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Robust Distributed Fusion with Labeled Random Finite Sets

Suqi Li, Wei Yi, Reza Hoseinnezhad, Giorgio Battistelli, Bailu Wang, Lingjiang Kong

arXiv:1710.00501v1eess.SYcs.SY

TL;DR

GCI fusion of labeled multi-object posteriors is sensitive to label inconsistencies across sensors. The paper analyzes this issue and proposes marginalizing posteriors to unlabeled form before fusion, then constructing a labeled fused density. Numerical experiments are reported to demonstrate robustness and effectiveness in challenging tracking scenarios.

  • Problem

    GCI fusion with labeled multi-object densities is highly sensitive to inconsistencies between label information in local posteriors.

    Source: §I · p. 2

  • Method

    The proposed solution marginalizes labeled posteriors to unlabeled form for GCI fusion, then constructs a labeled fused density and tractable algorithms for GLMB-family filters.

    Source: B. GCI Fusion with GMB Distributions · p. 9, D. Summary and Discussions · p. 11, §VII · p. 15

  • Results

    Numerical experiments in challenging tracking scenarios demonstrate the robustness and effectiveness of the proposed fusion algorithm with Gaussian mixture implementation.

    Source: §I · p. 2

  • Takeaways & Limitations

    The proposed distributed fusion solution is robust to label inconsistencies between sensor-node posteriors.

    Source: §VII · p. 15

Abstract

from arXiv · show

This paper considers the problem of the distributed fusion of multi-object posteriors in the labeled random finite set filtering framework, using Generalized Covariance Intersection (GCI) method. Our analysis shows that GCI fusion with labeled multi-object densities strongly relies on label consistencies between local multi-object posteriors at different sensor nodes, and hence suffers from a severe performance degradation when perfect label consistencies are violated. Moreover, we mathematically analyze this phenomenon from the perspective of Principle of Minimum Discrimination Information and the so called yes-object probability. Inspired by the analysis, we propose a novel and general solution for the distributed fusion with labeled multi-object densities that is robust to label inconsistencies between sensors. Specifically, the labeled multi-object posteriors are firstly marginalized to their unlabeled posteriors which are then fused using GCI method. We also introduce a principled method to construct the labeled fused density and produce tracks formally. Based on the developed theoretical framework, we present tractable algorithms for the family of generalized labeled multi-Bernoulli (GLMB) filters including $\delta$-GLMB, marginalized $\delta$-GLMB and labeled multi-Bernoulli filters. The robustness and efficiency of the proposed distributed fusion algorithm are demonstrated in challenging tracking scenarios via numerical experiments.

I. INTRODUCTION

Distributed multi-object tracking must fuse estimates without known cross-sensor correlations, while labeled filters also preserve object identities. This paper analyzes how label inconsistencies affect GCI fusion and proposes a robust fusion solution for labeled filters.

I. INTRODUCTION

  • GCI combines local posteriors while avoiding double counting, and can be computed distributively using suitable consensus algorithms.It is described as the Kullback-Leibler centroid of local posteriors.Source: §I · p. 1

B. Labeled Random Finite Set Distributions and Filters

  • Labeled RFS filters can track object identities, unlike the cited PHD, CPHD, and MB distributed filters, which estimate states without labels.GLMB includes the δ-GLMB tracker; LMB and Mδ-GLMB are computationally efficient approximations.Source: §I · p. 1

I. INTRODUCTION

  • GCI fusion with labeled multi-object densities is highly sensitive to label inconsistencies between sensor nodes.The paper analyzes this drawback in a theoretical framework and introduces GCI divergence, conditional multi-label distribution, and a label inconsistency indicator.Source: §I · p. 2
  • The proposed label-robust fusion solution also yields tractable distributed algorithms for δ-GLMB, Mδ-GLMB, and LMB filters.The paper reports numerical experiments demonstrating robustness and effectiveness in challenging tracking scenarios.Source: §I · p. 2

A GLMB RFS is completely characterized by the set of parameters {(ω(c)(I), p(c)(·)) : (I, c) ∈F(L) × C}.

GCI fuses local multi-object posteriors as a weighted geometric mean, with a Kullback-Leibler interpretation. For labeled densities, this fusion is performed label-wise, making its performance sensitive to inconsistencies between sensors' labels.

