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Noisy Road Network Matching

Noisy Road Network Matching. Mario A. López University of Denver . Yago Diez, J. Antoni Sellarès and Universitat de Girona . Motivation. “ Road Network Matching ”. In. Find. R’. R.

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Noisy Road Network Matching

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  1. Noisy Road Network Matching Mario A. López University of Denver Yago Diez, J. Antoni Sellarès and Universitat de Girona

  2. Motivation “Road Network Matching” In Find R’ R Known scale, unknown reference system (maps may appear rotated).

  3. Problem Formalization • We describe maps using road crossings • Adjacency degrees act as color cathegories.

  4. Problem Formalization Given two sets of road points A and B, |A| < |B|, find all the subsets B’ of B that can be expressed as rigid motions of A. • We want: • the points to approximately match(fuzzy nature ofreal data). • the adjacency degrees to coincide. • One-to-one matching! • (*) Rigid motion: composition of a translation and a rotation.

  5. Problem Formalization Let A, B be two road point sets of the same cardinality. An adjacency-degree preserving bijectivemapping f : SS’ maps each Road point P(a, r) to a distinct and unique road point f(P(a,r))=P(b,s) so that r = s. Let F be the set of all adjacency-degree preserving bijective mappings between S and S’. The Bottleneck Distance betweenS and S’ is is defined as: db(S , S’ ) = min f  F max P(a,r)  S d(P(a,r), f(P(a,r))).

  6. Given two road points sets A and B,n=|A|, m=|B|, n < m, and a real positive number ε, determine all the rigid motions τ for which there exists a subset B’ of B, |B’|=|A|, such that: db (τ(A),B’) ε (Bottleneck distance) Problem Formalization “Final Formulation”

  7. Example Find: Consider: A B

  8. Previous Work Previous Work On Road Network Matching Chen et Al.(STDBM’06): Similar problem with some differences: -Motions considered: - Chen et Al.: Translation + Scaling - Us: Translation + Rotation - Distance used: - Chen et Al.: Hausdorff - Us: Bottleneck

  9. Previous Work Previous Work On Point Set Matching Algorithms • Alt / Mehlhorn / Wagener / Welzl • (Discrete & Computational Geometry 88) • Efrat / Itai / Katz. (Comput. Geom. Theory Appl. 02) • Eppstein / Goodrich / Sun (SoCG 05) : Skip Quadtrees. • Diez / Sellarés (ICCSA 07)

  10. Matching Algorithm OUR APPROACH: • Tackle the problem from the COMPUTATIONAL GEOMETRY point of view. • Adapt the ideas in our paper at ICCSA 07 to the RNM problem. • Matching Algorithm: • Two main parts: • Enumeration • Testing

  11. Matching Algorithm Enumeration Generate all possible motions τ that may bring set A near some B’. We rule out all those pairs of points whose degrees do not coincide.

  12. Matching Algorithm Testing For every motion τ representative of an equivalence class, find a matching of cardinality n between τ(A) and S. Neighbor ( D(T), q ) A set of calls to Neighbor operation corresponds to one range search operation in a skip quadtree Delete ( D(T), s ) Corresponds to a deletion operation in a skip quadtree. Amortized cost of Neighbor, Delete: log n (Under adequate assumptions)

  13. Improving Running time Our main goal is to transform the problem into a series of smallerinstances. We will use a conservative strategy to discard, cheaply and at an early stage, those subsets of B where no match may happen. Our process consists on two main stages: 1. Losless Filtering Algorithm 2. Matching Algorithm (already presented!)

  14. Lossless Filtering Algorithm There cannot be any subset B‘ of B that approximately matches A fully contained in the four top-left quadrants, because A contains six points and the squares only five. • What geometric parameters, do we consider ? (rigid motion invariant ) • number of Road Points, • histogram of degrees, • max. and min. distance between points of the same degree, • CFCC codes.

  15. 2s (size s) Lossless Filtering Algorithm Initial step 1. Determine an adequate square bounding box of A. 2. Calculate associated geometric information.

  16. Lossless Filtering Algorithm . . . . . . Calculate quadtree of B with geometric parameters.

  17. . . . . . . Lossless Filtering Algorithm Points = 550 Points = 173 Points = 131 Points = 133 Points = 113 57 23 56 34 6 20 53 34 49 37 4 12 14 54 51 46 6 1 16 0 11 1 3 22 3 22 1 31 19 6 20 11 Example with geometric parameter: number of points

  18. Lossless Filtering Algorithm Search Algorithm a • Three search functions needed for every type of zone according to the current node: • Search type a zones. • -Search type b zones. • -Search type c zones. • The search begins at the root and continues until nodes of size s are reached. • Early discards will rule out of the search bigger subsets of B than later ones. c b b

  19. Lossless Filtering Algorithm points = 550 points = 173 points = 133 points = 131 points = 113 57 23 56 34 . . . 6 20 53 34 4 49 37 12 14 54 46 51 6 1 0 16 11 1 3 22 . . . 3 22 1 31 19 6 20 11 Search Algorithm • Search’s first step: • Target number of points = 25 • Launch search1?  yes • (in four sons) • Launch search2?  yes • (all possible couples) • Launch search3?  yes • (possible quartet)

  20. Lossless Filtering Algorithm points = 550 points = 173 points = 133 points = 131 points = 113 57 23 56 34 . . . 6 20 53 34 4 49 37 12 14 54 46 51 6 1 0 16 11 1 3 22 . . . 3 22 1 31 19 6 20 11 Search Algorithm • Target number of points = 25 • Launch search1?  yes • (in three sons) • Launch search2?  yes • (all possible couples) • Launch search3?  yes • (possible quartet)

  21. Lossless Filtering Algorithm

  22. points= 550 points = 173 points = 133 points = 131 points = 113 57 23 56 35 . . . 5 19 54 34 4 49 37 12 14 54 46 51 6 1 0 16 11 1 3 22 . . . 3 22 1 31 19 6 20 11 Lossless Filtering Algorithm Search Algorithm • Target number of points = 25 • Launch search1?  yes • (in two sons) • Launch search2?  yes • (three possible couples) • Launch search3?  yes • (possible quartet)

  23. Lossless Filtering Algorithm

  24. Lossless Filtering Algorithm Algorithm complexity: O(m2)

  25. Matching Algorithm Computational Cost Efrat, Itai, Katz: O( n4 m3 log m ) Our approach : ΣCand.ZonO( n4 n’ 3 log n’ )

  26. Implementation and Results Data used, Tiger/lines file from Arapahoe, Adams and Denver Counties:

  27. Experiments Experiment 1: Does the lossless filtering step help?

  28. Experiments Experiment 2: Filtering parameters comparison.

  29. Experiments Experiment 3: Computational Performance

  30. Experiments Experiment 3: Computational Performance

  31. Conclusions • First formalization of the NRNM problem in terms of the bottleneck distance. • Fast running times in light of the inherent complexity of the problem. • Experiments show how using the lossless filtering algorithm helps reduce the running time. • We have only used information that should be evident to all observers. • -We have also provided some examples on how the degree of noise in data influences the performance of the algorithm.

  32. Future Work • Other values of ε (for example, those that arise directly from the precision of measuring devices). • Maps with different levels of detail.

  33. Noisy Road Network Matching Mario A. López University of Denver Yago Diez, J. Antoni Sellarès and Universitat de Girona

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