We want to understand the camera synchronization's impact on the accuracy of stereo matching. For this purpose, we assume two static cameras take images at times $t_0$ and $t_1$, which are not necessarily the same. Two effects need to be looked at:

• The object changes shape between the time points when the images are taken, making it impossible to effectively match features between the stereo images. It happens primarily to objects that are not solid, such as humans moving their limbs while walking.
• The object moves parallel to the baseline, changing the measured disparity and, hence, resulting in an incorrect depth measurement.

Since we cannot do anything about the first issue, we focus exclusively on the second.

$$ \Delta t = t_1 - t_0, \quad \text{ time between the left and the right cameras take images.}\\ V \quad \text{ - speed with thich the object is moving parallel to the baseline}\\ D = V \Delta t \quad \text{ - distance travelled by the object}\\ R^* = \frac{fB}{d^} \quad \text{ - real range to the object}\\ d^ \quad \text{ - idealized disparity assuming perfect synchronization}\\ f, B \quad \text{ - cameras' focal length and stereo baseline} $$

Due to the object’s movement parallel to the baseline, its measured disparity differs from the correct $d^*$ expected at $t_0$ by

$$ \Delta d = \frac{D}{R^*} f $$

It results in the observed range measurement being incorrect and equal to

$$ R^{obs} = \frac{f B}{d^* \pm \Delta d} $$

Some simple manipulation of this equation based on the definitions of $R^*$ and $\Delta d$ listed above gives

$$ R^{obs} = R^* \frac{B}{B \pm V \Delta t}, $$

where the sign in the denominator depends on the direction of object’s motion relative to the cameras. If the object moves in the direction of the camera that took the image at the earlier time $t_0$, the observed range is further than in reality. Contrary, if the object moves to the camera that took the image at the later time point $t_1$ , the object is perceived to be closer than in reality.

Important: this equation shows that the range measurement error depends on the synchronization error $\Delta t$ and the object’s speed $V$. The faster the object is moving and the worse the synchronization - the bigger the error.

In particular, the error is independent of the cameras’ focal lengths, so it does not depend on the field of view!

Relative error

The difference between the true range and the measurement incorrect due to synchronization is equal to

$$ \Delta R = | R^* - R^{obs}| = \frac{V\Delta t}{B \pm V \Delta t} R^*. $$

So the relative range measurement error can be defined as

$$ \frac{\Delta R}{R^*} = \frac{V \Delta t}{B \pm V \Delta t}. $$