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Geodesic Fréchet Distance Inside a Simple Polygon

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

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

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

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

  5. Fréchet Distance • Fréchet Distance • Measures similarity of continuous shapes • Similar Different

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

  7. Fréchet Distance • Fréchet Distance Illustration: • Walk the dog Woof!

  8. Fréchet Distance • Fréchet Distance Illustration:

  9. Fréchet Distance dF • A different walk

  10. Fréchet Distance dF • Fréchet Idea: • Examine all possible walks. • Yields a set M of maximum leash lengths. • dF= shortestleash lengthinM.

  11. Fréchet Distance • Fréchet Distance: • Small dF curves are similar • LargedF curves NOT similar

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

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

  15. Calculating Fréchet Distance • ComputingdF: • Decision Problem • Optimization Problem

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

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

  18. Geodesic Fréchet Distance

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

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

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

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

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

  24. Geodesic Fréchet Distance • Time: Geodesic Decision Problem • Let N = complexity of Person & Dog curves • Let k = complexity of simple polygon • Time: O(N2 log k) versus O(N2) non-geodesic case • Compute cell boundaries • Test for monotone path YES YES NO

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

  26. Geodesic Fréchet Distance • Geodesic Optimization Problem • Traditional approach: • Parametric Search • Sort O(N2) constant-complexity cell boundary functions • Geodesic case: • Each cell boundary has O(k) complexity • Straightforward parametric search sorts O(kN2) values • Goal: Faster

  27. Geodesic Fréchet Distance dG • Randomized red-blue intersections • Practical alternative to parametric search • Critical Values • Potential solutions for dF • Resolve with red-blue intersections

  28. Geodesic Fréchet Distance dG • Critical Values • As e increases: • Free space changes monotonically

  29. Geodesic Fréchet Distance dG • Geodesic Optimization Problem • Critical Value • Intersection of monotone functions

  30. Geodesic Fréchet Distance dG • Red-Blue Intersections • Red function properties: • monotone decreasing & continuous • Blue function properties: • monotone increasing & continuous

  31. Geodesic Fréchet Distance dG • Red-Blue Intersections [Palazzi1994] e

  32. Geodesic Fréchet Distance dG • Counting Red-Blue Intersections • Sort the curve values at e = a and e = b • Count the number of blue curves below each red curve e

  33. Geodesic Fréchet Distance dG • Red-Blue Intersections • r3 lies above: • two blue curves at e = a. • one blue curve at e = b.  (2-1) intersections for r3 in a ≤ e≤ b. e

  34. Position on cell boundary Geodesic Fréchet Distance • Red-Blue Intersections: • Vertical slab: a ≤ e≤ b • Count number of intersections [arrays] • Report intersections [BST] • Get-random intersection [persistent BST] e

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

  36. Geodesic Fréchet Distance dG • 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

  37. Geodesic Fréchet Distance • Geodesic Optimization Problem • Parametric Search time: O(k+kN2 log kN) • Red-Blue expected runtime: O(k+(N2 log kN)log N)

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

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

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

  41. References: • [Cook2007] • Cook IV, A. F. & Wenk, C.Geodesic Fréchet Distance Inside a Simple PolygonProceedings 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 polygonJ. Comput. Syst. Sci., Academic Press, Inc., 1989, 39, 126-152

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

  43. Questions?