# Source code for kornia.geometry.epipolar.essential

```"""Module containing functionalities for the Essential matrix."""
from typing import Optional, Tuple

import torch

from kornia.utils import eye_like, vec_like
from kornia.utils.helpers import _torch_svd_cast

from .numeric import cross_product_matrix
from .projection import depth_from_point, projection_from_KRt
from .triangulation import triangulate_points

__all__ = [
"essential_from_fundamental",
"decompose_essential_matrix",
"essential_from_Rt",
"motion_from_essential",
"motion_from_essential_choose_solution",
"relative_camera_motion",
]

[docs]def essential_from_fundamental(F_mat: torch.Tensor, K1: torch.Tensor, K2: torch.Tensor) -> torch.Tensor:
r"""Get Essential matrix from Fundamental and Camera matrices.

Uses the method from Hartley/Zisserman 9.6 pag 257 (formula 9.12).

Args:
F_mat: The fundamental matrix with shape of :math:`(*, 3, 3)`.
K1: The camera matrix from first camera with shape :math:`(*, 3, 3)`.
K2: The camera matrix from second camera with shape :math:`(*, 3, 3)`.

Returns:
The essential matrix with shape :math:`(*, 3, 3)`.
"""
if not (len(F_mat.shape) >= 2 and F_mat.shape[-2:] == (3, 3)):
raise AssertionError(F_mat.shape)
if not (len(K1.shape) >= 2 and K1.shape[-2:] == (3, 3)):
raise AssertionError(K1.shape)
if not (len(K2.shape) >= 2 and K2.shape[-2:] == (3, 3)):
raise AssertionError(K2.shape)
if not len(F_mat.shape[:-2]) == len(K1.shape[:-2]) == len(K2.shape[:-2]):
raise AssertionError

return K2.transpose(-2, -1) @ F_mat @ K1

[docs]def decompose_essential_matrix(E_mat: torch.Tensor) -> Tuple[torch.Tensor, torch.Tensor, torch.Tensor]:
r"""Decompose an essential matrix to possible rotations and translation.

This function decomposes the essential matrix E using svd decomposition [96]
and give the possible solutions: :math:`R1, R2, t`.

Args:
E_mat: The essential matrix in the form of :math:`(*, 3, 3)`.

Returns:
A tuple containing the first and second possible rotation matrices and the translation vector.
The shape of the tensors with be same input :math:`[(*, 3, 3), (*, 3, 3), (*, 3, 1)]`.
"""
if not (len(E_mat.shape) >= 2 and E_mat.shape[-2:]):
raise AssertionError(E_mat.shape)

# decompose matrix by its singular values
U, _, V = _torch_svd_cast(E_mat)
Vt = V.transpose(-2, -1)

mask[..., -1:] *= -1.0  # fill last column with negative values

# avoid singularities
U = torch.where((torch.det(U) < 0.0)[..., None, None], U * mask, U)
Vt = torch.where((torch.det(Vt) < 0.0)[..., None, None], Vt * maskt, Vt)

W = cross_product_matrix(torch.tensor([[0.0, 0.0, 1.0]]).type_as(E_mat))
W[..., 2, 2] += 1.0

# reconstruct rotations and retrieve translation vector
U_W_Vt = U @ W @ Vt
U_Wt_Vt = U @ W.transpose(-2, -1) @ Vt

# return values
R1 = U_W_Vt
R2 = U_Wt_Vt
T = U[..., -1:]
return (R1, R2, T)

[docs]def essential_from_Rt(R1: torch.Tensor, t1: torch.Tensor, R2: torch.Tensor, t2: torch.Tensor) -> torch.Tensor:
r"""Get the Essential matrix from Camera motion (Rs and ts).

Reference: Hartley/Zisserman 9.6 pag 257 (formula 9.12)

Args:
R1: The first camera rotation matrix with shape :math:`(*, 3, 3)`.
t1: The first camera translation vector with shape :math:`(*, 3, 1)`.
