2IMG15 Algorithms for Geographic Data

1. 2IMG15 Algorithms for Geographic Data Spring 2017Lecture 8: Schematization

2. air pollution analysis earthquakes and break lines Automated cartography (recap) Automated cartography ... strivestoautomatealltaskthat are traditionallyperformed manually by a cartographer … • Visualize the results of an analysis • Make maps • general / special purpose • cartographic generalization • label placement

3. Cartographic generalization Topographic maps show spatial objects suitably for a target scale Scale: ratio of real size to map size • 1 : 20,000 large scale small extent • 1 : 150,000 • 1 : 800,000 small scale large extent Suitable: a legible map • Objects are distinguishable • Detail is perceivable • ...

4. Is everything to scale? Roads are 650m wide! Are they misleading us? • Yes • But with good reason

5. Do as the cartographers do Generalization • Deriving a small-scale map from a large-scale one Generalization operators • Different steps in the generalization process

6. An example Large-scale map Small-scale map

7. An example Select: choose what to show Large-scale map Small-scale map

8. An example Select: choose what to show Aggregate: combine things Large-scale map Small-scale map

9. An example Select: choose what to show Aggregate: combine things Typify: change representation Large-scale map Small-scale map

10. An example Select: choose what to show Aggregate: combine things Typify: change representation Simplify: remove detail Large-scale map Small-scale map

11. An example Select: choose what to show Aggregate: combine things Typify: change representation Simplify: remove detail Exaggerate: enlarge Large-scale map Small-scale map

12. An example Select: choose what to show Aggregate: combine things Typify: change representation Simplify: remove detail Exaggerate: enlarge Displace: make space Large-scale map Small-scale map

13. An example Select: choose what to show Aggregate: combine things Typify: change representation Simplify: remove detail Exaggerate: enlarge Displace: make space Large-scale map Small-scale map

14. A framework? Many classifications exist, the common operators are: Selection Aggregation Typification Displacement Simplification Exaggeration

15. Schematization

16. Schematic maps

17. Beyond a target scale Many thematic maps show far less than their scale permits • Special purpose: not everything is relevant • Only show functional detail • Even precise geography may not be functional

18. Ancient schematic maps

19. Ancient schematic maps

20. Ancient schematic maps

21. Not just for transport …

22. More schematic maps

23. Schematic?

24. Schematization What is schematization? a stylized, abstract representation • usually very simplified • iconic: few directions of lines, specific curves, … • might preserve topology • some visual resemblance to input Most commonly schematized • subdivisions • networks vertex-restricted or non-vertex-restricted area preserving, topology preserving, using curves or straight lines …

25. Network Schematization

26. Network schematization

27. Network schematization

28. Common criteria • many stations aligned horizontally, vertically or diagonally • sufficient spacing between different lines • connections have at most two bends • stations are not displaced too muchmaximum displacement can be different for different stations

29. Algorithmic solutions • many stations aligned horizontally, vertically or diagonally • sufficient spacing between different lines • connections have at most two bends • stations are not displaced too muchmaximum displacement can be different for different stations Many iterative approachessolution quality and convergence cannot be guaranteed

30. Algorithmic solutions • many stations aligned horizontally, vertically or diagonally • sufficient spacing between different lines • connections have at most two bends • stations are not displaced too muchmaximum displacement can be different for different stations [Cabello, de Berg, van Kreveld, 2005] combinatorial approach: • replace every connection by one of a schematic type • define a top-to-bottom placement order on connections • place each connection in its topmost position that gives no overlap

31. Algorithmic solutions [Cabello, de Berg, van Kreveld, 2005] combinatorial approach: • replace every connection by one of a schematic type • define a top-to-bottom placement order on connections • place each connection in its topmost position without overlap

32. Formalization DefinitionsA (polygonal) mapMis a set of simple polygonal paths {c1, …, cm} such that two paths do not intersect except at shared endpoints. A monotone map is a map where all paths are x-monotone.

