Bill Kramer kramer@nersc

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 Kramer kramer@nersc.gov CS 294-1

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 CS 294-1

3. 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 CS 294-1

4. 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 CS 294-1

5. 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. CS 294-1

6. 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 CS 294-1

7. 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 CS 294-1

8. The Object Movement • Approximate inside area defined by • xkj has to be outside plus and minus convex hulls • xkj is inside the circle of center S+ and of radius 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 CS 294-1

9. 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 CS 294-1

10. 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 CS 294-1

11. The Ultimate Goal CS 294-1

12. 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 CS 294-1

13. Algorithm for Two Bit Sensors CS 294-1

14. 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 CS 294-1

15. Experiments CS 294-1

16. 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 CS 294-1

17. 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 CS 294-1