MOFT-Multi Target Tracking Tutorials, Bayesian Inference and Filtering

1. MOFT Tutorials
Multi Object Filtering Multi Target Tracking
Bayesian Inference and Filtering

2. Bayesian Inference and Filtering
Bayesian Inference and Filtering
Classical Dynamic System
▪ system state evolves in the state space
▪ states are hidden & only partially observed in the observation space
▪ fundamental (dynamical) system problems, filtering, control, system
identification

3. Bayesian Inference and Filtering
Bayesian Inference and Filtering in MTT
Problem
▪ unknown state 𝐱
▪ position, velocity, acceleration
▪ measurements 𝐳
▪ sensor measurements of subset like position, bearing,etc
▪ estimate unknown state 𝐱 given the sequence of observations 𝐳
Bayesian Approach
▪ construct the posterior probability density function (pdf) of the state
using all available information from received noisy measurements

4. Bayesian Inference and Filtering
Overview of Bayesian Filtering and Bayes Filter
Objective
constructing the posterior density of the target state using the
observation history
𝑧1:𝑘 = 𝑧1, 𝑧2, ⋯ , 𝑧 𝑘
from time 1 to current time 𝑘,
by using
▪ dynamic model of the target (state transition equation)
▪ measurement model (observation equation)
▪ Bayes Filter

5. Bayesian Inference and Filtering
▪ general dynamic model

6. Bayesian Inference and Filtering
▪ general observation model

7. Bayesian Inference and Filtering
▪ State-space models

8. Bayesian Inference and Filtering
Dynamic Model and State Transition Equation
▪ target state 𝑥 𝑘 evolves in time according to the state transition
𝑥 𝑘 = 𝑓𝑘−1(𝑥 𝑘−1, 𝑣 𝑘−1)
▪ 𝑓𝑘−1 is a known, possible nonlinear function
▪ transforms any given state vector 𝑥 𝑘−1 and process noise 𝑣 𝑘−1 at time
𝑘 − 1 into a new state vector 𝑥 𝑘 at time 𝑘
▪ probabilistic description by a Markov transition density 𝑓𝑘|𝑘−1(𝑥 𝑘|𝑥 𝑘−1)
▪ the probability density that a target with state vector 𝑥 𝑘−1 at time 𝑘 −
1 moves to the state 𝑥 𝑘 at time 𝑘.

9. Bayesian Inference and Filtering
Measurement Model and Observation Equation
▪ relationship between the observation and the target state
𝑧 𝑘 = ℎ 𝑘(𝑥 𝑘, 𝑤 𝑘)
▪ ℎ 𝑘 is a known, possible nonlinear function
▪ transforms any given state vector 𝑥 𝑘 and process noise 𝑤 𝑘 at time
𝑘 into an observation vector 𝑧 𝑘
▪ probabilistic description by likelihood function 𝑔 𝑘(𝑧 𝑘|𝑥 𝑘)
▪ the probability density that a target with state vector 𝑥 𝑘 generates an
observation 𝑧 𝑘

10. Bayesian Inference and Filtering
Bayes Filter
▪ provides exact and complete characterization of the posterior density
(at each time step) in a recursive way
▪ estimates the state vector 𝑥 𝑘 recursively
▪ by using the sequence of all available observations 𝑧1:𝑘 up to time 𝑘
▪ constructs the posterior density 𝑝 𝑘(𝑥 𝑘−1|𝑧1:𝑘−1) of the state in two
steps
▪ Prediction Step
▪ Update Step
▪ a priori density 𝑝 𝑘−1(𝑥 𝑘−1|𝑧1:𝑘−1) at time 𝑘 − 1 is known

11. Bayesian Inference and Filtering

12. Bayesian Inference and Filtering
Bayes Filter Approximations
▪ Kalman Filter, KF
▪ Extended Kalman Filter, EKF
▪ Unscented Kalman Filter, UKF
▪ Particle Filter, PF

13. Bayesian Inference and Filtering
Kalman Filter

14. Bayesian Inference and Filtering
Kalman Filter

15. Bayesian Inference and Filtering
EKF - Extended Kalman Filter
▪ linearize model
▪ apply Kalman filtering equation to linearised model

16. Bayesian Inference and Filtering
UKF - Unscented Kalman Filter
▪ represent an 𝑛 𝑥–dimensional Gaussian by 2𝑛 𝑥 + 1 sigma-points
▪ predicted PDF: approximated by applying Φ 𝑘|𝑘−1 to sigma-points &
reconstruct Gaussian
▪ updated PDF: approximated by applying Ψ𝑘 to sigma-points &
reconstruct Gaussian.

17. Bayesian Inference and Filtering
Particle Filter

18. Bayesian Inference and Filtering
Particle Filter
▪ approximate posterior pdf by random samples (particles)
▪ recursively generate particle approximation of posterior pdfs
▪ no assumption about dynamic model or observation model

19. References
[1] Tutorial: Bayesian Filtering and Smoothing, Simo Sarkka, Aalto University,
Finland, EUSIPCO 2014, Lisbon, Portugal ,September 1, 2014
[2] A Tutorial on Bayesian Estimation and Tracking Techniques Applicable to
Nonlinear and Non-Gaussian Processes, A.J. Haug , MITRE TECHNICAL
REPORT January 2005
[3] Y. Bar-Shalom, X. Rong Li, T. Kirubarajan, “Estimation with Applications to
Tracking and Navigation”, Wiley, 2001
[4] U. Orguner, “Target Tracking”, Lecture notes, Linköpings University, 2010.
[5] S.S. Blackman, R. Popoli, “Design and Analysis of Modern Tracking
Systems”, Artech House, 1999
[6] Ba-Ngu Vo, Random Finite Set for Multi-object Dynamical System,
Department of ECE Curtin University Perth Western Australia

20. Other MOFT Tutorials – Lists and Links
Introduction to Multi Target Tracking
Bayesian Inference and Filtering
Kalman Filtering
Sequential Monte Carlo (SMC) Methods and Particle Filtering
Single Object Filtering Single Target Tracking
Nearest Neighbor(NN) and Probabilistic Data Association Filter(PDAF)
Multi Object Filtering Multi Target Tracking
Global Nearest Neighbor and Joint Probabilistic Data Association Filter
Data Association in Multi Target Tracking
Multiple Hypothesis Tracking, MHT

21. Other MOFT Tutorials – Lists and Links
Random Finite Sets, RFS
Random Finite Set Based RFS Filters
RFS Filters, Probability Hypothesis Density, PHD
RFS Filters, Cardinalized Probability Hypothesis Density, CPHD Filter
RFS Filters, Multi Bernoulli MemBer and Cardinality Balanced MeMBer, CBMemBer Filter
RFS Labeled Filters, Generalized Labeled Multi Bernoulli, GLMB and Labeled Multi Bernoulli, LMB Filters
Multiple Model Methods in Multi Target Tracking
Multi Target Tracking Implementation
Multi Target Tracking Performance and Metrics

22. http://www.egniya.com/EN/MOFT/Tutorials/
moft@egniya.com