1. 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):
[2] Fekete, Sándor P. "On simple polygonalizations with optimal area." Discrete & Computational Geometry 23.1 (2000):
[3] Shallow Survey
[4] Tran Kai Frank Da. 2D Alpha Shapes. In CGAL User and Reference Manual. CGAL Editorial Board, 4.7 edition,
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