Particle Filtering

1. Darryl Morrell Stochastic Modeling Seminar 1 Particle Filtering

2. Darryl Morrell Stochastic Modeling Seminar 2








10. Darryl Morrell Stochastic Modeling Seminar 10
















26. Darryl Morrell Stochastic Modeling Seminar 26 Particle Filters: a Solution to Hard Problems in Navigation, Target Tracking, and Perception Darryl Morrell & Ya Xue Department of Electrical Engineering Arizona State University Portions of this work supported by AFOSR under award number F49620-00-1-0124

27. Darryl Morrell Stochastic Modeling Seminar 27 References for More Information M. Arulampalam, S. Maskell, N. Gordon, and T. Clapp. A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking. IEEE Transactions on Signal Processing, 50(2):174-188, February 2002. This is an excellent tutorial paper-read this first. A. Doucet, N. de Freitas, and N. Gordon, editors. Sequential Monte Carlo Methods in Practice. Springer-Verlag, 2001. This is a broad ranging collection of articles that will introduce your to most of the important particle filter developments.

28. Darryl Morrell Stochastic Modeling Seminar 28 Introduction �The importance of Monte Carlo methods for inference in science and engineering problems has grown steadily over the past decade. This growth has largely been propelled by an explosive increase in accessible computing power. �it has become clear that Monte Carlo methods can significantly expand the class of problems that can be addressed practically.� (Introduction to Feb 2002 IEEE Transactions on Signal Processing special issue on Monte Carlo Methods)

29. Darryl Morrell Stochastic Modeling Seminar 29 Sequential Monte Carlo Techniques Sequential Monte Carlo techniques have been developed in a wide range of disciplines, and go under many names: Bootstrap filtering The condensation algorithm Particle filtering Interacting particle approximations Survival of the fittest

30. Darryl Morrell Stochastic Modeling Seminar 30 Presentation Outline Applications of particle filters Fundamental concepts Anatomy of a simple particle filter Variations on the simple particle filter Pros and Cons of particle filters Application to configuration of a foveal sensor Conclusions

31. Darryl Morrell Stochastic Modeling Seminar 31 Applications of Particle Filters Particle filters have provided solutions to problems from many disciplines: image processing and understanding tracking complex objects (e.g. people) in video sequences robot navigation tracking and identifying complex military targets (e.g. vehicle convoys)

32. Darryl Morrell Stochastic Modeling Seminar 32 Some Specific Applications Terrain aided navigation Car positioning using map information Robot navigation Tracking of articulated targets using video Tracking of complex targets using distributed sensors.

33. Darryl Morrell Stochastic Modeling Seminar 33 Terrain Aided Navigationhttp://www.control.isy.liu.se/research/sensorfusion/sensorfusion/sensorfusion.html Observations are measured ground clearance. Unknowns are aircraft position and velocity. The particle filter is needed because measured ground clearance does not uniquely determine position.

34. Darryl Morrell Stochastic Modeling Seminar 34 Car Positioning Using Map InfoGustafsson et al., �Particle Filters for Positioning, Navigation, and Tracking,� IEEE Transactions on SP, Feb 2002 Observations are yaw rate and speed information computed from wheel speed sensors. Vehicle position is unknown. The map provides constraints on the vehicle position.

35. Darryl Morrell Stochastic Modeling Seminar 35 Mobile Robot Localizationhttp://www.cs.washington.edu/ai/Mobile_Robotics/mcl/2 Observations are sensor data (image, video, sonar, laser rangefinder, etc.) Robot position is unknown. The robot�s position is estimated by correlating sensor data with known maps.

36. Darryl Morrell Stochastic Modeling Seminar 36 Tracking Articulated Objectshttp://www.dai.ed.ac.uk/CVonline/LOCAL_COPIES/RINGER1/mocap_overview.html Observations are video sequences from two cameras. Unknowns are positions and velocities of model components

37. Darryl Morrell Stochastic Modeling Seminar 37 Tracking with Networks of Distributed Sensorshttp://www.parc.xerox.com/spl/projects/cosense/ Targets are tracked using an ad hoc network of distributed micro-sensors.

38. Darryl Morrell Stochastic Modeling Seminar 38 Other Applications Channel equalization Estimation of parameters of multiple chirp signals Multiple target tracking Bearing�s-only target tracking Track before detect target tracking Image segmentation


40. Darryl Morrell Stochastic Modeling Seminar 40 Fundamental Concepts Bayesian inference Monte Carlo samples Importance Sampling Resampling

41. Darryl Morrell Stochastic Modeling Seminar 41 Bayesian Inference X is unknown-a random variable or set (vector) of random variables Z is observed-also a set of random variables We wish to infer X by observing Z. The probability distribution p(x) models our prior knowledge of X. The conditional probability distribution p(z|x) models the relationship between Z and X.