A GLMB RFS is completely characterized by the set of parameters {(ω(c)(I), p(c)(·)) : (I, c) ∈F(L) × C}.

  • LMB preserves only the posterior's first-order moment, while Mδ-GLMB preserves both the first-order moment and cardinality distribution; LMB is faster.The δ-GLMB posterior can grow exponentially in components because of explicit data associations.Source: A GLMB RFS is completely characterized… · p. 3

C. Generalized Covariance Intersection (GCI)

  • GCI fusion combines local posteriors as a weighted geometric mean and minimizes the weighted sum of their Kullback-Leibler divergences.This is also called Kullback-Leibler average fusion and is coherent with the Principle of Minimum Discrimination Information.Source: C. Generalized Covariance Intersection… · p. 3

III. DRAWBACKS OF GCI FUSION WITH LABELED MULTI-OBJECT DENSITIES

  • Labeled GCI fusion is performed label-wise, so it requires consistent labels across sensors and can perform poorly when the same object has different labels.Disparities in label-set weights or conditional densities can diminish the fused probability of a label-set hypothesis.Source: §III · p. 4

Using Adaptive Birth Processes

The section traces how birth-process and observation uncertainty can create label mismatches across sensors, then introduces tools for analyzing their effect on fusion. It also explains how conditional label information supports track production and how GCI divergence measures density similarity and information gain.

Using Adaptive Birth Processes

  • Adaptive birth processes tuned to each scan can assign different labels to the same newly born object at different sensors.This can occur when objects may appear anywhere in the surveillance area.Source: Using Adaptive Birth Processes · p. 4

Random Uncertainties in Observations

  • False measurements, missed detections, and excessive noise can shift estimated birth times, while pruning can remove a label at one sensor but retain it at another.In the example, the sensors estimate birth times of 6 s and 4 s; after pruning hypotheses below 10^-6, their surviving labels differ and GCI fusion is erroneous.Source: Random Uncertainties in Observations · pp. 4–5

IV. MOTIVATING ANALYSIS

  • The analysis examines how label inconsistencies affect fusion through PMDI and object-existence probabilities, motivating a robust solution.Source: §IV · p. 5

A. Mathematical Tools

1) Conditional Multi-label Distribution:

2) GCI Divergence:

  • GCI divergence is the minimal weighted average information gain and measures similarity among densities; larger divergence indicates less coherent inputs and greater PMDI violation.Source: 2) GCI Divergence · p. 5

3) Yes-Object Probability:

The analysis relates label inconsistency to GCI divergence and the fused yes-object probability. As inconsistency grows, the labeled fusion can lose object existence even when the corresponding unlabeled fusion assigns it high probability.

3) Yes-Object Probability:

  • Yes-object probability is 1 − π(∅); object states are extracted only when it exceeds a declaration threshold, usually 0.5 ≤ τ < 1.Source: 3) Yes-Object Probability · p. 5

B. Theoretical Analysis

  • The label inconsistency indicator dG measures disparities in sensor label information and reflects the difference between labeled and unlabeled GCI divergences.The divergence decomposition separates kinematic-state discrimination from label-distribution discrimination.Source: B. Theoretical Analysis · p. 6
  • For a fixed unlabeled yes-object probability, the labeled fused yes-object probability decreases monotonically as dG increases.The decline can be sharp when the unlabeled probability is near one.Source: B. Theoretical Analysis · p. 7
  • With unlabeled yes-object probability 0.999, dG above 6.2 reduces the labeled fused probability below 0.5, yielding no-object inference.A yes-object probability below the threshold is sufficient, but not necessary, for no-object inference.Source: B. Theoretical Analysis · p. 7

C. Summary and Motivation

The analysis finds that labeled GCI fusion is sensitive to disparities in local label information, which can arise even when local sensors work well. In the example, unlabeled fusion succeeds while labeled fusion fails.

