Compute the similarity between a fixed X-ray \(\mathbf I\) and a moving X-ray \(\mathbf{\hat I}\), where \(\mathbf{\hat I}\) is rendered from an estimated camera pose (registration) or volume (reconstruction).
We implement patchwise variants of the following metrics:
If patch_size=None, the similarity metric is computed over the entire image.
NormalizedCrossCorrelation2d (patch_size=None, eps=1e-05)
Compute Normalized Cross Correlation between two batches of images.
MultiscaleNormalizedCrossCorrelation2d (patch_sizes=[None], patch_weights=[1.0], eps=1e-05)
Compute Normalized Cross Correlation between two batches of images at multiple scales.
GradientNormalizedCrossCorrelation2d (patch_size=None, sigma=1.0, **kwargs)
Compute Normalized Cross Correlation between the image gradients of two batches of images.
MutualInformation (sigma=0.1, num_bins=256, epsilon=1e-10, normalize=True)
Mutual Information.
One can define geodesic pseudo-distances on \(\mathbf{SO}(3)\) and \(\mathbf{SE}(3)\).1 This let’s us measure registration error (in radians and millimeters, respectively) on poses, rather than needed to compute the projection of fiducials.
We implement two geodesics on \(\mathbf{SE}(3)\):
Given two rotation matrices \(\mathbf R_A, \mathbf R_B \in \mathbf{SO}(3)\), the angular distance between their axes of rotation is
\[ d_\theta(\mathbf R_A, \mathbf R_B) = \arccos \left( \frac{\mathrm{trace}(\mathbf R_A^T \mathbf R_B) - 1}{2} \right) = \| \log (\mathbf R_A^T \mathbf R_B) \| \,, \]
where \(\log(\cdot)\) is the logarithm map on \(\mathbf{SO}(3)\).2 Using the logarithm map on \(\mathbf{SE}(3)\), this generalizes to a geodesic loss function on camera poses \({\mathbf T}_A, {\mathbf T}_B \in \mathbf{SE}(3)\):
\[ \mathcal L_{\mathrm{log}}({\mathbf T}_A, {\mathbf T}_B) = \| \log({\mathbf T}_A^{-1} {\mathbf T}_B) \| \,. \]
LogGeodesicSE3 ()
Calculate the distance between transforms in the log-space of SE(3).
# SE(3) distance
geodesic_se3 = LogGeodesicSE3()
pose_1 = convert(
torch.tensor([[0.1, 1.0, torch.pi]]),
torch.ones(1, 3),
parameterization="euler_angles",
convention="ZYX",
)
pose_2 = convert(
torch.tensor([[0.1, 1.1, torch.pi]]),
torch.zeros(1, 3),
parameterization="euler_angles",
convention="ZYX",
)
geodesic_se3(pose_1, pose_2)tensor([1.7354])We can also formulate a geodesic distance on \(\mathbf{SE}(3)\) with units of length. Using the camera’s focal length \(f\), we convert the angular distance to an arc length:
\[ d_\theta(\mathbf R_A, \mathbf R_B; f) = \frac{f}{2} d_\theta(\mathbf R_A, \mathbf R_B) \,. \]
When combined with the Euclidean distance on the translations \(d_t(\mathbf t_A, \mathbf t_B) = \| \mathbf t_A - \mathbf t_B \|\), this yields the double geodesic loss on \(\mathbf{SE}(3)\):3
\[ \mathcal L_{\mathrm{geo}}({\mathbf T}_A, {\mathbf T}_B; f) = \sqrt{d^2_\theta(\mathbf R_A, \mathbf R_B; f) + d^2_t(\mathbf t_A, \mathbf t_B)} \,. \]
DoubleGeodesicSE3 (sdd:float, eps:float=1e-06)
Calculate the angular and translational geodesics between two SE(3) transformation matrices.
| Type | Default | Details | |
|---|---|---|---|
| sdd | float | Source-to-detector distance | |
| eps | float | 1e-06 | Avoid overflows in sqrt |
# Angular distance and translational distance both in mm
double_geodesic = DoubleGeodesicSE3(1020.0)
pose_1 = convert(
torch.tensor([[0.1, 1.0, torch.pi]]),
torch.ones(1, 3),
parameterization="euler_angles",
convention="ZYX",
)
pose_2 = convert(
torch.tensor([[0.1, 1.1, torch.pi]]),
torch.zeros(1, 3),
parameterization="euler_angles",
convention="ZYX",
)
double_geodesic(pose_1, pose_2) # Angular, translational, double geodesics(tensor([51.0000]), tensor([1.7321]), tensor([51.0294]))