04/19/23CS 294-1

Discussion of “Tracking a Moving Object

with a Binary Sensor Network”

Javed Aslam, Zack Butler, Florin Constantin, Valentino Crespi, George Cybenko, Daniela Rus

Bill Kramerkramer@nersc.gov

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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

Sj*V(t) < X(t) * V(t) < Si * V(t)

> /2 and < /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 huls are separated by the normal to the object’s 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 xk

j has to be outside plus and minus convex hulls xk

j is inside the circle of center S+ and of radius S+ to xk-1

j 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

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Experiments

Using MATLAB Random and grid

sensor alignment Linear, random turns

and mild turns (at most /6) directions

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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The Ultimate Goal

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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