Relative Positioning in Camera Coordinates and Sensor Coordinates

As already mentioned several times, to transform a camera reference system into a sensor reference system, it is sufficient to apply the transformation

$$\prescript{c}{}{\mathbf{R}}_w = \prescript{c}{}{\boldsymbol\... ...c}{}{\mathbf{R}}^{-1}_{w} \prescript{c}{}{\boldsymbol\Pi}_{b}$$ (9.5)

where $\prescript{c}{}{\mathbf{R}}_{w} = \mathbf{R}$ is the rotation matrix used in the equations of the pin-hole camera model.

For example, through these relationships, it is possible to obtain the relationships that connect the relative poses expressed in camera coordinates with those expressed in sensor coordinates as given by equation (1.66):

$$\begin{array}{l} \prescript{2b}{}{\mathbf{R}}_{1b} = \presc... ...thbf{R}} \left( \mathbf{t}_2 - \mathbf{t}_1 \right) \end{array}$$ (9.6)

with parameters $\prescript{1}{}{\mathbf{R}}$, $\mathbf {t}_1$, $\prescript{2}{}{\mathbf{R}}$, and $\mathbf{t}_2$ defined as in the pin-hole model, that is, matrices that transform from world to camera and vectors expressed in world coordinates. As can be seen, this result is fully compatible with that obtained in equation (9.4).

These relationships allow for the determination of the parameters of the relative pose $(\mathbf{R}, \mathbf{t})$ starting from the parameters of the pin-hole camera. These relationships enable the conversion of coordinates from sensor 1 to coordinates in sensor 2, as expressed in equation 1.64.