On Map-Matching Vehicle Tracking Data Sotiris Brakatsoulas Dieter Pfoser {sbrakats|pfoser}@cti.gr Carola Wenk Randall Salas {wenk|rsalas}@cs.utsa.edu.

On Map-Matching On Map-Matching Vehicle Tracking DataVehicle Tracking Data

Sotiris BrakatsoulasSotiris Brakatsoulas

Dieter PfoserDieter Pfoser

{sbrakats|pfoser}@cti.gr{sbrakats|pfoser}@cti.gr

Carola WenkCarola Wenk

Randall SalasRandall Salas

{wenk|rsalas}@cs.utsa.edu{wenk|rsalas}@cs.utsa.edu

VLDB '05 - D. Pfoser 2

MotivationMotivation

Moving Objects Data Moving Objects Data Vehicle Tracking DataVehicle Tracking Data TrajectoriesTrajectories


MotivationMotivation

Use of Use of Floating Car Data (FCD)Floating Car Data (FCD) generated by vehicle generated by vehicle fleet as samples to fleet as samples to assess to overall traffic conditionsassess to overall traffic conditions

Floating car data (FCD)Floating car data (FCD)– basic vehicle telemetry, e.g., speed, direction, ABS usebasic vehicle telemetry, e.g., speed, direction, ABS use– the the position of the vehicleposition of the vehicle ( ( tracking data) obtained by tracking data) obtained by

GPS trackingGPS tracking Traffic assessmentTraffic assessment

– data from one vehicle as a sample to assess to overall data from one vehicle as a sample to assess to overall traffic conditions – cork swimming in the river traffic conditions – cork swimming in the river

– large amounts of tracking datalarge amounts of tracking data (e.g., taxis, public (e.g., taxis, public transport, utility vehicles, private vehicles) transport, utility vehicles, private vehicles) accurate accurate picture of the traffic conditionspicture of the traffic conditions


Traffic Condition ParametersTraffic Condition Parameters

Traffic countTraffic count Travel timesTravel times

Relating tracking data to road network Relating tracking data to road network Map-MatchingMap-Matching


OutlineOutline

Vehicle Tracking Data, TrajectoriesVehicle Tracking Data, Trajectories– errors in the dataerrors in the data

Incremental MM TechniqueIncremental MM Technique– ““classical” approachclassical” approach

Global MM TechniqueGlobal MM Technique– curve – graph matchingcurve – graph matching

Quality of the Map-Matching Quality of the Map-Matching – MeasuresMeasures– Empirical EvaluationEmpirical Evaluation

Conclusions and future workConclusions and future work


Vehicle Tracking DataVehicle Tracking Data

Sampling the movementSampling the movement Sequence (temporal) of GPS pointsSequence (temporal) of GPS points

– affected by precision of GPS positioning erroraffected by precision of GPS positioning error– measurement errormeasurement error

Interpolating position samples Interpolating position samples trajectory trajectory– affected by frequency of position samplesaffected by frequency of position samples– sampling errorsampling error


Vehicle Tracking DataVehicle Tracking Data

Error exampleError example– vehicle speed 50km/h vehicle speed 50km/h

(max)(max)– sampling rate 30ssampling rate 30s

P1

P2

417m

208m

Map-matchingMap-matchingmatching trajectories matching trajectories to a path in the road to a path in the road networknetwork


Map MatchingMap Matching

Perception of the problemPerception of the problem– online vs. offline map-matchingonline vs. offline map-matching

Incremental methodIncremental method– incremental match of GPS points to road network incremental match of GPS points to road network

edgesedges– classical approachclassical approach

Global methodGlobal method– matching a curve to a graphmatching a curve to a graph– finding similar curve in graphfinding similar curve in graph


Incremental MethodIncremental Method

pi

c1

c2

c3

li

d1

d2

d3

αi,3αi,1

αi,2

pi-1

( , ) ( , ) dnd i j d i js p c a d p c

,( , ) cos( )ni j i js p c

ds s s

Position-by-position, edge-by-edge strategy to Position-by-position, edge-by-edge strategy to map-matchingmap-matching


Introducing globalityIntroducing globality Look-ahead to evaluate quality of different pathsLook-ahead to evaluate quality of different paths

– to match one edge consider its consequencesto match one edge consider its consequences Example: depth = 2 (depth = 1 Example: depth = 2 (depth = 1 no look-ahead) no look-ahead)


pi-1

pi

pi+1

c1

c2

c3



Actual map-matchingActual map-matching– evaluates for each trajectory edges (GPS point) a finite evaluates for each trajectory edges (GPS point) a finite

number of edges of the road network graphnumber of edges of the road network graph– O(O(nn) () (n – n – trajectory edges)trajectory edges)

