kornia/geometry/subpix/spatial_soft_argmax.py

from typing import Tuple, Union

import torch
import torch.nn as nn
import torch.nn.functional as F

from kornia.filters.sobel import spatial_gradient3d
from kornia.geometry.conversions import normalize_pixel_coordinates, normalize_pixel_coordinates3d
from kornia.utils import create_meshgrid, create_meshgrid3d
from kornia.utils._compat import torch_version_geq
from kornia.utils.helpers import safe_solve_with_mask

from .dsnt import spatial_expectation2d, spatial_softmax2d
from .nms import nms3d


def _get_window_grid_kernel2d(h: int, w: int, device: torch.device = torch.device('cpu')) -> torch.Tensor:
    r"""Helper function, which generates a kernel to with window coordinates,
       residual to window center.

    Args:
         h: kernel height.
         : kernel width.
         device: device, on which generate.

    Returns:
        conv_kernel [2x1xhxw]
    """
    window_grid2d = create_meshgrid(h, w, False, device=device)
    window_grid2d = normalize_pixel_coordinates(window_grid2d, h, w)
    conv_kernel = window_grid2d.permute(3, 0, 1, 2)
    return conv_kernel


def _get_center_kernel2d(h: int, w: int, device: torch.device = torch.device('cpu')) -> torch.Tensor:
    r"""Helper function, which generates a kernel to return center coordinates,
       when applied with F.conv2d to 2d coordinates grid.

    Args:
        h: kernel height.
        w: kernel width.
        device: device, on which generate.

    Returns:
        conv_kernel [2x2xhxw].
    """
    center_kernel = torch.zeros(2, 2, h, w, device=device)

    #  If the size is odd, we have one pixel for center, if even - 2
    if h % 2 != 0:
        h_i1 = h // 2
        h_i2 = (h // 2) + 1
    else:
        h_i1 = (h // 2) - 1
        h_i2 = (h // 2) + 1
    if w % 2 != 0:
        w_i1 = w // 2
        w_i2 = (w // 2) + 1
    else:
        w_i1 = (w // 2) - 1
        w_i2 = (w // 2) + 1
    center_kernel[(0, 1), (0, 1), h_i1:h_i2, w_i1:w_i2] = 1.0 / float((h_i2 - h_i1) * (w_i2 - w_i1))
    return center_kernel


def _get_center_kernel3d(d: int, h: int, w: int, device: torch.device = torch.device('cpu')) -> torch.Tensor:
    r"""Helper function, which generates a kernel to return center coordinates,
       when applied with F.conv2d to 3d coordinates grid.

    Args:
        d: kernel depth.
        h: kernel height.
        w: kernel width.
        device: device, on which generate.

    Returns:
        conv_kernel [3x3xdxhxw].
    """
    center_kernel = torch.zeros(3, 3, d, h, w, device=device)
    #  If the size is odd, we have one pixel for center, if even - 2
    if h % 2 != 0:
        h_i1 = h // 2
        h_i2 = (h // 2) + 1
    else:
        h_i1 = (h // 2) - 1
        h_i2 = (h // 2) + 1
    if w % 2 != 0:
        w_i1 = w // 2
        w_i2 = (w // 2) + 1
    else:
        w_i1 = (w // 2) - 1
        w_i2 = (w // 2) + 1
    if d % 2 != 0:
        d_i1 = d // 2
        d_i2 = (d // 2) + 1
    else:
        d_i1 = (d // 2) - 1
        d_i2 = (d // 2) + 1
    center_num = float((h_i2 - h_i1) * (w_i2 - w_i1) * (d_i2 - d_i1))
    center_kernel[(0, 1, 2), (0, 1, 2), d_i1:d_i2, h_i1:h_i2, w_i1:w_i2] = 1.0 / center_num
    return center_kernel


def _get_window_grid_kernel3d(d: int, h: int, w: int, device: torch.device = torch.device('cpu')) -> torch.Tensor:
    r"""Helper function, which generates a kernel to return coordinates,
       residual to window center.

