Perspective Projection

• The image point $I$ is the set of coordinates of the object as projected on the camera plane

$$I=\lambda \left[\begin{array}{c}x\\ y\\ 1\end{array}\right]$$

• The camera’s Intrinsic calibration matrix

$$IC=\left[\begin{array}{ccc}f& 0& 0\\ 0& f& 0\\ 0& 0& 1\end{array}\right]$$

Intrinsic Matrix

In reality, the intrinsic matrix will be more complicated than this!

• The projection matrix

$$P=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\end{array}\right]$$

• The Euclidean transformation between the world and camera frames called the extrinsic calibration matrix, which is often an identity matrix

$$EC=\left[\begin{array}{cccc}{r}_{11}& r_{12}& r_{13}& t_1\\ r_{21}& r_{22}& r_{23}& t_2\\ r_{31}& r_{32}& r_{33}& t_3\\ 0& 0& 0& 1\end{array}\right]$$

• The world point $X$ a.k.a the point that is being projected

The final projection equation is

$$I=IC\times P\times EC\times X$$