42. Darryl Morrell Stochastic Modeling Seminar 42 Bayes Theorem The conditional distribution p(x|z) represents posterior information about X given Z.

43. Darryl Morrell Stochastic Modeling Seminar 43 Monte Carlo Samples (Particles) The posterior distribution p(x|z) may be difficult or impossible to compute in closed form. An alternative is to represent p(x|z) using Monte Carlo samples (particles): Each particle has a value and a weight

44. Darryl Morrell Stochastic Modeling Seminar 44 Importance Sampling Ideally, the particles would represent samples drawn from the distribution p(x|z). In practice, we usually cannot get p(x|z) in closed form; in any case, it would usually be difficult to draw samples from p(x|z). We use importance sampling: Particles are drawn from an importance distribution. Particles are weighted by importance weights.

45. Darryl Morrell Stochastic Modeling Seminar 45 Resampling In inference problems, most weights tend to zero except a few (from particles that closely match observations), which become large. We resample to concentrate particles in regions where p(x|z) is larger.


47. Darryl Morrell Stochastic Modeling Seminar 47 Anatomy of a Simple Particle Filter A simple particle filter requires the following: A system state evolution model An observation model Particle computation processes: Propagate forward in time Compute weights given observations Resampling

48. Darryl Morrell Stochastic Modeling Seminar 48 System State The state represents the unknown whose value we want to infer. For example, Position (and velocity) of a robot, car, plane, ... Position of articulated model components. The system state at (discrete) time k is denoted xk. The state evolves according to the following dynamics equation: xk+1 = fk (xk, wk)

49. Darryl Morrell Stochastic Modeling Seminar 49 Observation Model The observation zk may be an image, a frame of video, a radar or sonar measurement, etc. The relationship between the observation and the state is given by the conditional probability distribution p(zk | xk). This distribution may be derived from a functional relationship between zk and xk : zk = hk (xk, vk)

50. Darryl Morrell Stochastic Modeling Seminar 50 Objective-Find p(xk|zk,�,z1) The objective of the particle filter is to compute the conditional distribution p(xk|zk,�,z1) To do this analytically, we would use the Chapman-Kolmogorov equation and Bayes Theorem along with Markov model assumptions. The particle filter gives us an approximate computational technique.

51. Darryl Morrell Stochastic Modeling Seminar 51 Particle Filter Algorithm Create particles as samples from the initial state distribution p(x0). For k going from 1 to K Sample each particle from a proposal distribution. Compute weights for each particle using the observation value. (Optionally) resample particles.

52. Darryl Morrell Stochastic Modeling Seminar 52 Initial State Distribution

53. Darryl Morrell Stochastic Modeling Seminar 53 State Update

54. Darryl Morrell Stochastic Modeling Seminar 54 Compute Weights

55. Darryl Morrell Stochastic Modeling Seminar 55 Resample

56. Darryl Morrell Stochastic Modeling Seminar 56 Particle Filter Demonstration A target moves from left to right. Two sensors: Each measures the distance from itself to the target. Sensors at (30,0) and (0,50) 4000 Particles were used to track the target. The animation on the following slide shows the particles, the true target position, and the estimated target position.

57. Darryl Morrell Stochastic Modeling Seminar 57 Particle Filter Demonstration


59. Darryl Morrell Stochastic Modeling Seminar 59 Variations on this Simple Implementation Use a different importance distribution: In this implementation, the importance distribution is the predicted state distribution p(xk+1|zk,�,z1). Several papers have pointed out that this distribution may not be the best one can use. If the observation at time k+1 is available, significant improvement in performance can be obtained.

60. Darryl Morrell Stochastic Modeling Seminar 60 Variations Use a different resampling technique: Resampling adds variance to the estimate; several resampling techniques are available that minimize this added variance. Our simple resampling leaves several particles with the same value; methods for spreading them are available.

61. Darryl Morrell Stochastic Modeling Seminar 61 Variations Reduce the resampling frequency: Our implementation resamples after every observation, which may add unneeded variance to the estimate. Alternatively, one can resample only when the particle weights warrant it. This can be determined by the effective sample size.

62. Darryl Morrell Stochastic Modeling Seminar 62 Variations Rao-Blackwellization: Some components of the model may have linear dynamics and can be well estimated using a conventional Kalman filter. The Kalman filter is combined with a particle filter to reduce the number of particles needed to obtain a given level of performance.