    Args:
        d: kernel depth.
        h: kernel height.
        w: kernel width.
        device: device, on which generate.

    Returns:
        conv_kernel [3x1xdxhxw]
    """
    grid2d = create_meshgrid(h, w, True, device=device)
    if d > 1:
        z = torch.linspace(-1, 1, d, device=device).view(d, 1, 1, 1)
    else:  # only onr channel with index == 0
        z = torch.zeros(1, 1, 1, 1, device=device)
    grid3d = torch.cat([z.repeat(1, h, w, 1).contiguous(), grid2d.repeat(d, 1, 1, 1)], dim=3)
    conv_kernel = grid3d.permute(3, 0, 1, 2).unsqueeze(1)
    return conv_kernel


class ConvSoftArgmax2d(nn.Module):
    r"""Module that calculates soft argmax 2d per window.

    See :func:`~kornia.geometry.subpix.conv_soft_argmax2d` for details.
    """

    def __init__(
        self,
        kernel_size: Tuple[int, int] = (3, 3),
        stride: Tuple[int, int] = (1, 1),
        padding: Tuple[int, int] = (1, 1),
        temperature: Union[torch.Tensor, float] = torch.tensor(1.0),
        normalized_coordinates: bool = True,
        eps: float = 1e-8,
        output_value: bool = False,
    ) -> None:
        super().__init__()
        self.kernel_size = kernel_size
        self.stride = stride
        self.padding = padding
        self.temperature = temperature
        self.normalized_coordinates = normalized_coordinates
        self.eps = eps
        self.output_value = output_value

    def __repr__(self) -> str:
        return (
            self.__class__.__name__
            + '('
            + 'kernel_size='
            + str(self.kernel_size)
            + ', '
            + 'stride='
            + str(self.stride)
            + ', '
            + 'padding='
            + str(self.padding)
            + ', '
            + 'temperature='
            + str(self.temperature)
            + ', '
            + 'normalized_coordinates='
            + str(self.normalized_coordinates)
            + ', '
            + 'eps='
            + str(self.eps)
            + ', '
            + 'output_value='
            + str(self.output_value)
            + ')'
        )

    def forward(self, x: torch.Tensor):  # type: ignore
        return conv_soft_argmax2d(
            x,
            self.kernel_size,
            self.stride,
            self.padding,
            self.temperature,
            self.normalized_coordinates,
            self.eps,
            self.output_value,
        )


class ConvSoftArgmax3d(nn.Module):
    r"""Module that calculates soft argmax 3d per window.

    See :func:`~kornia.geometry.subpix.conv_soft_argmax3d` for details.
    """

    def __init__(
        self,
        kernel_size: Tuple[int, int, int] = (3, 3, 3),
        stride: Tuple[int, int, int] = (1, 1, 1),
        padding: Tuple[int, int, int] = (1, 1, 1),
        temperature: Union[torch.Tensor, float] = torch.tensor(1.0),
        normalized_coordinates: bool = False,
        eps: float = 1e-8,
        output_value: bool = True,
        strict_maxima_bonus: float = 0.0,
    ) -> None:
        super().__init__()
        self.kernel_size = kernel_size
        self.stride = stride
        self.padding = padding
        self.temperature = temperature
        self.normalized_coordinates = normalized_coordinates
        self.eps = eps
        self.output_value = output_value
        self.strict_maxima_bonus = strict_maxima_bonus
        return

    def __repr__(self) -> str:
        return (
            self.__class__.__name__
            + '('
            + 'kernel_size='
            + str(self.kernel_size)
            + ', '
            + 'stride='
            + str(self.stride)
            + ', '
            + 'padding='
            + str(self.padding)
            + ', '
            + 'temperature='
            + str(self.temperature)
            + ', '
            + 'normalized_coordinates='
            + str(self.normalized_coordinates)
            + ', '
            + 'eps='
            + str(self.eps)
            + ', '
            + 'strict_maxima_bonus='
            + str(self.strict_maxima_bonus)
            + ', '
            + 'output_value='
            + str(self.output_value)
            + ')'
        )

