Relative Pose Between Two Cameras

Now consider the same point observed by two different cameras. Substituting the expression for the world coordinates obtained from the first camera into the equation for the second gives

$$\begin{array}{rl} \prescript{2}{}{\mathbf{m}} &= \prescript{... ...hbf{R}} \left( \mathbf{t}_2 - \mathbf{t}_1 \right). \end{array}$$ (10.3)

It is convenient to introduce the relative pose between the two camera coordinate systems by defining

$$\begin{array}{l} \prescript{2}{}{\mathbf{R}}_1 = \prescript{... ...thbf{R}} \left( \mathbf{t}_2 - \mathbf{t}_1 \right) \end{array}$$ (10.4)

where $\prescript{2}{}{\mathbf{R}}_1$ represents the rotation that transforms vectors expressed in the reference frame of camera 1 into the reference frame of camera 2, while $\prescript{2}{}{\mathbf{t}}$ represents the position of the optical center of camera 1 expressed in the coordinates of camera 2.

The relationship between the camera coordinates of the same observation in the two views therefore takes the compact form

$$\prescript{2}{}{\mathbf{m}} = \prescript{2}{}{\mathbf{R}}_1 \prescript{1}{}{\mathbf{m}} + \prescript{2}{}{\mathbf{t}}.$$ (10.5)

This equation will be used repeatedly throughout the remainder of the chapter. It is important to note that it depends exclusively on the relative pose of the two cameras and not on the characteristics of the observed scene.