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Open Problem 24Polygonal Curve Simplification

From the Open Problems Projecthttp://maven.smith.edu/~orourke/TOPP/

Jory Denny

Computational Geometry CSCE620-600Professor Nancy M. Amato

Outline

• Introduction• Problem Definition• Applications

• Results• Douglas and Peuker• Optimal Algorithms• Approximate Algorithms

• Conclusion• Summary of Results• Future Directions

IntroductionProblem Definition

• Given a polygonal curve , a polygonal curve is a simplification of if , and .

• Distance metric – minimum distance between and • Example: Euclidean, Manhattan, etc.

• Error criterion – method of calculating error • Example: Hausdorff criterion -

• Error of a simplification is defined as the maximum error of each line segment in

IntroductionProblem Definition

• Solid line is original curve • Dotted line is an example simplification

• is the error of the simplification (maximum distance of one of the original points to a line segment of the simplification)

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IntroductionProblem Definition

• Problems:• Min-# problem: Given a polygonal curve and a real number ,

compute an -approximation of that uses the smallest number of vertices among all -approximations of .

• Min- problem: Given a polygonal curve and an integer , compute an approximation of with at most vertices that minimizes the error over all approximations of that have at most vertices.

• Bounds dependent on the error criterion and distance metric used

IntroductionApplications

• Geographic Information Systems• Cartography• Computer Graphics• Medical Imaging• Data Compression

http://cs.joensuu.fi/~koles/approximation/Image42.gif

Outline

• Introduction• Problem Definition• Applications

• Results• Douglas and Peuker• Optimal Algorithms• Approximate Algorithms

• Conclusion• Summary of Results• Future Directions

ResultsDouglas and Peucker (Douglas and Peucker, 73)

• Input is • Recursive line simplification

• Approximate curve by line segment • Find the vertex furthest from this line• If the error of this vertex is within tolerance stop• Else recurs on each sub-chain and

• -time implementation (Hershberger and Snoeyink, 94)

• The algorithm is commonly used heuristic

ResultsDouglas and Peucker (Douglas and Peucker, 73)

Iteration 1: Intermediate point selected with largest error. Recurs on both halves.

Iteration 2: Left error within input. Recurs on right half.

Iteration 3: Approximation within limit.

ResultsOptimal Algorithms

• For optimal results for both problems with any error criterion and distance metric problem was formulated as a graph search problem (Imai and Iri, 88)

• Trivially -time• Graph constructed in -time for the min-# problem and -time for

the min- problem (Melkman and O’Rourke, 88)

• Further reduced to -time for the min-# problem and -time for the min- problem. Showed -time and -time algorithms for convex case (Chan and Chin, 96)

• The graph is an undirected graph of all possible approximation segments which lie within . The min-# problem is then a single source shortest path problem from to to minimize the number of segments

• Min- problem can then be solved with binary search on Min-# as subroutine

ResultsOptimal Algorithms

• First subquadratic time algorithms used Clique Cover Graph construction technique using distance metrics (other than Euclidean) to achieve -time (Agarwal and Varadarajan, 00)

• Special graph construction allowed for subquadratic running time• For min- problem this is expected time (i.e., randomized for this)

• Query based approach incrementally constructing a BFS obtained subquadratic time for Euclidean distance metric with infinite beam criterion (Daescu and Mi, 05)

• Only holds for specific input ()

ResultsApproximate Algorithms

• Other work in approximate algorithms under certain criterion show interesting results

• -time algorithms for two different error criterion (Hausdorff and Frechet) and a number of distance metrics error (Agarwal et. al., 05)

• Asymptotically optimal and linear time algorithm using the area of the domain between the two curves as error criterion based on local refining/coarsening strategy (Chen et. al., 05)

• Optimal simplification in near linear time for specific application of rendering, which mainly focuses on retaining the shape of the curve for visual purposes (Buzer, 07)

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Hausdorff criterion p1

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Frechet criterion (parametrically determined maximum)

Outline

• Introduction• Problem Definition• Applications

• Results• Douglas and Peuker• Optimal Algorithms• Approximate Algorithms

• Conclusion• Summary of Results• Future Directions

ConclusionSummary of Results

• For the general min-# problem (which is the main focus in the literature) -time

• For specific distance metrics and metric criterions exact algorithms are near linear in -time

• Approximate algorithms vary but all have near linear running time

ConclusionFuture Directions

• Furthur study into the min- problem is needed, as most work focuses on min-# problem

• Further extension of either the query based method or Clique Cover graph construction could be used to attain optimal results in near linear time?

• Open question: Can the general problem be solved in subquadratic time?

References

• D. H Douglas and T. K. Peucker. Algorithms for the reduction of the number of points required to represent a line or its caricature. The Canadian Cartographer, 10(2):112-122, 1973.

• H. Imai and M. Iri. Polygonal approximations of a curve – Formulations and algorithms. Computational Morphography, 1988.

• A. Melkman and J. O’Rourke. On polygonal chain approximation. Computational Morphography, 1988.

• J. Hershberger and J. Snoeyink. An O(nlogn) implementation of the Douglas-Peucker algorithm for line simplification. Proc. 10th Annu. ACM Sympos. Comput. Geom., 383-384,1994.

• W. S. Chan and F. Chin. Approximation of polygonal curves with minimum number of line segments or minimum error. Internat. J. Comput. Geom. Appl., 6:59-77, 1996.

• P. K. Agarwal and K. R. Varadarajan. Efficient algorithms for approximating polygonal chains. Discrete Comput. Geom., 23:273-291, 2000.

• O. Daescu and N. Mi. Polygonal chain approximation: a query based approach. Comput. Geom. Theory Appl., 30(1): 41-58, 2005.

• P. K. Agarwal and S. Har-Peled and N. H. Mustafa and Y. Wang. Near-linear time approximation algorithms for curve simplification. Proc. of the 10th Annual European Symp. on Algorithms, 29-41, 2005.

• L. Chen and J. Wang and J. Xu. Asymptotically optimal and linear-time algorithm for polygonal curve simplification. Tech. Report Penn. State Dept. of Mathematics, NO. AM274, 2005.

• L. Buzer. Optimal simplification of polygonal chain for rendering.  In Proc. of the 23rd annual symp. on Comp. geom,168-174, 2007.