Polygon Guarding with Orientation

ICRA 2014

Transcript

1. ROBOTIC SENSOR NETWORKS http://rsn.cs.umn.edu/ UNIVERSITY OF MINNESOTA Driven to Discover Polygon Guarding with Orientation Pratap Tokekar Volkan Isler
2. Polygon Guarding Problem What is the minimum number of “guards” (omnidirectional cameras) sufficient to see every point in an environment? 6/8/2014 2 "Polygon Guarding with Orientation." Pratap Tokekar and Volkan Isler.
3. Polygon Guarding Problem What is the minimum number of “guards” (omnidirectional cameras) sufficient to see every point in an environment? 6/8/2014 3 "Polygon Guarding with Orientation." Pratap Tokekar and Volkan Isler.
4. Polygon Guarding Problem • Applications – security, behavior analysis, trail cameras, forest-fire detection, etc. • Cameras are ubiquitous in robotics • Understanding limitations of visibility is important 6/8/2014 4 "Polygon Guarding with Orientation." Pratap Tokekar and Volkan Isler.
5. • Environment is an   sided 2D polygon   guards are sometimes necessary and always sufficient to see every point in the environment. – Any n-sided polygon can be guarded with   guards – Some n-sided polygon need   guards Art Gallery Theorem [Chvátal, 1975] 6/8/2014 5 "Polygon Guarding with Orientation." Pratap Tokekar and Volkan Isler.
6. • Environment is an   sided 2D polygon   guards are sometimes necessary and always sufficient to see every point in the environment. – Any n-sided polygon can be guarded with   guards – Some n-sided polygon need   guards Art Gallery Theorem [Chvátal, 1975] 6/8/2014 6 "Polygon Guarding with Orientation." Pratap Tokekar and Volkan Isler.
7. This Paper: Polygon Guarding with Target Orientation • Standard formulation does not handle self-occlusions 6/8/2014 7 "Polygon Guarding with Orientation." Pratap Tokekar and Volkan Isler.
8. This Paper: Polygon Guarding with Target Orientation • Standard formulation does not handle self-occlusions 6/8/2014 8 No front view of the person "Polygon Guarding with Orientation." Pratap Tokekar and Volkan Isler.
9. This Paper: Polygon Guarding with Target Orientation • Standard formulation does not handle self-occlusions • Many applications need a “good” view of the target 6/8/2014 9 No front view of the person "Polygon Guarding with Orientation." Pratap Tokekar and Volkan Isler.
10. This Paper: Polygon Guarding with Target Orientation • Standard formulation does not handle self-occlusions • Many applications need a “good” view of the target – Surveillance, video conferencing, casinos! 6/8/2014 10 No front view of the person "Polygon Guarding with Orientation." Pratap Tokekar and Volkan Isler.
11. This Paper: Polygon Guarding with Target Orientation • Standard formulation does not handle self-occlusions • Many applications need a “good” view of the target – Surveillance, video conferencing, casinos! 6/8/2014 11 No front view of the person "Polygon Guarding with Orientation." Pratap Tokekar and Volkan Isler.
12. Δ-guarding Constraint • Every point x is visible from a set of guards • x lies in the convex hull of guards that see x • Guards need not be visible from each other • Introduced by [Smith & Evans, 2003] 6/8/2014 12 x is Δ-guarded y is not "Polygon Guarding with Orientation." Pratap Tokekar and Volkan Isler.
13. Polygon Guarding with Orientation • If all points in the polygon are Δ-guarded, then the perimeter of any convex object is completely visible. 6/8/2014 13 "Polygon Guarding with Orientation." Pratap Tokekar and Volkan Isler.
14. Polygon Guarding with Orientation • If all points in the polygon are Δ-guarded, then the perimeter of any convex object is completely visible. 6/8/2014 14 Related Work • [Smith & Evans, 2003] • NP-hard • [Efrat, Har-Peled and Mitchell, 2005] • O(log OPT) randomized approximation algorithm for polygons without holes "Polygon Guarding with Orientation." Pratap Tokekar and Volkan Isler.
15. Polygon Guarding with Orientation • If all points in the polygon are Δ-guarded, then the perimeter of any convex object is completely visible. 6/8/2014 15 Our contributions 1. What is the minimum number of guards required to Δ-guard a polygon? 2. Algorithms to place guards for Δ-guarding polygons. "Polygon Guarding with Orientation." Pratap Tokekar and Volkan Isler.
