A Robust Strategy for Handling Linear Features in Topologically Consistent Polyline Simplification

1. A Robust Strategy for Handling Linear Features in Topologically Consistent Polyline Simplification da Silva, Adler C. G. Wu, Shin-Ting {acardoso,ting}@dca.fee.unicamp.br Department of Computer Engineering and Industrial Automation (DCA) School of Electrical and Computer Engineering (FEEC) State University of Campinas (UNICAMP) GEOINFO 2006

2. Topics • Motivation • Polyline Simplification • Consistent Simplification • Problem • Objective • Solution • Results • Concluding Remarks • Future Work GEOINFO 2006

3. Motivation • Create a topologically consistent simplification algorithm that • Handles all map features together • Generates better visual results • Achieves efficient processing • Produces scale independent maps GEOINFO 2006

4. Polyline Simplification Original Map Simplified Map 50,000 points 2,000 points Source: Digital Chart of the World Server (www.maproom.psu.edu/dcw) GEOINFO 2006

5. Polyline Simplification • Common problem in most algorithms • Loss of “Topological Consistency” • Cause: they take the polyline in isolation, without considering the features in its vicinity GEOINFO 2006

6. Example: RDP Algorithm • Maximum tolerable distance () • It adds the farthest vertex from line segment  GEOINFO 2006

7. Example: RDP Algorithm • Problem with big tolerance  GEOINFO 2006

8. Consistent Simplification • A topologically consistent polyline simplification algorithm must • Keep features in the correct side • Avoid intersections between features • Avoid self-intersections • The algorithm may • Simplify one polyline considering the features in its vicinity (simplification in context) • Simplify the complete collection of polylines together (global simplification) GEOINFO 2006

9. State of the Art • de Berg et al., 1998 • Simplification is viewed as an optimization problem • A single polyline is simplified in context • It handles only polylines that are part of a polygon • Saalfeld, 1999 • It is a improvement of RDP for recovering topology • A single polyline is simplified in context • It also handles polylines that are not part of a polygon • Inconsistency is removed by inserting more vertices • van der Poorten and Jones, 1999 / 2001 • The polylines of the map are simplified together • Based on Constrained Delaunay Triangulation • Topology is implicitly preserved • Relatively slow (10min for 30,000 vertices) GEOINFO 2006

10. Problem • de Berg et al. and Saalfeld handle a linear feature as a point feature • When handling a line segment, they consider that intersections can be avoided if the side of its vertices is preserved Problem with polygons Problem with polylines GEOINFO 2006

11. de Berg et al.’s Strategy • A polyline is part of a polygon • They formalize consistency of a point with respect to a polygon • de Berg et al.’s algorithm adds other restrictions that avoid the problematic cases GEOINFO 2006

12. Saalfeld’s Strategy • He generalizes the consistency of polygons to polylines • Compute sidedness: count the number of crossings of a ray from the point with P and P’ • Odd = wrong side • Even = correct side • Triangle Inversion Property • The insertion of a vertex changes only the sidedness of the points inside the triangle • Used to update sidedness of points GEOINFO 2006

13. Saalfeld’s Algorithm 1st step: RDP algorithm until  condition is satisfied 2nd Step: further insertions until sidedness and  conditions are satisfied GEOINFO 2006

14. Objective • General context • Develop a topologically consistent simpli-fication algorithm using Saalfeld’s strategy • Remove locally inconsistencies • Contribution of this work • Theoretical solution • Study on consistency to avoid (self-) intersections by taking into consideration only vertices of polylines • Practical solution • Replace the triangle inversion test by a robust test GEOINFO 2006

15. Theoretical Analysis • An inconsistency occurs whenever a subpolyline intersects the simplifying segment of another subpolyline • Example: Pkj intersects vivk, which is the simplifying segment of Pik Region with problem GEOINFO 2006

16. Theoretical Solution • Consider each subpolyline and its simplifying segment separately • Example: Sidedness of p1 is evaluated with respect to (Pik, vivk) and (Pkj, vkvj). GEOINFO 2006

17. Practical Solution • Pre-processed array of crossings with Pij • Number of crossings is very small • begin points to the first element • end points to the element after the last one • Number of crossings = (begin-end)+(crossing with segment vivj) GEOINFO 2006

18. Practical Solution • When inserting a vertex • Just update pointers begin and end (O(log n)) • Store a reference to original array GEOINFO 2006

19. Results: Synthetic Data • Intersections Original Data Triangle Inversion Array of Crossings Polylines Polygons GEOINFO 2006

20. Results: Synthetic Data • Self-intersections Original Data Triangle Inversion Array of Crossings Polylines Polygons GEOINFO 2006

21. Results: Processing Time Source: Digital Chart of the World Server (www.maproom.psu.edu/dcw) GEOINFO 2006

22. Results: Processing Time • Equivalent processing time • Insert a few more vertices for correcting inconsistencies GEOINFO 2006

23. Concluding Remarks • Mistake in consistent simplification algorithms • Handle linear features as point features • Theoretical solution • Handle separately each subpolyline and its simplifying line segment • Practical solution (for Saalfeld’s algorithm) • Pre-processed array of crossings • Complete elimination of inconsistencies • Equivalent processing time • A few more vertices are inserted to recover topology GEOINFO 2006

24. Future Work • The consistent simplification algorithm • Handles polylines in a global simplification • Considers only vertices that are currently in simplified polylines • Inserts less vertices  better visual results • Achieves faster processing • Can be used with many isolated algorithms • Produce scale independent maps  GEOINFO 2006

25. The End Thank You! GEOINFO 2006