Geodesic Fréchet Distance Inside a Simple Polygon Atlas F. Cook IV & Carola Wenk Proceedings of the 25th International Symposium on Theoretical Aspects.

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?