16. 6/8/2014 16 Without Δ-guarding "Polygon Guarding with Orientation." Pratap Tokekar and Volkan Isler.
17. 6/8/2014 17 Without Δ-guarding With Δ-guarding "Polygon Guarding with Orientation." Pratap Tokekar and Volkan Isler.
18. 6/8/2014 18 Without Δ-guarding With Δ-guarding Lemma 1: There exists a guard on every convex vertex in any valid solution for Δ-guarding any polygon. "Polygon Guarding with Orientation." Pratap Tokekar and Volkan Isler.
19. 6/8/2014 19 Lemma 1: There exists a guard on every convex vertex in any valid solution for Δ-guarding any polygon. "Polygon Guarding with Orientation." Pratap Tokekar and Volkan Isler.
20. 6/8/2014 20 Lemma 1: There exists a guard on every convex vertex in any valid solution for Δ-guarding any polygon. "Polygon Guarding with Orientation." Pratap Tokekar and Volkan Isler.
21. Lower Bound for Δ-guarding 6/8/2014 21   guards are always necessary for Δ-guarding any n-sided polygon (with or without holes). "Polygon Guarding with Orientation." Pratap Tokekar and Volkan Isler.
22. Lower Bound for Δ-guarding – All n-sided polygons need at least guards for Δ-guarding 6/8/2014 22   guards are always necessary for Δ-guarding any n-sided polygon (with or without holes). "Polygon Guarding with Orientation." Pratap Tokekar and Volkan Isler.
23. Proof Overview •   number of convex vertices •   number of reflex vertices •   total number of vertices • Case 1:   • Case 2:   6/8/2014 23 "Polygon Guarding with Orientation." Pratap Tokekar and Volkan Isler.
24. Case (1).   • Lemma 1: There exists a guard on every convex vertex in any valid solution for Δ-guarding any polygon. •   any valid Δ-guarding set •   6/8/2014 24 "Polygon Guarding with Orientation." Pratap Tokekar and Volkan Isler.
25. Case (2).   • No extension at convex vertices • Each edge introduces up to two edge extensions 6/8/2014 25 Edge extensions: Segments obtained by extending an edge on either side till they hit the boundary. "Polygon Guarding with Orientation." Pratap Tokekar and Volkan Isler.
26. Case (2).   • No extension at convex vertices • Each edge introduces up to two edge extensions 6/8/2014 26 Lemma 2: There exists a guard on every edge extension in any valid solution for Δ-guarding any polygon. Edge extensions: Segments obtained by extending an edge on either side till they hit the boundary. "Polygon Guarding with Orientation." Pratap Tokekar and Volkan Isler.
27. Case (2).   • m: number of edges incident with two reflex vertices • 2m: corresponding edge extensions 6/8/2014 27 Is |G| ≥ 2m? "Polygon Guarding with Orientation." Pratap Tokekar and Volkan Isler.
28. Case (2).   • m: number of edges incident with two reflex vertices • 2m: corresponding edge extensions 6/8/2014 28 Same guard may be present on multiple extensions if they intersect at a point. Is |G| ≥ 2m? No. "Polygon Guarding with Orientation." Pratap Tokekar and Volkan Isler.
29. Case (2).   • m: number of edges incident with two reflex vertices • 2m: corresponding edge extensions • k: max. number of such extensions that intersect at a point 6/8/2014 29 Same guard may be present on multiple extensions if they intersect at a point. Is |G| ≥ 2m? No. "Polygon Guarding with Orientation." Pratap Tokekar and Volkan Isler.
30. Case (2).   • m: number of edges incident with two reflex vertices • 2m: corresponding edge extensions • k: max. number of such extensions that intersect at a point • Any guard covers at most k extensions 6/8/2014 30 Same guard may be present on multiple extensions if they intersect at a point. Is |G| ≥ 2m? No. "Polygon Guarding with Orientation." Pratap Tokekar and Volkan Isler.
31. Case (2).   • m: number of edges incident with two reflex vertices • 2m: corresponding edge extensions • k: max. number of such extensions that intersect at a point • Any guard covers at most k extensions • Therefore, |G| ≥ 2m/k 6/8/2014 31 Same guard may be present on multiple extensions if they intersect at a point. Is |G| ≥ 2m? No. "Polygon Guarding with Orientation." Pratap Tokekar and Volkan Isler.
