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

## Presentation on theme: "1 Yago Diez, J. Antoni Sellarès and Universitat de Girona Noisy Road Network Matching Mario A. López University of Denver."— Presentation transcript:

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

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

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

4 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 of real data). the adjacency degrees to coincide. One-to-one matching! (*) Rigid motion: composition of a translation and a rotation. Problem Formalization

5 Let A, B be two road point sets of the same cardinality. An adjacency-degree preserving bijective mapping f : S S’ 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 between S and S’ is is defined as: d b (S, S’ ) = min f  F max P(a,r)  S d(P(a,r), f(P(a,r))). Problem Formalization

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: d b (τ(A),B’) ε (Bottleneck distance) Problem Formalization “Final Formulation”

7 Example Consider: A B Find:

8 Previous Work On Road Network Matching Previous Work 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 On Point Set Matching Algorithms Previous Work - 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 - 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 OUR APPROACH:

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

12 Matching Algorithm For every motion τ representative of an equivalence class, find a matching of cardinality n between τ(A) and S. Testing A set of calls to Neighbor operation corresponds to one range search operation in a skip quadtree Neighbor ( D(T), q ) 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 smaller instances. 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 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. 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.

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

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

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

18 Search Algorithm a b b c 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. Lossless Filtering Algorithm

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

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

21 Lossless Filtering Algorithm

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

23 Lossless Filtering Algorithm

24 Algorithm complexity: O(m 2 ) Lossless Filtering Algorithm

25 Matching Algorithm Efrat, Itai, Katz: O( n 4 m 3 log m ) Our approach : Σ Cand.Zon O( n 4 n’ 3 log n’ ) Computational Cost

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

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

Similar presentations