Initialization done using spatial range queryInitialization done using spatial range query Map-matching dominates initialization costMap-matching dominates initialization cost


Global MethodGlobal Method

Try to find a Try to find a curve in the road networkcurve in the road network (modeled as (modeled as a graph embedded in the plane with straight-line a graph embedded in the plane with straight-line edges) that is as edges) that is as close as possible to the vehicle close as possible to the vehicle trajectorytrajectory

Curves are compared usingCurves are compared using– Fréchet distance andFréchet distance and– Weak Fréchet distanceWeak Fréchet distance

Minimize over all possible curves in the road Minimize over all possible curves in the road networknetwork


Fréchet DistanceFréchet Distance

Dog walking exampleDog walking example– Person is walking his dog (person on one curve and the dog Person is walking his dog (person on one curve and the dog

on other)on other)– Allowed to control their speeds but not allowed to go Allowed to control their speeds but not allowed to go

backwards!backwards!– Fréchet distance of the curves: Fréchet distance of the curves: minimal minimal leashleash length length

necessary for both to walk the curves from beginning to endnecessary for both to walk the curves from beginning to end


Fréchet DistanceFréchet Distance–

– where where αα and and ββ range over continuous non-decreasing range over continuous non-decreasing reparametrizations onlyreparametrizations only

Weak Fréchet DistanceWeak Fréchet Distance– – drop the non-decreasing requirement for drop the non-decreasing requirement for αα and and ββ–

Well-suited for the comparison of trajectories since they Well-suited for the comparison of trajectories since they take the continuity of the curves into accounttake the continuity of the curves into account

, :[0,1] [0,1] [0,1]( , ) : inf max ( ( )) ( ( ))F tf g f t g t

Fréchet Distance Fréchet Distance

~

( , )F f g~

( , ) ( , )F Ff g f g


Free Space DiagramFree Space Diagram

Decision variant of the global map-matching problem Decision variant of the global map-matching problem – for a fixed for a fixed εε > 0 decide > 0 decide whether there exists a whether there exists a pathpath in the road in the road

network withnetwork with distance at most distance at most εε to the vehicle trajectory to the vehicle trajectory αα For each edge For each edge ((ii,,jj)) in in a grapha graph GG let its corresponding let its corresponding

Freespace Diagram FDFreespace Diagram FDi,ji,j = FD( = FD(αα, (, (i,ji,j))))

ε

(i,j)

i

j

α

1 2 3 4 5 6 α0

1(i,j)


Glue free space diagrams FDGlue free space diagrams FD i,ji,j together according together according

to adjacency information in the graph to adjacency information in the graph GG Free space surface of trajectory Free space surface of trajectory αα and the and the

graph graph GG

Free Space SurfaceFree Space Surface

GG

αα shown shown implicitely by implicitely by the free space the free space surfacesurface


TASKTASK:: Find Find monotone pathmonotone path in free space surface in free space surface– starting in some lower left corner, andstarting in some lower left corner, and– ending in some upper right cornerending in some upper right corner


GG



Sweep-line algorithmSweep-line algorithm– maintain points on sweep line that are reachable by maintain points on sweep line that are reachable by

some monotone path in the free space from some some monotone path in the free space from some lower-left cornerlower-left corner

– updating reachability information Dijkstra styleupdating reachability information Dijkstra style Minimization problemMinimization problem for for εε is solved using is solved using

parametric search or binary searchparametric search or binary search– Parametric search (binary search)Parametric search (binary search)– O(O(mnmn log log22((mnmn)) time)) time

((mm – graph edges, – graph edges, nn – trajectory edges) – trajectory edges)– Weak Weak Fréchet distance, drop monotone requirementFréchet distance, drop monotone requirement– O(O(mnmn log log mnmn) time) time


Comparing FrComparing Fréchet distance of original and matched échet distance of original and matched trajectorytrajectory

FrFréchet distances strongly affected by outliers, since they échet distances strongly affected by outliers, since they take the take the maximummaximum over a set of distances. over a set of distances.

How to fix it? Replace the maximum with a path integral How to fix it? Replace the maximum with a path integral over the reparametrization curve (over the reparametrization curve (αα(t),(t),ββ(t)):(t)):

– Remark: Dividing by the arclength of the reparametrization curve Remark: Dividing by the arclength of the reparametrization curve yields a normalization, and hence an „average“ of all distances.yields a normalization, and hence an „average“ of all distances.

