1. Geodesic Fréchet Distance Inside a Simple Polygon Atlas F. Cook IV & Carola Wenk Proceedings of the 25th International Symposium on Theoretical Aspects of Computer Science (STACS), Bordeaux, France, 2008 (Acceptance Rate: 27%)

2. 2 Overview Fréchet Distance  Importance  Intuition Geodesic Fréchet Distance  Decision Problem  Optimization Problem Red-Blue Intersections Conclusion References Questions

3. 3 Importance of Fréchet Distance ♫ It’s a beautiful day in the neighborhood…

4. 4 Importance of Fréchet Distance Distinguishing your neighbors:  Nose  Hairstyles

5. 5 Fréchet Distance  Measures similarity of continuous shapes  SimilarDifferent

6. 6 Fréchet Distance Comparison of geometric shapes  Computer Vision  Robotics  Medical Imaging Half-Full Half-Empty Same glass!

7. 7 Fréchet Distance Fréchet Distance Illustration:  Walk the dog

8. 8 Fréchet Distance Fréchet Distance Illustration:

9. 9 Fréchet Distance  F A different walk

10. 10 Fréchet Distance  F Fréchet Idea:  Examine all possible walks.  Yields a set M of maximum leash lengths.   F = shortest leash length in M.

11. 11 Fréchet Distance Fréchet Distance:  Small  F  curves are similar  Large  F  curves NOT similar

12. 12 Calculating Fréchet Distance Representing all walks: Position on blue curve  X-axis position “ “ red curve  Y-axis position Free Space Diagram

13. 13 Calculating Fréchet Distance Free Space Diagram  White:Person & dog are “close together” Leash length ≤ ε  Green:Person & dog are “far apart” Leash length > ε Free Space Diagram

14. 14 Calculating Fréchet Distance Free Space Diagram as ε is varied:

15. 15 Calculating Fréchet Distance Computing  F : 1.Decision Problem 2.Optimization Problem

16. 16 Calculating Fréchet Distance 1)Decision Problem  Given leash length: ε  Monotone path through free space? Answer: YES or NO Dynamic Programming [Alt1995] NOYES

17. 17 Calculating Fréchet Distance 2)Optimization Problem ε is too small ε is too big ε is as small as possible

18. 18 Geodesic Fréchet Distance

19. 19 Geodesic Fréchet Distance Defn: Geodesic in a simple polygon – shortest path that avoids obstacles [Mitchell1987].  Leash stays inside a simple polygon.

20. 20 Geodesic Fréchet Distance Computation: 1.Decision Problem  Geodesic Free Space Diagram 2.Optimization Problem ε is too smallε is too big ε is as small as possible

21. 21 Geodesics inside a simple polygon:  Funnel [Guibas1989] Horizontal/vertical line segment in a free space cell. Geodesic Fréchet Distance p d c

22. 22 Algorithm: Geodesic Decision Problem 1.Compute each cell boundary in logarithmic time. Geodesic Fréchet Distance Cell Funnel [Guibas1989]Funnel’s distance function Piecewise hyperbolic Bitonic Cell Free Space y = 

23. 23 Algorithm: Geodesic Decision Problem Compute each cell boundary in logarithmic time. 2.Test for monotone path: Cell free space  x-monotone, y-monotone, & connected  Only cell boundaries are required Geodesic Fréchet Distance

24. 24 Time: Geodesic Decision Problem  Let N = complexity of Person & Dog curves  Let k = complexity of simple polygon  Time: O(N 2 log k) versus O(N 2 ) non-geodesic case Compute cell boundaries Test for monotone path Geodesic Fréchet Distance NOYES

25. 25 Geodesic Optimization Problem Geodesic Fréchet Distance ε is too small ε is too big ε is as small as possible

26. 26 Geodesic Optimization Problem  Traditional approach: Parametric Search Sort O(N 2 ) constant-complexity cell boundary functions  Geodesic case: Each cell boundary has O(k) complexity Straightforward parametric search sorts O(kN 2 ) values Goal: Faster Geodesic Fréchet Distance

