the computation assuming it is the first, index 0."""
    preds = preds[:, 1:] if preds.shape[1] > 1 else preds
    target = target[:, 1:] if target.shape[1] > 1 else target
    return preds, target


def check_if_binarized(x: Tensor) -> None:
    """Check if tensor is binarized.

    Example:
        >>> from torchmetrics.functional.segmentation.utils import check_if_binarized
        >>> import torch
        >>> check_if_binarized(torch.tensor([0, 1, 1, 0]))

    """
    if not torch.all(x.bool() == x):
        raise ValueError("Input x should be binarized")


def _unfold(x: Tensor, kernel_size: tuple[int, ...]) -> Tensor:
    """Unfold the input tensor to a matrix. Function supports 3d images e.g. (B, C, D, H, W).

    Inspired by:
    https://github.com/f-dangel/unfoldNd/blob/main/unfoldNd/unfold.py

    Args:
        x: Input tensor to be unfolded.
        kernel_size: The size of the sliding blocks in each dimension.

    """
    batch_size, channels = x.shape[:2]
    n = x.ndim - 2
    if n == 2:
        return unfold(x, kernel_size)

    kernel_size_numel = kernel_size[0] * kernel_size[1] * kernel_size[2]
    repeat = [channels, 1] + [1 for _ in kernel_size]
    weight = torch.eye(kernel_size_numel, device=x.device, dtype=x.dtype)
    weight = weight.reshape(kernel_size_numel, 1, *kernel_size).repeat(*repeat)
    unfold_x = conv3d(x, weight=weight, bias=None)
    return unfold_x.reshape(batch_size, channels * kernel_size_numel, -1)


def generate_binary_structure(rank: int, connectivity: int) -> Tensor:
    """Translated version of the function from scipy.ndimage.morphology.

    Args:
        rank: The rank of the structuring element.
        connectivity: The number of neighbors connected to a given pixel.

    Returns:
        The structuring element.

    Examples::
        >>> from torchmetrics.functional.segmentation.utils import generate_binary_structure
        >>> import torch
        >>> generate_binary_structure(2, 1)
        tensor([[False,  True, False],
                [ True,  True,  True],
                [False,  True, False]])
        >>> generate_binary_structure(2, 2)
        tensor([[True,  True,  True],
                [True,  True,  True],
                [True,  True,  True]])
        >>> generate_binary_structure(3, 2)  # doctest: +NORMALIZE_WHITESPACE
        tensor([[[False,  True, False],
                 [ True,  True,  True],
                 [False,  True, False]],
                [[ True,  True,  True],
                 [ True,  True,  True],
                 [ True,  True,  True]],
                [[False,  True, False],
                 [ True,  True,  True],
                 [False,  True, False]]])

    """
    if connectivity < 1:
        connectivity = 1
    if rank < 1:
        return torch.tensor([1], dtype=torch.uint8)
    grids = torch.meshgrid([torch.arange(3) for _ in range(rank)], indexing="ij")
    output = torch.abs(torch.stack(grids, dim=0) - 1)
    output = torch.sum(output, dim=0)
    return output <= connectivity


def binary_erosion(
    image: Tensor, structure: Optional[Tensor] = None, origin: Optional[tuple[int, ...]] = None, border_value: int = 0
) -> Tensor:
    """Binary erosion of a tensor image.

    Implementation inspired by answer to this question: https://stackoverflow.com/questions/56235733/

    Args:
        image: The image to be eroded, must be a binary tensor with shape ``(batch_size, channels, height, width)``.
        structure: The structuring element used for the erosion. If no structuring element is provided, an element
            is generated with a square connectivity equal to one.
        origin: The origin of the structuring element.
        border_value: The value to be used for the border.

    Examples::
        >>> from torchmetrics.functional.segmentation.utils import binary_erosion
        >>> import torch
        >>> image = torch.tensor([[[[0, 0, 0, 0, 0],
        ...                         [0, 1, 1, 1, 0],
        ...                         [0, 1, 1, 1, 0],
        ...                         [0, 1, 1, 1, 0],
        ...                         [0, 0, 0, 0, 0]]]])
        >>> binary_erosion(image)
        tensor([[[[0, 0, 0, 0, 0],
                  [0, 0, 0, 0, 0],
                  [0, 0, 1, 0, 0],
                  [0, 0, 0, 0, 0],
                  [0, 0, 0, 0, 0]]]], dtype=torch.uint8)
        >>> binary_erosion(image, structure=torch.ones(4, 4))
        tensor([[[[0, 0, 0, 0, 0],
                  [0, 0, 0, 0, 0],
                  [0, 0, 0, 0, 0],
                  [0, 0, 0, 0, 0],
                  [0, 0, 0, 0, 0]]]], dtype=torch.uint8)

    """
    if not isinstance(image, Tensor):
        raise TypeError(f"Expected argument `image` to be of type Tensor but found {type(image)}")
    if image.ndim not in [4, 5]:
        raise ValueError(f"Expected argument `image` to be of rank 4 or 5 but found rank {image.ndim}")
    check_if_binarized(image)

    # construct the structuring element if not provided
    if structure is None:
        structure = generate_binary_structure(image.ndim - 2, 1).int().to(image.device)
    check_if_binarized(structure)

    if origin is None:
        origin = structure.ndim * (1,)

    # first pad the image to have correct unfolding; here is where the origins is used
    image_pad = pad(
        image,
        [x for i in range(len(origin)) for x in [origin[i], structure.shape[i] - origin[i] - 1]],
        mode="constant",
        value=border_value,
    )
    # Unfold the image to be able to perform operation on neighborhoods
    image_unfold = _unfold(image_pad.float(), kernel_size=structure.shape)

    strel_flatten = torch.flatten(structure).unsqueeze(0).unsqueeze(-1)
    sums = image_unfold - strel_flatten.int()

    # Take minimum over the neighborhood
    result, _ = sums.min(dim=1)

    # Reshape the image to recover initial shape
    return (torch.reshape(result, image.shape) + 1).byte()


def distance_transform(
    x: Tensor,
    sampling: Optional[Union[Tensor, list[float]]] = None,
    metric: Literal["euclidean", "chessboard", "taxicab"] = "euclidean",
    engine: Literal["pytorch", "scipy"] = "pytorch",
) -> Tensor:
    """Calculate distance transform of a binary tensor.

    This function calculates the distance transform of a binary tensor, replacing each foreground pixel with the
    distance to the closest background pixel. The distance is calculated using the euclidean, chessboard or taxicab
    distance.

    The memory consumption of this function is in the worst cast N/2**2 where N is the number of pixel. Since we need
    to compare all foreground pixels to all background pixels, the memory consumption is quadratic in the number of
    pixels. The memory consumption can be reduced by using the ``scipy`` engine, which is more memory efficient but
    should also be slower for larger images.

    Args:
        x: The binary tensor to calculate the distance transform of.
        sampling: The sampling refers to the pixel spacing in the image, i.e. the distance between two adjacent pixels.
            If not provided, the pixel spacing is assumed to be 1.
        metric: The distance to use for the distance transform. Can be one of ``"euclidean"``, ``"chessboard"``
            or ``"taxicab"``.
        engine: The engine to use for the distance transform. Can be one of ``["pytorch", "scipy"]``. In general,
            the ``pytorch`` engine is faster, but the ``scipy`` engine is more memory efficient.

