Extrinsic Parameters

Definition

The extrinsic parameters describe the rigid transformation (rotation + translation) from world coordinates to camera coordinates. They say where the camera is and how it is oriented, not how it is built.

Formula

$$P_C=R(P_W-C)=R{P}_W+T,T=-RC$$

First translate by $-C$ (align the origins), then rotate by $R$ (align the axes).

Trap

$C$ is the camera center in world coordinates. $T$ is not the camera position. Only for $R=I$ is $T=-C$.

Columns of R

The columns of $R$ are the axes of the source frame (world) expressed in the target frame (camera). Because $R$ is orthogonal, $R^{-1}=R^T$, and $R^T$ holds the camera axes in world coordinates.

Appears in

• Lecture 2.1, World to camera: $P_C=R(P_W-C)$, stereo example, figuring out rotations
• Lecture 2.1, External parameters as R, T

Related concepts

• Rotation Matrix
• Intrinsic Parameters
• Projection Matrix
• Homogeneous Coordinates