metrics

Image similarity metrics and geodesic distances for camera poses

Image similarity metrics

Used to quantify the similarity between ground truth X-rays (\(\mathbf I\)) and synthetic X-rays generated from estimated camera poses (\(\hat{\mathbf I}\)). If a metric is differentiable, it can be used to optimize camera poses with DiffDRR.

NCC and GradNCC are originally implemented in diffdrr.metrics. DiffPose provides torchmetrics wrappers for these functions.

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GradientNormalizedCrossCorrelation

 GradientNormalizedCrossCorrelation (patch_size=None)

torchmetric wrapper for GradNCC.

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MultiscaleNormalizedCrossCorrelation

 MultiscaleNormalizedCrossCorrelation (patch_sizes, patch_weights)

torchmetric wrapper for Multiscale NCC.

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NormalizedCrossCorrelation

 NormalizedCrossCorrelation (patch_size=None)

torchmetric wrapper for NCC.

Geodesic distances for SO(3) and SE(3)

One can define geodesic pseudo-distances on SO(3) and SE(3).1 This let’s us measure registration error (in radians and millimeters, respectively) on poses, rather than needed to compute the projection of fiducials.

  • For SO(3), the geodesic distance between two rotation matrices \(\mathbf R_A\) and \(\mathbf R_B\) is \[\begin{equation} d_\theta(\mathbf R_A, \mathbf R_B; r) = r \left| \arccos \left( \frac{\mathrm{trace}(\mathbf R_A^* \mathbf R_B) - 1}{2} \right ) \right| \,, \end{equation}\] where \(r\), the source-to-detector radius, is used to convert radians to millimeters.

  • For SE(3), we decompose the transformation matrix into a rotation and a translation, i.e., \(\mathbf T = (\mathbf R, \mathbf t)\). Then, we compute the geodesic on translations (just Euclidean distance), \[\begin{equation} d_t(\mathbf t_A, \mathbf t_B) = \| \mathbf t_A - \mathbf t_B \|_2 \,. \end{equation}\] Finally, we compute the double geodesic on the rotations and translations: \[\begin{equation} d(\mathbf T_A, \mathbf T_B) = \sqrt{d_\theta(\mathbf R_A, \mathbf R_B)^2 + d_t(\mathbf t_A, \mathbf t_B)^2} \,. \end{equation}\]

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GeodesicTranslation

 GeodesicTranslation ()

Calculate the angular distance between two rotations in SO(3).

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GeodesicSO3

 GeodesicSO3 ()

Calculate the angular distance between two rotations in SO(3).

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GeodesicSE3

 GeodesicSE3 ()

Calculate the distance between transforms in the log-space of SE(3).

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DoubleGeodesic

 DoubleGeodesic (sdr:float, eps:float=0.0001)

Calculate the angular and translational geodesics between two SE(3) transformation matrices.

Type Default Details
sdr float Source-to-detector radius
eps float 0.0001 Avoid overflows in sqrt 
# SO(3) distance
geodesic_so3 = GeodesicSO3()

pose_1 = RigidTransform(
    torch.tensor([[0.1, 1.0, torch.pi]]),
    torch.ones(1, 3),
    parameterization="euler_angles",
    convention="ZYX",
)
pose_2 = RigidTransform(
    torch.tensor([[0.1, 1.0, torch.pi]]),
    torch.ones(1, 3),
    parameterization="euler_angles",
    convention="ZYX",
)

print(geodesic_so3(pose_1, pose_2))  # Angular distance in radians

pose_1 = RigidTransform(
    torch.tensor([[0.1, 1.0, torch.pi]]),
    torch.ones(1, 3),
    parameterization="euler_angles",
    convention="ZYX",
)
pose_2 = RigidTransform(
    torch.tensor([[0.1, 1.1, torch.pi]]),
    torch.ones(1, 3),
    parameterization="euler_angles",
    convention="ZYX",
)

print(geodesic_so3(pose_1, pose_2))  # Angular distance in radians
tensor([0.])
tensor([0.1000])
# SE(3) distance
geodesic_se3 = GeodesicSE3()

pose_1 = RigidTransform(
    torch.tensor([[0.1, 1.0, torch.pi]]),
    torch.ones(1, 3),
    parameterization="euler_angles",
    convention="ZYX",
)
pose_2 = RigidTransform(
    torch.tensor([[0.1, 1.1, torch.pi]]),
    torch.zeros(1, 3),
    parameterization="euler_angles",
    convention="ZYX",
)

geodesic_se3(pose_1, pose_2)
tensor([1.7355])
# Angular distance and translational distance both in mm
double_geodesic = DoubleGeodesic(1020 / 2)

pose_1 = RigidTransform(
    torch.tensor([[0.1, 1.0, torch.pi]]),
    torch.ones(1, 3),
    parameterization="euler_angles",
    convention="ZYX",
)
pose_2 = RigidTransform(
    torch.tensor([[0.1, 1.1, torch.pi]]),
    torch.zeros(1, 3),
    parameterization="euler_angles",
    convention="ZYX",
)

double_geodesic(pose_1, pose_2)
(tensor([51.0000]), tensor([1.7321]), tensor([51.0294]))