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MSc Geomatics thesis presentation
Validation and automatic repair of planar partitions using a constrained triangulation
Ken Arroyo Ohori
Friday, 27 August 2010 at 10:00
Grote VergaderzaalOTB Research Institute
Ja!alaan 9, Delft
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Representing thematic information
• Pierre Charles Dupin’s 1819 map
• Subdividing the feature space of that theme
• Distinct visual representation for each class
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Digital thematic maps
• CGIS, SYMAP (Late ‘60s)
• Polygons to represent boundaries
• Well suited for a computer
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Planar partitions CORINE 2000
CORINE E39N32
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Why planar partitions?
• Simple constraints: no gaps, no overlaps, (no disjoint regions)
• Easy to answer questions:
• (Aggregation) What is the total area of the features of type A and B?
• (Topology) Are features A and B adjacent?
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Gaps and overlaps10 cm 0.3 mm
CORINE E41N27 CORINE E39N33 CORINE E40N33
CORINE E39N32 CORINE E40N32
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The problem
• Validate a planar partition
• If it’s invalid, automatically repair it.
Spain - Portugal border
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The problem
• Validate a planar partition
• If it’s invalid, automatically repair it.
Spain - Portugal border
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Why is it hard? Discrete and approximate representations
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Errors in computer operations Orientation test
(17.300000000000001, 17.300000000000001)
(24.00000000000005, 24.0000000000000517765)
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Valid polygons
ESRI ISO
Polygons with holes
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More ambiguityDifferent sourcesSpain - Portugal border
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• Rings:
• Closed, not self-intersecting, not zero area, correct winding, ...
• Polygons:
• Nested rings, connected interior, not zero area, ...
• Planar Partitions:
• No gaps, no overlaps, no disjoint regions...
Validation process
CORINE E30N33
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Polygon validation constraints
• ArcGIS: too short line segments, unclosed rings, self-intersections, incorrect ring ordering.
• JTS/GEOS: self-touching rings, zero area rings, zero area polygons, improperly nested rings, duplicate vertices, spikes and gores, touching parts, crossing rings.
• Oracle Spatial: polygon with fewer than 4 vertices, unclosed rings, self-intersections, touching rings, overlapping rings, points too close together, wrong orientation.
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ArcGIS and the zero area polygonProblems are unavoidable
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• Nearly impossible without topology: finding gaps
• Creation of a planar graph based representation
• Plümer and Groger: no dangling edges, no zero-length edges, planarity, no holes, no self-intersections, no overlaps, connectivity.
Validation of planar partitions
Andorra in CORINE
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ThresholdsRepair of planar partitions
using snapping
minimumdistancethreshold
maximumdistancethreshold
minimumdistancethreshold
maximumdistancethreshold
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Repair using snapping
• Extensively available: ArcGIS, FME, GRASS, Radius Topology
• Possibilities: point to point, point to line, line to line
0.5 mmCORINE E30N20
CORINE E30N19
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Repair using snapping Not a complete solution
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High resolution data
Low resolution data
Repair using snapping Clean-up required
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After snapping:constraints for planar partition repair
• Radius Topology: share node, node-split-edge, edge-split-edge
• GRASS: break at intersections, remove duplicate line segments, remove dangling edges, remove bridges, remove vertices within a threshold of a line segment, remove too small areas, remove too small angles
• But still, no guarantees
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Topological planar partition repair
• Use topological constraints instead
• Snapping can still be performed
• Available in ArcGIS: must not overlap, must not have gaps, must not overlap with, must not have dangles, must not have pseudonodes, must not self intersect
CORINE E41N27
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Topological repair in ArcGIS Manual editing of topology
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Topological repair in ArcGIS Manual editing of topology
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The solution
• Repair individual polygons.
• Create a triangulation containing every edge of every polygon.
• Tag every triangle with the polygons it belongs to.
• Re-tag areas with multiple or no tags, according to predefined criteria.
• Reconstruct the polygons in the triangulation.
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Triangulations Point sets
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Triangulations Polygons
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The constrained Delaunay triangulation
• Delaunay triangulation: empty circle property, uniqueness
• Constrained edges
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Repair an individual polygon
• Use the same techniques devised to repair planar partitions.
• Create a triangulation from the polygon.
• Iteratively define exterior and interior when passing a constrained edge.
• Reconstruct polygon.
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Create triangulation
• Add every edge of every (now valid) polygon to the triangulation as a constrained edge.
• Track whenever constrained edges are split.
CORINE E39N32
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Tag triangulation
• Mark each triangle with the polygons that it belongs to.
• No tags = gap
• Multiple tags = overlap
CORINE E39N32 & E40N32
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Repair operations
• At the end, ensure that each triangle has exactly one tag.
• Some possible options:
• Assign triangle or region to the neighbour present on most sides.
• Assign triangle or region to the neighbour with the longest boundary.
• Assign region to the class with the highest priority.
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Repair operations Spain - Portugal border
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3 repair operations
Triangle with longest boundary
Random region Region with longest boundary
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Polygon reconstruction
• Recursively create a chain of edges representing all boundaries (and some connecting segments).
• Cut where more than two edges join.
• Join small chains in the correct order to form rings.
CORINE E40N31pol. 1752
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The prototype
• C++ with CGAL and OGR
• Open source and freely available
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Comparisons with other software
• Test polygons with specific problems (ArcGIS)
• Significant differences in interpretation
• Standards specify how to define a certain polygon, not how to interpret an existing one
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Comparisons with other software
• Large “normal” data sets (ArcGIS, FME, GRASS)
• Good performance, considering that it does much more
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Conclusions
• Planar partition validation and automatic repair with a constrained triangulation is theoretically simple, yet powerful.
• It keeps a valid topology throughout, without a complex set of rules to check every step of the way.
• Changes that are made to the triangulation have only a local effect.
• New repair operations, based on different criteria, can be easily implemented without breaking the validity of the planar partition.
• Snapping is possible, but not required.
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Future work
• Optimisations for simpler polygons
• Improved algorithms for extracting polygons from a triangulation
• Eliminating memory limitations
• Improving the order of point insertions
• Extension to 3D
• Implementation in a database
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Questions?
http://www.gdmc.nl/~ken/thesis.pdf
Department of GIS Technology, OTB Research Institute for the Built Environment
MSc thesis in Geomatics
Validation and automatic repair of planar partitions using a constrained triangulation
Ken Arroyo Ohori
Aug
ust 2
010