Discussion of

1
Discussion of Tracking a Moving Object with a
Binary Sensor NetworkJaved Aslam, Zack Butler,
Florin Constantin, Valentino Crespi, George
Cybenko, Daniela Rus

• Corey Miller

2
One Bit Sensors

• Sensors with a small number of bits save
  communications and energy
• Three assumptions
• Sensors can identify a target approaching or
  moving away
• The sense bits are available to a centralized
  processor
• Can be done with a broadcast or other ways
• For precise location, sensors have another sense
  bit that provides proximity information
• Sensors indicate plus if object is approaching
  and minus if object is moving away

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The Basic Idea

• A convex hull of a set of points is defined as
• Formally It is the smallest convex set
  containing the points.
• Informally It is a rubber band wrapped around
  the "outside" points.
• Plus and Minus sensors each have a convex hull
• Current position of the object is between the
  convex hull of the plus sensors and the convex
  hull of the minus sensors
• The object is moving towards the convex hull of
  the plus sensors

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Diagram of the Basic Idea

• Sj is the minus sensor
• Si is the plus sensor
• X is the position of the object
• V is direction of movement X(t)
• dl is the increment of movement
• From Lemma 1
• SjV(t) lt X(t) V(t) lt Si V(t)
• ? gt ?/2 and ? lt ?/2

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Limits of the method

• Coarse approximation
• the object is outside the minus and plus convex
  hulls. (Theorem 2)
• C(plus) ? C(minus) ?
• X(t) ? C(plus) ? C(minus)
• The plus and minus hulls are separated by the
  normal to the objects velocity (Theorem 2)
• V points towards C(plus)
• Can translate this into linear programming
  equations.

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Using history

• Future positions of the object have to lie inside
  all the circles whose center is located at a plus
  sensor and
• Outside all the circles whose center is located
  at a minus sensor
• Each sensor has a radius d(S,X) the distance
  between S and X

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Algorithm for a One Bit Sensor

• Uses particle filtering
• Translates continuous probability density
  function into a discrete probability vector
• Allows non-Guassian errors
• Predictive and update cycles
• A new set of particles is created for each sensor
  reading
• Previous position is chosen according to the old
  weights
• A possible successor position is chosen
• If the successor position meets acceptance
  criteria, add it to the set of new particles and
  compute a weight

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The Object Movement

• Approximate inside area defined by
• xkj (new particle) has to be outside plus and
  minus convex hulls
• xkj is inside the circle of center S with radius
  the distance from S to xk-1j
• S is any plus sensor at time k and k-1
• xkj is outside the circle of center S- and of
  radius S- to xk-1j
• S- is any plus sensor at time k and k-1
• Probability of particles is used to determine
  which position is the predicted one
• All particles with probability above a threshold
  are used
• Low threshold increases estimation error
• High threshold increases running time

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Experiments

• Using MATLAB
• Random and grid sensor alignment
• Linear, random turns and mild turns (direction
  change of at most ?/6)
• Used root mean square error
• Particles with equal weight and
• Particles with weight according to their
  probabilities
• Not clear why trend of probability weighed
  answers changes for random, linear

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Limitations of the model

• Can only distinguish direction of motion not
  location
• Trajectories that have parallel velocities with a
  constant distance apart cannot be separated.
• The paper formally proves this

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Limitations of the model
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The Proximity Bit

• In addition to the plus/minus bit, sensors can
  have a proximity bit
• For example an IR sensor
• Range can be different
• Useful to set so proximity bits do not overlap
• Algorithm 1 is extended
• When a sensor detects an object the ancestors of
  every particle that has not been inside the range
  are shifted as far as the last time the object
  was spotted by proportional amounts.
• This is algorithm 1 when no proximity sensor is
  triggered

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Algorithm for Two Bit Sensors
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Experiments

• Metric is relative position error after the
  object is detected by a proximity sensor
• How many trajectories out of 10,000 are detected
  after k steps.
• The distribution of the amount of time that
  passes until an object is first spotted is
  exponential

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Experiments
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Experiments

• Algorithm 2 greatly improves the accuracy of
  location estimation.
• Down to a RMSE of .02 for a 64 sensor network
• Grid layout somewhat better than random
• Sufficient for many tracking applications

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Summary

• Basically the approach asks each sensor
• Is the object moving toward or away from you?
• Calculates velocity
• Is the sensor in your proximity?
• Determines likely position
• Several open questions
• How to handle noise
• Report a 0 if signal is below a threshold?
• Or declare the sensor untrustworthy through a
  central approximation
• Use of only frontier sensors those that are
  visible from the convex hull
• Decentralize the computation