Review of Cooperative Localization with Factor Graphs. Aggelos Bletsas ECE TUC. Noptilus Project Sept. 2011
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1 Review of Cooperative Localization with Factor Graphs Aggelos Bletsas ECE TUC Noptilus Project Sept. 2011
2 Acknowledgments Material of this presentation from: [1] H. Wymeersch, J. Lien, M.Z. Win, Cooperative Localization in Wireless Networks, Proceedings of the IEEE, Vol. 97, No. 2, Feb [2] F. R. Kschichang, B. J. Frey and H.A. Loeliger, Factor Graphs and the Sum- Product Algorithm, IEEE Transactions on Information Theory, Vol. 47, No. 2, Feb Noptilus - Sept Cooperative Localization with Factor Graphs 2
3 Summary 1. The Cooperative Localization Problem 2. The Factor Graph Approach 3. Short Tutorial on Factor Graphs (FG) and Sum-Product Algorithm (SPA) 4. Cooperative Localization FG and run example 5. Results 6. Thoughts and Comments Noptilus - Sept Cooperative Localization with Factor Graphs 3
4 The problem Ambiguous locations of agents 2 and 4. Cooperation can resolve such ambiguity. Noptilus - Sept Cooperative Localization with Factor Graphs 4
5 Basic Approaches in the EE community... Exchange messages between neighboring nodes: Gradient descent minimization based on distributed LS (2000)... Gradient descent minimization based on distributed MDS (2005)... Barycentric-weighted sum of neighbor s coordinates (2009)... Probabilistic approach: Message Passing with SPA on a FGs (2009)... Noptilus - Sept Cooperative Localization with Factor Graphs 5
6 The probabilistic approach (PA) (Marcovian) Mobility Model Basic idea: inference based on posterior probability... Posterior probability: MMSE or MAP estimator. Noptilus - Sept Cooperative Localization with Factor Graphs 6
7 PA Further Assumptions Independent movement and measurements across different nodes... Posterior as a product over time and number of terminals! Noptilus - Sept Cooperative Localization with Factor Graphs 7
8 Cooperative Localization with Factor Graphs (FG) Unknown location x i of agent node i: random variable... The FG explicitly models the posterior and the interconnections between agents and anchors! The FG will estimate the density function of x i for each agent i... How?...by marginalization of the posterior above... Algorithm?...Sum Product Algorithm (SPA)! Noptilus - Sept Cooperative Localization with Factor Graphs 8
9 Brief Tutorial on FG and SPA (1) FGs: map a product of factors to a graph of variables related to edges and factors as vertices... FGs: have been extensively used in the Turbo Codes/LDPC community... FGs: related to Markov Random Fields and Bayesian Networks (more on this later) Noptilus - Sept Cooperative Localization with Factor Graphs 9
10 Brief Tutorial on FG and SPA (2) Sum Product Sum Product Algorithm (SPA): Targets at marginalization i.e. find out the density/distribution function of a specific r.v. on the FG! Messages exchanged (passed) on each edge: functions of the corresponding r.v. Each message can be seen as a generalized belief/density/distribution for that r.v. Noptilus - Sept Cooperative Localization with Factor Graphs 10
11 Brief Tutorial on FG and SPA (3) Noptilus - Sept Cooperative Localization with Factor Graphs 11
12 Brief Tutorial on FG and SPA (4) Noptilus - Sept Cooperative Localization with Factor Graphs 12
13 Brief Tutorial on FG and SPA (5) Various ways to calculate the marginal distributions from messages... Noptilus - Sept Cooperative Localization with Factor Graphs 13
14 Brief Tutorial on FG and SPA (6) For cycle-free FGs, the posterior can be written as a function of the factors, the number of variables and the multiplicity of each r.v. For FGs with cycles, the SPA may provide marginal approximations, not the marginal itself. FGs vs Bayesian Nets: FGs are not directed, Bayesian Nets are acyclic. FGs vs Bayesian Nets: SPA is simpler than belief propagation. FGs vs Markov Random Fields: posterior is mapped to a single FGs, while a single posterior can be mapped to different MRV graphs. Noptilus - Sept Cooperative Localization with Factor Graphs 14
15 The Localization Problem FG (1) Exploiting the independence of mobility and internal measurements 15
16 The Localization Problem FG (1) SPA: Prediction 16
17 The Localization Problem FG (2) Noptilus - Sept Cooperative Localization with Factor Graphs 17
18 The Localization Problem FG (2) Noptilus - Sept Cooperative Localization with Factor Graphs 18
19 The Localization Problem FG (3) Ensuring locality of information... Each outer box is a node (agent or anchor) Noptilus - Sept Cooperative Localization with Factor Graphs 19
20 The Localization Problem FG (3) SPA: Correction Noptilus - Sept Cooperative Localization with Factor Graphs 20
21 Example: 1st iteration μ >Χ ( 2) φ Χ 1-> 2 2 μ >Χ ( 2) φ Χ 1 3-> 2 2 b 2 X ) ( 2 Noptilus - Sept Cooperative Localization with Factor Graphs 21
22 Example: 1st iteration (reproduced) μ >Χ ( 2) φ Χ 1-> 2 2 μ >Χ ( 2) φ Χ 1 3-> 2 2 b 2 X ) ( 2 Noptilus - Sept Cooperative Localization with Factor Graphs 22
23 Example: 2nd iteration μ φ ( Χ 2) 1-> 2 >Χ 2 μ >Χ ( 2) μφ ( Χ ) 4-> 2 >Χ2 2 2 φ Χ b ) 3-> X ( 2 Noptilus - Sept Cooperative Localization with Factor Graphs 23
24 More involved example Non-cooperative LS Cooperative MMSE Noptilus - Sept Cooperative Localization with Factor Graphs 24
25 Cooperative LS dwmds Noptilus - Sept Cooperative Localization with Factor Graphs 25
26 Numerical Results Noptilus - Sept Cooperative Localization with Factor Graphs 26
27 Some comments on FG/SPA Localization Promising, simple computations... The model encompasses mobility... Solutions are obtained after fixed number of iterations... However, Needs distribution of measurements... Message passing (SPA) involves communication of functions... not realistic for resource-constrained acoustic modems (5-14kbps)... Could we parameterize the communicated messages (e.g. via EM)? Noptilus - Sept Cooperative Localization with Factor Graphs 27
28 A few additional remarks... Mapping measurements to range estimates is benefited by LOS vs NLOS estimation. LOS/NLOS estimation is based on signal features, such as RMS delay spread, mean excess delay... Thus, localization is tightly connected to PHY layer... E.g. variance of range measurement error is a function of distance and environment. If location is seen as a parameter, the Cramer-Rao bound is not tight... The Ziv-Zakai bound (ZZB) is tighter for a wider range of SNRs. Noptilus - Sept Cooperative Localization with Factor Graphs 28
29 References M.Z. Win, A. Conti, S. Mazuelas, Y. Shen, W. M. Gifford, D. Dardari and M. Chiani, Network Localization and Navigation via Cooperation, IEEE Communications Magazine, May H. Wymeersch, J. Lien, M.Z. Win, Cooperative Localization in Wireless Networks, Proceedings of the IEEE, Vol. 97, No. 2, Feb F. R. Kschichang, B. J. Frey and H.A. Loeliger, Factor Graphs and the Sum-Product Algorithm, IEEE Transactions on Information Theory, Vol. 47, No. 2, Feb Noptilus - Sept Cooperative Localization with Factor Graphs 29
30 Thank You Noptilus - Sept Cooperative Localization with Factor Graphs 30
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