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## Geodesic Fréchet Distance Inside a Simple Polygon

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**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**Overview**• Fréchet Distance • Importance • Intuition • Geodesic Fréchet Distance • Decision Problem • Optimization Problem • Red-Blue Intersections • Conclusion • References • Questions**Importance of Fréchet Distance**• ♫ It’s a beautiful day in the neighborhood…**Importance of Fréchet Distance**• Distinguishing your neighbors: • Nose • Hairstyles**Fréchet Distance**• Fréchet Distance • Measures similarity of continuous shapes • Similar Different**Fréchet Distance**• Comparison of geometric shapes • Computer Vision • Robotics • Medical Imaging Half-Full Same glass! Half-Empty**Fréchet Distance**• Fréchet Distance Illustration: • Walk the dog Woof!**Fréchet Distance**• Fréchet Distance Illustration:**Fréchet Distance dF**• A different walk**Fréchet Distance dF**• Fréchet Idea: • Examine all possible walks. • Yields a set M of maximum leash lengths. • dF= shortestleash lengthinM.**Fréchet Distance**• Fréchet Distance: • Small dF curves are similar • LargedF curves NOT similar**Calculating Fréchet Distance**• Representing all walks: Free Space Diagram Position on blue curve X-axis position “ “ red curve Y-axis position**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**Calculating Fréchet Distance**• Free Space Diagram as ε is varied:**Calculating Fréchet Distance**• ComputingdF: • Decision Problem • Optimization Problem**Calculating Fréchet Distance**• Decision Problem • Given leash length: ε • Monotone path through free space? • Answer: YES or NO • Dynamic Programming [Alt1995] YES YES NO**Calculating Fréchet Distance**• Optimization Problem ε is as small as possible ε is too small ε is too big**Geodesic Fréchet Distance**• Defn: Geodesic in a simple polygon – shortest path that avoids obstacles [Mitchell1987]. • Leash stays inside a simple polygon.**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**Geodesic Fréchet Distance**• Geodesics inside a simple polygon: • Funnel [Guibas1989] • Horizontal/vertical line segment in a free space cell. d c p**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**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**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**Geodesic Fréchet Distance**• Geodesic Optimization Problem ε is as small as possible ε is too small ε is too big**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**Geodesic Fréchet Distance dG**• Randomized red-blue intersections • Practical alternative to parametric search • Critical Values • Potential solutions for dF • Resolve with red-blue intersections**Geodesic Fréchet Distance dG**• Critical Values • As e increases: • Free space changes monotonically**Geodesic Fréchet Distance dG**• Geodesic Optimization Problem • Critical Value • Intersection of monotone functions**Geodesic Fréchet Distance dG**• Red-Blue Intersections • Red function properties: • monotone decreasing & continuous • Blue function properties: • monotone increasing & continuous**Geodesic Fréchet Distance dG**• Red-Blue Intersections [Palazzi1994] e**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**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**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**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**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**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)**Future Work**• Geodesic Fréchet Distance • Applications • Faster solution • Randomized alternatives to parametric search • Surfaces • Piecewise-smooth curves**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**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**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**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