    Returns:
        The distance transform of the input tensor.

    Examples::
        >>> from torchmetrics.functional.segmentation.utils import distance_transform
        >>> import torch
        >>> x = torch.tensor([[0, 0, 0, 0, 0],
        ...                   [0, 1, 1, 1, 0],
        ...                   [0, 1, 1, 1, 0],
        ...                   [0, 1, 1, 1, 0],
        ...                   [0, 0, 0, 0, 0]])
        >>> distance_transform(x)
        tensor([[0., 0., 0., 0., 0.],
                [0., 1., 1., 1., 0.],
                [0., 1., 2., 1., 0.],
                [0., 1., 1., 1., 0.],
                [0., 0., 0., 0., 0.]])

    """
    if not isinstance(x, Tensor):
        raise ValueError(f"Expected argument `x` to be of type `torch.Tensor` but got `{type(x)}`.")
    if x.ndim != 2:
        raise ValueError(f"Expected argument `x` to be of rank 2 but got rank `{x.ndim}`.")
    if sampling is not None and not isinstance(sampling, list):
        raise ValueError(
            f"Expected argument `sampling` to either be `None` or of type `list` but got `{type(sampling)}`."
        )
    if metric not in ["euclidean", "chessboard", "taxicab"]:
        raise ValueError(
            f"Expected argument `metric` to be one of `['euclidean', 'chessboard', 'taxicab']` but got `{metric}`."
        )
    if engine not in ["pytorch", "scipy"]:
        raise ValueError(f"Expected argument `engine` to be one of `['pytorch', 'scipy']` but got `{engine}`.")

    if sampling is None:
        sampling = [1, 1]
    else:
        if len(sampling) != 2:
            raise ValueError(f"Expected argument `sampling` to have length 2 but got length `{len(sampling)}`.")

    if engine == "pytorch":
        x = x.float()
        # calculate distance from every foreground pixel to every background pixel
        i0, j0 = torch.where(x == 0)
        i1, j1 = torch.where(x == 1)
        dis_row = (i1.view(-1, 1) - i0.view(1, -1)).abs()
        dis_col = (j1.view(-1, 1) - j0.view(1, -1)).abs()

        # # calculate distance
        h, _ = x.shape
        if metric == "euclidean":
            dis = ((sampling[0] * dis_row) ** 2 + (sampling[1] * dis_col) ** 2).sqrt()
        if metric == "chessboard":
            dis = torch.max(sampling[0] * dis_row, sampling[1] * dis_col).float()
        if metric == "taxicab":
            dis = (sampling[0] * dis_row + sampling[1] * dis_col).float()

        # select only the closest distance
        mindis, _ = torch.min(dis, dim=1)
        z = torch.zeros_like(x).view(-1)
        z[i1 * h + j1] = mindis
        return z.view(x.shape)

    if not _SCIPY_AVAILABLE:
        raise ValueError(
            "The `scipy` engine requires `scipy` to be installed. Either install `scipy` or use the `pytorch` engine."
        )
    from scipy import ndimage

    if metric == "euclidean":
        return ndimage.distance_transform_edt(x.cpu().numpy(), sampling)
    return ndimage.distance_transform_cdt(x.cpu().numpy(), sampling, metric=metric)


def mask_edges(
    preds: Tensor,
    target: Tensor,
    crop: bool = True,
    spacing: Optional[Union[tuple[int, int], tuple[int, int, int]]] = None,
) -> Union[tuple[Tensor, Tensor], tuple[Tensor, Tensor, Tensor, Tensor]]:
    """Get the edges of binary segmentation masks.

    Args:
        preds: The predicted binary segmentation mask
        target: The ground truth binary segmentation mask
        crop: Whether to crop the edges to the region of interest. If ``True``, the edges are cropped to the bounding
        spacing: The pixel spacing of the input images. If provided, the edges are calculated using the euclidean

    Returns:
        If spacing is not provided, a 2-tuple containing the edges of the predicted and target mask respectively is
        returned. If spacing is provided, a 4-tuple containing the edges and areas of the predicted and target mask
        respectively is returned.

    """
    _check_same_shape(preds, target)
    if preds.ndim not in [2, 3]:
        raise ValueError(f"Expected argument `preds` to be of rank 2 or 3 but got rank `{preds.ndim}`.")
    check_if_binarized(preds)
    check_if_binarized(target)

    if crop:
        or_val = preds | target
        if not or_val.any():
            p, t = torch.zeros_like(preds), torch.zeros_like(target)
            return p, t, p, t
        # this seems to be working but does not seem to be right
        preds, target = pad(preds, preds.ndim * [1, 1]), pad(target, target.ndim * [1, 1])

    if spacing is None:
        # no spacing, use binary erosion
        be_pred = binary_erosion(preds.unsqueeze(0).unsqueeze(0)).squeeze() ^ preds
        be_target = binary_erosion(target.unsqueeze(0).unsqueeze(0)).squeeze() ^ target
        return be_pred, be_target

    # use neighborhood to get edges
    table, kernel = get_neighbour_tables(spacing, device=preds.device)
    spatial_dims = len(spacing)
    conv_operator = conv2d if spatial_dims == 2 else conv3d
    volume = torch.stack([preds.unsqueeze(0), target.unsqueeze(0)], dim=0).float()
    code_preds, code_target = conv_operator(volume, kernel.to(volume))

    # edges
    all_ones = len(table) - 1
    edges_preds = (code_preds != 0) & (code_preds != all_ones)
    edges_target = (code_target != 0) & (code_target != all_ones)

    # # areas of edges
    areas_preds = torch.index_select(table, 0, code_preds.view(-1).int()).view_as(code_preds)
    areas_target = torch.index_select(table, 0, code_target.view(-1).int()).view_as(code_target)
    return edges_preds[0], edges_target[0], areas_preds[0], areas_target[0]


def surface_distance(
    preds: Tensor,
    target: Tensor,
    distance_metric: Literal["euclidean", "chessboard", "taxicab"] = "euclidean",
    spacing: Optional[Union[Tensor, list[float]]] = None,
) -> Tensor:
    """Calculate the surface distance between two binary edge masks.

    May return infinity if the predicted mask is empty and the target mask is not, or vice versa.

    Args:
        preds: The predicted binary edge mask.
        target: The target binary edge mask.
        distance_metric: The distance metric to use. One of `["euclidean", "chessboard", "taxicab"]`.
        spacing: The spacing between pixels along each spatial dimension.

    Returns:
        A tensor with length equal to the number of edges in predictions e.g. `preds.sum()`. Each element is the
        distance from the corresponding edge in `preds` to the closest edge in `target`.

    Example::
        >>> import torch
        >>> from torchmetrics.functional.segmentation.utils import surface_distance
        >>> preds = torch.tensor([[1, 1, 1, 1, 1],
        ...                       [1, 0, 0, 0, 1],
        ...                       [1, 0, 0, 0, 1],
        ...                       [1, 0, 0, 0, 1],
        ...                       [1, 1, 1, 1, 1]], dtype=torch.bool)
        >>> target = torch.tensor([[1, 1, 1, 1, 0],
        ...                        [1, 0, 0, 1, 0],
        ...                        [1, 0, 0, 1, 0],
        ...                        [1, 0, 0, 1, 0],
        ...                        [1, 1, 1, 1, 0]], dtype=torch.bool)
        >>> surface_distance(preds, target, distance_metric="euclidean", spacing=[1, 1])
        tensor([0., 0., 0., 0., 1., 0., 1., 0., 1., 0., 1., 0., 0., 0., 0., 1.])

