Optimal Reconstruction with a Small Number of Views

Estimating positions of world points from features observed in images is a key problem in 3D reconstruction, image mosaicking, simultaneous localization and mapping and structure from motion. We consider a special instance in which there is a dominant ground plane $\mathcal{G}$ viewed from a parallel viewing plane $\mathcal{S}$ above it. Such instances commonly arise, for example, in aerial photography. Consider a world point $g \in \mathcal{G}$ and its worst case reconstruction uncertainty $\varepsilon(g,\mathcal{S})$ obtained by merging \emph{all} possible views of $g$ chosen from $\mathcal{S}$. We first show that one can pick two views $s_p$ and $s_q$ such that the uncertainty $\varepsilon(g,\{s_p,s_q\})$ obtained using only these two views is almost as good as (i.e. within a small constant factor of) $\varepsilon(g,\mathcal{S})$. Next, we extend the result to the entire ground plane $\mathcal{G}$ and show that one can pick a small subset of $\mathcal{S'} \subseteq \mathcal{S}$ (which grows only linearly with the area of $\mathcal{G}$) and still obtain a constant factor approximation, for every point $g \in \mathcal{G}$, to the minimum worst case estimate obtained by merging all views in $\mathcal{S}$. Our results provide a view selection mechanism with provable performance guarantees which can drastically increase the speed of scene reconstruction algorithms. In addition to theoretical results, we demonstrate their effectiveness in an application where aerial imagery is used for monitoring farms and orchards.