27. 27 Randomized red-blue intersections  Practical alternative to parametric search  Critical Values Potential solutions for  F Resolve with red-blue intersections Geodesic Fréchet Distance  G

28. 28 Critical Values  As  increases: Free space changes monotonically Geodesic Fréchet Distance  G

29. 29 Geodesic Optimization Problem  Critical Value Intersection of monotone functions Geodesic Fréchet Distance  G

30. 30 Red-Blue Intersections  Red function properties: monotone decreasing & continuous  Blue function properties: monotone increasing & continuous Geodesic Fréchet Distance  G

31. 31 Red-Blue Intersections [Palazzi1994] Geodesic Fréchet Distance  G 

32. 32 Counting Red-Blue Intersections  Sort the curve values at  =  and  =   Count the number of blue curves below each red curve Geodesic Fréchet Distance  G 

33. 33 Red-Blue Intersections  r 3 lies above: two blue curves at  =  one blue curve at  = .  (2-1) intersections for r 3 in  ≤  ≤ . Geodesic Fréchet Distance  G 

34. 34 Geodesic Fréchet Distance  Red-Blue Intersections:  Vertical slab:  ≤  ≤  Count number of intersections [arrays] Report intersections [BST] Get-random intersection [persistent BST] Position on cell boundary

35. 35 Geodesic Fréchet Distance Geodesic Optimization Problem  Goal: Make  as small as possible  Repeatedly find a random critical value and use the idea of binary search to converge. ε is too small ε is too big ε is as small as possible

36. 36 Parametric Search vs. Randomization:  Parametric Search [traditional] Sorting cell boundary functions Huge constant factors [Cole1987]  Randomized Red-Blue Intersections Practical alternative to parametric search  Not previously applied to Fréchet distance Faster expected runtime Straightforward implementation Geodesic Fréchet Distance  G

37. 37 Geodesic Optimization Problem  Parametric Search time:O(k+kN 2 log kN)  Red-Blue expected runtime: O(k+(N 2 log kN)log N) Geodesic Fréchet Distance

38. 38 Geodesic Fréchet Distance  Applications  Faster solution Randomized alternatives to parametric search  Surfaces  Piecewise-smooth curves Future Work

39. 39 Conclusion Fréchet Distance  Measures similarity of continuous shapes  SimilarDifferent Geodesic Fréchet Distance: Simple Polygon  Obstacles affect similarity  Red-Blue intersections Practical alternative to parametric search

40. 40 References: [Alt1995]  Alt, H. & Godau, M. Computing the Fréchet Distance Between Two Polygonal Curves International Journal of Computational Geometry and Applications, 1995, 5, 75-91 [Cole1987]  Cole, R. Slowing down sorting networks to obtain faster sorting algorithms J. ACM, ACM Press, 1987, 34, 200-208

41. 41 References: [Cook2007]  Cook IV, A. F. & Wenk, C. Geodesic Fréchet Distance Inside a Simple Polygon Proceedings of the 25th International Symposium on Theoretical Aspects of Computer Science (STACS), Bordeaux, France, 2008 [Guibas1989]  Guibas, L. J. & Hershberger, J. Optimal shortest path queries in a simple polygon J. Comput. Syst. Sci., Academic Press, Inc., 1989, 39, 126-152

42. 42 References: [Mitchell1987]  Mitchell, J. S. B.; Mount, D. M. & Papadimitriou, C. H. The discrete geodesic problem SIAM J. Comput., Society for Industrial and Applied Mathematics, 1987, 16, 647-668 [Palazzi1994]  Palazzi, L. & Snoeyink, J. Counting and reporting red/blue segment intersections CVGIP: Graph. Models Image Process., Academic Press, Inc., 1994, 56, 304-310

43. 43 Questions?