Quality of Matching ResultQuality of Matching Result

, :[0,1] [0,1] [0,1]( , ) : inf max ( ( )) ( ( ))F tf g f t g t

, :[0,1] [0,1]( , )

( , ) : inf ( ( )) ( ( ))F f g f t g t


Unfortunate drawbacksUnfortunate drawbacks– we do not know how to compute this integral.we do not know how to compute this integral.

Approximate integral by Approximate integral by sampling the curves sampling the curves and computing a sumand computing a sum instead of an integral. instead of an integral. – 22mm– very costly and gives no approximation guarantee or very costly and gives no approximation guarantee or

convergence rateconvergence rate

Quality of Matching ResultQuality of Matching Result


Empirical EvaluationEmpirical Evaluation

GPS vehicle tracking dataGPS vehicle tracking data– 45 trajectories 45 trajectories

(~4200 GPS points)(~4200 GPS points)– sampling rate 30 secondssampling rate 30 seconds

Road network dataRoad network data– vector map of Athens, Greecevector map of Athens, Greece

(10 x 10km) (10 x 10km) Evaluating matching qualityEvaluating matching quality

– results from results from incremental incremental vs.vs. global global method method– FrFréchet distance vs. averaged échet distance vs. averaged FrFréchet distance (worst-échet distance (worst-

case vs. average measure)case vs. average measure)



FrFréchet vs. Weak échet vs. Weak FrFréchet distance produces échet distance produces same matching resultsame matching result– no backing-up on trajectories (course sampling rate) or no backing-up on trajectories (course sampling rate) or – road network (on edge between intersections)road network (on edge between intersections)



Global algorithm produces better resultsGlobal algorithm produces better results Quality advantage reduced when using avg. Quality advantage reduced when using avg.

Fréchet measureFréchet measure

FrFréchet distanceéchet distance Avg. FrAvg. Fréchet distanceéchet distance


ConclusionsConclusions

Offline map-matching algorithmsOffline map-matching algorithms– FrFréchet distanceéchet distance based algorithm vs. based algorithm vs. incrementalincremental

algorithmalgorithm– accuracyaccuracy vs. vs. speedspeed– no difference between no difference between FrFréchetéchet and and weak weak FrFréchetéchet

algorithms in terms of matching results (data algorithms in terms of matching results (data dependent)dependent)

Matching qualityMatching quality– FrFréchet distance échet distance strict measurestrict measure– Average FrAverage Fréchet distance échet distance tolerates outlierstolerates outliers


Future WorkFuture Work

Pathfinder Projecthttp://dke.cti.gr/chorochronos

Making the Making the FrFréchet algorithm faster!échet algorithm faster!– Exploit trajectory data properties (error ellipse) to limit Exploit trajectory data properties (error ellipse) to limit

the graphthe graph– introduce localityintroduce locality

Other types of tracking dataOther types of tracking data– positioning technology (wireless networks, GSM, positioning technology (wireless networks, GSM,

microwave positioning)microwave positioning)– type of moving objects (planes, people)type of moving objects (planes, people)

Data management for traffic management and Data management for traffic management and controlcontrol


QuestionsQuestions

|| open norm|| open norm reparametrizationsreparametrizations dynamic programmingdynamic programming DijkstraDijkstra parametric search, binary searchparametric search, binary search complexity of the graphcomplexity of the graph


• Directed Hausdorff distance d

(A,B) = max min || a-b ||

• Undirected Hausdorff distance d(A,B) = max (d

(A,B) , d

(B,A) )

But:• Small Hausdorff distance

• When considered as curves the distance should be large

• The Fréchet distance takes continuity of curves into account

AB

(B,A)

(A,B)

What does „similar“ mean?What does „similar“ mean?

( , ) max mind A B a b


Free Space DiagramFree Space Diagram



Depending on the type of projection/match of Depending on the type of projection/match of ppii

to to ccjj , i.e., , i.e.,

– (i) its projection is between the end points of (i) its projection is between the end points of ccjj , or, , or,

– (ii) it is projected onto the end points of the line (ii) it is projected onto the end points of the line segment, segment,

the algorithm does, or does not advance to the the algorithm does, or does not advance to the next position sample.next position sample.



Introducing globalityIntroducing globality Look-ahead to evaluate quality of different pathsLook-ahead to evaluate quality of different paths Example: depth = 2 (depth = 1 Example: depth = 2 (depth = 1 no look-ahead) no look-ahead)

pi-1

pi

pi+1

On Map-Matching Vehicle Tracking Data Sotiris Brakatsoulas Dieter Pfoser {sbrakats|pfoser}@cti.gr Carola Wenk Randall Salas {wenk|rsalas}@cs.utsa.edu.

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