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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
November 20 GEOINFO 2006 2/25
Topics
Motivation Polyline Simplification Consistent Simplification Problem Objective Solution Results Concluding Remarks Future Work
November 20 GEOINFO 2006 3/25
Motivation
Create a topologically consistent simplification algorithm that• Handles all map features together• Generates better visual results• Achieves efficient processing• Produces scale independent maps
November 20 GEOINFO 2006 4/25
Polyline Simplification
Original Map
50,000 points 2,000 points
Simplified Map
Source: Digital Chart of the World Server (www.maproom.psu.edu/dcw)
November 20 GEOINFO 2006 5/25
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
November 20 GEOINFO 2006 6/25
Example: RDP Algorithm Maximum tolerable distance () It adds the farthest vertex from line segment
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Example: RDP Algorithm Problem with big tolerance
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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)
November 20 GEOINFO 2006 9/25
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)
November 20 GEOINFO 2006 10/25
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
November 20 GEOINFO 2006 11/25
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
November 20 GEOINFO 2006 12/25
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
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1st step: RDP algorithm until condition is satisfied
2nd Step: further insertions until sidedness and conditions are satisfied
Saalfeld’s Algorithm
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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
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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
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Consider each subpolyline and its simplifying segment separately• Example: Sidedness of p1 is evaluated with
respect to (Pik, vivk) and (Pkj, vkvj).
Theoretical Solution
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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)
November 20 GEOINFO 2006 18/25
Practical Solution When inserting a vertex
• Just update pointers begin and end (O(log n))• Store a reference to original array
November 20 GEOINFO 2006 19/25
Results: Synthetic Data Intersections
Original Data
Triangle Inversion
Array of Crossings
Polylines
Polygons
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Results: Synthetic Data Self-intersections
Original Data
Triangle Inversion
Array of Crossings
Polylines
Polygons
November 20 GEOINFO 2006 21/25
Results: Processing Time
Source: Digital Chart of the World Server (www.maproom.psu.edu/dcw)
November 20 GEOINFO 2006 22/25
Results: Processing Time
Equivalent processing time Insert a few more vertices for correcting inconsistencies
November 20 GEOINFO 2006 23/25
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
November 20 GEOINFO 2006 24/25
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
November 20 GEOINFO 2006 25/25
The End
Thank You!