Topological Distance Between Nonplanar Transportation Networks
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1 Topological Distance Between Nonplanar Transportation Networks Ahmed Abdelkader 1, Geoff Boeing 2, Brittany Terese Fasy 3, David Millman 3 1 University of Maryland, College Park 2 Northeastern University 3 Montana State University Fall Workshop on Computational Geometry Queens College, City University of New York Flushing, NY October 26th, 2018 Ahmed Abdelkader (CS@UMD) FWCG / 15
2 Motivation Map Comparison Problem Two maps of Berlin: 2007 (left) vs (right) [Ahmad, Fasy and Wenk, SigSpatial 14] How to compare two such maps? Ahmed Abdelkader FWCG / 15
3 Motivation Why Map Comparison? Ground truth vs. density plot [AFW, SigSpatial 15] Applications Evaluation and comparison of map reconstruction algorithms Urban morphology Ahmed Abdelkader FWCG / 15
4 Motivation Why Topology? Two street intersections [AFW, SigSpatial 14] Two types of error Hausdorff distance: low Topology: different Ahmed Abdelkader FWCG / 15
5 Related Work Topology-based Map Comparison A topological signature at x [AFW, SigSpatial 14] Compute the local persistent homology in the neighborhood of x Define offsets by thickening street segments Compute a filtration from overlapping offsets Compute the persistent homology of the filtration.. relative to the boundary of the neighborhood Compare the barcodes for different maps Ahmed Abdelkader (CS@UMD) FWCG / 15
6 Related Work Assumption of Planarity.... can lead to drastic errors in street network analysis [Boeing, Environment & Planning B 18] Left: High Five Interchange, Dallas, USA Right: Magdeburg Water Bridge, Magdeburg, Germany Ahmed Abdelkader (CS@UMD) FWCG / 15
7 Related Work Multi-Layered Environments [van Toll, Cook, van Kreveld and Geraerts, TSAS 18] Nearly 2D For many planning problems, 3D is an overkill Extend algorithms and data structures from 2D Ahmed Abdelkader FWCG / 15
8 Model Grade-separated Street Networks Abstract model for street networks Allowing non-planarity Multiple copies of R 2 (layers) identified at special vertices (portals) Segment offsets grow through portals onto connected layers Ahmed Abdelkader (CS@UMD) FWCG / 15
9 Model Segment Offsets Layer 0 Layer 1 Layer 2 t = 0 Offset of a segment uv across three layers Ahmed Abdelkader (CS@UMD) FWCG / 15
10 Model Segment Offsets Layer 0 Layer 1 Layer 2 t = 0 t = 40 Offset of a segment uv across three layers Ahmed Abdelkader (CS@UMD) FWCG / 15
11 Model Segment Offsets Layer 0 Layer 1 Layer 2 t = 0 t = 40 t = 82 Offset of a segment uv across three layers Ahmed Abdelkader (CS@UMD) FWCG / 15
12 Model Segment Offsets Layer 0 Layer 1 Layer 2 t = 0 t = 40 t = 82 t = 136 Offset of a segment uv across three layers Ahmed Abdelkader (CS@UMD) FWCG / 15
13 Overlap Computations Overlap Computation - Two-way A C B Determine the earliest time when offsets intersect Case(1): segments in the same layer, e.g., A & C = old school Case(2): segments in different layers, e.g., A & B = via portals.. Ahmed Abdelkader (CS@UMD) FWCG / 15
14 Overlap Computations Overlap Computation - Two-way: Case(2) A B Shortest path through portal graph Precompute all-pairs shortest paths on a portal graph Connect A and B to portals on respective layers Find the shortest path from A to B Ahmed Abdelkader (CS@UMD) FWCG / 15
15 Overlap Computations Overlap Computation - Three-way Which layer realizes the overlap? Same as segment = offset is a segment offset (S) Different layer = offset is a disk centered at some portal (D) Earliest time when offsets intersect.. but at which layer? Case(1): SSS = segment Voronoi diagram Case(2): SSD = disk + segment bisector (straight & parabolic arcs) Case(3): SDD = weighted segment + disk bisector (hyperbola) Case(4): DDD = Apollonius diagram Ahmed Abdelkader (CS@UMD) FWCG / 15
16 Overlap Computations Overlap Computation - Three-way Which layer realizes the overlap? Same as segment = offset is a segment offset (S) Different layer = offset is a disk centered at some portal (D) Earliest time when offsets intersect.. but at which layer? Case(1): SSS = segment Voronoi diagram Case(2): SSD = disk + segment bisector (straight & parabolic arcs) Case(3): SDD = weighted segment + disk bisector (hyperbola) Case(4): DDD = Apollonius diagram Ahmed Abdelkader (CS@UMD) FWCG / 15
17 Overlap Computations Overlap Computation - Three-way: Case(1) SSS Case(1.1): earliest overlap at a Voronoi vertex Ahmed Abdelkader (CS@UMD) FWCG / 15
18 Overlap Computations Overlap Computation - Three-way: Case(1) SSS Case(1.2): earliest overlap in the interior of a cell Ahmed Abdelkader (CS@UMD) FWCG / 15
19 Overlap Computations Overlap Computation - Three-way: Case(4) DDD Case(4.1): earliest overlap at an Apollonius vertex Ahmed Abdelkader (CS@UMD) FWCG / 15
20 Overlap Computations Overlap Computation - Three-way: Case(4) DDD Case(4.2): earliest overlap in the interior of a cell Ahmed Abdelkader (CS@UMD) FWCG / 15
21 Overlap Computations Summary Topological Distance for Non-planar Street Networks Model grade-separated networks as multi-layered graphs Define segment offsets and filtrations across layers Next steps Implement LPH computation (based on CGAL & Dionysus) Run experiments, e.g., data from OpenStreetMap Formalize the new map distance Thanks for listening Questions? Ahmed Abdelkader FWCG / 15
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