calibration

Rigid transforms with camera calibration matrices

An X-ray C-arm can be modeled as a pinhole camera with its own extrinsic and intrinsic matrices. This module provides utilities for parsing these matrices and working with rigid transforms.

Rigid transformations

We represent rigid transforms as \(4 \times 4\) matrices (following the right-handed convention of PyTorch3D),

\[\begin{equation} \begin{bmatrix} \mathbf R^T & \mathbf 0 \\ \mathbf t^T & 1 \end{bmatrix} \in \mathbf{SE}(3) \,, \end{equation}\]

where \(\mathbf R \in \mathbf{SO}(3)\) is a rotation matrix and \(\mathbf t\in \mathbb R^3\) represents a translation.

Note that since rotation matrices are orthogonal, we have a simple closed-form equation for the inverse: \[\begin{equation} \begin{bmatrix} \mathbf R^T & \mathbf 0 \\ \mathbf t^T & 1 \end{bmatrix}^{-1} = \begin{bmatrix} \mathbf R & \mathbf 0 \\ -\mathbf R \mathbf t & 1 \end{bmatrix} \,. \end{equation}\]

For convenience, we add a wrapper of pytorch3d.transforms.Transform3d that can be construced from a (batched) rotation matrix and translation vector. This module also includes the closed-form inverse specific to rigid transforms.

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RigidTransform

 RigidTransform (R:jaxtyping.Float[Tensor,'...'],
                 t:jaxtyping.Float[Tensor,'...3'],
                 parameterization:str='matrix',
                 convention:Optional[str]=None, device=None,
                 dtype=torch.float32)

Wrapper of pytorch3d.transforms.Transform3d with extra functionalities.

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convert

 convert (transform, input_parameterization, output_parameterization,
          input_convention=None, output_convention=None)

Convert between representations of SE(3).

Computing a perspective projection

Given an extrinsic and intrinsic camera matrix, we can compute the perspective projection of a batch of points. This is used for computing where fiducials in world coordinates get mapped onto the image plane.

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perspective_projection

 perspective_projection (extrinsic:__main__.RigidTransform,
                         intrinsic:jaxtyping.Float[Tensor,'33'],
                         x:jaxtyping.Float[Tensor,'bn3'])

	Type	Details
extrinsic	RigidTransform	Extrinsic camera matrix (world to camera)
intrinsic	Float[Tensor, ‘3 3’]	Intrinsic camera matrix (camera to image)
x	Float[Tensor, ‘b n 3’]	World coordinates
Returns	Float[Tensor, ‘b n 2’]