Rigid Transformations and SE(3)

Camera calibration is fundamentally about estimating rigid transformations — rotations and translations that relate different coordinate frames: camera, world, robot gripper, calibration target, rig, etc. This chapter covers the rotation representations and transformation algebra used throughout calibration-rs.

Rotation Representations

Rotation Matrix $R\in \text{SO}(3)$

A $3\times 3$ orthogonal matrix with determinant $+1$:

$$R^{T}R=I,\det (R)=1$$

The set of all such matrices forms the special orthogonal group $\text{SO}(3)$. It has 3 degrees of freedom (9 entries minus 6 constraints).

Unit Quaternion

A 4-element vector $\mathbf{q}=[q_x,q_y,q_z,q_w{]}^T$ with unit norm $\Vert \mathbf{q}\Vert =1$. The rotation of a vector $\mathbf{v}$ is:

$${\mathbf{v}}^{\prime }=\mathbf{q}\otimes \mathbf{v}\otimes {\mathbf{q}}^{\ast }$$

where $\otimes$ is quaternion multiplication and ${\mathbf{q}}^{\ast }$ is the conjugate.

Quaternions are preferred for optimization because they avoid gimbal lock and have simpler interpolation properties than Euler angles. calibration-rs (via nalgebra) stores quaternions as [i, j, k, w] internally, which maps to $[q_x,q_y,q_z,q_w]$.

Axis-Angle

A rotation of angle $\theta$ around unit axis $\hat{\mathbf{a}}$ is represented as:

$$\mathbf{\omega }=\theta \hat{\mathbf{a}}\in {\mathbb{R}}^3$$

with $\theta =\Vert \mathbf{\omega }\Vert$ and $\hat{\mathbf{a}}=\mathbf{\omega }/\theta$.

The Rodrigues formula converts axis-angle to rotation matrix:

$$R=I+\frac{\sin \theta }{\theta }[\mathbf{\omega }{]}_{\times }+\frac{1-\cos \theta }{{\theta }^2}[\mathbf{\omega }{]}_{\times }^2$$

where $[\mathbf{\omega }{]}_{\times }$ is the skew-symmetric matrix of $\mathbf{\omega }$:

$$[\mathbf{\omega }{]}_{\times }=\left[\begin{array}{ccc}0& -{\omega }_z& {\omega }_y\\ {\omega }_z& 0& -{\omega }_x\\ -{\omega }_y& {\omega }_x& 0\end{array}\right]$$

Rigid Transformations: SE(3)

A rigid body transformation combines a rotation $R$ and translation $\mathbf{t}$:

$$T=\left[\begin{array}{cc}R& \mathbf{t}\\ 0^T& 1\end{array}\right]\in \text{SE}(3)$$

The special Euclidean group $\text{SE}(3)$ has 6 degrees of freedom (3 rotation + 3 translation).

Applying a Transform

A point $\mathbf{P}$ transformed by $T$:

$${\mathbf{P}}^{\prime }=R\mathbf{P}+\mathbf{t}$$

or in homogeneous coordinates:

$$\left[\begin{array}{c}{\mathbf{P}}^{\prime }\\ 1\end{array}\right]=T\left[\begin{array}{c}\mathbf{P}\\ 1\end{array}\right]$$

Composition

Transforms compose by matrix multiplication:

$$T_{A,C}=T_{A,B}\cdot T_{B,C}$$

Inverse

$$T^{-1}=\left[\begin{array}{cc}{R}^T& -R^T\mathbf{t}\\ 0^T& 1\end{array}\right]$$

Naming Convention

calibration-rs uses the notation $T_{A,B}$ to mean "the transform from frame $B$ to frame $A$":

$${\mathbf{P}}_A=T_{A,B}\cdot {\mathbf{P}}_B$$

Common transforms:

nalgebra Type: Iso3

calibration-rs uses nalgebra's Isometry3<f64> (aliased as Iso3) for SE(3) transforms:

#![allow(unused)]
fn main() {
pub type Iso3 = nalgebra::Isometry3<f64>;
}

This stores a UnitQuaternion (rotation) and a Translation3 (translation) separately, avoiding the redundancy of a full $4\times 4$ matrix.

SE(3) Storage for Optimization

In the optimization IR, SE(3) parameters are stored as a 7-element vector:

$$[q_x,q_y,q_z,q_w,t_x,t_y,t_z]$$

The quaternion $[q_x,q_y,q_z,q_w]$ is unit-constrained. The optimizer uses the SE(3) manifold (see Manifold Optimization) to maintain this constraint during parameter updates.

The Exponential Map

The Lie algebra $\mathfrak{se}(3)$ provides a 6-dimensional tangent space at the identity. A tangent vector $\mathbf{\xi }=[\mathbf{\omega },\mathbf{v}{]}^T\in {\mathbb{R}}^6$ maps to an SE(3) element via the exponential map:

$$T=\exp ({\mathbf{\xi }}^{\wedge })=\left[\begin{array}{cc}\exp ([\mathbf{\omega }{]}_{\times })& V\mathbf{v}\\ 0^T& 1\end{array}\right]$$

where $V$ is the left Jacobian of SO(3):

$$V=I+\frac{1-\cos \theta }{{\theta }^2}[\mathbf{\omega }{]}_{\times }+\frac{\theta -\sin \theta }{{\theta }^3}[\mathbf{\omega }{]}_{\times }^2$$

The exponential map is central to manifold optimization: parameter updates are computed in the tangent space and then retracted to the manifold (see Manifold Optimization).