Relationship with the Relative Pose of the Sensors

In Chapter 1.10, the relative pose was expressed using reference frames rigidly attached to the sensors. In computer vision applications, however, it is generally more convenient to work directly in camera coordinates.

The two formalisms are completely equivalent and differ only by the fixed rigid transformation relating the camera reference frame to that of the sensor.

Let $\prescript{c}{}{\boldsymbol\Pi}_{b}$ denote the rotation that transforms sensor coordinates into camera coordinates. Then

$$\prescript{c}{}{\mathbf{R}}_w = \prescript{c}{}{\boldsymbol\Pi}_{b} \prescript{w}{}{\mathbf{R}}^{-1}_{b}$$ (10.6)

and consequently

$$\prescript{w}{}{\mathbf{R}}_{b} = \prescript{c}{}{\mathbf{R}}^{-1}_{w} \prescript{c}{}{\boldsymbol\Pi}_{b}.$$ (10.7)

This relationship makes it possible to convert a relative pose expressed in camera coordinates into the corresponding relative pose expressed in sensor coordinates:

$$\begin{array}{l} \prescript{2b}{}{\mathbf{R}}_{1b} = \prescr... ...oldsymbol\Pi}^{-1}_{b} \prescript{2}{}{\mathbf{t}}. \end{array}$$ (10.8)

In the remainder of the chapter, the relative pose parameters expressed in camera coordinates will be used almost exclusively, namely the pair

$$\left( \prescript{2}{}{\mathbf{R}}_1, \prescript{2}{}{\mathbf{t}} \right)$$

defined in equation (10.4), since all epipolar relationships, triangulation techniques, and three-dimensional reconstruction algorithms are naturally formulated in this reference frame.