    """
    if not (preds.dtype == torch.bool and target.dtype == torch.bool):
        raise ValueError(f"Expected both inputs to be of type `torch.bool`, but got {preds.dtype} and {target.dtype}.")

    if not torch.any(target):
        dis = torch.inf * torch.ones_like(target)
    else:
        if not torch.any(preds):
            dis = torch.inf * torch.ones_like(preds)
            return dis[target]
        dis = distance_transform(~target, sampling=spacing, metric=distance_metric)
    return dis[preds]


def edge_surface_distance(
    preds: Tensor,
    target: Tensor,
    distance_metric: Literal["euclidean", "chessboard", "taxicab"] = "euclidean",
    spacing: Optional[Union[Tensor, list[float]]] = None,
    symmetric: bool = False,
) -> Union[Tensor, tuple[Tensor, Tensor]]:
    """Extracts the edges from the input masks and calculates the surface distance between them.

    Args:
        preds: The predicted binary edge mask.
        target: The target binary edge mask.
        distance_metric: The distance metric to use. One of `["euclidean", "chessboard", "taxicab"]`.
        spacing: The spacing between pixels along each spatial dimension.
        symmetric: Whether to calculate the symmetric distance between the edges.

    Returns:
        A tensor with length equal to the number of edges in predictions e.g. `preds.sum()`. Each element is the
        distance from the corresponding edge in `preds` to the closest edge in `target`. If `symmetric` is `True`, the
        function returns a tuple containing the distances from the predicted edges to the target edges and vice versa.

    """
    output = mask_edges(preds, target)
    edges_preds, edges_target = output[0].bool(), output[1].bool()
    if symmetric:
        return (
            surface_distance(edges_preds, edges_target, distance_metric=distance_metric, spacing=spacing),
            surface_distance(edges_target, edges_preds, distance_metric=distance_metric, spacing=spacing),
        )
    return surface_distance(edges_preds, edges_target, distance_metric=distance_metric, spacing=spacing)


@functools.lru_cache
def get_neighbour_tables(
    spacing: Union[tuple[int, int], tuple[int, int, int]], device: Optional[torch.device] = None
) -> tuple[Tensor, Tensor]:
    """Create a table that maps neighbour codes to the contour length or surface area of the corresponding contour.

    Args:
        spacing: The spacing between pixels along each spatial dimension.
        device: The device on which the table should be created.

    Returns:
        A tuple containing as its first element the table that maps neighbour codes to the contour length or surface
        area of the corresponding contour and as its second element the kernel used to compute the neighbour codes.

    """
    if isinstance(spacing, tuple) and len(spacing) == 2:
        return table_contour_length(spacing, device)
    if isinstance(spacing, tuple) and len(spacing) == 3:
        return table_surface_area(spacing, device)
    raise ValueError("The spacing must be a tuple of length 2 or 3.")


def table_contour_length(spacing: tuple[int, int], device: Optional[torch.device] = None) -> tuple[Tensor, Tensor]:
    """Create a table that maps neighbour codes to the contour length of the corresponding contour.

    Adopted from:
    https://github.com/deepmind/surface-distance/blob/master/surface_distance/lookup_tables.py

    Args:
        spacing: The spacing between pixels along each spatial dimension. Should be a tuple of length 2.
        device: The device on which the table should be created.

    Returns:
        A tuple containing as its first element the table that maps neighbour codes to the contour length of the
        corresponding contour and as its second element the kernel used to compute the neighbour codes.

    Example::
        >>> from torchmetrics.functional.segmentation.utils import table_contour_length
        >>> table, kernel = table_contour_length((2,2))
        >>> table
        tensor([0.0000, 1.4142, 1.4142, 2.0000, 1.4142, 2.0000, 2.8284, 1.4142, 1.4142,
                2.8284, 2.0000, 1.4142, 2.0000, 1.4142, 1.4142, 0.0000])
        >>> kernel
        tensor([[[[8, 4],
                  [2, 1]]]])

    """
    if not isinstance(spacing, tuple) and len(spacing) != 2:
        raise ValueError("The spacing must be a tuple of length 2.")

    first, second = spacing  # spacing along the first and second spatial dimension respectively
    diag = 0.5 * math.sqrt(first**2 + second**2)
    table = torch.zeros(16, dtype=torch.float32, device=device)
    for i in [1, 2, 4, 7, 8, 11, 13, 14]:
        table[i] = diag
    for i in [3, 12]:
        table[i] = second
    for i in [5, 10]:
        table[i] = first
    for i in [6, 9]:
        table[i] = 2 * diag
    kernel = torch.as_tensor([[[[8, 4], [2, 1]]]], device=device)
    return table, kernel


@functools.lru_cache
def table_surface_area(spacing: tuple[int, int, int], device: Optional[torch.device] = None) -> tuple[Tensor, Tensor]:
    """Create a table that maps neighbour codes to the surface area of the corresponding surface.

    Adopted from:
    https://github.com/deepmind/surface-distance/blob/master/surface_distance/lookup_tables.py

    Args:
        spacing: The spacing between pixels along each spatial dimension. Should be a tuple of length 3.
        device: The device on which the table should be created.

    Returns:
        A tuple containing as its first element the table that maps neighbour codes to the surface area of the
        corresponding surface and as its second element the kernel used to compute the neighbour codes.

