The multi–object filtering problem is a generalization of the well–known single– object filtering problem. In essence, multi–object filtering is concerned with the joint estimation of the unknown and time–varying number of objects and the state of each of these objects. The filtering problem becomes particular challenging when the number of objects cannot be inferred from the collected observations and when no association between an observation and an object is possible.

1. Random Finite Set Filters for
Superpositional Sensors
Application to Multi-Object Filtering
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2. Background
Multi-Object Filtering, Finite Set Statistics & Superposition-Type Sensors
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3. Single-Object Filtering
• Estimate the state of a single object given a single measurement
• Kalman Filter is the most common way to approach this problem
• Bayes Filtering is a generalization of the Kalman filter
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4. Multi-Object Filtering
• Joint estimation of the unknown and time-varying number of multiple
objects and the state of each with multiple measurements
• Challenging when
• the number of objects cannot be inferred by number of measurements
directly
• no association between a measurement and an object can be made
• Need to account for additional effects like missed detections or clutter
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5. Multi-Object Filtering
• Heuristic approaches exists that leverage classic single-object filters
• Multi Hypothesis Tracking (MHT),
• Joint Probability Data Association (JPDA)
• …
• Non-heuristic approaches exists that are based on Finite Set Statistics
• Probability Hypothesis Density (PHD) filter,
• Multi-Object Multi-Bernoulli (MeMBer) filter
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6. Finite Set Statistics
• Main building block are random finite sets (rfs)
• Finite-set valued random variables
• Random in the values of entries and the number of entries in the set
• Multi-object state can be modeled as a single rfs
• Allows the formulation of the multi-object filtering problem in a
mathematical rigorous way
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7. Measurement Generation
Process

8. Detection-Type Sensors
• Measurements are not directly
generated by sensor
• Raw sensor data will be a single
measurement
• Extraction of measurements by
preprocessing / peak detection
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Principle of measurement generation process. 16
measurements are taken in a range of +/- 90°. Here three
measurements would be extract

9. Superposition-Type Sensors
• Govern superposition (sps) principle
• Comprised out of all individual
measurements that would be
generated by each object
individually
• Raw data is not easily separable
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Concept of sps principle. Raw signal is generated
by adding signals of three source signals.

10. Scope
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11. Research Question
Can we derive effective and efficient multi-object Bayes filters for
superposition-type sensors?
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12. Research Methodology
• Derive multi-object Bayes filter for sps-type sensors in a systematic and
formal way
• Employ Finite Set Statistics as a systematic technique for modeling the
problem
• Multi-object state is modeled as Finite Set
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13. Key Results
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14. Key Results
• Theoretical Bayes filter equations for sps-type sensors
• Practical Bernoulli filter realizations for sps-type sensors
(Σ-MeMBer filters)
• Computationally tractable approximations
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15. Bayes Filter equations
• Recursively estimates the multi-object pdf with each step
• Probability of state is independent of non-parent states
• Separable into Prediction and Correction step
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PredictionInitialization Correction

16. Bayes Filter Steps
• Prediction of the next multi-object state with system dynamics
• Not special for sps-type sensors
• Correction of the multi-object state given the sensor measurements
• Special for sps-type sensors
➡ Fokus on Correction step
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17. Set-Integrals
• Complexity is hidden behind set—integral
• Expands to an infinite sum over all sets of length zero to infinity
➡ Set-Integrals are computational expensive
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18. Bayes Filter Corrector
• Measurement likelihood is used
• Measurement is used to correct the knowledge about the multi-object
state
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19. Measurement Likelihood
• Likelihood that the measurement is the result of the additive contributions of
the objects
• Sensor noise is assumed to be additive
• Objects might not be visible to the sensor
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20. Measurement Likelihood
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• Likelihood that the measurement is the result of the additive contributions of
the objects
• Sensor noise is assumed to be additive
• Objects might not be visible to the sensor

21. Superposition-type Corrector
• Theoretical correct but practically not applicable yet
• Practical implementation requires fixation of underlying distribution
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22. Practical Realization (Σ-MeMBer)
• Predicted and corrected pdf need to be from the same type to be useful
• Bayes filter equations need to be solved in closed form
• Popular choices for initial multi-object probability distributions are
• Poisson
• Multi-Bernoulli
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23. Multi Bernoulli
• Fully described by multiple Bernoulli components each having two parameters
• Probability of existence and single-object density
• Modelled with combination of single-object densities
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24. Σ-MeMBer Filter
• Propagates only the Bernoulli parameters over time
• Predictor shown to be solvable in closed form
• Only corrector needs to be determined
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PredictionInitialization Correction

