1. Polygonal Curve Simplification
Open Problem 24
Polygonal Curve Simplification
From the Open Problems Project
Jory Denny
Computational Geometry CSCE
Professor Nancy M. Amato 1 1

2. Outline Introduction Results Conclusion Problem Definition
Applications
Results
Douglas and Peuker
Optimal Algorithms
Approximate Algorithms
Conclusion
Summary of Results
Future Directions 2 2

3. Introduction Problem Definition
Given a polygonal curve 𝑃= 𝑝 0 , 𝑝 1 , 𝑝 2 ,…, 𝑝 𝑛−1 , a polygonal curve 𝑃 ′ ={ 𝑝 𝑖 0 = 𝑝 0 ,…, 𝑝 𝑖 𝑘 = 𝑝 𝑛 } is a simplification of 𝑃 if 𝑘<𝑛, and 𝑃 ′ ⊂𝑃.
Distance metric 𝑑( 𝑝 1 𝑝 2 , 𝑝 𝑘 ) – minimum distance between 𝑝 𝑘 and 𝑝 1 𝑝 2
Example: Euclidean, Manhattan, etc.
Error criterion – method of calculating error 𝑒( 𝑝 𝑟 𝑝 𝑠 )
Example: Hausdorff criterion - max 𝑟≤𝑘≤𝑠 {𝑑 𝑝 𝑟 𝑝 𝑠 , 𝑝 𝑘 }
Error 𝜖 of a simplification 𝑃′ is defined as the maximum error of each line segment in 𝑃′

4. Introduction Problem Definition𝛼 pn p1
Solid line is original curve
Dotted line is an example simplification
𝛼 is the error of the simplification (maximum distance of one of the original points to a line segment of the simplification)

5. Introduction Problem Definition
Problems:
Min-# problem: Given a polygonal curve 𝑃 and a real number 𝜖≥0, compute an 𝜖-approximation of 𝑃 that uses the smallest number of vertices among all 𝜖-approximations of 𝑃.
Min-𝝐 problem: Given a polygonal curve 𝑃 and an integer 𝑘, compute an approximation of 𝑃 with at most 𝑘 vertices that minimizes the error over all approximations of 𝑃 that have at most 𝑘 vertices.
Bounds dependent on the error criterion and distance metric used

6. Introduction Applications
Geographic Information Systems
Cartography
Computer Graphics
Medical Imaging
Data Compression

7. Outline Introduction Results Conclusion Problem Definition
Applications
Results
Douglas and Peuker
Optimal Algorithms
Approximate Algorithms
Conclusion
Summary of Results
Future Directions 7 7

8. Results Douglas and Peucker (Douglas and Peucker, 73)
Input is 𝜖≥0
Recursive line simplification
Approximate curve by line segment 𝑝 0 𝑝 𝑛
Find the vertex 𝑝 𝑘 furthest from this line
If the error of this vertex is within tolerance stop
Else recurs on each sub-chain 𝑃 1 ={ 𝑝 0 ,…, 𝑝 𝑘 } and 𝑃 2 = 𝑝 𝑘+1 , …, 𝑝 𝑛−1
𝑂(𝑛 log 𝑛) -time implementation (Hershberger and Snoeyink, 94)
The algorithm is commonly used heuristic

9. Results Douglas and Peucker (Douglas and Peucker, 73)
Iteration 1: Intermediate point selected with largest error. Recurs on both halves.
Iteration 2: Left error within 𝜖 input. Recurs on right half.
Iteration 3: Approximation within 𝜖 limit.

10. Results Optimal 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 𝑂( 𝑛 3 )-time
Graph constructed in 𝑂( 𝑛 2 log 𝑛 )-time for the min-# problem and 𝑂( 𝑛 2 log 2 𝑛 )-time for the min-𝜖 problem (Melkman and O’Rourke, 88)
Further reduced to 𝑂( 𝑛 2 )-time for the min-# problem and 𝑂( 𝑛 2 log 𝑛 )-time for the min-𝜖 problem. Showed 𝑂 𝑛 -time and 𝑂( 𝑛 2 )-time algorithms for convex case (Chan and Chin, 96)
The graph is an undirected graph of all possible approximation segments 𝑝 𝑟 𝑝 𝑠 which lie within 𝜖. The min-# problem is then a single source shortest path problem from 𝑝 1 to 𝑝 𝑛 to minimize the number of segments
Min-𝜖 problem can then be solved with binary search on Min-# as subroutine

11. Results Optimal Algorithms
First subquadratic time algorithms used Clique Cover Graph construction technique using distance metrics (other than Euclidean) to achieve 𝑂( 𝑛 𝛿 )-time (Agarwal and Varadarajan, 00)
Special graph construction allowed for subquadratic running time
For min-𝜖 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 (𝑑 𝑝 𝑖 , 𝑝 𝑗 <𝜖 𝑎𝑛𝑑 𝑑 𝑝 𝑖 , 𝑝 𝑗 > 𝜖 2 𝑓𝑜𝑟 𝑎𝑙𝑙 0≤𝑖<𝑗≤𝑛−1)

12. Results Approximate Algorithms
Other work in approximate algorithms under certain criterion show interesting results
𝑂( 𝑛 𝛿 )-time algorithms for two different error criterion (Hausdorff and Frechet) and a number of distance metrics error 𝜖 2 (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) p1 pn 𝛼 p1 pn 𝛼
Hausdorff criterion
Frechet criterion (parametrically determined maximum)

13. Outline Introduction Results Conclusion Problem Definition
Applications
Results
Douglas and Peuker
Optimal Algorithms
Approximate Algorithms
Conclusion
Summary of Results
Future Directions 13 13

14. Conclusion Summary of Results
For the general min-# problem (which is the main focus in the literature) Ω( 𝑛 2 )-time
For specific distance metrics and metric criterions exact algorithms are near linear in 𝑂( 𝑛 𝛿 )-time
Approximate algorithms vary but all have near linear running time

15. Conclusion Future Directions
Furthur study into the min-𝜖 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?

16. 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): , 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., ,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: , 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, , 2007.