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

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

yielding,

$\displaystyle {\bf M}_{wp} = \frac{f}{Z_{ave}}
\left[ \begin{array}{ccc}
1 & 0 & 0 \\
0 & 1 & 0 \\
\end{array}\right]$   and$\displaystyle \left[ \begin{array}{c}
x \\ y
\end{array}\right]
= \frac{f}{Z_{ave}}
\left[ \begin{array}{c}
X \\ Y
\end{array}\right]
$

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

$\displaystyle {\bf x} = \frac{f}{Z_{ave} + \Delta Z}
\left[ \begin{array}{c}
X ...
...\right)^{2} - \ldots
\right)
\left[ \begin{array}{c}
X \\ Y
\end{array}\right]
$

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}$ :

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

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 2008-01-20