# Source code for kornia.geometry.transform.imgwarp

import warnings
from typing import Optional, Tuple

import torch
import torch.nn.functional as F

from kornia.geometry.conversions import (
convert_affinematrix_to_homography,
convert_affinematrix_to_homography3d,
normalize_pixel_coordinates,
)
from kornia.geometry.linalg import transform_points
from kornia.geometry.transform.homography_warper import normalize_homography
from kornia.geometry.transform.projwarp import get_projective_transform
from kornia.utils import create_meshgrid
from kornia.utils.helpers import _torch_inverse_cast, _torch_solve_cast

__all__ = [
"warp_perspective",
"warp_affine",
"get_perspective_transform",
"get_rotation_matrix2d",
"remap",
"invert_affine_transform",
"angle_to_rotation_matrix",
"get_affine_matrix2d",
"get_affine_matrix3d",
"get_shear_matrix2d",
"get_shear_matrix3d",
]

[docs]def warp_perspective(
src: torch.Tensor,
M: torch.Tensor,
dsize: Tuple[int, int],
mode: str = 'bilinear',
padding_mode: str = 'zeros',
align_corners: Optional[bool] = None,
) -> torch.Tensor:
r"""Applies a perspective transformation to an image.

The function warp_perspective transforms the source image using
the specified matrix:

.. math::
\text{dst} (x, y) = \text{src} \left(
\frac{M^{-1}_{11} x + M^{-1}_{12} y + M^{-1}_{13}}{M^{-1}_{31} x + M^{-1}_{32} y + M^{-1}_{33}} ,
\frac{M^{-1}_{21} x + M^{-1}_{22} y + M^{-1}_{23}}{M^{-1}_{31} x + M^{-1}_{32} y + M^{-1}_{33}}
\right )

Args:
src: input image with shape :math:(B, C, H, W).
M: transformation matrix with shape :math:(B, 3, 3).
dsize: size of the output image (height, width).
mode: interpolation mode to calculate output values 'bilinear' | 'nearest'.
padding_mode: padding mode for outside grid values 'zeros' | 'border' | 'reflection'.
align_corners(bool, optional): interpolation flag.

Returns:
the warped input image :math:(B, C, H, W).

Example:
>>> img = torch.rand(1, 4, 5, 6)
>>> H = torch.eye(3)[None]
>>> out = warp_perspective(img, H, (4, 2), align_corners=True)
>>> print(out.shape)
torch.Size([1, 4, 4, 2])

.. note::
This function is often used in conjunction with :func:get_perspective_transform.

.. note::
See a working example here <https://kornia-tutorials.readthedocs.io/en/
latest/warp_perspective.html>_.
"""
if not isinstance(src, torch.Tensor):
raise TypeError(f"Input src type is not a torch.Tensor. Got {type(src)}")

if not isinstance(M, torch.Tensor):
raise TypeError(f"Input M type is not a torch.Tensor. Got {type(M)}")

if not len(src.shape) == 4:
raise ValueError(f"Input src must be a BxCxHxW tensor. Got {src.shape}")

if not (len(M.shape) == 3 and M.shape[-2:] == (3, 3)):
raise ValueError(f"Input M must be a Bx3x3 tensor. Got {M.shape}")

# TODO: remove the statement below in kornia v0.6
if align_corners is None:
message: str = (
"The align_corners default value has been changed. By default now is set True "
"in order to match cv2.warpPerspective. In case you want to keep your previous "
"behaviour set it to False. This warning will disappear in kornia > v0.6."
)
warnings.warn(message)
# set default value for align corners
align_corners = True

B, C, H, W = src.size()
h_out, w_out = dsize

# we normalize the 3x3 transformation matrix and convert to 3x4
dst_norm_trans_src_norm: torch.Tensor = normalize_homography(M, (H, W), (h_out, w_out))  # Bx3x3

src_norm_trans_dst_norm = _torch_inverse_cast(dst_norm_trans_src_norm)  # Bx3x3

# this piece of code substitutes F.affine_grid since it does not support 3x3
grid = (
create_meshgrid(h_out, w_out, normalized_coordinates=True, device=src.device).to(src.dtype).repeat(B, 1, 1, 1)
)
grid = transform_points(src_norm_trans_dst_norm[:, None, None], grid)

[docs]def warp_affine(
src: torch.Tensor,
M: torch.Tensor,
dsize: Tuple[int, int],
mode: str = 'bilinear',
padding_mode: str = 'zeros',
align_corners: Optional[bool] = None,
) -> torch.Tensor:
r"""Applies an affine transformation to a tensor.