    def forward(self, x: torch.Tensor):  # type: ignore
        return conv_soft_argmax3d(
            x,
            self.kernel_size,
            self.stride,
            self.padding,
            self.temperature,
            self.normalized_coordinates,
            self.eps,
            self.output_value,
            self.strict_maxima_bonus,
        )


def conv_soft_argmax2d(
    input: torch.Tensor,
    kernel_size: Tuple[int, int] = (3, 3),
    stride: Tuple[int, int] = (1, 1),
    padding: Tuple[int, int] = (1, 1),
    temperature: Union[torch.Tensor, float] = torch.tensor(1.0),
    normalized_coordinates: bool = True,
    eps: float = 1e-8,
    output_value: bool = False,
) -> Union[torch.Tensor, Tuple[torch.Tensor, torch.Tensor]]:
    r"""Compute the convolutional spatial Soft-Argmax 2D over the windows of a given heatmap.

    .. math::
        ij(X) = \frac{\sum{(i,j)} * exp(x / T)  \in X} {\sum{exp(x / T)  \in X}}

    .. math::
        val(X) = \frac{\sum{x * exp(x / T)  \in X}} {\sum{exp(x / T)  \in X}}

    where :math:`T` is temperature.

    Args:
        input: the given heatmap with shape :math:`(N, C, H_{in}, W_{in})`.
        kernel_size: the size of the window.
        stride: the stride of the window.
        padding: input zero padding.
        temperature: factor to apply to input.
        normalized_coordinates: whether to return the coordinates normalized in the range of :math:`[-1, 1]`.
            Otherwise, it will return the coordinates in the range of the input shape.
        eps: small value to avoid zero division.
        output_value: if True, val is output, if False, only ij.

    Returns:
        Function has two outputs - argmax coordinates and the softmaxpooled heatmap values themselves.
        On each window, the function computed returns with shapes :math:`(N, C, 2, H_{out},
        W_{out})`, :math:`(N, C, H_{out}, W_{out})`,

        where

         .. math::
             H_{out} = \left\lfloor\frac{H_{in}  + 2 \times \text{padding}[0] -
               (\text{kernel\_size}[0] - 1) - 1}{\text{stride}[0]} + 1\right\rfloor

         .. math::
             W_{out} = \left\lfloor\frac{W_{in}  + 2 \times \text{padding}[1] -
               (\text{kernel\_size}[1] - 1) - 1}{\text{stride}[1]} + 1\right\rfloor

    Examples:
        >>> input = torch.randn(20, 16, 50, 32)
        >>> nms_coords, nms_val = conv_soft_argmax2d(input, (3,3), (2,2), (1,1), output_value=True)
    """
    if not torch.is_tensor(input):
        raise TypeError(f"Input type is not a torch.Tensor. Got {type(input)}")

    if not len(input.shape) == 4:
        raise ValueError(f"Invalid input shape, we expect BxCxHxW. Got: {input.shape}")

    if temperature <= 0:
        raise ValueError(f"Temperature should be positive float or tensor. Got: {temperature}")

    b, c, h, w = input.shape
    kx, ky = kernel_size
    device: torch.device = input.device
    dtype: torch.dtype = input.dtype
    input = input.view(b * c, 1, h, w)

    center_kernel: torch.Tensor = _get_center_kernel2d(kx, ky, device).to(dtype)
    window_kernel: torch.Tensor = _get_window_grid_kernel2d(kx, ky, device).to(dtype)

    # applies exponential normalization trick
    # https://timvieira.github.io/blog/post/2014/02/11/exp-normalize-trick/
    # https://github.com/pytorch/pytorch/blob/bcb0bb7e0e03b386ad837015faba6b4b16e3bfb9/aten/src/ATen/native/SoftMax.cpp#L44
    x_max = F.adaptive_max_pool2d(input, (1, 1))

    # max is detached to prevent undesired backprop loops in the graph
    x_exp = ((input - x_max.detach()) / temperature).exp()