C. Summary and Motivation

  • Larger dG corresponds to larger labeled GCI divergence and a smaller fused yes-object probability, indicating less optimal fusion and reduced object-existence probability.Source: C. Summary and Motivation · p. 7
  • Labeled GCI fusion has little tolerance for inconsistencies in label information across sensors.Source: C. Summary and Motivation · p. 7
  • Label inconsistency can arise from adaptive birth, observation uncertainty, or local pruning even when each sensor works well.Source: C. Summary and Motivation · p. 7
  • In the example, local filters and unlabeled fusion return one object state estimate, whereas labeled fusion fails.The figure plots GCI divergences and yes-object probabilities over time.Source: C. Summary and Motivation · p. 7, Figure 3 · p. 7

V. ROBUST DISTRIBUTED FUSION: PROPOSED METHOD

The proposed method fuses the unlabeled versions of local labeled posteriors, then constructs a labeled fused density for track production. Its state-space fusion can accommodate different label spaces across sensors.

V. ROBUST DISTRIBUTED FUSION: PROPOSED METHOD

  • Track production uses the fusing sensor’s conditional multi-label distribution, combined by GCI with uniform distributions when label statistics differ substantially across sensors.Source: §V · p. 8
  • The method marginalizes local labeled posteriors to unlabeled densities, fuses those densities with GCI, and augments the result to produce a labeled fused density.Source: §V · p. 8
  • Because fusion is performed on state space X, the method can work when sensors’ actual label spaces differ.Source: §V · p. 8

A. The Unlabeled Version of Labeled RFS

2) Unlabeled Versions of Common Labeled Densities:

The section relates labeled GLMB and LMB densities to their unlabeled versions, then derives tractable GCI fusion for GMB densities. The fusion uses a first-order-moment-preserving MB approximation and yields a GMB density suitable for pairwise fusion.

2) Unlabeled Versions of Common Labeled Densities:

  • The unlabeled versions of GLMB and LMB densities are represented as GMB densities, with the LMB case reducing to an MB density.The section relates their parameters to those of the unlabeled distributions.Source: 2) Unlabeled Versions of Common… · p. 9

B. GCI Fusion with GMB Distributions

  • Direct GCI fusion of GMB densities is computationally intractable because it involves sums over discrete spaces and permutations.Source: B. GCI Fusion with GMB Distributions · p. 9
  • A two-step approximation preserves the GMB density’s first-order moment with an MB approximation, then applies GCI fusion to the approximations.The fractional-power simplification assumes mutually separated Bernoulli components, with the separation level close to 1.Source: B. GCI Fusion with GMB Distributions · pp. 9–10
  • The resulting GCI fusion of two GMB densities is itself GMB, enabling fusion across more than two sensors through pairwise fusion.The ordering of pairwise fusion is not addressed in this section’s derivation.Source: B. GCI Fusion with GMB Distributions · p. 10

C. Construction of the Labeled Fused Posterior

The paper constructs a labeled fused density from the fused unlabeled density when object identities and trajectories are needed. Replacing the local conditional multi-label distribution with that from a first-order-moment-preserving LMB approximation simplifies the construction and yields another GLMB.

C. Construction of the Labeled Fused Posterior

D. Summary and Discussions

The discussion presents R-GCI-GLMB as a robust distributed fusion solution for GLMB-family filters, with marginalization, GMB fusion, and labeled-density construction as its main stages. The performance assessment compares Gaussian-mixture implementations in a two-sensor, three-object scenario under two birth procedures.

D. Summary and Discussions

  • R-GCI-GLMB marginalizes local GLMB posteriors, fuses their unlabeled GMB versions, and constructs a labeled fused posterior; it also applies to LMB and Mδ-GLMB filters.For LMB filters, the unlabeled posterior is already MB, so the GMB-to-MB approximation is unnecessary.Source: D. Summary and Discussions · p. 11
  • The main computational cost lies in GMB fusion; marginalizing GLMB densities and constructing the fused GLMB are conceptual operations requiring no calculations.The implementation prunes negligible Gaussian components and uses ranked-assignment truncation to moderate computation.Source: D. Summary and Discussions · p. 11

VI. PERFORMANCE ASSESSMENT

  • The assessment uses Gaussian-mixture implementations and evaluates OSPA error and cardinality statistics over 200 Monte Carlo runs.The scenario has two sensors tracking three objects; the experiments use adaptive and prior-information-based birth procedures.Source: §VI · p. 12, Figure 5 · p. 13

A. Scenario 1

In the prior-information birth experiment, R-GCI-GLMB significantly outperforms the other methods in OSPA error and gives more accurate cardinality estimates with less deviation than C-GCI-LMB. Under adaptive births, C-GCI-LMB fails when local label assignments become inconsistent, while R-GCI-GLMB errors are lower than local-filter errors after transients.