.. image:: _static/img/warp_affine.png

The function warp_affine transforms the source tensor using
the specified matrix:

.. math::
\text{dst}(x, y) = \text{src} \left( M_{11} x + M_{12} y + M_{13} ,
M_{21} x + M_{22} y + M_{23} \right )

Args:
src: input tensor of shape :math:(B, C, H, W).
M: affine transformation of shape :math:(B, 2, 3).
dsize: size of the output image (height, width).
mode: interpolation mode to calculate output values 'bilinear' | 'nearest'.
padding_mode (str): padding mode for outside grid values 'zeros' | 'border' | 'reflection'.
align_corners : mode for grid_generation.

Returns:
the warped tensor with shape :math:(B, C, H, W).

.. note::
This function is often used in conjunction with :func:get_rotation_matrix2d,
:func:get_shear_matrix2d, :func:get_affine_matrix2d, :func:invert_affine_transform.

.. note::
See a working example here <https://kornia-tutorials.readthedocs.io/en/latest/
rotate_affine.html>__.

Example:
>>> img = torch.rand(1, 4, 5, 6)
>>> A = torch.eye(2, 3)[None]
>>> out = warp_affine(img, A, (4, 2), align_corners=True)
>>> print(out.shape)
torch.Size([1, 4, 4, 2])
"""
if not isinstance(src, torch.Tensor):
raise TypeError(f"Input src type is not a torch.Tensor. Got {type(src)}")

if not isinstance(M, torch.Tensor):
raise TypeError(f"Input M type is not a torch.Tensor. Got {type(M)}")

if not len(src.shape) == 4:
raise ValueError(f"Input src must be a BxCxHxW tensor. Got {src.shape}")

if not (len(M.shape) == 3 or M.shape[-2:] == (2, 3)):
raise ValueError(f"Input M must be a Bx2x3 tensor. Got {M.shape}")

# TODO: remove the statement below in kornia v0.6
if align_corners is None:
message: str = (
"The align_corners default value has been changed. By default now is set True "
"in order to match cv2.warpAffine. In case you want to keep your previous "
"behaviour set it to False. This warning will disappear in kornia > v0.6."
)
warnings.warn(message)
# set default value for align corners
align_corners = True

B, C, H, W = src.size()

# we generate a 3x3 transformation matrix from 2x3 affine
M_3x3: torch.Tensor = convert_affinematrix_to_homography(M)
dst_norm_trans_src_norm: torch.Tensor = normalize_homography(M_3x3, (H, W), dsize)

# src_norm_trans_dst_norm = torch.inverse(dst_norm_trans_src_norm)
src_norm_trans_dst_norm = _torch_inverse_cast(dst_norm_trans_src_norm)

grid = F.affine_grid(src_norm_trans_dst_norm[:, :2, :], [B, C, dsize, dsize], align_corners=align_corners)

[docs]def get_perspective_transform(src, dst):
r"""Calculates a perspective transform from four pairs of the corresponding
points.

The function calculates the matrix of a perspective transform so that:

.. math ::

\begin{bmatrix}
t_{i}x_{i}^{'} \\
t_{i}y_{i}^{'} \\
t_{i} \\
\end{bmatrix}
=
\textbf{map_matrix} \cdot
\begin{bmatrix}
x_{i} \\
y_{i} \\
1 \\
\end{bmatrix}

where

.. math ::
dst(i) = (x_{i}^{'},y_{i}^{'}), src(i) = (x_{i}, y_{i}), i = 0,1,2,3

Args:
src: coordinates of quadrangle vertices in the source image with shape :math:(B, 4, 2).
dst: coordinates of the corresponding quadrangle vertices in
the destination image with shape :math:(B, 4, 2).

Returns:
the perspective transformation with shape :math:(B, 3, 3).

