Aligned Cameras and Triangulation in Camera Coordinates

The equations given above refer to a “sensor” or “world” reference frame. For completeness, and to introduce relations that will be used later, the equations for a “camera” reference frame are also given here.

To keep the sign of the baseline positive, let $b=\tilde{x}_2 - \tilde{x}_1$, $\tilde{y}_1=\tilde{y}_2=0$, and $\tilde{z}_1=\tilde{z}_2=0$. In this case, the left camera (with subscript 1) is at the origin of the reference frame.

In camera coordinates, the relations between the two images are written as

$$d = u_1 - u_2 = k_u \frac{ b }{ \tilde{z} }$$ (10.22)

for the disparity, and

$$\begin{array}{l} \tilde{x} = (u_1 - u_0) \dfrac{b}{d} \\ ... ... \dfrac{b}{d} \\ \tilde{z} = k_u \dfrac{b}{d} \\ \end{array}$$ (10.23)

for the equation of the three-dimensional point projected onto the point $(u_1, v)$ of the left camera, with disparity $d$.