When discrete meets differential : AAAssessing the stability of structure from small motion

We provide a theoretical proof showing that under a proportional noise model, the discrete eight point algorithm behaves similarly to the differential eight point algorithm when the motion is small. This implies that the discrete algorithm can handle arbitrarily small motion for a general scene, as long as the noise decreases proportionally with the amount of image motion and the proportionality constant is small enough. This stability result extends to all normalized variants of the eight point algorithm. Using simulations, we show that given arbitrarily small motions and proportional noise regime, the normalized eight point algorithms outperform their differential counterparts by a large margin. Using real data, we show that in practical small motion problems involving optical flow, these discrete structure from motion (SFM) algorithms also provide better estimates than their differential counterparts, even when the motion magnitudes reach sub-pixel level. The better performance of these normalized discrete variants means that there is much to recommend them as differential SFM algorithms that are linear and normalized. © 2009 Springer Science+Business Media, LLC.