Aligned Cameras and Triangulation in Camera Coordinates

The equations presented earlier refer to a "sensor" or "world" reference frame. For completeness, and to introduce relationships that will be used later, the equations in the case of a "camera" reference frame are now provided.

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

In camera coordinates, the relationships between the two images can be expressed as

$$d = u_1 - u_2 = k_u \frac{ b }{ \tilde{z} }$$ (9.20)

for the disparity and

$$\begin{array}{l} \tilde{x} = (u_1 - u_0) \dfrac{b}{d} \\ ... ... \dfrac{b}{d} \\ \tilde{z} = k_u \dfrac{b}{d} \\ \end{array}$$ (9.21)

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