Size Limited Bounding Polygons for Planar Point Sets Stuart A. MacGillivray and Bradford G. Nickerson Faculty of Computer Science, University of New Brunswick, Fredericton, New Brunswick, Canada 1 1 1 1 Problem Definition Non-Convex Approaches Massive point sets arising from geo-referenced observations Polygonal approximation of their boundary is necessary for data management; e.g. updating determining overlap to resolve spatial conflict area approximation Ideal polygon minimizes two 'size' metrics: complexity, i.e. number of vertices/edges area covered by polygon Given a set S of N points in the plane, the k-MIN AREA problem is defined as: Minimum Area k-sided Boundary (k-MIN AREA): Find a (weakly) simple k-sided polygon P with minimum area containing all points in S. Concavities can give better definitions of shape of a point set α-shape algorithm [1] computes concave boundary using Delaunay triangulation Parameter α defines the radius of a circle where no points can exist; α approaching -∞ produces isolated points. Can be computed in O(N log N) time on N points Minimum complexity of shape starts at convex hull Fig. 3(b) [1] [4] Convex Approaches Convex polygons are easier to compute Lower bounds on area required due to convexity Bounding box: Fixed complexity, computed in O(N) time Convex Hull: Worst-case O(N) complexity, O(N log N) time Worst-case area can be large, e.g. the example below with convex hull covering approximately five times the minimal area. Triangulation of a subset of the 2,124,850 points from the preceding example. Computing k-sided Boundaries Determine the best bounding polygon with at most k sides. Approach 1: Simplification Start with a convex hull or alpha-shape Incrementally remove vertices/edges by extending edges to meet at new points, producing a simpler polygon containing its predecessor. Choose the extension that minimizes additional area. Repeat until the shape or hull has been reduced to k sides. Approach 2: Building Start with simple bounding box Incrementally 'slice' off largest triangle containing no points Repeat until polygon containing P has k sides. Result is convex; can we add concavities efficiently? Point set S sampled from Shallow Survey 2012 Common Dataset [3]. |S| = N = 2,124,850 points Extents: 860 meters by 278 meters Bounding Box Area: 238,343 square meters (using k = 4 points) Convex Hull Area: 147,075 square meters (using k = 56 points) Approximate point coverage: 29,682 square meters Open Questions Is there a minimum area k-sided polygonal boundary? Yes, for an axis aligned bounding rectangle, k = 4 Unknown for any bounding triangle or rectangle? k-sided bounding polygon with axis aligned edges? Is the k-MIN AREA problem NP-complete? Fekete [2] proved that finding the N-sided polygon with minimal area cannot be done in polynomial time, and is NP-complete. [1] Edelsbrunner, Herbert, David G. Kirkpatrick, and Raimund Seidel. "On the shape of a set of points in the plane." Information Theory, IEEE Transactions on 29.4 (1983): 551-559. [2] Fekete, Sándor P. "On simple polygonalizations with optimal area." Discrete & Computational Geometry 23.1 (2000): 73-110. [3] Shallow Survey 2012. http://www.conference.co.nz/shallowsurvey [4] Tran Kai Frank Da. 2D Alpha Shapes. In CGAL User and Reference Manual. CGAL Editorial Board, 4.7 edition, 2015. http://doc.cgal.org/latest/Alpha_shapes_2/index.html Sponsored by: