# Source code for kornia.geometry.epipolar.metrics

```"""Module including useful metrics for Structure from Motion."""

import torch

import kornia

[docs]def sampson_epipolar_distance(
pts1: torch.Tensor, pts2: torch.Tensor, Fm: torch.Tensor, squared: bool = True, eps: float = 1e-8
) -> torch.Tensor:
r"""Returns Sampson distance for correspondences given the fundamental matrix.

Args:
pts1: correspondences from the left images with shape
(B, N, 2 or 3). If they are not homogeneous, converted automatically.
pts2: correspondences from the right images with shape
(B, N, 2 or 3). If they are not homogeneous, converted automatically.
Fm: Fundamental matrices with shape :math:`(B, 3, 3)`. Called Fm to
avoid ambiguity with torch.nn.functional.
squared: if True (default), the squared distance is returned.
eps: Small constant for safe sqrt.

Returns:
the computed Sampson distance with shape :math:`(B, N)`.

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

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

if pts1.size(-1) == 2:
pts1 = kornia.convert_points_to_homogeneous(pts1)

if pts2.size(-1) == 2:
pts2 = kornia.convert_points_to_homogeneous(pts2)

# From Hartley and Zisserman, Sampson error (11.9)
# sam =  (x'^T F x) ** 2 / (  (((Fx)_1**2) + (Fx)_2**2)) +  (((F^Tx')_1**2) + (F^Tx')_2**2)) )

# line1_in_2: torch.Tensor = (F @ pts1.permute(0,2,1)).permute(0,2,1)
# line2_in_1: torch.Tensor = (F.permute(0,2,1) @ pts2.permute(0,2,1)).permute(0,2,1)
# Instead we can just transpose F once and switch the order of multiplication
F_t: torch.Tensor = Fm.permute(0, 2, 1)
line1_in_2: torch.Tensor = pts1 @ F_t
line2_in_1: torch.Tensor = pts2 @ Fm

# numerator = (x'^T F x) ** 2
numerator: torch.Tensor = (pts2 * line1_in_2).sum(2).pow(2)

# denominator = (((Fx)_1**2) + (Fx)_2**2)) +  (((F^Tx')_1**2) + (F^Tx')_2**2))
denominator: torch.Tensor = line1_in_2[..., :2].norm(2, dim=2).pow(2) + line2_in_1[..., :2].norm(2, dim=2).pow(2)
out: torch.Tensor = numerator / denominator
if squared:
return out
return (out + eps).sqrt()

[docs]def symmetrical_epipolar_distance(
pts1: torch.Tensor, pts2: torch.Tensor, Fm: torch.Tensor, squared: bool = True, eps: float = 1e-8
) -> torch.Tensor:
r"""Returns symmetrical epipolar distance for correspondences given the fundamental matrix.

Args:
pts1: correspondences from the left images with shape
(B, N, 2 or 3). If they are not homogeneous, converted automatically.
pts2: correspondences from the right images with shape
(B, N, 2 or 3). If they are not homogeneous, converted automatically.
Fm: Fundamental matrices with shape :math:`(B, 3, 3)`. Called Fm to
avoid ambiguity with torch.nn.functional.
squared: if True (default), the squared distance is returned.
eps: Small constant for safe sqrt.

Returns:
the computed Symmetrical distance with shape :math:`(B, N)`.

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

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

if pts1.size(-1) == 2:
pts1 = kornia.convert_points_to_homogeneous(pts1)

if pts2.size(-1) == 2:
pts2 = kornia.convert_points_to_homogeneous(pts2)

# From Hartley and Zisserman, symmetric epipolar distance (11.10)
# sed = (x'^T F x) ** 2 /  (((Fx)_1**2) + (Fx)_2**2)) +  1/ (((F^Tx')_1**2) + (F^Tx')_2**2))

# line1_in_2: torch.Tensor = (F @ pts1.permute(0,2,1)).permute(0,2,1)
# line2_in_1: torch.Tensor = (F.permute(0,2,1) @ pts2.permute(0,2,1)).permute(0,2,1)

# Instead we can just transpose F once and switch the order of multiplication
F_t: torch.Tensor = Fm.permute(0, 2, 1)
line1_in_2: torch.Tensor = pts1 @ F_t
line2_in_1: torch.Tensor = pts2 @ Fm

# numerator = (x'^T F x) ** 2
numerator: torch.Tensor = (pts2 * line1_in_2).sum(2).pow(2)

# denominator_inv =  1/ (((Fx)_1**2) + (Fx)_2**2)) +  1/ (((F^Tx')_1**2) + (F^Tx')_2**2))
denominator_inv: torch.Tensor = 1.0 / (line1_in_2[..., :2].norm(2, dim=2).pow(2)) + 1.0 / (
line2_in_1[..., :2].norm(2, dim=2).pow(2)
)
out: torch.Tensor = numerator * denominator_inv
if squared:
return out
return (out + eps).sqrt()
```