MAXIMUM LIKELIHOOD JOINT ASSOCIATION, TRACKING, AND FUSION IN STRONG CLUTTER

Leonid PerlovskyHarvard University and the AF Research Lab

Seminar Department of Electrical and Computer Engineering, University of Connecticut

Storr, 6 Mar., 2009

OUTLINE

• Related research

• Combinatorial complexity and logic

• Dynamic logic

• Joint likelihood, math. formulation

• Examples

• Publications, recognition

RELATED RESEARCH

• > 50 publications by Perlovsky and co-authors on concurrent association, tracking, and fusion (+ > 200 other applications)

– Perlovsky, L. I. (1991). Model Based Target Tracker with Fuzzy Logic. 25th Annual Asilomar Conference on Signals, Systems and Computers, Pacific Grove, CA.

– Perlovsky, L.I., Schoendorf, W.H., Tye, D.M., Chang, W. (1995). Concurrent Classification and Tracking Using Maximum Likelihood Adaptive Neural System. Journal of Underwater Acoustics, 45(2), pp.399-414.

• Many publications by Bar-Shalom, Streit, Luginbuhl, Willett, Avitzour, and co-authors

• Similarity: algorithms related to EM• Differences:

– Formulation of likelihood– Maximization procedures– Performance: linear complexity, Cramer-Rao Bound

• Cramer-Rao Bound for joint association and tracking– Perlovsky, L.I. (1997). Cramer-Rao Bound for Tracking in Clutter and Tracking Multiple

Objects. Pattern Recognition Letters, 18(3), pp.283-288.

COMBINATORIAL COMPLEXITY 50 years of difficulties

• Detect signal in noise and clutter at the farthest possible distance

• SP, detection, exploitation, fusion, tracking, etc. in noise/clutter– Requires association (pixels<->objects) before detection

If 1 object, no noise: (1) detect pixels, (2) detect objects, (3) recognize targets– Joint detection-discrimination-classification…

• Combinatorial Complexity (CC) – Need to evaluate large numbers of combinations (pixels<->objects) ,

operations: ~MN

– A general problem (since the 1950s) SP, detection, recognition, tracking, fusion, exploitation, situational awareness,… Pattern recognition, neural networks, rule systems…

• Combinations of 100 elements are 100100

– Larger than the number of particles in known Universe Greater than all the elementary events in the Universe during its entire life

• CC affects many SP algorithms– Our sensors under-utilize signals– Work much worse than Cramer-Rao Bound information-theoretic limit

CC vs. LOGIC

• CC is related to formal logic– Gödel proved that logic is “illogical,” “inconsistent” (1930s)

– CC is Gödel's “incompleteness” in a finite system

• Fuzzy logic – How to select degree of fuzziness?– The mind fits fuzziness for every process => CC

• Logic pervades all algorithms and neural networks – Rule systems, fuzzy systems (degree of fuzziness), pattern recognition, neural networks (training uses logic)

• Probabilistic association (Bar-Shalom) – Overcame logic in association– Where all logical steps overcome?

DYNAMIC LOGIC overcame logic limitations

•CC is related to logic– CC is Gödel's “incompleteness” in a finite system – Logic pervaded all algorithms and neural

networks in the pastrule systems, fuzzy systems (degree of fuzziness), pattern

recognition, neural networks (training uses logical statements)

•Dynamic Logic is a process-logic– “from vague to crisp” (statements, targets,

decisions…)

•Overcomes CC– Fast algorithms

OUTLINE

• Related research

• Combinatorial complexity and logic

• Dynamic logic

• Joint likelihood, math. formulation

• Examples

• Publications, recognition

JOINT LIKELIHOOD for tracks and clutter

• Total likelihood– L = l ({x}) = l (x(n))

• no assumption of “independence”

• Conditional likelihoods

– l (x(n)) = r(m) l (x(n) | Mm(Sm,n))

– l (x(n) | Mm(Sm,n)) is a conditional likelihood for x(n) given m

• {x(n)} are not independent, M(n) may depend on n’