.. note::
This function is often used in conjunction with :func:warp_perspective.
"""
if not isinstance(src, torch.Tensor):
raise TypeError(f"Input type is not a torch.Tensor. Got {type(src)}")

if not isinstance(dst, torch.Tensor):
raise TypeError(f"Input type is not a torch.Tensor. Got {type(dst)}")

if not src.shape[-2:] == (4, 2):
raise ValueError(f"Inputs must be a Bx4x2 tensor. Got {src.shape}")

if not src.shape == dst.shape:
raise ValueError(f"Inputs must have the same shape. Got {dst.shape}")

if not (src.shape == dst.shape):
raise ValueError(f"Inputs must have same batch size dimension. Expect {src.shape} but got {dst.shape}")

# we build matrix A by using only 4 point correspondence. The linear
# system is solved with the least square method, so here
# we could even pass more correspondence
p = []
for i in [0, 1, 2, 3]:
p.append(_build_perspective_param(src[:, i], dst[:, i], 'x'))
p.append(_build_perspective_param(src[:, i], dst[:, i], 'y'))

# A is Bx8x8
A = torch.stack(p, dim=1)

# b is a Bx8x1
b = torch.stack(
[
dst[:, 0:1, 0],
dst[:, 0:1, 1],
dst[:, 1:2, 0],
dst[:, 1:2, 1],
dst[:, 2:3, 0],
dst[:, 2:3, 1],
dst[:, 3:4, 0],
dst[:, 3:4, 1],
],
dim=1,
)

# solve the system Ax = b
X, LU = _torch_solve_cast(b, A)

# create variable to return
batch_size = src.shape
M = torch.ones(batch_size, 9, device=src.device, dtype=src.dtype)
M[..., :8] = torch.squeeze(X, dim=-1)

return M.view(-1, 3, 3)  # Bx3x3

def _build_perspective_param(p: torch.Tensor, q: torch.Tensor, axis: str) -> torch.Tensor:
ones = torch.ones_like(p)[..., 0:1]
zeros = torch.zeros_like(p)[..., 0:1]
if axis == 'x':
[p[:, 0:1], p[:, 1:2], ones, zeros, zeros, zeros, -p[:, 0:1] * q[:, 0:1], -p[:, 1:2] * q[:, 0:1]], dim=1
)

if axis == 'y':
[zeros, zeros, zeros, p[:, 0:1], p[:, 1:2], ones, -p[:, 0:1] * q[:, 1:2], -p[:, 1:2] * q[:, 1:2]], dim=1
)

raise NotImplementedError(f"perspective params for axis {axis} is not implemented.")

def angle_to_rotation_matrix(angle: torch.Tensor) -> torch.Tensor:
r"""Create a rotation matrix out of angles in degrees.

Args:
angle: tensor of angles in degrees, any shape.

Returns:
tensor of *x2x2 rotation matrices.

Shape:
- Input: :math:(*)
- Output: :math:(*, 2, 2)

Example:
>>> input = torch.rand(1, 3)  # Nx3
>>> output = angle_to_rotation_matrix(input)  # Nx3x2x2
"""
cos_a: torch.Tensor = torch.cos(ang_rad)
sin_a: torch.Tensor = torch.sin(ang_rad)
return torch.stack([cos_a, sin_a, -sin_a, cos_a], dim=-1).view(*angle.shape, 2, 2)

[docs]def get_rotation_matrix2d(center: torch.Tensor, angle: torch.Tensor, scale: torch.Tensor) -> torch.Tensor:
r"""Calculates an affine matrix of 2D rotation.

The function calculates the following matrix:

.. math::
\begin{bmatrix}
\alpha & \beta & (1 - \alpha) \cdot \text{x}
- \beta \cdot \text{y} \\
-\beta & \alpha & \beta \cdot \text{x}
+ (1 - \alpha) \cdot \text{y}
\end{bmatrix}

where

.. math::
\alpha = \text{scale} \cdot cos(\text{angle}) \\
\beta = \text{scale} \cdot sin(\text{angle})

The transformation maps the rotation center to itself
If this is not the target, adjust the shift.

Args:
center: center of the rotation in the source image with shape :math:(B, 2).
angle: rotation angle in degrees. Positive values mean
counter-clockwise rotation (the coordinate origin is assumed to
be the top-left corner) with shape :math:(B).
scale: scale factor for x, y scaling with shape :math:(B, 2).

Returns:
the affine matrix of 2D rotation with shape :math:(B, 2, 3).

