1. Target Tracking with Binary Proximity Sensors: Fundamental Limits, Minimal Descriptions, and Algorithms N. Shrivastava, R. Mudumbai, U. Madhow, and S. Suri Univ. of California Santa Barbara SENSYS 2006 Presented by Jeffrey Hsiao

2. Outline Introduction Geometry of Binary Sensing Fundamental Limits Tracking Algorithms Simulation Results Mote Experiments Closing Remarks Comments

3. Outline Introduction Geometry of Binary Sensing Fundamental Limits Tracking Algorithms Simulation Results Mote Experiments Closing Remarks Comments

4. Introduction Binary Proximity Sensors –outputs a 1 when the target of interest is within its sensing range R –0 otherwise 1 0 R

5. Fundamental Limit Of Spatial Resolution Spatial Resolution –worst-case deviation between the estimated and the actual paths Ideal achievable resolution  is of the order of 1/  R –R is the sensing range of individual sensors –  is the sensor density per unit area

6. Minimal Representations For The Target’s Trajectory Analogy between binary sensing and the sampling theory and quantization –“high-frequency” variations in the target’s trajectory are invisible to the sensor field –estimate the shape or velocity for a “low-pass” version of the trajectory. –consider piecewise linear approximations to the trajectory that can be described economically

7. Occam’s Razor Approach Explanation of any phenomenon should make as few assumptions as possible Entities should not be multiplied beyond necessity All things being equal, the simplest solution tends to be the best one

8. Outline Introduction Geometry of Binary Sensing Fundamental Limits Tracking Algorithms Simulation Results Mote Experiments Closing Remarks Comments

9. Geometry of Binary Sensing

10. Signature

11. Geometry of Binary Sensing Localization Patch

12. Geometry of Binary Sensing Localization Arc

13. Outline Introduction Geometry of Binary Sensing Fundamental Limits Tracking Algorithms Simulation Results Mote Experiments Closing Remarks Comments

14. Fundamental Limits An Upper Bound on Spatial Resolution Achievability of Spatial Resolution Bound Remarks on Spatial Resolution Theorems Sampling and Velocity Estimation

15. An Upper Bound on Spatial Resolution THEOREM 1 If a network of binary proximity sensors has average sensor density  and each sensor has sensing radius R Then the worst-case error in localizing the target is at least W(1/  R)

16. An Upper Bound on Spatial Resolution

17. Achievability of Spatial Resolution Bound THEOREM 2 –Consider a network of binary proximity sensors, distributed according to the Poisson distribution of density , where each sensor has sensing radius R. –Then the localization error at any point in the plane is of order 1/  R.

18. Remarks on Spatial Resolution Theorems The more sensors we have, the better the spatial accuracy one should be able to achieve Having a large sensing radius may seem like a disadvantage A quadratic increase in the number of patches into which the sensor field is partitioned by the sensing disks

19. Sampling and Velocity Estimation Low-pass Trajectories

20. Velocity Estimation Error THEOREM 3 –Suppose a portion of the trajectory is approximated by a straight line segment of length L to within spatial resolution  –Then, the maximum variation in the velocity estimate due to the choice of different candidate straight line approximations is at most –Furthermore, this also bounds the relative velocity error if the true trajectory is well approximated as a straight line over the segment under consideration

21. Outline Introduction Geometry of Binary Sensing Fundamental Limits Tracking Algorithms Simulation Results Mote Experiments Closing Remarks Comments

22. Tracking Algorithms I – be the subset of sensors whose binary output is 1 during the relevant interval Z –be the remaining sensors whose binary output is 0 during this interval

23. Spatial Band B

27. Analysis of OCCAMTRACK THEOREM 4 –The algorithm OCCAMTRACK computes a piecewise linear path that visits the localization arcs in order and uses at most twice the optimal number of segments in the worst-case –If there are m arcs in the sequence, then the worst-case time complexity of OCCAMTRACK is O(m 3 )

28. Robust Tracking With Non-ideal Sensors

29. Particle Filtering Algorithm At any time n, we have K particles (or candidate trajectories) –with the current location for the kth particle denoted by x k [n] At the next time instant n+1, suppose that the localization patch is F –Choose m candidates for x k [n+1] uniformly at random from F We now have mK candidate trajectories Pick the K particles with the best cost functions

30. Cost Function Chose an additive cost function that penalizes changes in the vector velocity

31. Geometric Postprocessing Particle filtering algorithm described above gives a robust estimate of the trajectory consistent with the sensor observations But it provides no guarantees of a “clean” or minimal description

32. Geometric Postprocessing

33. Outline Introduction Geometry of Binary Sensing Fundamental Limits Tracking Algorithms Simulation Results Mote Experiments Closing Remarks Comments

34. Simulation Results The code for OCCAMTRACK was Written in C and C++ The code for PARTICLE-FILTER was written in Matlab The experiments were performed on an AMD Athlon 1.8 Ghz PC with 350 MB RAM

35. OCCAMTRACK with ideal sensing A 1000×1000 unit field Containing 900 sensors in a regular 30×30 grid Sensing range for each sensor was set to 100 units Geometric random walks to generate a variety of trajectories –Each walk consists of 10 to 50 steps, where each step chooses a random direction and walks in that direction for some length, before making the next turn Each trajectory has the same total length Generated 50 such trajectories randomly

36. Quality Of Trajectory Approximation

37. Velocity Estimation Performance

38. Spatial Resolution As A Function Of Density

39. Spatial Resolution As A Function Of Sensing Range

40. Tracking with Non-Ideal Sensing

42. Outline Introduction Geometry of Binary Sensing Fundamental Limits Tracking Algorithms Simulation Results Mote Experiments Closing Remarks Comments

43. Mote Experiments 16 MICA2 motes arranged in a 4×4 grid with 30 centimeter separation

44. Probability Of Target Detection With Distance

45. Mote Experiments

46. Outline Introduction Geometry of Binary Sensing Fundamental Limits Tracking Algorithms Simulation Results Mote Experiments Closing Remarks Comments

47. Closing Remarks Have identified the fundamental limits of tracking performance possible with binary proximity sensors Have provided algorithms that approach these limits

48. Closing Remarks Results show that the binary proximity model, despite its minimalism, does indeed provide enough information to achieve respectable tracking accuracy –assuming that the product of the sensing radius and sensor density is large enough

49. Future Works An in-depth understanding, and accompanying algorithms, for multiple targets is therefore an important topic for future investigation To incorporate additional information (e.g., velocity, distance) if available To embed Occam’s razor criteria in the particle filtering algorithm

50. Outline Introduction Geometry of Binary Sensing Fundamental Limits Tracking Algorithms Simulation Results Mote Experiments Closing Remarks Comments

51. Strength –A simple yet robust algorithm proposed for target tracking in sensor networks –Complete work Analysis Simulation Experiments Weakness –For non-ideal case, performance for particle filtering and geometric postprocessing is not mentioned Time complexity could be high

52. Thank You Very Much For Your Attention! Any More Questions?