    Example::
        >>> from torchmetrics.functional.segmentation.utils import table_surface_area
        >>> table, kernel = table_surface_area((2,2,2))
        >>> table
        tensor([0.0000, 0.8660, 0.8660, 2.8284, 0.8660, 2.8284, 1.7321, 4.5981, 0.8660,
                1.7321, 2.8284, 4.5981, 2.8284, 4.5981, 4.5981, 4.0000, 0.8660, 2.8284,
                1.7321, 4.5981, 1.7321, 4.5981, 2.5981, 5.1962, 1.7321, 3.6945, 3.6945,
                6.2925, 3.6945, 6.2925, 5.4641, 4.5981, 0.8660, 1.7321, 2.8284, 4.5981,
                1.7321, 3.6945, 3.6945, 6.2925, 1.7321, 2.5981, 4.5981, 5.1962, 3.6945,
                5.4641, 6.2925, 4.5981, 2.8284, 4.5981, 4.5981, 4.0000, 3.6945, 6.2925,
                5.4641, 4.5981, 3.6945, 5.4641, 6.2925, 4.5981, 5.6569, 3.6945, 3.6945,
                2.8284, 0.8660, 1.7321, 1.7321, 3.6945, 2.8284, 4.5981, 3.6945, 6.2925,
                1.7321, 2.5981, 3.6945, 5.4641, 4.5981, 5.1962, 6.2925, 4.5981, 2.8284,
                4.5981, 3.6945, 6.2925, 4.5981, 4.0000, 5.4641, 4.5981, 3.6945, 5.4641,
                5.6569, 3.6945, 6.2925, 4.5981, 3.6945, 2.8284, 1.7321, 2.5981, 3.6945,
                5.4641, 3.6945, 5.4641, 5.6569, 3.6945, 2.5981, 3.4641, 5.4641, 2.5981,
                5.4641, 2.5981, 3.6945, 1.7321, 4.5981, 5.1962, 6.2925, 4.5981, 6.2925,
                4.5981, 3.6945, 2.8284, 5.4641, 2.5981, 3.6945, 1.7321, 3.6945, 1.7321,
                1.7321, 0.8660, 0.8660, 1.7321, 1.7321, 3.6945, 1.7321, 3.6945, 2.5981,
                5.4641, 2.8284, 3.6945, 4.5981, 6.2925, 4.5981, 6.2925, 5.1962, 4.5981,
                1.7321, 3.6945, 2.5981, 5.4641, 2.5981, 5.4641, 3.4641, 2.5981, 3.6945,
                5.6569, 5.4641, 3.6945, 5.4641, 3.6945, 2.5981, 1.7321, 2.8284, 3.6945,
                4.5981, 6.2925, 3.6945, 5.6569, 5.4641, 3.6945, 4.5981, 5.4641, 4.0000,
                4.5981, 6.2925, 3.6945, 4.5981, 2.8284, 4.5981, 6.2925, 5.1962, 4.5981,
                5.4641, 3.6945, 2.5981, 1.7321, 6.2925, 3.6945, 4.5981, 2.8284, 3.6945,
                1.7321, 1.7321, 0.8660, 2.8284, 3.6945, 3.6945, 5.6569, 4.5981, 6.2925,
                5.4641, 3.6945, 4.5981, 5.4641, 6.2925, 3.6945, 4.0000, 4.5981, 4.5981,
                2.8284, 4.5981, 6.2925, 5.4641, 3.6945, 5.1962, 4.5981, 2.5981, 1.7321,
                6.2925, 3.6945, 3.6945, 1.7321, 4.5981, 2.8284, 1.7321, 0.8660, 4.5981,
                5.4641, 6.2925, 3.6945, 6.2925, 3.6945, 3.6945, 1.7321, 5.1962, 2.5981,
                4.5981, 1.7321, 4.5981, 1.7321, 2.8284, 0.8660, 4.0000, 4.5981, 4.5981,
                2.8284, 4.5981, 2.8284, 1.7321, 0.8660, 4.5981, 1.7321, 2.8284, 0.8660,
                2.8284, 0.8660, 0.8660, 0.0000])
        >>> kernel
        tensor([[[[[128,  64],
                   [ 32,  16]],
                  [[  8,   4],
                   [  2,   1]]]]])

    """
    if not isinstance(spacing, tuple) and len(spacing) != 3:
        raise ValueError("The spacing must be a tuple of length 3.")