Example:
>>> center = torch.zeros(1, 2)
>>> scale = torch.ones((1, 2))
>>> angle = 45. * torch.ones(1)
>>> get_rotation_matrix2d(center, angle, scale)
tensor([[[ 0.7071,  0.7071,  0.0000],
[-0.7071,  0.7071,  0.0000]]])

.. note::
This function is often used in conjunction with :func:warp_affine.
"""
if not isinstance(center, torch.Tensor):
raise TypeError(f"Input center type is not a torch.Tensor. Got {type(center)}")

if not isinstance(angle, torch.Tensor):
raise TypeError(f"Input angle type is not a torch.Tensor. Got {type(angle)}")

if not isinstance(scale, torch.Tensor):
raise TypeError(f"Input scale type is not a torch.Tensor. Got {type(scale)}")

if not (len(center.shape) == 2 and center.shape == 2):
raise ValueError(f"Input center must be a Bx2 tensor. Got {center.shape}")

if not len(angle.shape) == 1:
raise ValueError(f"Input angle must be a B tensor. Got {angle.shape}")

if not (len(scale.shape) == 2 and scale.shape == 2):
raise ValueError(f"Input scale must be a Bx2 tensor. Got {scale.shape}")

if not (center.shape == angle.shape == scale.shape):
raise ValueError(
"Inputs must have same batch size dimension. Got center {}, angle {} and scale {}".format(
center.shape, angle.shape, scale.shape
)
)

if not (center.device == angle.device == scale.device) or not (center.dtype == angle.dtype == scale.dtype):
raise ValueError(
"Inputs must have same device Got center ({}, {}), angle ({}, {}) and scale ({}, {})".format(
center.device, center.dtype, angle.device, angle.dtype, scale.device, scale.dtype
)
)

# convert angle and apply scale
rotation_matrix: torch.Tensor = angle_to_rotation_matrix(angle)
scaling_matrix: torch.Tensor = torch.eye(2, device=rotation_matrix.device, dtype=rotation_matrix.dtype).repeat(
rotation_matrix.size(0), 1, 1
)

scaling_matrix = scaling_matrix * scale.unsqueeze(dim=2).repeat(1, 1, 2)
scaled_rotation: torch.Tensor = rotation_matrix @ scaling_matrix
alpha: torch.Tensor = scaled_rotation[:, 0, 0]
beta: torch.Tensor = scaled_rotation[:, 0, 1]

# unpack the center to x, y coordinates
x: torch.Tensor = center[..., 0]
y: torch.Tensor = center[..., 1]

# create output tensor
batch_size: int = center.shape
one = torch.tensor(1.0, device=center.device, dtype=center.dtype)
M: torch.Tensor = torch.zeros(batch_size, 2, 3, device=center.device, dtype=center.dtype)

M[..., 0:2, 0:2] = scaled_rotation
M[..., 0, 2] = (one - alpha) * x - beta * y
M[..., 1, 2] = beta * x + (one - alpha) * y
return M

[docs]def remap(
tensor: torch.Tensor,
map_x: torch.Tensor,
map_y: torch.Tensor,
mode: str = 'bilinear',
padding_mode: str = 'zeros',
align_corners: Optional[bool] = None,
normalized_coordinates: bool = False,
) -> torch.Tensor:
r"""Applies a generic geometrical transformation to a tensor.

.. image:: _static/img/remap.png

The function remap transforms the source tensor using the specified map:

.. math::
\text{dst}(x, y) = \text{src}(map_x(x, y), map_y(x, y))

Args:
tensor: the tensor to remap with shape (B, D, H, W).
Where D is the number of channels.
map_x: the flow in the x-direction in pixel coordinates.
The tensor must be in the shape of (B, H, W).
map_y: the flow in the y-direction in pixel coordinates.
The tensor must be in the shape of (B, H, W).
mode: interpolation mode to calculate output values
'bilinear' | 'nearest'.
padding_mode: padding mode for outside grid values
'zeros' | 'border' | 'reflection'.
align_corners: mode for grid_generation.
normalized_coordinates: whether the input coordinates are
normalised in the range of [-1, 1].

Returns:
the warped tensor with same shape as the input grid maps.

Example:
>>> from kornia.utils import create_meshgrid
>>> grid = create_meshgrid(2, 2, False)  # 1x2x2x2
>>> grid += 1  # apply offset in both directions
>>> input = torch.ones(1, 1, 2, 2)
>>> remap(input, grid[..., 0], grid[..., 1], align_corners=True)   # 1x1x2x2
tensor([[[[1., 0.],
[0., 0.]]]])

