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Subsections

Projective formalism

 

Pinhole camera

 
  
Figure 1: Pinhole Model. Projection of an objet of the 3-D space on the retinian plane
\begin{figure}
 \begin{center}
 \leavevmode

\setlength {\unitlength}{ 1.5cm}
 
...
 ...1.3){$C_z$}\end{picture}
\setlength {\unitlength}{1in}
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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.

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

 
 \begin{displaymath}
 P = AP_0{\cal R}_c\end{displaymath} (2)

where:

The parameters of ${\cal R}_c$ (i.e. rx,ry,rz,tx,ty,tz) 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].


next up previous
Next: Perspective projection into two views Up: Mosaic Tutorial Previous: Mosaic Tutorial
Imad Zoghlami
3/13/1998