November 20GEOINFO 20061/25 A Robust Strategy for Handling Linear Features in Topologically Consistent Polyline Simplification Department of Computer Engineering and Industrial Automation (DCA) School of Electrical and Computer Engineering (FEEC) State University of Campinas (UNICAMP) da Silva, Adler C. G. Wu, Shin-Ting

November 20GEOINFO 20062/25 Topics l Motivation l Polyline Simplification l Consistent Simplification l Problem l Objective l Solution l Results l Concluding Remarks l Future Work

November 20GEOINFO 20063/25 Motivation l Create a topologically consistent simplification algorithm that Handles all map features together Generates better visual results Achieves efficient processing Produces scale independent maps

November 20GEOINFO 20064/25 Polyline Simplification Original Map 50,000 points2,000 points Simplified Map Source: Digital Chart of the World Server (

November 20GEOINFO 20065/25 Polyline Simplification l Common problem in most algorithms Loss of Topological Consistency l Cause: they take the polyline in isolation, without considering the features in its vicinity

November 20GEOINFO 20066/25 Example: RDP Algorithm l Maximum tolerable distance ( ) l It adds the farthest vertex from line segment

November 20GEOINFO 20067/25 Example: RDP Algorithm l Problem with big tolerance

November 20GEOINFO 20068/25 Consistent Simplification l A topologically consistent polyline simplification algorithm must Keep features in the correct side Avoid intersections between features Avoid self-intersections l The algorithm may Simplify one polyline considering the features in its vicinity (simplification in context) Simplify the complete collection of polylines together (global simplification)

November 20GEOINFO 20069/25 State of the Art l 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 l 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 l 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)

November 20GEOINFO /25 Problem l 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 polygonsProblem with polylines

November 20GEOINFO /25 de Berg et al.s Strategy l A polyline is part of a polygon They formalize consistency of a point with respect to a polygon l de Berg et al.s algorithm adds other restrictions that avoid the problematic cases

November 20GEOINFO /25 Saalfelds Strategy l 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 l Triangle Inversion Property The insertion of a vertex changes only the sidedness of the points inside the triangle Used to update sidedness of points

November 20GEOINFO /25 1 st step: RDP algorithm until condition is satisfied 2 nd Step: further insertions until sidedness and conditions are satisfied Saalfelds Algorithm

November 20GEOINFO /25 Objective l General context Develop a topologically consistent simpli- fication algorithm using Saalfelds strategy Remove locally inconsistencies l 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

November 20GEOINFO /25 Theoretical Analysis l An inconsistency occurs whenever a subpolyline intersects the simplifying segment of another subpolyline Example: P kj intersects v i v k, which is the simplifying segment of P ik Region with problem

November 20GEOINFO /25 l Consider each subpolyline and its simplifying segment separately Example: Sidedness of p 1 is evaluated with respect to ( P ik, v i v k ) and ( P kj, v k v j ). Theoretical Solution

November 20GEOINFO /25 Practical Solution Pre-processed array of crossings with P ij Number of crossings is very small begin points to the first element end points to the element after the last one l Number of crossings = (begin-end) +(crossing with segment v i v j )

November 20GEOINFO /25 Practical Solution l When inserting a vertex Just update pointers begin and end ( O(log n) ) Store a reference to original array

November 20GEOINFO /25 Results: Synthetic Data l Intersections Original DataTriangle InversionArray of Crossings Polylines Polygons

November 20GEOINFO /25 Results: Synthetic Data l Self-intersections Original DataTriangle InversionArray of Crossings Polylines Polygons

November 20GEOINFO /25 Results: Processing Time Source: Digital Chart of the World Server (

November 20GEOINFO /25 Results: Processing Time l Equivalent processing time l Insert a few more vertices for correcting inconsistencies

November 20GEOINFO /25 Concluding Remarks l Mistake in consistent simplification algorithms Handle linear features as point features l Theoretical solution Handle separately each subpolyline and its simplifying line segment l Practical solution (for Saalfelds algorithm) Pre-processed array of crossings Complete elimination of inconsistencies Equivalent processing time A few more vertices are inserted to recover topology

November 20GEOINFO /25 Future Work l 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

November 20GEOINFO /25 The End Thank You!