.. note::
This function is often used in conjunction with :func:kornia.utils.create_meshgrid.
"""
if not isinstance(tensor, torch.Tensor):
raise TypeError(f"Input tensor type is not a torch.Tensor. Got {type(tensor)}")

if not isinstance(map_x, torch.Tensor):
raise TypeError(f"Input map_x type is not a torch.Tensor. Got {type(map_x)}")

if not isinstance(map_y, torch.Tensor):
raise TypeError(f"Input map_y type is not a torch.Tensor. Got {type(map_y)}")

if not tensor.shape[-2:] == map_x.shape[-2:] == map_y.shape[-2:]:
raise ValueError("Inputs last two dimensions must match.")

batch_size, _, height, width = tensor.shape

# grid_sample need the grid between -1/1
map_xy: torch.Tensor = torch.stack([map_x, map_y], dim=-1)

# normalize coordinates if not already normalized
if not normalized_coordinates:
map_xy = normalize_pixel_coordinates(map_xy, height, width)

# simulate broadcasting since grid_sample does not support it
map_xy_norm: torch.Tensor = map_xy.expand(batch_size, -1, -1, -1)

# warp ans return
tensor_warped: torch.Tensor = F.grid_sample(
)
return tensor_warped

[docs]def invert_affine_transform(matrix: torch.Tensor) -> torch.Tensor:
r"""Inverts an affine transformation.

The function computes an inverse affine transformation represented by
2×3 matrix:

.. math::
\begin{bmatrix}
a_{11} & a_{12} & b_{1} \\
a_{21} & a_{22} & b_{2} \\
\end{bmatrix}

The result is also a 2×3 matrix of the same type as M.

Args:
matrix: original affine transform. The tensor must be
in the shape of :math:(B, 2, 3).

Return:
the reverse affine transform with shape :math:(B, 2, 3).

.. note::
This function is often used in conjunction with :func:warp_affine.
"""
if not isinstance(matrix, torch.Tensor):
raise TypeError(f"Input matrix type is not a torch.Tensor. Got {type(matrix)}")

if not (len(matrix.shape) == 3 and matrix.shape[-2:] == (2, 3)):
raise ValueError(f"Input matrix must be a Bx2x3 tensor. Got {matrix.shape}")

matrix_tmp: torch.Tensor = convert_affinematrix_to_homography(matrix)
matrix_inv: torch.Tensor = torch.inverse(matrix_tmp)

return matrix_inv[..., :2, :3]

[docs]def get_affine_matrix2d(
translations: torch.Tensor,
center: torch.Tensor,
scale: torch.Tensor,
angle: torch.Tensor,
sx: Optional[torch.Tensor] = None,
sy: Optional[torch.Tensor] = None,
) -> torch.Tensor:
r"""Composes affine matrix from the components.

Args:
translations: tensor containing the translation vector with shape :math:(B, 2).
center: tensor containing the center vector with shape :math:(B, 2).
scale: tensor containing the scale factor with shape :math:(B, 2).
angle: tensor of angles in degrees :math:(B).
sx: tensor containing the shear factor in the x-direction with shape :math:(B).
sy: tensor containing the shear factor in the y-direction with shape :math:(B).

Returns:
the affine transformation matrix :math:(B, 3, 3).

.. note::
This function is often used in conjunction with :func:warp_affine, :func:warp_perspective.
"""
transform: torch.Tensor = get_rotation_matrix2d(center, -angle, scale)
transform[..., 2] += translations  # tx/ty

# pad transform to get Bx3x3
transform_h = convert_affinematrix_to_homography(transform)

if any(s is not None for s in [sx, sy]):
shear_mat = get_shear_matrix2d(center, sx, sy)
transform_h = transform_h @ shear_mat

return transform_h

[docs]def get_shear_matrix2d(center: torch.Tensor, sx: Optional[torch.Tensor] = None, sy: Optional[torch.Tensor] = None):
r"""Composes shear matrix Bx4x4 from the components.

Note: Ordered shearing, shear x-axis then y-axis.

.. math::
\begin{bmatrix}
1 & b \\
a & ab + 1 \\
\end{bmatrix}

Args:
center: shearing center coordinates of (x, y).
sx: shearing degree along x axis.
sy: shearing degree along y axis.

Returns:
params to be passed to the affine transformation with shape :math:(B, 3, 3).

