Post on 01-Apr-2015
transcript
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Yago Diez, J. Antoni Sellarès and
Universitat de Girona
Noisy Road Network Matching
Mario A. López
University of Denver
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“Road Network Matching”
Motivation
Known scale, unknown reference system (maps may appear
rotated).
Find
R’
In
R
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Problem Formalization
-We describe maps using road crossings
- Adjacency degrees act as color cathegories.
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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
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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:
db(S , S’ ) = min f F max P(a,r) S d(P(a,r), f(P(a,r))).
Problem Formalization
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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”
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Example
Consider:
A
B
Find:
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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
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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)
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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:
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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.
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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)
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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!)
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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.
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Initial step
1. Determine an adequate square bounding box of A.
2s (size s)
2. Calculate associated geometric information.
Lossless Filtering Algorithm
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Calculate quadtree of B with geometric parameters.
.
.
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Lossless Filtering Algorithm
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...
...
Points = 550
Points = 173Points = 113 Points = 131 Points = 133
23 5756
3720 6 53 34
54 12 1451 49 46 34 4
0 6 1 16
1 3 22 313 11 1 22
20 19 6 11
Example with geometric parameter: number of points
Lossless Filtering Algorithm
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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
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- Search’s first step:
Search Algorithm
...
...
points = 550
points = 173points = 113 points = 131 points = 133
23 5756
3720 6 53 34
54 12 1451 49 46 34 4
0 6 1 16
1 3 22 313 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
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Search Algorithm
...
...
points = 550
points = 173points = 113 points = 131 points = 133
23 5756
3720 6 53 34
54 12 1451 49 46 34 4
0 6 1 16
1 3 22 313 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
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Lossless Filtering Algorithm
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Search Algorithm
...
...
points= 550
points = 173points = 113 points = 131 points = 133
23 5756
3719 5 54 35
54 12 1451 49 46 34 4
0 6 1 16
1 3 22 313 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
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Lossless Filtering Algorithm
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Algorithm complexity:
O(m2)
Lossless Filtering Algorithm
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Matching Algorithm
Efrat, Itai, Katz:
O( n4 m3 log m )
Our approach :
ΣCand.Zon O( n4 n’ 3 log n’ )
Computational Cost
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Implementation and Results
Data used, Tiger/lines file from Arapahoe, Adams and Denver Counties:
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Experiments
Experiment 1: Does the lossless filtering step help?
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Experiments
Experiment 2: Filtering parameters comparison.
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Experiments
Experiment 3: Computational Performance
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Experiments
Experiment 3: Computational Performance
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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.
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Future Work
- Other values of ε (for example, those that arise directly from the precision of measuring devices).
- Maps with different levels of detail.
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Yago Diez, J. Antoni Sellarès and
Universitat de Girona
Noisy Road Network Matching
Mario A. López
University of Denver