33. Potential issues • keep cyclic order of paths around endpoints • each path of the schematic map is a deformation of the original path, without passing over endpoints • or, each path in the original map is a deformation of a path in the schematic map, without passing over the endpoints

34. Formalization Definitions Two maps M and M’are equivalentif an only if • they have the same endpoints • each path of M can be continuously deformed to a path of M’ without passing over the endpoints schematic path: axis-aligned, x-monotone, at most 3-links … Formal problem statement Given a polygonal map M, compute an equivalent map M’ whose paths are schematic. optional: minimum vertical distance, shared (pieces of) paths …

35. Formalization Formal problem statement Given a polygonal map M, compute an equivalent map M’ whose paths are schematic. optional: minimum vertical distance, shared (pieces of) paths … [Cabello, de Berg, van Kreveld, 2005] combinatorial approach: • replace every connection by one of a schematic type • define a top-to-bottom placement order on connections • place each connection in its topmost position without overlap

36. Formalization Formal problem statement Given a polygonal map M, compute an equivalent map M’ whose paths are schematic. optional: minimum vertical distance, shared (pieces of) paths … [Cabello, de Berg, van Kreveld, 2005] combinatorial approach: • replace every connection by one of a schematic type • define a top-to-bottom placement order on connections • place each connection in its topmost position without overlap

37. Below and above a curve Definition Point p is below/above path cif any continuous deformation of c, that does not pass over p, intersects the vertical upper/lowerray from p. • to decide whether a point is above or below a path, we do not consider other points c p1 p2

38. Below and above a curve Definition Point p is below/above path c if any continuous deformation of c, that does not pass over p, intersects the vertical upper/lower ray from p. c c c p p p p and c have no relation p is both above and below c p is below c

39. Order among paths Definition Path c is above path c’ if any endpoint of c is above c’ or any endpoint of c’ is below c. Lemma The above-below relation among paths is invariant between equivalent maps. not related below below

40. Order among paths Lemma The above-below relation among paths is invariant between equivalent maps. ➨ the above-below relation is preserved in the schematic map ➨ if the above-below relation is acyclic, extend to order and use to place schematic connections topmost Remaining questions Is there always an order? If there is an order, can we can compute it efficiently? Does an order imply a schematic map exists? At least for certain types of connections?

41. Order among paths Is there always an order? Lemma For a monotone map M, the above-below relation among paths is acyclic. Furthermore, if M has complexity n, a total order extending this relation can be computed in O(n log n) time. Definition Path c is above path c’ if any endpoint of c is above c’ or any endpoint of c’ is below c. for x-monotone paths equivalent to Path a is above path b (denoted a►b) if and only if there are points (x, ya) ∈ a and (x, yb) ∈ b with ya> yb. No

42. Computing above-below relationships If there is an order, can we can compute it efficiently? Theorem For a map M of total complexity n, we can decide in O(n log n) time whether an equivalent, monotone map exists.

43. Computing above-below relationships Theorem For a map M of total complexity n, we can decide in O(n log n) time whether an equivalent, monotone map exists. Proof • compute a rectified map M’in O(n log n) time make monotone use previous lemma to define order construct rectified map

44. Computing above-below relationships Theorem For a map M of total complexity n, we can decide in O(n log n) time whether an equivalent, monotone map exists. Proof • compute a rectified map M’in O(n log n) time • transform rectified map M’ into monotone map N if possible • for each path in M’, compute rectified canonical paths [Cabello et al, 2004] • N is monotone iff M admits an equivalent monotone map

45. Computing above-below relationships Theorem For a map M of total complexity n, we can decide in O(n log n) time whether an equivalent, monotone map exists. Proof • compute a rectified map M’in O(n log n) time • transform rectified map M’ into monotone map N if possible • if N is monotone, compute order in O(n log n) time

46. Computing above-below relationships Theorem For a map M of total complexity n, we can decide in O(n log n) time whether an equivalent, monotone map exists. Schematization of networksSergio Cabello, Mark de Berg, and Marc van Kreveld Computational Geometry 30:223–238, 2005 Testing Homotopy for Paths in the Plane Sergio Cabello, Yuanxin Liu, Andrea Mantler, and Jack Snoeyink Discrete & Computational Geometry 31(1):61-81, 2004

47. Schematic maps Does an order imply a schematic map exists? At least for certain types of connections? But if we specify the types of schematic connections, we can test … Intuition we can only use schematic connections together that have a clear topmost placement … x-monotone ordered schematic map model No No

48. Schematic map models x-monotone ordered schematic map models

49. Algorithm Input: a map M and an x-monotone ordered schematic map model Output: an equivalent schematic map M’ or “does not exist” • compute above-below relations among paths of M • if acyclic, complete to order, otherwise return “does not exist” • place paths (topmost each) in order return “does not exist” if placement is not possible Running time: O(n log n) where n is the complexity of M maintain lower envelop of already-placed connections …

50. Results ➨