Examples:
>>> rng = torch.manual_seed(0)
>>> sx = torch.randn(1)
>>> sx
tensor([1.5410])
>>> center = torch.tensor([[0., 0.]])  # Bx2
>>> get_shear_matrix2d(center, sx=sx)
tensor([[[  1.0000, -33.5468,   0.0000],
[ -0.0000,   1.0000,   0.0000],
[  0.0000,   0.0000,   1.0000]]])

.. note::
This function is often used in conjunction with :func:warp_affine, :func:warp_perspective.
"""
sx = torch.tensor([0.0]).repeat(center.size(0)) if sx is None else sx
sy = torch.tensor([0.0]).repeat(center.size(0)) if sy is None else sy

x, y = torch.split(center, 1, dim=-1)
x, y = x.view(-1), y.view(-1)

sx_tan = torch.tan(sx)  # type: ignore
sy_tan = torch.tan(sy)  # type: ignore
ones = torch.ones_like(sx)  # type: ignore
shear_mat = torch.stack(
[
ones,
-sx_tan,
sx_tan * y,  # type: ignore   # noqa: E241
-sy_tan,
ones + sx_tan * sy_tan,
sy_tan * (sx_tan * y + x),  # noqa: E241
],
dim=-1,
).view(-1, 2, 3)

shear_mat = convert_affinematrix_to_homography(shear_mat)
return shear_mat

[docs]def get_affine_matrix3d(
translations: torch.Tensor,
center: torch.Tensor,
scale: torch.Tensor,
angles: torch.Tensor,
sxy: Optional[torch.Tensor] = None,
sxz: Optional[torch.Tensor] = None,
syx: Optional[torch.Tensor] = None,
syz: Optional[torch.Tensor] = None,
szx: Optional[torch.Tensor] = None,
szy: Optional[torch.Tensor] = None,
) -> torch.Tensor:
r"""Composes 3d affine matrix from the components.

Args:
translations: tensor containing the translation vector (dx,dy,dz) with shape :math:(B, 3).
center: tensor containing the center vector (x,y,z) with shape :math:(B, 3).
scale: tensor containing the scale factor with shape :math:(B).
angle: angle axis vector containing the rotation angles in degrees in the form
of (rx, ry, rz) with shape :math:(B, 3). Internally it calls Rodrigues to compute
the rotation matrix from axis-angle.
sxy: tensor containing the shear factor in the xy-direction with shape :math:(B).
sxz: tensor containing the shear factor in the xz-direction with shape :math:(B).
syx: tensor containing the shear factor in the yx-direction with shape :math:(B).
syz: tensor containing the shear factor in the yz-direction with shape :math:(B).
szx: tensor containing the shear factor in the zx-direction with shape :math:(B).
szy: tensor containing the shear factor in the zy-direction with shape :math:(B).

Returns:
the 3d affine transformation matrix :math:(B, 3, 3).

.. note::
This function is often used in conjunction with :func:warp_perspective.
"""
transform: torch.Tensor = get_projective_transform(center, -angles, scale)
transform[..., 3] += translations  # tx/ty/tz

# pad transform to get Bx3x3
transform_h = convert_affinematrix_to_homography3d(transform)
if any(s is not None for s in [sxy, sxz, syx, syz, szx, szy]):
shear_mat = get_shear_matrix3d(center, sxy, sxz, syx, syz, szx, szy)
transform_h = transform_h @ shear_mat

return transform_h

[docs]def get_shear_matrix3d(
center: torch.Tensor,
sxy: Optional[torch.Tensor] = None,
sxz: Optional[torch.Tensor] = None,
syx: Optional[torch.Tensor] = None,
syz: Optional[torch.Tensor] = None,
szx: Optional[torch.Tensor] = None,
szy: Optional[torch.Tensor] = None,
):
r"""Composes shear matrix Bx4x4 from the components.
Note: Ordered shearing, shear x-axis then y-axis then z-axis.

