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The Weak-Perspective Camera

The affine camera becomes a weak-perspective camera when the rows of ${\bf M}$ form a uniformly scaled rotation matrix. The simplest form is

\begin{displaymath}
{\bf T}_{wp} =
\left[ \begin{array}{cccc}
1 & 0 & 0 & 0 \...
...1 & 0 & 0 \\
0 & 0 & 0 & Z_{ave}/f \\
\end{array} \right]
\end{displaymath}

yielding,

\begin{displaymath}
{\bf M}_{wp} = \frac{f}{Z_{ave}}
\left[ \begin{array}{ccc}...
..._{ave}}
\left[ \begin{array}{c}
X \\ Y
\end{array} \right]
\end{displaymath}

This is simply the perspective equation with individual point depths $Z_{i}$ replaced by an average constant depth $Z_{ave}$

The weak-perspective model is valid when the average variation of the depth of the object $(\Delta Z)$ along the line of sight is small compared to the $Z_{ave}$ and the field of view is small. We see this as follows.

Expanding the perspective projection equation using a Taylor series, we obtain

\begin{displaymath}
{\bf x} = \frac{f}{Z_{ave} + \Delta Z}
\left[ \begin{array...
...\right)
\left[ \begin{array}{c}
X \\ Y
\end{array} \right]
\end{displaymath}

When $\vert \Delta Z \vert << Z_{ave}$ only the zero-order term remains giving the weak-perspective projection. The error in image position is then ${\bf x}_{err} = {\bf x}_{p} - {\bf x}_{wp}$:

\begin{displaymath}
{\bf x}_{err} = - \frac{f}{Z_{ave}}
\left(
\frac{\Delta Z...
...\right)
\left[ \begin{array}{c}
X \\ Y
\end{array} \right]
\end{displaymath}

showing that a small focal length $(f)$, small field of view $(X/Z_{ave}$ and $(Y/Z_{ave}$) and small depth variation $(\Delta Z)$ contribute to the validity of the model.



Subhashis Banerjee 2002-02-18