R2: The second camera rotation matrix with shape :math:`(*, 3, 3)`.
t2: The second camera translation vector with shape :math:`(*, 3, 1)`.

Returns:
The Essential matrix with the shape :math:`(*, 3, 3)`.
"""
if not (len(R1.shape) >= 2 and R1.shape[-2:] == (3, 3)):
raise AssertionError(R1.shape)
if not (len(t1.shape) >= 2 and t1.shape[-2:] == (3, 1)):
raise AssertionError(t1.shape)
if not (len(R2.shape) >= 2 and R2.shape[-2:] == (3, 3)):
raise AssertionError(R2.shape)
if not (len(t2.shape) >= 2 and t2.shape[-2:] == (3, 1)):
raise AssertionError(t2.shape)

# first compute the camera relative motion
R, t = relative_camera_motion(R1, t1, R2, t2)

# get the cross product from relative translation vector
Tx = cross_product_matrix(t[..., 0])

return Tx @ R

[docs]def motion_from_essential(E_mat: torch.Tensor) -> Tuple[torch.Tensor, torch.Tensor]:
r"""Get Motion (R's and t's ) from Essential matrix.

Computes and return four possible poses exist for the decomposition of the Essential
matrix. The possible solutions are :math:`[R1,t], [R1,−t], [R2,t], [R2,−t]`.

Args:
E_mat: The essential matrix in the form of :math:`(*, 3, 3)`.

Returns:
The rotation and translation containing the four possible combination for the retrieved motion.
The tuple is as following :math:`[(*, 4, 3, 3), (*, 4, 3, 1)]`.
"""
if not (len(E_mat.shape) >= 2 and E_mat.shape[-2:] == (3, 3)):
raise AssertionError(E_mat.shape)

# decompose the essential matrix by its possible poses
R1, R2, t = decompose_essential_matrix(E_mat)

# compbine and returns the four possible solutions
Rs = torch.stack([R1, R1, R2, R2], dim=-3)
Ts = torch.stack([t, -t, t, -t], dim=-3)

return (Rs, Ts)

[docs]def motion_from_essential_choose_solution(
E_mat: torch.Tensor,
K1: torch.Tensor,
K2: torch.Tensor,
x1: torch.Tensor,
x2: torch.Tensor,
) -> Tuple[torch.Tensor, torch.Tensor, torch.Tensor]:
r"""Recover the relative camera rotation and the translation from an estimated essential matrix.

The method checks the corresponding points in two images and also returns the triangulated
3d points. Internally uses :py:meth:`~kornia.geometry.epipolar.decompose_essential_matrix` and then chooses
the best solution based on the combination that gives more 3d points in front of the camera plane from
:py:meth:`~kornia.geometry.epipolar.triangulate_points`.

Args:
E_mat: The essential matrix in the form of :math:`(*, 3, 3)`.
K1: The camera matrix from first camera with shape :math:`(*, 3, 3)`.
K2: The camera matrix from second camera with shape :math:`(*, 3, 3)`.
x1: The set of points seen from the first camera frame in the camera plane
coordinates with shape :math:`(*, N, 2)`.
x2: The set of points seen from the first camera frame in the camera plane
coordinates with shape :math:`(*, N, 2)`.
mask: A boolean mask which can be used to exclude some points from choosing
the best solution. This is useful for using this function with sets of points of
different cardinality (for instance after filtering with RANSAC) while keeping batch
semantics. Mask is of shape :math:`(*, N)`.

Returns:
The rotation and translation plus the 3d triangulated points.
The tuple is as following :math:`[(*, 3, 3), (*, 3, 1), (*, N, 3)]`.