    zeros = [0.0, 0.0, 0.0]
    table = torch.tensor(
        [
            [zeros, zeros, zeros, zeros],
            [[0.125, 0.125, 0.125], zeros, zeros, zeros],
            [[-0.125, -0.125, 0.125], zeros, zeros, zeros],
            [[-0.25, -0.25, 0.0], [0.25, 0.25, -0.0], zeros, zeros],
            [[0.125, -0.125, 0.125], zeros, zeros, zeros],
            [[-0.25, -0.0, -0.25], [0.25, 0.0, 0.25], zeros, zeros],
            [[0.125, -0.125, 0.125], [-0.125, -0.125, 0.125], zeros, zeros],
            [[0.5, 0.0, -0.0], [0.25, 0.25, 0.25], [0.125, 0.125, 0.125], zeros],
            [[-0.125, 0.125, 0.125], zeros, zeros, zeros],
            [[0.125, 0.125, 0.125], [-0.125, 0.125, 0.125], zeros, zeros],
            [[-0.25, 0.0, 0.25], [-0.25, 0.0, 0.25], zeros, zeros],
            [[0.5, 0.0, 0.0], [-0.25, -0.25, 0.25], [-0.125, -0.125, 0.125], zeros],
            [[0.25, -0.25, 0.0], [0.25, -0.25, 0.0], zeros, zeros],
            [[0.5, 0.0, 0.0], [0.25, -0.25, 0.25], [-0.125, 0.125, -0.125], zeros],
            [[-0.5, 0.0, 0.0], [-0.25, 0.25, 0.25], [-0.125, 0.125, 0.125], zeros],
            [[0.5, 0.0, 0.0], [0.5, 0.0, 0.0], zeros, zeros],
            [[0.125, -0.125, -0.125], zeros, zeros, zeros],
            [[0.0, -0.25, -0.25], [0.0, 0.25, 0.25], zeros, zeros],
            [[-0.125, -0.125, 0.125], [0.125, -0.125, -0.125], zeros, zeros],
            [[0.0, -0.5, 0.0], [0.25, 0.25, 0.25], [0.125, 0.125, 0.125], zeros],
            [[0.125, -0.125, 0.125], [0.125, -0.125, -0.125], zeros, zeros],
            [[0.0, 0.0, -0.5], [0.25, 0.25, 0.25], [-0.125, -0.125, -0.125], zeros],
            [[-0.125, -0.125, 0.125], [0.125, -0.125, 0.125], [0.125, -0.125, -0.125], zeros],
            [[-0.125, -0.125, -0.125], [-0.25, -0.25, -0.25], [0.25, 0.25, 0.25], [0.125, 0.125, 0.125]],
            [[-0.125, 0.125, 0.125], [0.125, -0.125, -0.125], zeros, zeros],
            [[0.0, -0.25, -0.25], [0.0, 0.25, 0.25], [-0.125, 0.125, 0.125], zeros],
            [[-0.25, 0.0, 0.25], [-0.25, 0.0, 0.25], [0.125, -0.125, -0.125], zeros],
            [[0.125, 0.125, 0.125], [0.375, 0.375, 0.375], [0.0, -0.25, 0.25], [-0.25, 0.0, 0.25]],
            [[0.125, -0.125, -0.125], [0.25, -0.25, 0.0], [0.25, -0.25, 0.0], zeros],
            [[0.375, 0.375, 0.375], [0.0, 0.25, -0.25], [-0.125, -0.125, -0.125], [-0.25, 0.25, 0.0]],
            [[-0.5, 0.0, 0.0], [-0.125, -0.125, -0.125], [-0.25, -0.25, -0.25], [0.125, 0.125, 0.125]],
            [[-0.5, 0.0, 0.0], [-0.125, -0.125, -0.125], [-0.25, -0.25, -0.25], zeros],
            [[0.125, -0.125, 0.125], zeros, zeros, zeros],
            [[0.125, 0.125, 0.125], [0.125, -0.125, 0.125], zeros, zeros],
            [[0.0, -0.25, 0.25], [0.0, 0.25, -0.25], zeros, zeros],
            [[0.0, -0.5, 0.0], [0.125, 0.125, -0.125], [0.25, 0.25, -0.25], zeros],
            [[0.125, -0.125, 0.125], [0.125, -0.125, 0.125], zeros, zeros],
            [[0.125, -0.125, 0.125], [-0.25, -0.0, -0.25], [0.25, 0.0, 0.25], zeros],
            [[0.0, -0.25, 0.25], [0.0, 0.25, -0.25], [0.125, -0.125, 0.125], zeros],
            [[-0.375, -0.375, 0.375], [-0.0, 0.25, 0.25], [0.125, 0.125, -0.125], [-0.25, -0.0, -0.25]],
            [[-0.125, 0.125, 0.125], [0.125, -0.125, 0.125], zeros, zeros],
            [[0.125, 0.125, 0.125], [0.125, -0.125, 0.125], [-0.125, 0.125, 0.125], zeros],
            [[-0.0, 0.0, 0.5], [-0.25, -0.25, 0.25], [-0.125, -0.125, 0.125], zeros],
            [[0.25, 0.25, -0.25], [0.25, 0.25, -0.25], [0.125, 0.125, -0.125], [-0.125, -0.125, 0.125]],
            [[0.125, -0.125, 0.125], [0.25, -0.25, 0.0], [0.25, -0.25, 0.0], zeros],
            [[0.5, 0.0, 0.0], [0.25, -0.25, 0.25], [-0.125, 0.125, -0.125], [0.125, -0.125, 0.125]],
            [[0.0, 0.25, -0.25], [0.375, -0.375, -0.375], [-0.125, 0.125, 0.125], [0.25, 0.25, 0.0]],
            [[-0.5, 0.0, 0.0], [-0.25, -0.25, 0.25], [-0.125, -0.125, 0.125], zeros],
            [[0.25, -0.25, 0.0], [-0.25, 0.25, 0.0], zeros, zeros],
            [[0.0, 0.5, 0.0], [-0.25, 0.25, 0.25], [0.125, -0.125, -0.125], zeros],
            [[0.0, 0.5, 0.0], [0.125, -0.125, 0.125], [-0.25, 0.25, -0.25], zeros],
            [[0.0, 0.5, 0.0], [0.0, -0.5, 0.0], zeros, zeros],
            [[0.25, -0.25, 0.0], [-0.25, 0.25, 0.0], [0.125, -0.125, 0.125], zeros],
            [[-0.375, -0.375, -0.375], [-0.25, 0.0, 0.25], [-0.125, -0.125, -0.125], [-0.25, 0.25, 0.0]],
            [[0.125, 0.125, 0.125], [0.0, -0.5, 0.0], [-0.25, -0.25, -0.25], [-0.125, -0.125, -0.125]],
            [[0.0, -0.5, 0.0], [-0.25, -0.25, -0.25], [-0.125, -0.125, -0.125], zeros],
            [[-0.125, 0.125, 0.125], [0.25, -0.25, 0.0], [-0.25, 0.25, 0.0], zeros],
            [[0.0, 0.5, 0.0], [0.25, 0.25, -0.25], [-0.125, -0.125, 0.125], [-0.125, -0.125, 0.125]],
            [[-0.375, 0.375, -0.375], [-0.25, -0.25, 0.0], [-0.125, 0.125, -0.125], [-0.25, 0.0, 0.25]],
            [[0.0, 0.5, 0.0], [0.25, 0.25, -0.25], [-0.125, -0.125, 0.125], zeros],
            [[0.25, -0.25, 0.0], [-0.25, 0.25, 0.0], [0.25, -0.25, 0.0], [0.25, -0.25, 0.0]],
            [[-0.25, -0.25, 0.0], [-0.25, -0.25, 0.0], [-0.125, -0.125, 0.125], zeros],
            [[0.125, 0.125, 0.125], [-0.25, -0.25, 0.0], [-0.25, -0.25, 0.0], zeros],
            [[-0.25, -0.25, 0.0], [-0.25, -0.25, 0.0], zeros, zeros],
            [[-0.125, -0.125, 0.125], zeros, zeros, zeros],
            [[0.125, 0.125, 0.125], [-0.125, -0.125, 0.125], zeros, zeros],
            [[-0.125, -0.125, 0.125], [-0.125, -0.125, 0.125], zeros, zeros],
