Camera Coordinate Transformation

The geometric relationships introduced in the following sections will be expressed directly in camera coordinates. Before addressing the problem of stereoscopic vision, it is therefore useful to derive the transformation relating the coordinates of a point observed in one view to the coordinates of the same point observed in a second view.

Let $\mathbf {x}$ be a point expressed in the world reference frame. In the pin-hole model, the coordinates of the point in the reference frame of camera i are

$$\prescript{i}{}{\mathbf{m}} = \prescript{i}{}{\mathbf{R}} \left( \mathbf{x} - \mathbf{t}_i \right)$$ (10.1)

where:

• $\prescript{i}{}{\mathbf{R}}$ is the rotation matrix that transforms world coordinates into camera coordinates;
• $\mathbf{t}_i$ is the position of the pin-hole of camera i, expressed in world coordinates;
• $\prescript{i}{}{\mathbf{m}}$ is the point expressed in the reference frame of camera i.

The inverse transformation is

$$\mathbf{x} = \prescript{i}{}{\mathbf{R}}^{-1} \prescript{i}{}{\mathbf{m}} + \mathbf{t}_i$$ (10.2)

and makes it possible to convert camera coordinates into the corresponding world coordinates.