Geodesic Fréchet Distance Inside a Simple PolygonAtlas F. Cook IV & Carola Wenk
Proceedings of the 25th International Symposium onTheoretical Aspects of Computer Science (STACS), Bordeaux, France, 2008 (Acceptance Rate: 27%)
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Overview
Fréchet Distance Importance Intuition
Geodesic Fréchet Distance Decision Problem Optimization Problem
Red-Blue Intersections Conclusion References Questions
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Importance of Fréchet Distance
♫ It’s a beautiful day in the neighborhood…
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Importance of Fréchet Distance
Distinguishing your neighbors: Nose Hairstyles
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Fréchet Distance
Fréchet Distance Measures similarity of continuous shapes
Similar Different
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Fréchet Distance
Comparison of geometric shapes Computer Vision Robotics Medical Imaging
Half-Full
Half-Empty
Same glass!
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Fréchet Distance
Fréchet Distance Illustration: Walk the dog
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Fréchet Distance
Fréchet Distance Illustration:
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Fréchet Distance F
A different walk
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Fréchet Distance F
Fréchet Idea: Examine all possible walks. Yields a set M of maximum leash lengths. F = shortest leash length in M.
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Fréchet Distance Fréchet Distance:
Small F curves are similar
Large F curves NOT similar
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Calculating Fréchet Distance Representing all walks:
Position on blue curve X-axis position “ “ red curve Y-axis position
Free Space Diagram
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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
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Calculating Fréchet Distance
Free Space Diagram as ε is varied:
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Calculating Fréchet Distance
Computing F:
1. Decision Problem
2. Optimization Problem
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Calculating Fréchet Distance
1) Decision Problem Given leash length: ε Monotone path through free space?
Answer: YES or NO Dynamic Programming [Alt1995]
NOYESYES
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Calculating Fréchet Distance
2) Optimization Problem
ε is too smallε is too big ε is as small as possible
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Geodesic Fréchet Distance
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Geodesic Fréchet Distance
Defn: Geodesic in a simple polygon – shortest path that avoids obstacles [Mitchell1987].
Leash stays inside a simple polygon.
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Geodesic Fréchet Distance Computation:
1. Decision Problem Geodesic Free Space Diagram
2. Optimization Problem
ε is too smallε is too bigε is as small as possible
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Geodesics inside a simple polygon: Funnel [Guibas1989]
Horizontal/vertical line segment in a free space cell.
Geodesic Fréchet Distance
p
d
c
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Algorithm: Geodesic Decision Problem1. Compute each cell boundary in logarithmic time.
Geodesic Fréchet Distance
Cell
Funnel [Guibas1989] Funnel’s distance function• Piecewise hyperbolic• Bitonic
Cell Free Space
y =
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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
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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
Geodesic Fréchet Distance
NOYESYES
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Geodesic Optimization Problem
Geodesic Fréchet Distance
ε is too smallε is too big ε is as small as possible
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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
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Randomized red-blue intersections Practical alternative to parametric search
Critical Values Potential solutions for F
Resolve with red-blue intersections
Geodesic Fréchet Distance G
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Critical Values As increases:
Free space changes monotonically
Geodesic Fréchet Distance G
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Geodesic Optimization Problem Critical Value
Intersection of monotone functions
Geodesic Fréchet Distance G
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Red-Blue Intersections Red function properties:
monotone decreasing & continuous
Blue function properties: monotone increasing & continuous
Geodesic Fréchet Distance G
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Red-Blue Intersections [Palazzi1994]
Geodesic Fréchet Distance G
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Counting Red-Blue Intersections Sort the curve values at = and = Count the number of blue curves below each red
curve
Geodesic Fréchet Distance G
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Red-Blue Intersections r3 lies above:
two blue curves at = one blue curve at = .
(2-1) intersections for r3 in ≤ ≤ .
Geodesic Fréchet Distance G
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Geodesic Fréchet Distance
Red-Blue Intersections: Vertical slab: ≤ ≤
Count number of intersections [arrays] Report intersections [BST] Get-random intersection [persistent BST]
Positionon cell
boundary
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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 smallε is too bigε is as small as possible
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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
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Geodesic Optimization Problem Parametric Search time: O(k+kN2 log kN) Red-Blue expected runtime: O(k+(N2 log
kN)log N)
Geodesic Fréchet Distance
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Geodesic Fréchet Distance Applications Faster solution
Randomized alternatives to parametric search
Surfaces Piecewise-smooth curves
Future Work
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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
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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
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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
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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
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Questions?