            [[-0.125, -0.125, 0.125], [-0.25, -0.25, 0.0], [0.25, 0.25, -0.0], zeros],
            [[0.0, -0.25, 0.25], [0.0, -0.25, 0.25], zeros, zeros],
            [[0.0, 0.0, 0.5], [0.25, -0.25, 0.25], [0.125, -0.125, 0.125], zeros],
            [[0.0, -0.25, 0.25], [0.0, -0.25, 0.25], [-0.125, -0.125, 0.125], zeros],
            [[0.375, -0.375, 0.375], [0.0, -0.25, -0.25], [-0.125, 0.125, -0.125], [0.25, 0.25, 0.0]],
            [[-0.125, -0.125, 0.125], [-0.125, 0.125, 0.125], zeros, zeros],
            [[0.125, 0.125, 0.125], [-0.125, -0.125, 0.125], [-0.125, 0.125, 0.125], zeros],
            [[-0.125, -0.125, 0.125], [-0.25, 0.0, 0.25], [-0.25, 0.0, 0.25], zeros],
            [[0.5, 0.0, 0.0], [-0.25, -0.25, 0.25], [-0.125, -0.125, 0.125], [-0.125, -0.125, 0.125]],
            [[-0.0, 0.5, 0.0], [-0.25, 0.25, -0.25], [0.125, -0.125, 0.125], zeros],
            [[-0.25, 0.25, -0.25], [-0.25, 0.25, -0.25], [-0.125, 0.125, -0.125], [-0.125, 0.125, -0.125]],
            [[-0.25, 0.0, -0.25], [0.375, -0.375, -0.375], [0.0, 0.25, -0.25], [-0.125, 0.125, 0.125]],
            [[0.5, 0.0, 0.0], [-0.25, 0.25, -0.25], [0.125, -0.125, 0.125], zeros],
            [[-0.25, 0.0, 0.25], [0.25, 0.0, -0.25], zeros, zeros],
            [[-0.0, 0.0, 0.5], [-0.25, 0.25, 0.25], [-0.125, 0.125, 0.125], zeros],
            [[-0.125, -0.125, 0.125], [-0.25, 0.0, 0.25], [0.25, 0.0, -0.25], zeros],
            [[-0.25, -0.0, -0.25], [-0.375, 0.375, 0.375], [-0.25, -0.25, 0.0], [-0.125, 0.125, 0.125]],
            [[0.0, 0.0, -0.5], [0.25, 0.25, -0.25], [-0.125, -0.125, 0.125], zeros],
            [[-0.0, 0.0, 0.5], [0.0, 0.0, 0.5], zeros, zeros],
            [[0.125, 0.125, 0.125], [0.125, 0.125, 0.125], [0.25, 0.25, 0.25], [0.0, 0.0, 0.5]],
            [[0.125, 0.125, 0.125], [0.25, 0.25, 0.25], [0.0, 0.0, 0.5], zeros],
            [[-0.25, 0.0, 0.25], [0.25, 0.0, -0.25], [-0.125, 0.125, 0.125], zeros],
            [[-0.0, 0.0, 0.5], [0.25, -0.25, 0.25], [0.125, -0.125, 0.125], [0.125, -0.125, 0.125]],
            [[-0.25, 0.0, 0.25], [-0.25, 0.0, 0.25], [-0.25, 0.0, 0.25], [0.25, 0.0, -0.25]],
            [[0.125, -0.125, 0.125], [0.25, 0.0, 0.25], [0.25, 0.0, 0.25], zeros],
            [[0.25, 0.0, 0.25], [-0.375, -0.375, 0.375], [-0.25, 0.25, 0.0], [-0.125, -0.125, 0.125]],
            [[-0.0, 0.0, 0.5], [0.25, -0.25, 0.25], [0.125, -0.125, 0.125], zeros],
            [[0.125, 0.125, 0.125], [0.25, 0.0, 0.25], [0.25, 0.0, 0.25], zeros],
            [[0.25, 0.0, 0.25], [0.25, 0.0, 0.25], zeros, zeros],
            [[-0.125, -0.125, 0.125], [0.125, -0.125, 0.125], zeros, zeros],
            [[0.125, 0.125, 0.125], [-0.125, -0.125, 0.125], [0.125, -0.125, 0.125], zeros],
            [[-0.125, -0.125, 0.125], [0.0, -0.25, 0.25], [0.0, 0.25, -0.25], zeros],
            [[0.0, -0.5, 0.0], [0.125, 0.125, -0.125], [0.25, 0.25, -0.25], [-0.125, -0.125, 0.125]],
            [[0.0, -0.25, 0.25], [0.0, -0.25, 0.25], [0.125, -0.125, 0.125], zeros],
            [[0.0, 0.0, 0.5], [0.25, -0.25, 0.25], [0.125, -0.125, 0.125], [0.125, -0.125, 0.125]],
            [[0.0, -0.25, 0.25], [0.0, -0.25, 0.25], [0.0, -0.25, 0.25], [0.0, 0.25, -0.25]],
            [[0.0, 0.25, 0.25], [0.0, 0.25, 0.25], [0.125, -0.125, -0.125], zeros],
            [[-0.125, 0.125, 0.125], [0.125, -0.125, 0.125], [-0.125, -0.125, 0.125], zeros],
            [[-0.125, 0.125, 0.125], [0.125, -0.125, 0.125], [-0.125, -0.125, 0.125], [0.125, 0.125, 0.125]],
            [[-0.0, 0.0, 0.5], [-0.25, -0.25, 0.25], [-0.125, -0.125, 0.125], [-0.125, -0.125, 0.125]],
            [[0.125, 0.125, 0.125], [0.125, -0.125, 0.125], [0.125, -0.125, -0.125], zeros],
            [[-0.0, 0.5, 0.0], [-0.25, 0.25, -0.25], [0.125, -0.125, 0.125], [0.125, -0.125, 0.125]],
            [[0.125, 0.125, 0.125], [-0.125, -0.125, 0.125], [0.125, -0.125, -0.125], zeros],
            [[0.0, -0.25, -0.25], [0.0, 0.25, 0.25], [0.125, 0.125, 0.125], zeros],
            [[0.125, 0.125, 0.125], [0.125, -0.125, -0.125], zeros, zeros],
            [[0.5, 0.0, -0.0], [0.25, -0.25, -0.25], [0.125, -0.125, -0.125], zeros],
            [[-0.25, 0.25, 0.25], [-0.125, 0.125, 0.125], [-0.25, 0.25, 0.25], [0.125, -0.125, -0.125]],
            [[0.375, -0.375, 0.375], [0.0, 0.25, 0.25], [-0.125, 0.125, -0.125], [-0.25, 0.0, 0.25]],
            [[0.0, -0.5, 0.0], [-0.25, 0.25, 0.25], [-0.125, 0.125, 0.125], zeros],
            [[-0.375, -0.375, 0.375], [0.25, -0.25, 0.0], [0.0, 0.25, 0.25], [-0.125, -0.125, 0.125]],
            [[-0.125, 0.125, 0.125], [-0.25, 0.25, 0.25], [0.0, 0.0, 0.5], zeros],
            [[0.125, 0.125, 0.125], [0.0, 0.25, 0.25], [0.0, 0.25, 0.25], zeros],
            [[0.0, 0.25, 0.25], [0.0, 0.25, 0.25], zeros, zeros],
            [[0.5, 0.0, -0.0], [0.25, 0.25, 0.25], [0.125, 0.125, 0.125], [0.125, 0.125, 0.125]],
            [[0.125, -0.125, 0.125], [-0.125, -0.125, 0.125], [0.125, 0.125, 0.125], zeros],
            [[-0.25, -0.0, -0.25], [0.25, 0.0, 0.25], [0.125, 0.125, 0.125], zeros],
            [[0.125, 0.125, 0.125], [0.125, -0.125, 0.125], zeros, zeros],
            [[-0.25, -0.25, 0.0], [0.25, 0.25, -0.0], [0.125, 0.125, 0.125], zeros],
            [[0.125, 0.125, 0.125], [-0.125, -0.125, 0.125], zeros, zeros],
            [[0.125, 0.125, 0.125], [0.125, 0.125, 0.125], zeros, zeros],
            [[0.125, 0.125, 0.125], zeros, zeros, zeros],
            [[0.125, 0.125, 0.125], zeros, zeros, zeros],
            [[0.125, 0.125, 0.125], [0.125, 0.125, 0.125], zeros, zeros],
            [[0.125, 0.125, 0.125], [-0.125, -0.125, 0.125], zeros, zeros],
