Recognizing Visibility Graphs of Polygons with Holes and Internal-External Visibility Graphs of Polygons

Visibility graph of a polygon corresponds to its internal diagonals and boundary edges. For each vertex on the boundary of the polygon, we have a vertex in this graph and if two vertices of the polygon see each other there is an edge between their corresponding vertices in the graph. Two vertices of a polygon see each other if and only if their connecting line segment completely lies inside the polygon, and they are externally visible if and only if this line segment completely lies outside the polygon. Recognizing visibility graphs is the problem of deciding whether there is a simple polygon whose visibility graph is isomorphic to a given input graph. This problem is well-known and well-studied, but yet widely open in geometric graphs and computational geometry. Existential Theory of the Reals is the complexity class of problems that can be reduced to the problem of deciding whether there exists a solution to a quantifier-free formula F(X1,X2,...,Xn), involving equalities and inequalities of real polynomials with real variables. The complete problems for this complexity class are called Existential Theory of the Reals Complete. In this paper we show that recognizing visibility graphs of polygons with holes is Existential Theory of the Reals Complete. Moreover, we show that recognizing visibility graphs of simple polygons when we have the internal and external visibility graphs, is also Existential Theory of the Reals Complete.Comment: Sumbitted to COCOON2018 Conferenc

Obstacle Numbers of Planar Graphs

Given finitely many connected polygonal obstacles $O_1,\dots,O_k$ in the plane and a set $P$ of points in general position and not in any obstacle, the {\em visibility graph} of $P$ with obstacles $O_1,\dots,O_k$ is the (geometric) graph with vertex set $P$, where two vertices are adjacent if the straight line segment joining them intersects no obstacle. The obstacle number of a graph $G$ is the smallest integer $k$ such that $G$ is the visibility graph of a set of points with $k$ obstacles. If $G$ is planar, we define the planar obstacle number of $G$ by further requiring that the visibility graph has no crossing edges (hence that it is a planar geometric drawing of $G$). In this paper, we prove that the maximum planar obstacle number of a planar graph of order $n$ is $n-3$, the maximum being attained (in particular) by maximal bipartite planar graphs. This displays a significant difference with the standard obstacle number, as we prove that the obstacle number of every bipartite planar graph (and more generally in the class PURE-2-DIR of intersection graphs of straight line segments in two directions) of order at least $3$ is $1$.Comment: Appears in the Proceedings of the 25th International Symposium on Graph Drawing and Network Visualization (GD 2017

View Selection with Geometric Uncertainty Modeling

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 $\mathcal{S}$. Finally, we present a multi-resolution view selection method which extends our techniques to non-planar scenes. We show that the method can produce rich and accurate dense reconstructions with a small number of views. 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