.. math::
\begin{bmatrix}
1 & o & r & oy + rz \\
m & p & s & mx + py + sz -y \\
n & q & t & nx + qy + tz -z \\
0 & 0 & 0 & 1  \\
\end{bmatrix}
Where:
m = S_{xy}
n = S_{xz}
o = S_{yx}
p = S_{xy}S_{yx} + 1
q = S_{xz}S_{yx} + S_{yz}
r = S_{zx} + S_{yx}S_{zy}
s = S_{xy}S_{zx} + (S_{xy}S_{yx} + 1)S_{zy}
t = S_{xz}S_{zx} + (S_{xz}S_{yx} + S_{yz})S_{zy} + 1

Params:
center: shearing center coordinates of (x, y, z).
sxy: shearing degree along x axis, towards y plane.
sxz: shearing degree along x axis, towards z plane.
syx: shearing degree along y axis, towards x plane.
syz: shearing degree along y axis, towards z plane.
szx: shearing degree along z axis, towards x plane.
szy: shearing degree along z axis, towards y plane.

Returns:
params to be passed to the affine transformation.

Examples:
>>> rng = torch.manual_seed(0)
>>> sxy, sxz, syx, syz = torch.randn(4, 1)
>>> sxy, sxz, syx, syz
(tensor([1.5410]), tensor([-0.2934]), tensor([-2.1788]), tensor([0.5684]))
>>> center = torch.tensor([[0., 0., 0.]])  # Bx3
>>> get_shear_matrix3d(center, sxy=sxy, sxz=sxz, syx=syx, syz=syz)
tensor([[[  1.0000,  -1.4369,   0.0000,   0.0000],
[-33.5468,  49.2039,   0.0000,   0.0000],
[  0.3022,  -1.0729,   1.0000,   0.0000],
[  0.0000,   0.0000,   0.0000,   1.0000]]])

.. note::
This function is often used in conjunction with :func:warp_perspective3d.
"""
sxy = torch.tensor([0.0]).repeat(center.size(0)) if sxy is None else sxy
sxz = torch.tensor([0.0]).repeat(center.size(0)) if sxz is None else sxz
syx = torch.tensor([0.0]).repeat(center.size(0)) if syx is None else syx
syz = torch.tensor([0.0]).repeat(center.size(0)) if syz is None else syz
szx = torch.tensor([0.0]).repeat(center.size(0)) if szx is None else szx
szy = torch.tensor([0.0]).repeat(center.size(0)) if szy is None else szy

x, y, z = torch.split(center, 1, dim=-1)
x, y, z = x.view(-1), y.view(-1), z.view(-1)
# Prepare parameters
sxy_tan = torch.tan(sxy)  # type: ignore
sxz_tan = torch.tan(sxz)  # type: ignore
syx_tan = torch.tan(syx)  # type: ignore
syz_tan = torch.tan(syz)  # type: ignore
szx_tan = torch.tan(szx)  # type: ignore
szy_tan = torch.tan(szy)  # type: ignore

# compute translation matrix
m00, m10, m20, m01, m11, m21, m02, m12, m22 = _compute_shear_matrix_3d(
sxy_tan, sxz_tan, syx_tan, syz_tan, szx_tan, szy_tan
)

m03 = m01 * y + m02 * z
m13 = m10 * x + m11 * y + m12 * z - y
m23 = m20 * x + m21 * y + m22 * z - z

# shear matrix is implemented with negative values
sxy_tan, sxz_tan, syx_tan, syz_tan, szx_tan, szy_tan = -sxy_tan, -sxz_tan, -syx_tan, -syz_tan, -szx_tan, -szy_tan
m00, m10, m20, m01, m11, m21, m02, m12, m22 = _compute_shear_matrix_3d(
sxy_tan, sxz_tan, syx_tan, syz_tan, szx_tan, szy_tan
)

shear_mat = torch.stack([m00, m01, m02, m03, m10, m11, m12, m13, m20, m21, m22, m23], dim=-1).view(-1, 3, 4)
shear_mat = convert_affinematrix_to_homography3d(shear_mat)

return shear_mat

def _compute_shear_matrix_3d(sxy_tan, sxz_tan, syx_tan, syz_tan, szx_tan, szy_tan):
zeros = torch.zeros_like(sxy_tan)  # type: ignore
ones = torch.ones_like(sxy_tan)  # type: ignore

m00, m10, m20 = ones, sxy_tan, sxz_tan
m01, m11, m21 = syx_tan, sxy_tan * syx_tan + ones, sxz_tan * syx_tan + syz_tan
m02 = syx_tan * szy_tan + szx_tan
m12 = sxy_tan * szx_tan + szy_tan * m11
m22 = sxz_tan * szx_tan + szy_tan * m21 + ones
return m00, m10, m20, m01, m11, m21, m02, m12, m22