            [[-0.25, -0.25, 0.0], [0.25, 0.25, -0.0], [0.125, 0.125, 0.125], zeros],
            [[0.125, 0.125, 0.125], [0.125, -0.125, 0.125], zeros, zeros],
            [[-0.25, -0.0, -0.25], [0.25, 0.0, 0.25], [0.125, 0.125, 0.125], zeros],
            [[0.125, -0.125, 0.125], [-0.125, -0.125, 0.125], [0.125, 0.125, 0.125], zeros],
            [[0.5, 0.0, -0.0], [0.25, 0.25, 0.25], [0.125, 0.125, 0.125], [0.125, 0.125, 0.125]],
            [[0.0, 0.25, 0.25], [0.0, 0.25, 0.25], zeros, zeros],
            [[0.125, 0.125, 0.125], [0.0, 0.25, 0.25], [0.0, 0.25, 0.25], zeros],
            [[-0.125, 0.125, 0.125], [-0.25, 0.25, 0.25], [0.0, 0.0, 0.5], zeros],
            [[-0.375, -0.375, 0.375], [0.25, -0.25, 0.0], [0.0, 0.25, 0.25], [-0.125, -0.125, 0.125]],
            [[0.0, -0.5, 0.0], [-0.25, 0.25, 0.25], [-0.125, 0.125, 0.125], zeros],
            [[0.375, -0.375, 0.375], [0.0, 0.25, 0.25], [-0.125, 0.125, -0.125], [-0.25, 0.0, 0.25]],
            [[-0.25, 0.25, 0.25], [-0.125, 0.125, 0.125], [-0.25, 0.25, 0.25], [0.125, -0.125, -0.125]],
            [[0.5, 0.0, -0.0], [0.25, -0.25, -0.25], [0.125, -0.125, -0.125], zeros],
            [[0.125, 0.125, 0.125], [0.125, -0.125, -0.125], zeros, zeros],
            [[0.0, -0.25, -0.25], [0.0, 0.25, 0.25], [0.125, 0.125, 0.125], zeros],
            [[0.125, 0.125, 0.125], [-0.125, -0.125, 0.125], [0.125, -0.125, -0.125], zeros],
            [[-0.0, 0.5, 0.0], [-0.25, 0.25, -0.25], [0.125, -0.125, 0.125], [0.125, -0.125, 0.125]],
            [[0.125, 0.125, 0.125], [0.125, -0.125, 0.125], [0.125, -0.125, -0.125], zeros],
            [[-0.0, 0.0, 0.5], [-0.25, -0.25, 0.25], [-0.125, -0.125, 0.125], [-0.125, -0.125, 0.125]],
            [[-0.125, 0.125, 0.125], [0.125, -0.125, 0.125], [-0.125, -0.125, 0.125], [0.125, 0.125, 0.125]],
            [[-0.125, 0.125, 0.125], [0.125, -0.125, 0.125], [-0.125, -0.125, 0.125], zeros],
            [[0.0, 0.25, 0.25], [0.0, 0.25, 0.25], [0.125, -0.125, -0.125], zeros],
            [[0.0, -0.25, -0.25], [0.0, 0.25, 0.25], [0.0, 0.25, 0.25], [0.0, 0.25, 0.25]],
            [[0.0, 0.0, 0.5], [0.25, -0.25, 0.25], [0.125, -0.125, 0.125], [0.125, -0.125, 0.125]],
            [[0.0, -0.25, 0.25], [0.0, -0.25, 0.25], [0.125, -0.125, 0.125], zeros],
            [[0.0, -0.5, 0.0], [0.125, 0.125, -0.125], [0.25, 0.25, -0.25], [-0.125, -0.125, 0.125]],
            [[-0.125, -0.125, 0.125], [0.0, -0.25, 0.25], [0.0, 0.25, -0.25], zeros],
            [[0.125, 0.125, 0.125], [-0.125, -0.125, 0.125], [0.125, -0.125, 0.125], zeros],
            [[-0.125, -0.125, 0.125], [0.125, -0.125, 0.125], zeros, zeros],
            [[0.25, 0.0, 0.25], [0.25, 0.0, 0.25], zeros, zeros],
            [[0.125, 0.125, 0.125], [0.25, 0.0, 0.25], [0.25, 0.0, 0.25], zeros],
            [[-0.0, 0.0, 0.5], [0.25, -0.25, 0.25], [0.125, -0.125, 0.125], zeros],
            [[0.25, 0.0, 0.25], [-0.375, -0.375, 0.375], [-0.25, 0.25, 0.0], [-0.125, -0.125, 0.125]],
            [[0.125, -0.125, 0.125], [0.25, 0.0, 0.25], [0.25, 0.0, 0.25], zeros],
            [[-0.25, -0.0, -0.25], [0.25, 0.0, 0.25], [0.25, 0.0, 0.25], [0.25, 0.0, 0.25]],
            [[-0.0, 0.0, 0.5], [0.25, -0.25, 0.25], [0.125, -0.125, 0.125], [0.125, -0.125, 0.125]],
            [[-0.25, 0.0, 0.25], [0.25, 0.0, -0.25], [-0.125, 0.125, 0.125], zeros],
            [[0.125, 0.125, 0.125], [0.25, 0.25, 0.25], [0.0, 0.0, 0.5], zeros],
            [[0.125, 0.125, 0.125], [0.125, 0.125, 0.125], [0.25, 0.25, 0.25], [0.0, 0.0, 0.5]],
            [[-0.0, 0.0, 0.5], [0.0, 0.0, 0.5], zeros, zeros],
            [[0.0, 0.0, -0.5], [0.25, 0.25, -0.25], [-0.125, -0.125, 0.125], zeros],
            [[-0.25, -0.0, -0.25], [-0.375, 0.375, 0.375], [-0.25, -0.25, 0.0], [-0.125, 0.125, 0.125]],
            [[-0.125, -0.125, 0.125], [-0.25, 0.0, 0.25], [0.25, 0.0, -0.25], zeros],
            [[-0.0, 0.0, 0.5], [-0.25, 0.25, 0.25], [-0.125, 0.125, 0.125], zeros],
            [[-0.25, 0.0, 0.25], [0.25, 0.0, -0.25], zeros, zeros],
            [[0.5, 0.0, 0.0], [-0.25, 0.25, -0.25], [0.125, -0.125, 0.125], zeros],
            [[-0.25, 0.0, -0.25], [0.375, -0.375, -0.375], [0.0, 0.25, -0.25], [-0.125, 0.125, 0.125]],
            [[-0.25, 0.25, -0.25], [-0.25, 0.25, -0.25], [-0.125, 0.125, -0.125], [-0.125, 0.125, -0.125]],
            [[-0.0, 0.5, 0.0], [-0.25, 0.25, -0.25], [0.125, -0.125, 0.125], zeros],
            [[0.5, 0.0, 0.0], [-0.25, -0.25, 0.25], [-0.125, -0.125, 0.125], [-0.125, -0.125, 0.125]],
            [[-0.125, -0.125, 0.125], [-0.25, 0.0, 0.25], [-0.25, 0.0, 0.25], zeros],
            [[0.125, 0.125, 0.125], [-0.125, -0.125, 0.125], [-0.125, 0.125, 0.125], zeros],
            [[-0.125, -0.125, 0.125], [-0.125, 0.125, 0.125], zeros, zeros],
            [[0.375, -0.375, 0.375], [0.0, -0.25, -0.25], [-0.125, 0.125, -0.125], [0.25, 0.25, 0.0]],
            [[0.0, -0.25, 0.25], [0.0, -0.25, 0.25], [-0.125, -0.125, 0.125], zeros],
            [[0.0, 0.0, 0.5], [0.25, -0.25, 0.25], [0.125, -0.125, 0.125], zeros],
            [[0.0, -0.25, 0.25], [0.0, -0.25, 0.25], zeros, zeros],
            [[-0.125, -0.125, 0.125], [-0.25, -0.25, 0.0], [0.25, 0.25, -0.0], zeros],
            [[-0.125, -0.125, 0.125], [-0.125, -0.125, 0.125], zeros, zeros],
            [[0.125, 0.125, 0.125], [-0.125, -0.125, 0.125], zeros, zeros],
            [[-0.125, -0.125, 0.125], zeros, zeros, zeros],
            [[-0.25, -0.25, 0.0], [-0.25, -0.25, 0.0], zeros, zeros],
            [[0.125, 0.125, 0.125], [-0.25, -0.25, 0.0], [-0.25, -0.25, 0.0], zeros],
