Tracking Moving Objects in Anonymized Trajectories

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Tracking Moving Objects in Anonymized Trajectories
Nikolay Vyahhi1, Spiridon Bakiras2, Panos
Kalnis3, and Gabriel Ghinita3 1St. Petersburg
State University 2John Jay College, City Univ. of
New York 3National University of Singapore
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Motivation

• Collection of Trajectory Data
• Example Traffic monitoring system
• GPS or Sensors deployed across a city
• Queries Predict traffic conditions
• Data expected to be anonymous
• Remove ID
• Reconstruction of original trajectories
• E.g., Police tracking a suspect

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Problem Statement

• Given a large database with anonymized
  spatio-temporal measurements, reconstruct the
  original object trajectories
• Requirements
• Efficiency (large databases)
• Accuracy (useful results)

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Problem Statement

• Input A series of M snapshots Si, each
  containing exactly N measurements from timestamp
  ti
• Output A set of N trajectories
• Each measurement can be associated with a single
  trajectory

M N 3
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Related work Multiple Target Tracking

• This problem is closely related to multiple
  target tracking (MTT) algorithms
• Studied in the field of radar technology
• Three major categories
• Nearest neighbor (NN)
• Joint probabilistic data association (JPDA)
• Multiple hypothesis tracking (MHT)

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Related work NN and JPDA

• They work in a single scan of the dataset
• Greedy approach in each timestamp, every sample
  is associated with a single track
• Objective minimize the error across all
  associations in the current timestamp
• Performance
• Efficient can work in polynomial time
• Greedy approach results in many false associations

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Related work MHT

• Multiple hypotheses are maintained
• Joint probabilities are calculated recursively
  when new measurements are received
• Each association is based on both previous and
  subsequent data (multiple scans)
• Unfeasible hypotheses are eventually eliminated
• Performance
• Very accurate
• Computational and space complexity is exponential
  to the number of measurements

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Comparison

• Very accurate
• Very slow

• Large errors
• Fast

• Very accurate
• Much faster than MHT

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Our ApproachMCMF Min-cost Max-flow

• Transform the tracking problem into a min-cost
  max-flow problem
• Min-cost max-flow (graph algorithm)
• Input a weighted graph G with two special nodes
  (source s and destination t)
• Objective find the maximum flow that can be sent
  from s to t that results in the minimum cost
• Well-known algorithms exist that work in
  polynomial time

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Transformation

• All edges have capacity 1
• Node id (ti, pi, pj) the object moves from
  location pi
• in timestamp ti to location pj in timestamp
  ti1

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Calculating the Cost Values

• Assume two successive measurements (pi and pj)
  belong to the same track
• Use these values to predict the next location
• Calculate the error (i.e., cost) for every
  possible location pk

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Limitation of this Approach

• Problem A single measurement can be associated
  with multiple tracks!

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SolutionCreate a Block for each Measurement
Block for kth measurement of mth timestamp (pm,k)

• Corresponds to all partial tracks pm-1,i ? pm,k ?
  pm1,j
• A block containing a flow is marked as active
• The only possible route inside an active block,
  is through the reverse path of the existing flow

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Block Functionality
Block for p3,1
Block for p2,1
Original track p1,1 ? p2,1 ? p3,1 New track
p1,1 ? p2,1 ? p3,2
Original track p2,1 ? p3,1 ? p4,1 New track
p2,2 ? p3,1 ? p4,1
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Improving the Running Time

• Flow network is too large
• Inefficient, since solution requires multiple
  shortest path calculations
• Assume any object can travel at most Rmax
  distance between two consecutive timestamps. Rmax
  depends on
• The maximum speed of the objects
• The time interval between two timestamps
• This reduces significantly the number of vertices
  and edges inside each block

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The Tracking Algorithm

• Successive Shortest Path Algorithm
• At each iteration, send a single flow unit across
  the shortest path from s to t
• Total of N iterations in our case
• Most efficient implementation
• Dijkstra with Fibonacci heap for priority queue
• Graph contains negative weights, but can utilize
  vertex potentials to avoid this (provided that
  there are no negative weight cycles)
• Bellman-Ford also works very well

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Dealing with Negative Weight Cycles

• Negative weight cycles do appear in MCMF
  calculations
• In this case, follow a greedy approach
• Output all the tracks that are discovered so far
• they might not be optimal
• Remove all vertices and edges associated with
  these tracks from the flow network
• Start a new min-cost max-flow calculation on the
  reduced graph

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Complexity

• Computational
• N iterations of a shortest path algorithm
• O(MN2K(log(MNK) K)) for Dijkstra with Fibonacci
  heap
• K is the average number of feasible associations
  (due to Rmax) per measurement
• Space
• O(MNK2) for storing the graph

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Experimental Evaluation

• Data generator
• Road map of San Francisco city
• For each object, randomly select a starting point
  and a destination point
• The object then follows the shortest path between
  the two points
• At each timestamp, every object i covers a
  distance di ? 0,Rmax
• Number of measurements 50,000 to 500,000

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Experimental Evaluation

• Competitor Global Nearest Neighbor (GNN)
• Employs clustering within each snapshot
• Considered the best single scan algorithm runs
  in O(MNC2) time (C is the average cluster size)
• Performance metrics
• CPU time
• Success rate percentage of partial tracks
  (triplets) that agree with original data

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Variable N
Success rate
CPU time sec
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Variable Rmax (speed)
CPU time sec
Success rate
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Points to Remember

• Multiple-Target Tracking
• Large Anonymized Trajectory Databases
• Existing methods are either inefficient or
  inaccurate
• We proposed a polynomial time solution based on a
  novel transformation of the MTT problem into a
  min-cost max-flow problem
• Very accurate
• Need to improve the running time

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Bibliography on LBS Privacy

• http//anonym.comp.nus.edu.sg

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