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

November 20 GEOINFO 2006 1/25

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{acardoso,ting}@dca.fee.unicamp.br


Topics

Motivation Polyline Simplification Consistent Simplification Problem Objective Solution Results Concluding Remarks Future Work


Motivation

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


Polyline Simplification

Original Map

50,000 points 2,000 points

Simplified Map

Source: Digital Chart of the World Server (www.maproom.psu.edu/dcw)


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


Example: RDP Algorithm Maximum tolerable distance () It adds the farthest vertex from line segment


Example: RDP Algorithm Problem with big tolerance


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)


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)


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


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


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


1st step: RDP algorithm until condition is satisfied

2nd Step: further insertions until sidedness and conditions are satisfied

Saalfeld’s Algorithm


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


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 withproblem


Consider each subpolyline and its simplifying segment separately• Example: Sidedness of p1 is evaluated with

respect to (Pik, vivk) and (Pkj, vkvj).

Theoretical Solution


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)


Practical Solution When inserting a vertex

• Just update pointers begin and end (O(log n))• Store a reference to original array


Results: Synthetic Data Intersections

Original Data

Triangle Inversion

Array of Crossings

Polylines

Polygons


Results: Synthetic Data Self-intersections

Original Data

Triangle Inversion

Array of Crossings

Polylines

Polygons


Results: Processing Time

Source: Digital Chart of the World Server (www.maproom.psu.edu/dcw)


Results: Processing Time

Equivalent processing time Insert a few more vertices for correcting inconsistencies


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


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


The End

Thank You!

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

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