            [[-0.25, -0.25, 0.0], [-0.25, -0.25, 0.0], [-0.125, -0.125, 0.125], zeros],
            [[-0.25, -0.25, 0.0], [-0.25, -0.25, 0.0], [-0.25, -0.25, 0.0], [0.25, 0.25, -0.0]],
            [[0.0, 0.5, 0.0], [0.25, 0.25, -0.25], [-0.125, -0.125, 0.125], zeros],
            [[-0.375, 0.375, -0.375], [-0.25, -0.25, 0.0], [-0.125, 0.125, -0.125], [-0.25, 0.0, 0.25]],
            [[0.0, 0.5, 0.0], [0.25, 0.25, -0.25], [-0.125, -0.125, 0.125], [-0.125, -0.125, 0.125]],
            [[-0.125, 0.125, 0.125], [0.25, -0.25, 0.0], [-0.25, 0.25, 0.0], zeros],
            [[0.0, -0.5, 0.0], [-0.25, -0.25, -0.25], [-0.125, -0.125, -0.125], zeros],
            [[0.125, 0.125, 0.125], [0.0, -0.5, 0.0], [-0.25, -0.25, -0.25], [-0.125, -0.125, -0.125]],
            [[-0.375, -0.375, -0.375], [-0.25, 0.0, 0.25], [-0.125, -0.125, -0.125], [-0.25, 0.25, 0.0]],
            [[0.25, -0.25, 0.0], [-0.25, 0.25, 0.0], [0.125, -0.125, 0.125], zeros],
            [[0.0, 0.5, 0.0], [0.0, -0.5, 0.0], zeros, zeros],
            [[0.0, 0.5, 0.0], [0.125, -0.125, 0.125], [-0.25, 0.25, -0.25], zeros],
            [[0.0, 0.5, 0.0], [-0.25, 0.25, 0.25], [0.125, -0.125, -0.125], zeros],
            [[0.25, -0.25, 0.0], [-0.25, 0.25, 0.0], zeros, zeros],
            [[-0.5, 0.0, 0.0], [-0.25, -0.25, 0.25], [-0.125, -0.125, 0.125], zeros],
            [[0.0, 0.25, -0.25], [0.375, -0.375, -0.375], [-0.125, 0.125, 0.125], [0.25, 0.25, 0.0]],
            [[0.5, 0.0, 0.0], [0.25, -0.25, 0.25], [-0.125, 0.125, -0.125], [0.125, -0.125, 0.125]],
            [[0.125, -0.125, 0.125], [0.25, -0.25, 0.0], [0.25, -0.25, 0.0], zeros],
            [[0.25, 0.25, -0.25], [0.25, 0.25, -0.25], [0.125, 0.125, -0.125], [-0.125, -0.125, 0.125]],
            [[-0.0, 0.0, 0.5], [-0.25, -0.25, 0.25], [-0.125, -0.125, 0.125], zeros],
            [[0.125, 0.125, 0.125], [0.125, -0.125, 0.125], [-0.125, 0.125, 0.125], zeros],
            [[-0.125, 0.125, 0.125], [0.125, -0.125, 0.125], zeros, zeros],
            [[-0.375, -0.375, 0.375], [-0.0, 0.25, 0.25], [0.125, 0.125, -0.125], [-0.25, -0.0, -0.25]],
            [[0.0, -0.25, 0.25], [0.0, 0.25, -0.25], [0.125, -0.125, 0.125], zeros],
            [[0.125, -0.125, 0.125], [-0.25, -0.0, -0.25], [0.25, 0.0, 0.25], zeros],
            [[0.125, -0.125, 0.125], [0.125, -0.125, 0.125], zeros, zeros],
            [[0.0, -0.5, 0.0], [0.125, 0.125, -0.125], [0.25, 0.25, -0.25], zeros],
            [[0.0, -0.25, 0.25], [0.0, 0.25, -0.25], zeros, zeros],
            [[0.125, 0.125, 0.125], [0.125, -0.125, 0.125], zeros, zeros],
            [[0.125, -0.125, 0.125], zeros, zeros, zeros],
            [[-0.5, 0.0, 0.0], [-0.125, -0.125, -0.125], [-0.25, -0.25, -0.25], zeros],
            [[-0.5, 0.0, 0.0], [-0.125, -0.125, -0.125], [-0.25, -0.25, -0.25], [0.125, 0.125, 0.125]],
            [[0.375, 0.375, 0.375], [0.0, 0.25, -0.25], [-0.125, -0.125, -0.125], [-0.25, 0.25, 0.0]],
            [[0.125, -0.125, -0.125], [0.25, -0.25, 0.0], [0.25, -0.25, 0.0], zeros],
            [[0.125, 0.125, 0.125], [0.375, 0.375, 0.375], [0.0, -0.25, 0.25], [-0.25, 0.0, 0.25]],
            [[-0.25, 0.0, 0.25], [-0.25, 0.0, 0.25], [0.125, -0.125, -0.125], zeros],
            [[0.0, -0.25, -0.25], [0.0, 0.25, 0.25], [-0.125, 0.125, 0.125], zeros],
            [[-0.125, 0.125, 0.125], [0.125, -0.125, -0.125], zeros, zeros],
            [[-0.125, -0.125, -0.125], [-0.25, -0.25, -0.25], [0.25, 0.25, 0.25], [0.125, 0.125, 0.125]],
            [[-0.125, -0.125, 0.125], [0.125, -0.125, 0.125], [0.125, -0.125, -0.125], zeros],
            [[0.0, 0.0, -0.5], [0.25, 0.25, 0.25], [-0.125, -0.125, -0.125], zeros],
            [[0.125, -0.125, 0.125], [0.125, -0.125, -0.125], zeros, zeros],
            [[0.0, -0.5, 0.0], [0.25, 0.25, 0.25], [0.125, 0.125, 0.125], zeros],
            [[-0.125, -0.125, 0.125], [0.125, -0.125, -0.125], zeros, zeros],
            [[0.0, -0.25, -0.25], [0.0, 0.25, 0.25], zeros, zeros],
            [[0.125, -0.125, -0.125], zeros, zeros, zeros],
            [[0.5, 0.0, 0.0], [0.5, 0.0, 0.0], zeros, zeros],
            [[-0.5, 0.0, 0.0], [-0.25, 0.25, 0.25], [-0.125, 0.125, 0.125], zeros],
            [[0.5, 0.0, 0.0], [0.25, -0.25, 0.25], [-0.125, 0.125, -0.125], zeros],
            [[0.25, -0.25, 0.0], [0.25, -0.25, 0.0], zeros, zeros],
            [[0.5, 0.0, 0.0], [-0.25, -0.25, 0.25], [-0.125, -0.125, 0.125], zeros],
            [[-0.25, 0.0, 0.25], [-0.25, 0.0, 0.25], zeros, zeros],
            [[0.125, 0.125, 0.125], [-0.125, 0.125, 0.125], zeros, zeros],
            [[-0.125, 0.125, 0.125], zeros, zeros, zeros],
            [[0.5, 0.0, -0.0], [0.25, 0.25, 0.25], [0.125, 0.125, 0.125], zeros],
            [[0.125, -0.125, 0.125], [-0.125, -0.125, 0.125], zeros, zeros],
            [[-0.25, -0.0, -0.25], [0.25, 0.0, 0.25], zeros, zeros],
            [[0.125, 0.125, 0.125], zeros, zeros, zeros],
            [[-0.25, -0.25, 0.0], [0.25, 0.25, -0.0], zeros, zeros],
            [[0.125, 0.125, 0.125], zeros, zeros, zeros],
            [[0.125, 0.125, 0.125], zeros, zeros, zeros],
            [zeros, zeros, zeros, zeros],
        ],
        dtype=torch.float32,
        device=device,
    )

    space = torch.as_tensor(
        [[[spacing[1] * spacing[2], spacing[0] * spacing[2], spacing[0] * spacing[1]]]],
        device=device,
        dtype=table.dtype,
    )
    norm = torch.linalg.norm(table * space, dim=-1)
    table = norm.sum(-1)
    kernel = torch.as_tensor([[[[[128, 64], [32, 16]], [[8, 4], [2, 1]]]]], device=device)
    return table, kernel