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

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

yielding,

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

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

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

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 ${\bf x}_{err} = {\bf x}_{p} - {\bf x}_{wp}$:

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

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.