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

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

yielding,

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

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

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

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

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

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.