Projective formalism

 

Pinhole camera

 

  

Figure 1: Pinhole Model. Projection of an objet of the 3-D space on the retinian plane

The pinhole camera is the most common camera model in Computer Vision.

We represent a pinhole camera by a plane $\Pi^r$ called retinian plane and a point C called optical center. Let M be a 3D point with the coordinates (X Y Z T)^t in the projective space ${\cal P}^n$.In figure 1 we show how we project the point M in m ( T u, T v,T), its image on the retinian plane $\Pi^r$. The formula 1 gives us the relation between M and m, P being the projection matrix. The projection matrix P can be decomposed into a product of three matrixes as we show in formula 2.

 

$$\left ( \begin {array}{c} T u \ T v \ T \ \end{array... ... \begin {array}{c} X \ Y \ Z \ T \ \end{array} \right)$$ (1)

 

$$P = AP_0{\cal R}_c$$ (2)

where:

• ${\cal R}_c$, the 3D coordinates change matrix:  

  $${\cal R}_c = \left ( \begin {array}{c c} R & t\ 0^t & 1\ \end{array} \right)$$ (3)

• P₀, the standard projection :

  $$P_0 = \left({I}_3 ~\vert~ 0 \right)$$ (4)

• A, the 2D coordinates change matrix:

  $${A} = \left( \begin{array} {ccc} \alpha_u & -\alpha_u\cot\... ... & \alpha_v\sin\theta & v_0 \ 0 & 0 & 1 \end{array} \right)$$ (5)

The parameters of ${\cal R}_c$ (i.e. r_x,r_y,r_z,t_x,t_y,t_z) are called the extrinsic parameters and the parameters of A (i.e. $\alpha_u,\alpha_v,u_0,v_0,\theta$) are called the intrinsic parameters. The parameter $\theta$ (the angle between the rows and columns of the camera) is usually set to $\frac{\pi}{2}$[Vai90].