    # F.avg_pool2d(.., divisor_override = 1.0) - proper way for sum pool in PyTorch 1.2.
    # Not available yet in version 1.0, so let's do manually
    pool_coef: float = float(kx * ky)

    # softmax denominator
    den = pool_coef * F.avg_pool2d(x_exp, kernel_size, stride=stride, padding=padding) + eps

    x_softmaxpool = pool_coef * F.avg_pool2d(x_exp * input, kernel_size, stride=stride, padding=padding) / den
    x_softmaxpool = x_softmaxpool.view(b, c, x_softmaxpool.size(2), x_softmaxpool.size(3))

    # We need to output also coordinates
    # Pooled window center coordinates
    grid_global: torch.Tensor = create_meshgrid(h, w, False, device).to(dtype).permute(0, 3, 1, 2)

    grid_global_pooled = F.conv2d(grid_global, center_kernel, stride=stride, padding=padding)

    # Coordinates of maxima residual to window center
    # prepare kernel
    coords_max: torch.Tensor = F.conv2d(x_exp, window_kernel, stride=stride, padding=padding)

    coords_max = coords_max / den.expand_as(coords_max)
    coords_max = coords_max + grid_global_pooled.expand_as(coords_max)
    # [:,:, 0, ...] is x
    # [:,:, 1, ...] is y

    if normalized_coordinates:
        coords_max = normalize_pixel_coordinates(coords_max.permute(0, 2, 3, 1), h, w)
        coords_max = coords_max.permute(0, 3, 1, 2)

    # Back B*C -> (b, c)
    coords_max = coords_max.view(b, c, 2, coords_max.size(2), coords_max.size(3))

    if output_value:
        return coords_max, x_softmaxpool
    return coords_max


def conv_soft_argmax3d(
    input: torch.Tensor,
    kernel_size: Tuple[int, int, int] = (3, 3, 3),
    stride: Tuple[int, int, int] = (1, 1, 1),
    padding: Tuple[int, int, int] = (1, 1, 1),
    temperature: Union[torch.Tensor, float] = torch.tensor(1.0),
    normalized_coordinates: bool = False,
    eps: float = 1e-8,
    output_value: bool = True,
    strict_maxima_bonus: float = 0.0,
) -> Union[torch.Tensor, Tuple[torch.Tensor, torch.Tensor]]:
    r"""Compute the convolutional spatial Soft-Argmax 3D over the windows of a given heatmap.

    .. math::
             ijk(X) = \frac{\sum{(i,j,k)} * exp(x / T)  \in X} {\sum{exp(x / T)  \in X}}

    .. math::
             val(X) = \frac{\sum{x * exp(x / T)  \in X}} {\sum{exp(x / T)  \in X}}

    where ``T`` is temperature.

    Args:
        input: the given heatmap with shape :math:`(N, C, D_{in}, H_{in}, W_{in})`.
        kernel_size:  size of the window.
        stride: stride of the window.
        padding: input zero padding.
        temperature: factor to apply to input.
        normalized_coordinates: whether to return the coordinates normalized in the range of :math:[-1, 1]`.
            Otherwise, it will return the coordinates in the range of the input shape.
        eps: small value to avoid zero division.
        output_value: if True, val is output, if False, only ij.
        strict_maxima_bonus: pixels, which are strict maxima will score (1 + strict_maxima_bonus) * value.
          This is needed for mimic behavior of strict NMS in classic local features

    Returns:
        Function has two outputs - argmax coordinates and the softmaxpooled heatmap values themselves.
        On each window, the function computed returns with shapes :math:`(N, C, 3, D_{out}, H_{out}, W_{out})`,
        :math:`(N, C, D_{out}, H_{out}, W_{out})`,

        where

         .. math::
             D_{out} = \left\lfloor\frac{D_{in}  + 2 \times \text{padding}[0] -
             (\text{kernel\_size}[0] - 1) - 1}{\text{stride}[0]} + 1\right\rfloor

