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

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

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

Environment & Planning B 18] Left: High Five Interchange, Dallas,

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

Sampling Conditions for Conforming Voronoi Meshing by the VoroCrust Algorithm

Sampling Conditions for Conforming Voronoi Meshing by the VoroCrust Algorithm Ahmed Abdelkader 1, Chandrajit Bajaj 2, Mohamed Ebeida 3, Ahmed Mahmoud 4, Scott Mitchell 3, John Owens 4 and Ahmad Rushdi