# Camera Extrinsics

First introduced through Visual SLAM book. Really understood through Cyrill Stachniss.

Extrinsics

In robots or autonomous vehicles,

extrinsicsis often defined as the transform between the camera coordinate system and the robot body coordinate system, describing where the camera is installed.

Difference with intrinsics?

Extrinsics change whenever the the camera is moved, where intrinsics is a 1-time thing.

#### External Parameters

Notice that we defined $P$ as in the camera coordinate system, but in fact, the coordinates of $P$ should be its world coordinates because the camera is moving (we use the symbol $P_{w}$). It should be converted to the camera coordinate system based on the current pose of the camera.

The camera’s pose is described by its rotation matrix $R$ and the translation vector $t$. Then:

Z \mathbf{P}_{uv}= Z \left[ \begin{array}{l} u\\ v\\ 1 \end{array} \right] = \mathbf{K} \left( {\mathbf{R}{ \mathbf{P}_w} + \mathbf{t}} \right) = \mathbf{K} \mathbf{T} \mathbf{P}_w \end{equation}$$ Note that the latter formula implies a conversion from homogeneous to non-homogeneous coordinates (can you see it?) We use homogeneous coordinates in $\mathbf{T}\mathbf{P}$, then convert to non-homogeneous coordinates, and then multiply it by $\mathbf{K}$. It describes the projection relationship of world coordinates to pixel coordinates of $P$. Among them, the camera's pose $\mathbf{R}, \mathbf{t}$ is also called the camera's **extrinsics**. ### Related - [[notes/Camera Intrinsics|Camera Intrinsics]]