A. Scenario 1

  • With adaptive births, C-GCI-LMB fails due to inconsistencies between local label assignments, while R-GCI-GLMB errors are significantly lower than local-filter errors after transients.Source: A. Scenario 1 · p. 13, Figure 6 · p. 13
  • Under prior-information births, R-GCI-GLMB produces more accurate cardinality estimates with less deviation than C-GCI-LMB.Source: A. Scenario 1 · p. 13, Figure 7 · p. 13
  • R-GCI-GLMB significantly outperforms the other methods in OSPA error under prior-information births.After object appearances or disappearances, its tracking errors gradually retract; C-GCI-LMB errors increase, especially after births.Source: A. Scenario 1 · p. 13, Figure 7 · p. 13
  • Each new birth can increase disparity between sensors’ label information, raising the label inconsistency indicator and degrading C-GCI-LMB performance.Source: A. Scenario 1 · p. 13

B. Scenario 2

Scenario 2 evaluates R-GCI-GLMB in a three-sensor, eight-object network and compares its tracking accuracy and runtime with GCI-CPHD. The conclusion summarizes the label-inconsistency problem, robust fusion approach, and supporting simulations.

B. Scenario 2

  • At sensor 2, R-GCI-GLMB produces more accurate cardinality estimates with less variation than GCI-CPHD.The comparison is shown for PD,k = 0.98.Source: B. Scenario 2 · p. 14, Figure 9 · p. 14
  • In a three-sensor, eight-object scenario, the reported R-GCI-GLMB tracks maintain track lock, estimate positions accurately, and identify objects correctly.The tracking display is a single run under PD,k = 0.98.Source: B. Scenario 2 · pp. 13–14, Figure 8 · p. 14
  • R-GCI-GLMB's average execution time is only slightly longer than GCI-CPHD's while providing enhanced performance and automatically accounting for track labeling.The execution-time comparison is for PD,k = 0.98.Source: B. Scenario 2 · p. 14, Figure 10 · p. 15
  • Across detection probabilities, R-GCI-GLMB has remarkably better OSPA performance than GCI-CPHD, with a stable performance difference.Table III reports averaged OSPA errors across detection-probability values.Source: B. Scenario 2 · p. 14, Table III · p. 14
  • With one, two, or three sensors, R-GCI-GLMB's estimation accuracy improves as the number of sensors increases.The comparison uses post-transient OSPA errors averaged over 200 Monte Carlo runs and 100 time steps.Source: B. Scenario 2 · p. 15, Table IV · p. 15

VII. CONCLUSION

  • The conclusion reports that labeled GCI fusion is sensitive to inconsistent labels and presents a robust fusion solution with tractable algorithms for the GLMB filter family.Gaussian-mixture simulations in challenging tracking scenarios demonstrate the algorithms' robustness and effectiveness.Source: §VII · p. 15

APPENDIX A PROOF OF PROPOSITION 2

The appendix excerpts outline algebraic derivations for unlabeled GLMB and LMB densities and a reformulation using the unlabeled GLMB representation.

APPENDIX A PROOF OF PROPOSITION 2

  • According to (17), the unlabeled version of a GLMB density of form (4) is distributed according to an expression; σ denotes a permutation of I, σ(i) its ith element, and Iv a vector constructed by sorting I.Source: Appendix A · p. 15

APPENDIX B PROOF OF PROPOSITION 3

  • The passage states that, according to (17), the unlabeled version of an LMB density of form (6) is distributed according to (17), and that Equation (81) can be further represented.Source: Appendix B · p. 15

APPENDIX C PROOF OF PROPOSITION 4

  • The Appendix C proof uses the unlabeled GLMB representation as a GMB density and combines earlier equations to rewrite the expressions.The passage identifies the correspondence as Is0 = Ls0 and Φs0 = Cs0.Source: Appendix C · pp. 15–16

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