64. Darryl Morrell Stochastic Modeling Seminar 64 Advantages of Particle Filters Under general conditions, the particle filter estimate becomes asymptotically optimal as the number of particles goes to infinity. Non-linear, non-Gaussian state update and observation equations can be used. Multi-modal distributions are not a problem. Particle filter solutions to inference problems are often easy to formulate.

65. Darryl Morrell Stochastic Modeling Seminar 65 Disadvantages of Particle Filters Na�ve formulations of problems usually result in significant computation times. It is hard to tell if you have enough particles. The best importance distribution and/or resampling methods may be very problem specific.


67. Darryl Morrell Stochastic Modeling Seminar 67 A Foveal Sensor A foveal sensor has a high acuity area (similar to the fovea of the eye) that can be steered towards a desired location.

68. Darryl Morrell Stochastic Modeling Seminar 68 Mathematical Model The foveal sensor is modeled mathematically as: zk = tan-1(Ck(xk-dk)) xk is the target position. dk controls the location of the center of the foveal region. Ck controls the width of the foveal region.

69. Darryl Morrell Stochastic Modeling Seminar 69 Before 1st Observation

70. Darryl Morrell Stochastic Modeling Seminar 70 Collecting 1st Observation

71. Darryl Morrell Stochastic Modeling Seminar 71 Collecting 2nd Observation

72. Darryl Morrell Stochastic Modeling Seminar 72 Collecting 3rd Observation

73. Darryl Morrell Stochastic Modeling Seminar 73 Implementation We implemented a particle filter to estimate the target position from observations. The foveal region is centered on the predicted target position. The gain is either set to a constant value or adjusted to include a certain percentage of the particles in the foveal region. The implementation took a few hours. Tuning the filter has taken a few weeks.

74. Darryl Morrell Stochastic Modeling Seminar 74 Comparison with Previous Foveal Sensor A two dimensional linear system dynamics model is used. The system state transition matrix is stable. The following plot shows curves of constant estimation error as a function of process and observation noise variance: Stat fixed gain-Kalman filter implementation of fixed gain sensor PF fixed gain-Particle filter implementation of a fixed gain sensor PF Var. gain-Particle filter implementation of an adaptive gain sensor

75. Darryl Morrell Stochastic Modeling Seminar 75 Curves of Constant Error

76. Darryl Morrell Stochastic Modeling Seminar 76 Discussion of Results Adaptive acuity gives better performance than fixed acuity. The particle filter implementations do not perform well with very small observation noise variances. The number of particles is too small for very sharply peaked observation densities-few particles fall within the peaks. Several approaches to improve the performance for small observation variances are currently under investigation.

77. Darryl Morrell Stochastic Modeling Seminar 77 Fixed vs. Adaptive Acuity The foveal sensor collects observations of position. The acuity of the foveal region is adjusted so that 80% of the predicted particle positions fall into the foveal region. The gain of the foveal region is smoothed using a low-pass filter with an exponentially decaying impulse response. We show plots of the average squared estimate error as a function of time for Fixed acuity foveal sensor for gains of 0.25, 1, and 4 Adaptive acuity foveal sensor

78. Darryl Morrell Stochastic Modeling Seminar 78 Average Estimate Error


80. Darryl Morrell Stochastic Modeling Seminar 80 Conclusions Particle filters (and other Monte Carlo methods) are a powerful tool to solve difficult inference problems. Formulating a filter is now a tractable exercise for many previously difficult or impossible problems. Implementing a filter effectively may require significant creativity and expertise to keep the computational requirements tractable.

Particle Filtering

Particle Filtering

Presentation Transcript

Particle Filtering a brief introductory tutorial

Particle Filtering for Diagnosis, Prognosis and On Condition Maintenance

An introduction to Particle filtering

Particle Filtering

Particle filtering

Rao-Blackwellized Particle Filtering Pieter Abbeel UC Berkeley EECS

Filtering

Filtering

Particle Filtering

Annealed Particle Filtering

Auxiliary particle filtering: recent developments

Filtering

Rao-Blackwellised Particle Filtering

Dynamic Bayesian Networks and Particle Filtering

Rao-Blackwellised Particle Filtering for Dynamic Bayesian Networks

6.891 Computer Experiments for Particle Filtering

4. Particle Filtering

Filtering

Particle Filtering in MEG: from single dipole filtering to Random Finite Sets

Introduction To Particle Filtering:

Fixed-lag Sampling Strategies for Particle Filtering SLAM