         .. math::
             H_{out} = \left\lfloor\frac{H_{in}  + 2 \times \text{padding}[1] -
             (\text{kernel\_size}[1] - 1) - 1}{\text{stride}[1]} + 1\right\rfloor

         .. math::
             W_{out} = \left\lfloor\frac{W_{in}  + 2 \times \text{padding}[2] -
             (\text{kernel\_size}[2] - 1) - 1}{\text{stride}[2]} + 1\right\rfloor

    Examples:
        >>> input = torch.randn(20, 16, 3, 50, 32)
        >>> nms_coords, nms_val = conv_soft_argmax3d(input, (3, 3, 3), (1, 2, 2), (0, 1, 1))
    """
    if not torch.is_tensor(input):
        raise TypeError(f"Input type is not a torch.Tensor. Got {type(input)}")

    if not len(input.shape) == 5:
        raise ValueError(f"Invalid input shape, we expect BxCxDxHxW. Got: {input.shape}")

    if temperature <= 0:
        raise ValueError(f"Temperature should be positive float or tensor. Got: {temperature}")

    b, c, d, h, w = input.shape
    kx, ky, kz = kernel_size
    device: torch.device = input.device
    dtype: torch.dtype = input.dtype
    input = input.view(b * c, 1, d, h, w)

    center_kernel: torch.Tensor = _get_center_kernel3d(kx, ky, kz, device).to(dtype)
    window_kernel: torch.Tensor = _get_window_grid_kernel3d(kx, ky, kz, device).to(dtype)

    # applies exponential normalization trick
    # https://timvieira.github.io/blog/post/2014/02/11/exp-normalize-trick/
    # https://github.com/pytorch/pytorch/blob/bcb0bb7e0e03b386ad837015faba6b4b16e3bfb9/aten/src/ATen/native/SoftMax.cpp#L44
    x_max = F.adaptive_max_pool3d(input, (1, 1, 1))

    # max is detached to prevent undesired backprop loops in the graph
    x_exp = ((input - x_max.detach()) / temperature).exp()

    pool_coef: float = float(kx * ky * kz)

    # softmax denominator
    den = pool_coef * F.avg_pool3d(x_exp.view_as(input), kernel_size, stride=stride, padding=padding) + eps

    # We need to output also coordinates
    # Pooled window center coordinates
    grid_global: torch.Tensor = create_meshgrid3d(d, h, w, False, device=device).to(dtype).permute(0, 4, 1, 2, 3)

    grid_global_pooled = F.conv3d(grid_global, center_kernel, stride=stride, padding=padding)

    # Coordinates of maxima residual to window center
    # prepare kernel
    coords_max: torch.Tensor = F.conv3d(x_exp, window_kernel, stride=stride, padding=padding)

    coords_max = coords_max / den.expand_as(coords_max)
    coords_max = coords_max + grid_global_pooled.expand_as(coords_max)
    # [:,:, 0, ...] is depth (scale)
    # [:,:, 1, ...] is x
    # [:,:, 2, ...] is y

    if normalized_coordinates:
        coords_max = normalize_pixel_coordinates3d(coords_max.permute(0, 2, 3, 4, 1), d, h, w)
        coords_max = coords_max.permute(0, 4, 1, 2, 3)

    # Back B*C -> (b, c)
    coords_max = coords_max.view(b, c, 3, coords_max.size(2), coords_max.size(3), coords_max.size(4))

    if not output_value:
        return coords_max

    x_softmaxpool = (
        pool_coef * F.avg_pool3d(x_exp.view(input.size()) * input, kernel_size, stride=stride, padding=padding) / den
    )
    if strict_maxima_bonus > 0:
        in_levels: int = input.size(2)
        out_levels: int = x_softmaxpool.size(2)
        skip_levels: int = (in_levels - out_levels) // 2
        strict_maxima: torch.Tensor = F.avg_pool3d(nms3d(input, kernel_size), 1, stride, 0)
        strict_maxima = strict_maxima[:, :, skip_levels: out_levels - skip_levels]
        x_softmaxpool *= 1.0 + strict_maxima_bonus * strict_maxima
    x_softmaxpool = x_softmaxpool.view(b, c, x_softmaxpool.size(2), x_softmaxpool.size(3), x_softmaxpool.size(4))
    return coords_max, x_softmaxpool


def spatial_soft_argmax2d(
    input: torch.Tensor,
    temperature: torch.Tensor = torch.tensor(1.0),
    normalized_coordinates: bool = True,
) -> torch.Tensor:
    r"""Compute the Spatial Soft-Argmax 2D of a given input heatmap.