32. Case (2).   • |G| ≥ 2m/k • If red extensions intersect, blue cannot – Therefore, separate guards for all blue extensions • |G| ≥ k • Combining, |G|2 ≥ 2m • m: reflex-reflex edges – m ≥ n-2nc ≥ n/2 6/8/2014 32 Each edge corresponding to these k extensions (red) also has another extension (blue) "Polygon Guarding with Orientation." Pratap Tokekar and Volkan Isler.
33. Recap • Case 1:   – Guard on every convex vertex –   • Case 2:   – Guard on every edge extension –   • Done! 6/8/2014 33 "Polygon Guarding with Orientation." Pratap Tokekar and Volkan Isler.
34. Guard Placement • Placement algorithms are difficult – only a few problems have constant-factor approximation algorithms 6/8/2014 34 If optimal algorithm uses k guards, c-approx. algorithm uses at most ck guards. "Polygon Guarding with Orientation." Pratap Tokekar and Volkan Isler.
35. Guard Placement • Placement algorithms are difficult – only a few problems have constant-factor approximation algorithms •   approximation with greedy algorithm (guards restricted to vertices) 6/8/2014 35 If optimal algorithm uses k guards, c-approx. algorithm uses at most ck guards. "Polygon Guarding with Orientation." Pratap Tokekar and Volkan Isler.
36. Guard Placement • Placement algorithms are difficult – only a few problems have constant-factor approximation algorithms •   approximation with greedy algorithm (guards restricted to vertices) •   is too high – Guarding all corners and edges may not be necessary – Guard only the region of interest 6/8/2014 36 If optimal algorithm uses k guards, c-approx. algorithm uses at most ck guards. "Polygon Guarding with Orientation." Pratap Tokekar and Volkan Isler.
37. Guarding Paths • Chord: line segment between two visible points on the boundary – e.g. Target paths 6/8/2014 37 "Polygon Guarding with Orientation." Pratap Tokekar and Volkan Isler.
38. Guarding Paths • Chord: line segment between two visible points on the boundary – e.g. Target paths • Given: Set of chords in simply- connected polygon • Objective: Δ-guard at least one point on each chord 6/8/2014 38 "Polygon Guarding with Orientation." Pratap Tokekar and Volkan Isler.
39. Guarding Paths • Chord: line segment between two visible points on the boundary – e.g. Target paths • Given: Set of chords in simply- connected polygon • Objective: Δ-guard at least one point on each chord • Result:12-approximation algorithm. 6/8/2014 39 "Polygon Guarding with Orientation." Pratap Tokekar and Volkan Isler.
40. Main Idea (1/3) • Each chord partitions polygon into two subpolygons • If chord s3 t3 is Δ-guarded, then there exists a guard used in Δ-guarding s3 t3 in both subpolygons – There exists a guard in P3 6/8/2014 40 "Polygon Guarding with Orientation." Pratap Tokekar and Volkan Isler.
41. Main Idea (2/3) 6/8/2014 41 • Find maximum set of chords whose subpolygons are disjoint • |G| ≥ |maximum number of disjoint subpolygons| • Place four guards per disjoint subpolygon to guard a subset of chords "Polygon Guarding with Orientation." Pratap Tokekar and Volkan Isler.
42. Main Idea (3/3) • Guarding remaining chords is more involved 1. Partitioned into groups 2. Impose partial ordering on the groups – Represent as a tree 6/8/2014 42 A B PB PA A B "Polygon Guarding with Orientation." Pratap Tokekar and Volkan Isler.
43. Recap • All n-sided polygons need at least   guards for Δ- guarding •   approximation for Δ-guarding any polygon with holes (guards restricted to vertices) • Constant-factor approximation algorithm for Δ-guarding a set of chords • Ongoing Work: • Guarding chords with standard notion of visibility • Other types of regions of interest 6/8/2014 43 "Polygon Guarding with Orientation." Pratap Tokekar and Volkan Isler.
44. ROBOTIC SENSOR NETWORKS http://rsn.cs.umn.edu/ UNIVERSITY OF MINNESOTA Driven to Discover Thank you! tokekar@cs.umn.edu Work supported by NSF awards #0916209, #0917676, #0936710, #0934327, #1111638. Frederick R. Weisman Art Museum University of Minnesota
45. Δ-guarding Constraint If all points in the polygon are Δ-guarded, then the perimeter of any convex object is completely visible. 6/2/2014 25 "Polygon Guarding with Orientation." Pratap Tokekar and Volkan Isler.
46.   sometimes sufficient • Polygons with holes • Polygon without holes? 6/1/2014 26 "Polygon Guarding with Orientation." Pratap Tokekar and Volkan Isler.