• CC: L contains MN items: all associations of pixels and models (LOGIC)

n

m

EXAMPLES OF MODELS

• Linear track model– Mm(Sm,n) = Xm + Vm*t; Sm = (Xm, Vm, rm, Cm

-1)

• Gaussian conditional likelihoods

– l (x(n) | Mm(Sm,n)) =

(2) -d/2 (detC)-1/2 exp{ -0.5 [ x(n) - Mm(Sm,n) ]T Cm-1 [ x(n) - Mm(Sm,n) ] }

– No “Gaussian” assumption • errors are Gaussian • mixture of any pdfs can be used

• Uniform clutter model– rm, l (x(n) | Mm(Sm,n)) = 1/ volume(x)

DYNAMIC LOGIC (DL) non-combinatorial solution

• Start with a set of signals and unknown models– any parameter values Sm – associate models with signals (vague)– (1) f(m|n) = r(m) l (n|m) / r(m') l (n|m')

• Improve parameter estimation– (2) Sm = Sm + f(m|n) [ln l

(n|m)/Mm]*[Mm/Sm]

• Continue iterations (1)-(2). Theorem: DL is a converging system- likelihood increases on each iteration

'm

n

OUTLINE

• Related research

• Combinatorial complexity and logic

• Dynamic logic

• Joint likelihood, math. formulation

• Examples

• Publications, recognition

TRACKING AND DETECTION BELOW CLUTTER

yDL starts with uncertain knowledge and converges rapidly on exact solution

Performance achieves joint CRB for association and estimation

0 1 km

TRACKING AND DETECTION BELOW CLUTTER

Cross-Range

Ra

ng

e1

km0

(a)True

Tracks

detections

Ra

ng

e1

km0

c d

(b)

e f g h

Multiple Hypothesis Testing “logical” complexity ~ 101800; DL complexity ~ 106; S/C ~ 18 dB improvement

NUMBER OF TARGETS

•Active models and one dormant model - Only r(m) is estimated for the dormant model- The dormant model is activated if r(m) > threshold- An active model is deactivated if r(m) < threshold- In this example threshold = 0.001 of the total signal

- threshold = 0.001 x(n)

n

LOCAL MAXIMA

•Practically it is not a problem

•Reasons- Vague initial states smooth local maxima- Activation and deactivation eliminates local convergences

- In system applications, new data are coming all the time

local maxima come and go, real tracks persist

JOINT FUSION, ASSOCIATION, TRACKING, AND NAVIGATION

•3 platforms-sensors•Targets cannot be detected or tracked with one sensor

•All data are processed simultaneously•GPS is inadequate for triangulation

- Relative platform positions have to be estimated jointly with target tracks

•Multiple Hypothesis Testing “logical” complexity ~ 1017000

Sensor 1 (of 3): Model Evolves to Locate Target Tracks in Image Data

UNCLASSIFIED

truth data Initial uncertain model

Models converged to the truthImproved model after few iterations Few more iterations

Sensor 2 (of 3): Model Evolves to Locate Target Tracks in Image Data

UNCLASSIFIED

Sensor 3 (of 3): Model Evolves to Locate Target Tracks in Image Data

UNCLASSIFIED

NAVIGATION, FUSION, TRACKING, AND DETECTION this is the basis for the previous 3 figures, all fused in x,y,z, coordinates

OUTLINE

• Related research

• Combinatorial complexity and logic

• Dynamic logic

• Joint likelihood, math. formulation

• Examples

• Publications, recognition

PUBLICATIONS

300 publications

OXFORD UNIVERSITY PRESS(2001; 3rd printing)

Neurodynamics of High Cognitive Functionswith Prof. Kozma, Springer, 2007

Sapient Systemswith Prof. Mayorga, Springer, 2007

RECOGNITION

• 2007 Gabor Award- The top engineering award from

International Neural Network Society (INNS)

• Elected to the Board of Governors of INNS

• 2007 John L. McLucas Award- The top scientific award from the US Air

Force

CONCLUSION

• Dynamic Logic – an approach to improve algorithms and developing new ones

–Being developed since late 1980s–Proven breakthrough in several areas

• More can be done

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