    Args:
        input: the given heatmap with shape :math:`(B, N, H, W)`.
        temperature: factor to apply to input.
        normalized_coordinates: whether to return the coordinates normalized in the range of :math:`[-1, 1]`.
            Otherwise, it will return the coordinates in the range of the input shape.

    Returns:
        the index of the maximum 2d coordinates of the give map :math:`(B, N, 2)`.
        The output order is x-coord and y-coord.

    Examples:
        >>> input = torch.tensor([[[
        ... [0., 0., 0.],
        ... [0., 10., 0.],
        ... [0., 0., 0.]]]])
        >>> spatial_soft_argmax2d(input, normalized_coordinates=False)
        tensor([[[1.0000, 1.0000]]])
    """
    input_soft: torch.Tensor = spatial_softmax2d(input, temperature)
    output: torch.Tensor = spatial_expectation2d(input_soft, normalized_coordinates)
    return output


class SpatialSoftArgmax2d(nn.Module):
    r"""Compute the Spatial Soft-Argmax 2D of a given heatmap.

    See :func:`~kornia.geometry.subpix.spatial_soft_argmax2d` for details.
    """

    def __init__(
        self, temperature: torch.Tensor = torch.tensor(1.0), normalized_coordinates: bool = True
    ) -> None:
        super().__init__()
        self.temperature: torch.Tensor = temperature
        self.normalized_coordinates: bool = normalized_coordinates

    def __repr__(self) -> str:
        return (
            self.__class__.__name__
            + '(temperature='
            + str(self.temperature)
            + ', '
            + 'normalized_coordinates='
            + str(self.normalized_coordinates)
            + ')'
        )

    def forward(self, input: torch.Tensor) -> torch.Tensor:
        return spatial_soft_argmax2d(input, self.temperature, self.normalized_coordinates)


def conv_quad_interp3d(
    input: torch.Tensor, strict_maxima_bonus: float = 10.0, eps: float = 1e-7
) -> Tuple[torch.Tensor, torch.Tensor]:
    r"""Compute the single iteration of quadratic interpolation of the extremum (max or min).

    Args:
        input: the given heatmap with shape :math:`(N, C, D_{in}, H_{in}, W_{in})`.
        strict_maxima_bonus: pixels, which are strict maxima will score (1 + strict_maxima_bonus) * value.
          This is needed for mimic behavior of strict NMS in classic local features
        eps: parameter to control the hessian matrix ill-condition number.

    Returns:
        the location and value per each 3x3x3 window which contains strict extremum, similar to one done is SIFT.
        :math:`(N, C, 3, D_{out}, H_{out}, W_{out})`, :math:`(N, C, D_{out}, H_{out}, W_{out})`,

        where

         .. math::
             D_{out} = \left\lfloor\frac{D_{in}  + 2 \times \text{padding}[0] -
             (\text{kernel\_size}[0] - 1) - 1}{\text{stride}[0]} + 1\right\rfloor

         .. math::
             H_{out} = \left\lfloor\frac{H_{in}  + 2 \times \text{padding}[1] -
             (\text{kernel\_size}[1] - 1) - 1}{\text{stride}[1]} + 1\right\rfloor

         .. math::
             W_{out} = \left\lfloor\frac{W_{in}  + 2 \times \text{padding}[2] -
             (\text{kernel\_size}[2] - 1) - 1}{\text{stride}[2]} + 1\right\rfloor

