Finding a Hausdorff Core of a Polygon: On Convex Polygon Containment with Bounded Hausdorff Distance Reza Dorrigiv, Stephane Durocher, Arash Farzan, Robert Fraser, Alejandro López-Ortiz, J. Ian Munro, Alejandro Salinger, and Matthew Skala

Outline Hausdorff Distance Polygonal Approximation Hausdorff vs. MACS (potato, skull) Polygonal Approximation Enclosure vs. Inclusion Disk-based Approximate Solution Decision Version Optimization Version (via parametric search) Analysis

Hausdorff Distance Hausdorff Example

Our Problem Given a simple polygon P and a metric d, a d-core of P is a convex polygon Q contained in P that minimizes d(P,Q). We are interested in Hausdorff distance, which we denote as H(P,Q). Does there exist a convex polygon Q contained in P such that H(P,Q) <= k? Note: the 1-centre of a polygon P is the point c which minimizes H(P,c).

Application Approximating Contours for Navigation Given: a set of contours representing the ocean floor; a proposed path for a ship. Want to quickly determine whether the path is feasible. Use a hierarchical decomposition of convex regions, which are monotonic with respect to the original curve.

Why Hausdorff? There are many measures. Hausdorff distance is intended to capture the “closeness” of two objects.

Polygonal Approximation Enclosure Problems Convex Hull Smallest Enclosing k-gon Inclusion Problems Convex Skull (potato peeling, MACS) Largest Included k-gon Hausdorff Approximation

Hausdorff Approximation Approximating a convex polygon solved by Lopez and Reisner. min-# min-ε Hausdorff approximation when 1-centre is contained in P solved by Chassery and Coeurjolly.

Hausdorff Core Properties . Only need to consider points on the convex hull of P. Circles of radius k centred on the vertices of P will intersect the solution if H(P,Q) <= k

One Point in the Solution Can we say anything definite about what the solution may contain? If the solution does not contain the 1-centre of the polygon, then it can intersect at least one vertex of P.

Proposed Solution Sketch Given a radius r, if it is possible to form a simple closed convex path touching each circle, a solution exists for H(P,Q) = r. Explore path using a “right-most first” approach. Finally, determine the minimum disk radius for which a solution exists.

Why is this challenging? Tri Star

Why is this challenging? (2) Evil E

Approximation Algorithm No obvious way to solve exact problem. We propose an approximation scheme based upon a discretization of space. Intuition: Divide the disks into discrete segments. Grow the disks slightly such that at least one segment of each expanded disk will be contained in the optimal solution. Check for a solution among all intervals on all disks.

Decision Version Given r. Does a solution Q’ exist where H(P,Q’)<=r+2ε’? We grow the circles by ε’, and search for a convex path contacting each disk and contained in P. What is ε’? Suppose we are given ε as input, then we compute ε’=ε∙dvf, where dvf is the distance from the 1-centre to the farthest vertex.

Discrete Disk Intervals Disks have radius r+ε’, what length should the intervals on the disks be? Required: If an exact solution Q exists for radius r, then we must find an approximate solution. Therefore, at least one interval on each expanded disk must be contained in Q. Need to know minimum angle in solution.

Minimum Angle in Solution Can we find a straight line segment that would serve as a solution? Consider a circular segment C of radius 2∙dvf and arc length ε’. The interior angle of this segment is ε/2. If Q’ can be covered by C, then a straight line segment Qℓ exists such that H(Q’,Qℓ)<ε’.

How to find an approximating line? Assume that an approximating line exists, i.e. H(Q’,Qℓ)<ε’. Grow all disks by ε’ so that they have radius r+2ε’. The approximating line must intersect each disk. The line must also intersect p, which may be the 1-centre or a vertex of P. Build an visibility interval graph from p to each disk. Given an interval graph containing an intersection of all disks, we have a solution. Big Picture: If there exists a solution Q s.t. H(P,Q)<=r, and if Q can be approximated s.t. H(Q,Qℓ)<2ε’, we find Qℓ .

Else If we cannot approximate with a line, we try to approximate with a convex polygon. We know the interior angles of Q are all greater than ε/2. The intervals of the expanded disks can be defined such that at least one interval of each disk is contained in Q if it exists. Intervals are of length 12π/(ε2∙dvf). Find a solution by finding a set of intervals with mutual strong visibility (using dynamic programming). Solution is of the form Q={q1,q2,…qk}, where qi is a point on the ith disk in an interval contained in the solution.

Approximation Properties If there exists a solution Q, we are guaranteed to find an approximate solution Q’. Conversely, if we do not find an approximate solution, we know that no solution exists for the given radius r. If we find an approximate solution Q’, it does not imply that a solution exists for disks of radius r. Given these properties, it is straightforward to create a minimization version of the algorithm.

Minimization version Perform binary search over possible values for the radius of the disks in [0…dvf]. At each iteration, execute decision algorithm. If a solution is found at radius ri, shrink the disks s.t. ri+1=ri-dvf/2i. Otherwise, grow the disks s.t. ri+1=ri+dvf/2i. Our algorithm finishes when a solution is found for a radius ri, and none exists for ri-ε’. O(log(ε-1)) iterations to find solution. Solution Q’ is found s.t. H(P,Q’)<rOPT+3ε’.

Space and Time Find constrained 1-centre(s) - O(n2) Test for line solution - O(n3) Intervals on each disk - O(ε-2) Pairwise combinations of intervals - O(ε-4) Dynamic programming table entries - O(nε-4) DP table requires O(n2ε-4) space (each entry is O(n)) DP algorithm takes O(nε-6) time (checking O(ε-2) intervals for each entry in the table) Possibly O(n) 1-centres, so decision version takes O(n3+n2ε-6) time Minimization version - O((n3+n2ε-6)∙log(ε-1)) time

Conclusions and Context Presented an approximate Hausdorff core algorithm for simple polygons, leaving the exact version open. Polynomial, but impractical. Improved versions may be used as part of a system for the hierarchical decomposition of contour regions.

Future Work Explore different error metrics. Allow some degree of non-convexity in the approximate polygon. Determine better approximation technique or exact technique if possible.

Thanks NSERC Strategic Grant Shameless promotion: (for optimal data structures for organization and retrieval of spatial data) Shameless promotion: LATIN 2010 submission deadline Sept. 21st http://latin10.org April 19-23, 2010 in Oaxaca, Mexico