25. Σ-MeMBer Filter
• Propagates only the Bernoulli parameters over time
• Predictor shown to be solvable in closed form
• Only corrector needs to be determined
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PredictionInitialization Correction

26. Σ-MeMBer Corrector
• When predicted distribution is Multi-Bernoulli
➡ Resulting distribution is not a Multi-Bernoulli
• Filter not recursively applicable
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27. Approximations
Is it possible to reformulate the Σ-MeMBer corrector such that
we can derive at least approximate Multi-Bernoulli parameters ?
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28. Factorization
• Split the Σ-MeMBer corrector into two parts
• Missed part that is not dependent on measurement
➡ Results in a true Multi-Bernoulli
• Detected part that is dependent on measurement
➡ Results not in a Multi-Bernoulli
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Missed Part Detected Part

29. Factorization: Approximation
• Approximate by its probability hypothesis density (phd)
• Infer parameters by comparing with the phd of a Multi-Bernoulli
➡ Results in an invalid probability density due to negative values
• Limiting values to be always positive
➡ Results in an overconfident estimate of the probability density
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PHD of corrector detected PHD of Multi-Bernoulli

30. Factorization Equation
• Each Bernoulli component is propagated individually
• Each Bernoulli component leads to two
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Bernoulli Parameters for missed part Bernoulli Parameters for detected part

31. Intensity Approximation
• Directly approximate corrector by its probability hypothesis density (phd)
• Infer parameters by comparing with phd of a Multi-Bernoulli
• Does not require further approximations
• Does not change the number of components
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PHD of corrector PHD of Multi-Bernoulli

32. Intensity Equation
• Each Bernoulli component is propagated individually
• Number of components stays constant
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33. Results
• Provided two possible ways to approximate Bernoulli parameters
• Allows the application of the predictor and corrector recursively
• Potential effective/usable filters
• What about the efficiency/computational tractability?
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34. Pseudo-Likelihood
• Most terms are easy to compute
• Pseudo-likelihood is hard to compute
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Factorization
Intensity approximation

35. Pseudo-Likelihood
• Convolutions lead to many combinations
• Computable but very demanding
• Approximation needed to make it computationally tractable
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36. Computationally Tractable Approximations
• Pseudo-Likelihood is an ordinary probability density function (pdf)
• Approximate pdf by replacing with a density that is more easily to
compute
• Gaussian Mixture
• Gaussian
• Poisson Binomial
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37. Gaussian Pseudo Likelihood
• Pseudo-likelihood is comprised out of convolution of individual pdfs
• Determine the mean and variance
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38. Gaussian Pseudo Likelihood
• Assume that the additive noise is zero-mean Gaussian
• Simplifies to single Gaussian
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39. Numerical Studies
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40. Setup
• Comparison of all filter variants
overall
• Up to 6 objects at the same time
• Objects move linear in a 2D area
• Non-linear superposition-type
sensor model
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Example of object movement. Squares denote the position of object
entering the area. Triangles mark the position of the objects
disappearance. Sensors are placed in the corners.

41. Monte Carlo Verification
• Use Sequential Monte Carlo (SMC) implementations
• All filter parameters have been fixed over multiple runs
• Measurements are generated individually each run
• 5% of objects are assumed to be not visible on average
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42. • Exemplary comparison between Factorized and Intensity approximation
• Dotted and gray lines are the ground truth
Example Run
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43. • Cardinality estimates over all MC trials
• Dotted line is the ground truth. Solid line is the average cardinality.
Cardinality Error
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44. Testing of Limitations
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OSPA(c=16andp=1)
AQG Σ-MeMBer
IQG Σ-MeMBer
CB-MeMBer
TNC-MeMBer
• OSPA metric for varying probability of visibility (lower is better)
• Benchmarked against detection-type (CB) and TNC MeMBer filter

45. Summary & Conclusion
Implications on the domain
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46. Summary
• Derived multi-object Bayes filters for sps-type sensors in mathematical
formal way
• Provided practical implementations by choosing multi-Bernoulli
distribution as underlying distribution
• Provided computationally tractable approximations
• Analyzed performance in a numerical study
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47. Conclusion
• It is possible to derive an effective and efficient multi-object filter for
sps-type sensor
• Σ-MeMBer filter provide an usable and computationally tractable
way to estimate the state of multiple objects with sps-type sensors
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48. The End
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49. Appendix
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50. Multi-Object Predictor
• Solely dependent on system dynamics
• Objects move independently of each other
• Objects may enter
• Objects may disappear monitored area
I could throw it out
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51. • If Initial and Birth distribution is Multi-Bernoulli
➡ Resulting distribution is also Multi-Bernoulli
• Only need to propagate predicted Bernoulli components
MeMBer Predictor
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52. Functionals