    Examples:
        >>> input = torch.randn(20, 16, 3, 50, 32)
        >>> nms_coords, nms_val = conv_quad_interp3d(input, 1.0)
    """
    if not torch.is_tensor(input):
        raise TypeError(f"Input type is not a torch.Tensor. Got {type(input)}")

    if not len(input.shape) == 5:
        raise ValueError(f"Invalid input shape, we expect BxCxDxHxW. Got: {input.shape}")

    B, CH, D, H, W = input.shape
    grid_global: torch.Tensor = create_meshgrid3d(D, H, W, False, device=input.device).permute(0, 4, 1, 2, 3)
    grid_global = grid_global.to(input.dtype)

    # to determine the location we are solving system of linear equations Ax = b, where b is 1st order gradient
    # and A is Hessian matrix
    b: torch.Tensor = spatial_gradient3d(input, order=1, mode='diff')  #
    b = b.permute(0, 1, 3, 4, 5, 2).reshape(-1, 3, 1)
    A: torch.Tensor = spatial_gradient3d(input, order=2, mode='diff')
    A = A.permute(0, 1, 3, 4, 5, 2).reshape(-1, 6)
    dxx = A[..., 0]
    dyy = A[..., 1]
    dss = A[..., 2]
    dxy = 0.25 * A[..., 3]  # normalization to match OpenCV implementation
    dys = 0.25 * A[..., 4]  # normalization to match OpenCV implementation
    dxs = 0.25 * A[..., 5]  # normalization to match OpenCV implementation

    Hes = torch.stack([dxx, dxy, dxs, dxy, dyy, dys, dxs, dys, dss], dim=-1).view(-1, 3, 3)
    if not torch_version_geq(1, 10):
        # The following is needed to avoid singular cases
        Hes += torch.rand(Hes[0].size(), device=Hes.device).abs()[None] * eps

    nms_mask: torch.Tensor = nms3d(input, (3, 3, 3), True)
    x_solved: torch.Tensor = torch.zeros_like(b)
    x_solved_masked, _, solved_correctly = safe_solve_with_mask(b[nms_mask.view(-1)], Hes[nms_mask.view(-1)])

    #  Kill those points, where we cannot solve
    new_nms_mask = nms_mask.masked_scatter(nms_mask, solved_correctly)

    x_solved.masked_scatter_(new_nms_mask.view(-1, 1, 1), x_solved_masked[solved_correctly])
    dx: torch.Tensor = -x_solved

    # Ignore ones, which are far from window center
    mask1 = dx.abs().max(dim=1, keepdim=True)[0] > 0.7
    dx.masked_fill_(mask1.expand_as(dx), 0)
    dy: torch.Tensor = 0.5 * torch.bmm(b.permute(0, 2, 1), dx)
    y_max = input + dy.view(B, CH, D, H, W)
    if strict_maxima_bonus > 0:
        y_max += strict_maxima_bonus * new_nms_mask.to(input.dtype)

    dx_res: torch.Tensor = dx.flip(1).reshape(B, CH, D, H, W, 3).permute(0, 1, 5, 2, 3, 4)
    coords_max: torch.Tensor = grid_global.repeat(B, 1, 1, 1, 1).unsqueeze(1)
    coords_max = coords_max + dx_res

    return coords_max, y_max


class ConvQuadInterp3d(nn.Module):
    r"""Calculate soft argmax 3d per window

    See :func:`~kornia.geometry.subpix.conv_quad_interp3d` for details.
    """

    def __init__(self, strict_maxima_bonus: float = 10.0, eps: float = 1e-7) -> None:
        super().__init__()
        self.strict_maxima_bonus = strict_maxima_bonus
        self.eps = eps
        return

    def __repr__(self) -> str:
        return self.__class__.__name__ + '(' + 'strict_maxima_bonus=' + str(self.strict_maxima_bonus) + ')'

    def forward(self, x: torch.Tensor):  # type: ignore
        return conv_quad_interp3d(x, self.strict_maxima_bonus, self.eps)