Target Tracking with Binary Proximity Sensors: Fundamental Limits, Minimal Descriptions, and Algorithms

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 0 R 1

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 • Ris 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. Geometry of Binary Sensing • 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(m3)

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 xk[n] • At the next time instant n+1, suppose that the localization patch is F • Choose m candidates for xk[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

41. 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