Finite Set Statistics for Sensor Data Fusion

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# Finite Set Statistics for Sensor Data Fusion - PowerPoint PPT Presentation

Finite Set Statistics for Sensor Data Fusion. Bahador Khaleghi Pattern Analysis and Machine Intelligence Lab. Outline. FISST Introduction Bayes Filter (Single Target) Multi-target Estimation FISST for Multi-Target Estimation Pros and Cons Approximating MT Bayes Filter

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### Finite Set Statistics for Sensor Data Fusion

Pattern Analysis and Machine Intelligence Lab

Outline
• FISST Introduction
• Bayes Filter (Single Target)
• Multi-target Estimation
• FISST for Multi-Target Estimation
• Pros and Cons
• Approximating MT Bayes Filter
• Practical Implementations
• Demos
• Conclusion and Future Work
What is FISST?
• Random sets theory was first proposed to study integral geometry in 1970s
• IDEA: treat sets as random elements within probabilistic framework
• FISST is formulated for finite sets
• FISST has not been popular for sensor data fusion as it has not been well understood
• It comes from applied math community rather than computer science
Why FISST?
• FISST enables unification of much of data fusion under a single Bayesian umbrella
• Major potential contributions
• A unified expert systems theory: fusion of imperfect and highly disparate data
• A unified level 1 fusion: detection, tracking, and identification as a single problem
• A unified performance evaluation framework
• A Unified and optimal single-target and multi-target detection and estimation
Optimal Bayes Filtering
• Relies on Bayes theorem to provide a solution to recursive target estimation problem

Likelihood

Prior Distribution

System State

Measurements

Normalization Factor

Optimal Bayes Filtering
• Both Prior and Normalizing Factor involve integrals that generally can not be evaluated analytically
• Kalman filter is fortunate exception for linear measurement and motion models
• Approximation to Bayes filter
• Approximations based on KF (e.g. EKF, UKF)
• Approximate the integrals using particles (e.g. SMC)
• Multi-target estimation is even more challenging
Multi-target Estimation
• Objective: estimate, at each time step, the multi-object state from a sequence of noisy and cluttered observation sets
• Conventional Stages
• Detection => Association => Tracking => Identification
Multi-target Estimation
• Challenges
• Noise
• Missed and false detections (clutter)
• Target dynamics (linear, nonlinear, manoeuvring)
• Imperfect and disparate observations
• Non-standard targets and scenarios, e.g. extended and unresolved targets
• Varying number of targets, sensors
• Data association problem
Multi-target Estimation
• Conventional approaches
• Solve the data association problem (e.g. MHT, JPDA) then apply Bayes (KF) filtering to them
• Drawbacks
• Measurements and motion models are approximate
• Bayes filtering results are approximate
• Data association is usually solve only in approximate
• Many challenging aspects of problem can not be modeled explicitly in a principled manner
FISST for Multi-target Estimation
• FISST provides a natural yet rigorous mathematical tool to represent and compute in multi-target estimation systems
• IDEA: represent the multi-target state and observations as finite sets instead of vectors
• Rationale: vector representation
• Does not admit a mathematically consistent notion of estimation error
• Cannot represent all occurrences of multi-target state
FISST for Multi-target Estimation
• Finite set representation
• Casts estimation error as well-established concept of set distance
• Can represent all possible occurrences
• There is no inherent ordering of measurements or targets
• Allows explicit modeling of many challenging aspects of (multi)target estimation
Multi-target Bayes Filtering
• It is not a straightforward generalization
• Requires a novel calculus (derivative and integral) for finite random sets, which is a generalization of vector calculus (statistics 101)
• Recent book by Mahler provides just this!
• Results in closed form solutions for priors and likelihoods, which are not analytical ones as expected
Sample FISST Formulations

Surviving Targets

Spawned Targets

Appearing Targets

Clutter

Measurement Model

• Pros
• Obviates data association problem
• Formulates detection, tracking, and identification as a unified problem
• A natural way of treating challenging effects involved in realistic tracking scenarios
• Cons
• Mathematically complex and less understood
• Generally intractable (combinatorial complexity)
• Track continuity problem
Approximating MT Bayes Filter
• In single-target case successful approximations are based on moment matching
• e.g. Kalman filter: match the first two moments (i.e. mean and covariance)
• PHD filter: propagates the first moment in MT case, i.e. Probability Hypothesis Density
• Cardinalized PHD filter: propagates the PHD as well as pdf for number of targets (better estimation accuracy and adaptation)
Approximating (C)PHD Filter
• SMC method(s)
• Require additional computations, i.e. particle clustering to extract multi-target state estimates
• Higher flexibility
• Gaussian Mixture
• More computationally efficient
• More restrictive, i.e. linear and mildly non-linear models
• Both compare favourably with MHT and JPDA with less computational load
Sample Applications
Sample Applications
• SMC-PHD Filter on Video
Conclusion
• FISST is a mathematically principled and natural solution to multi-target estimation
• FISST provides a unified framework for explicit modeling of all challenging aspects of problem with promising results
• FISST potential to serve as a unifying framework for expect systems in both single-target and multi-target case is yet to be investigated
• Many other aspects of data fusion systems may potentially be unified using FISST!