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A simple model

In the simplest case where the projection center is placed at the origin of the world frame and the image plane is the plane $Z=1$, the projection process can be modeled as follows:

\begin{displaymath}
\begin{array}{cc}
x= \frac{X}{Z} & y= \frac{Y}{Z}
\end{array}\end{displaymath} (C1)

For a world point $(X,Y,Z)$ and the corresponding image point $(x,y)$. Using the homogeneous representation of the points a linear projection equation is obtained:
\begin{displaymath}
\left[\begin{array}{c} x \\ y\\ 1 \end{array} \right]
\sim
\...
...t]
\left[\begin{array}{c} X \\ Y \\ Z \\ 1 \end{array} \right]
\end{displaymath} (C2)

This projection is illustrated in Figure 3.2. The optical axis passes through the center of projection ${\tt C}$ and is orthogonal to the retinal plane ${\cal R}$. It's intersection with the retinal plane is defined as the principal point ${\tt c}$.
Figure 3.2: Perspective projection
\begin{figure}\centerline{
\psfig{figure=geometry/figures/PerspectiveCamera.ps, width=6cm}}\end{figure}



Marc Pollefeys 2002-11-22