"""
if not (len(E_mat.shape) >= 2 and E_mat.shape[-2:] == (3, 3)):
raise AssertionError(E_mat.shape)
if not (len(K1.shape) >= 2 and K1.shape[-2:] == (3, 3)):
raise AssertionError(K1.shape)
if not (len(K2.shape) >= 2 and K2.shape[-2:] == (3, 3)):
raise AssertionError(K2.shape)
if not (len(x1.shape) >= 2 and x1.shape[-1] == 2):
raise AssertionError(x1.shape)
if not (len(x2.shape) >= 2 and x2.shape[-1] == 2):
raise AssertionError(x2.shape)
if not len(E_mat.shape[:-2]) == len(K1.shape[:-2]) == len(K2.shape[:-2]):
raise AssertionError

unbatched = len(E_mat.shape) == 2

if unbatched:
# add a leading batch dimension. We will remove it at the end, before
# returning the results
E_mat = E_mat[None]
K1 = K1[None]
K2 = K2[None]
x1 = x1[None]
x2 = x2[None]

# compute four possible pose solutions
Rs, ts = motion_from_essential(E_mat)

# set reference view pose and compute projection matrix
R1 = eye_like(3, E_mat)  # Bx3x3
t1 = vec_like(3, E_mat)  # Bx3x1

# compute the projection matrices for first camera
R1 = R1[:, None].expand(-1, 4, -1, -1)
t1 = t1[:, None].expand(-1, 4, -1, -1)
K1 = K1[:, None].expand(-1, 4, -1, -1)
P1 = projection_from_KRt(K1, R1, t1)  # 1x4x4x4

# compute the projection matrices for second camera
R2 = Rs
t2 = ts
K2 = K2[:, None].expand(-1, 4, -1, -1)
P2 = projection_from_KRt(K2, R2, t2)  # Bx4x4x4

# triangulate the points
x1 = x1[:, None].expand(-1, 4, -1, -1)
x2 = x2[:, None].expand(-1, 4, -1, -1)
X = triangulate_points(P1, P2, x1, x2)  # Bx4xNx3

# project points and compute their depth values
d1 = depth_from_point(R1, t1, X)
d2 = depth_from_point(R2, t2, X)

# verify the point values that have a positive depth value
depth_mask = (d1 > 0.0) & (d2 > 0.0)

# get pose and points 3d and return
R_out = Rs[:, mask_indices][:, 0, 0]
t_out = ts[:, mask_indices][:, 0, 0]
points3d_out = X[:, mask_indices][:, 0, 0]

if unbatched:
R_out = R_out[0]
t_out = t_out[0]
points3d_out = points3d_out[0]

return R_out, t_out, points3d_out

[docs]def relative_camera_motion(
R1: torch.Tensor, t1: torch.Tensor, R2: torch.Tensor, t2: torch.Tensor
) -> Tuple[torch.Tensor, torch.Tensor]:
r"""Compute the relative camera motion between two cameras.

Given the motion parameters of two cameras, computes the motion parameters of the second
one assuming the first one to be at the origin. If :math:`T1` and :math:`T2` are the camera motions,
the computed relative motion is :math:`T = T_{2}T^{−1}_{1}`.

Args:
R1: The first camera rotation matrix with shape :math:`(*, 3, 3)`.
t1: The first camera translation vector with shape :math:`(*, 3, 1)`.
R2: The second camera rotation matrix with shape :math:`(*, 3, 3)`.
t2: The second camera translation vector with shape :math:`(*, 3, 1)`.

Returns:
A tuple with the relative rotation matrix and
translation vector with the shape of :math:`[(*, 3, 3), (*, 3, 1)]`.
"""
if not (len(R1.shape) >= 2 and R1.shape[-2:] == (3, 3)):
raise AssertionError(R1.shape)
if not (len(t1.shape) >= 2 and t1.shape[-2:] == (3, 1)):
raise AssertionError(t1.shape)
if not (len(R2.shape) >= 2 and R2.shape[-2:] == (3, 3)):
raise AssertionError(R2.shape)
if not (len(t2.shape) >= 2 and t2.shape[-2:] == (3, 1)):
raise AssertionError(t2.shape)

# compute first the relative rotation
R = R2 @ R1.transpose(-2, -1)

# compute the relative translation vector
t = t2 - R @ t1

return (R, t)
```