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On Pooling-Based Track Fusion Strategies : Harmonic Mean Density

Nikhil Sharma, Shovan Bhaumik, Ratnasingham Tharmarasa, Thiagalingam Kirubarajan

TL;DR

This work tackles cross-correlation in distributed track fusion, where naive fusion double-counts common information. It introduces Harmonic Mean Density (HMD) as a pooling-based fusion rule that can fuse uni-modal and multi-modal Gaussian mixtures without non-integer-power densities, using a Gaussian-approximation of the denominator to yield closed-form updates. Key theoretical results include the convexity of the normalization term, monotonicity, and KL-divergence bounds that justify pooling, along with a recursive formulation for multiple sensors. Empirical and simulation results in 2D and 3D maneuvering scenarios show that HMD achieves lower RMSE and good consistency compared with AMD and GMD, with computation comparable to existing conservative methods, making it practically attractive for real-time distributed tracking.

Abstract

In a distributed sensor fusion architecture, using standard Kalman filter (naive fusion) can lead to degraded results as track correlations are ignored and conservative fusion strategies are employed as a sub-optimal alternative to the problem. Since, Gaussian mixtures provide a flexible means of modeling any density, therefore fusion strategies suitable for use with Gaussian mixtures are needed. While the generalized covariance intersection (CI) provides a means to fuse Gaussian mixtures, the procedure is cumbersome and requires evaluating a non-integer power of the mixture density. In this paper, we develop a pooling-based fusion strategy using the harmonic mean density (HMD) interpolation of local densities and show that the proposed method can handle both Gaussian and mixture densities without much changes to the framework. Mathematical properties of the proposed fusion strategy are studied and simulated on 2D and 3D maneuvering target tracking scenarios. The simulations suggest that the proposed HMD fusion performs better than other conservative strategies in terms of root-mean-squared error while being consistent.

On Pooling-Based Track Fusion Strategies : Harmonic Mean Density

TL;DR

This work tackles cross-correlation in distributed track fusion, where naive fusion double-counts common information. It introduces Harmonic Mean Density (HMD) as a pooling-based fusion rule that can fuse uni-modal and multi-modal Gaussian mixtures without non-integer-power densities, using a Gaussian-approximation of the denominator to yield closed-form updates. Key theoretical results include the convexity of the normalization term, monotonicity, and KL-divergence bounds that justify pooling, along with a recursive formulation for multiple sensors. Empirical and simulation results in 2D and 3D maneuvering scenarios show that HMD achieves lower RMSE and good consistency compared with AMD and GMD, with computation comparable to existing conservative methods, making it practically attractive for real-time distributed tracking.

Abstract

In a distributed sensor fusion architecture, using standard Kalman filter (naive fusion) can lead to degraded results as track correlations are ignored and conservative fusion strategies are employed as a sub-optimal alternative to the problem. Since, Gaussian mixtures provide a flexible means of modeling any density, therefore fusion strategies suitable for use with Gaussian mixtures are needed. While the generalized covariance intersection (CI) provides a means to fuse Gaussian mixtures, the procedure is cumbersome and requires evaluating a non-integer power of the mixture density. In this paper, we develop a pooling-based fusion strategy using the harmonic mean density (HMD) interpolation of local densities and show that the proposed method can handle both Gaussian and mixture densities without much changes to the framework. Mathematical properties of the proposed fusion strategy are studied and simulated on 2D and 3D maneuvering target tracking scenarios. The simulations suggest that the proposed HMD fusion performs better than other conservative strategies in terms of root-mean-squared error while being consistent.

Paper Structure

This paper contains 18 sections, 6 theorems, 74 equations, 16 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Review of Arithmetic and Geometric Mean Density
  4. Geometric Mean Density (GMD) Fusion
  5. Arithmetic Mean Density (AMD) Fusion
  6. Proposed Harmonic Mean Density
  7. Properties of Harmonic Average Density
  8. Harmonic Averaging as a Recursive Fusion Strategy
  9. On the K-L Divergence of Harmonic Mean Density
  10. On the Monotonicity of Harmonic Mean Density
  11. Implementation
  12. Implications of the spread of means term in eqn. \ref{['GaussApproxCov']}
  13. An Empirical Study
  14. Simulation
  15. Three Sensor NCV Scenario
  16. ...and 3 more sections

Key Result

Theorem 1

The harmonic fusion avoids double counting of information.

Figures (16)

  • Figure 1: A non-normalized Gaussian mixture raised to a non-integer power $w \in [0,1]$. The original mixture is shown with dashed line, as $\omega$ approaches 0, the density becomes relatively more diffused.
  • Figure 2: Normalization constant for various values of $\omega$ in case of GMD (red) and HMD (green).
  • Figure 3: Fused density using various methods for a correlation coefficient of $\rho = 0.5$ among local densities.
  • Figure 4: (a) Average run time v/s dimension of the Gaussian density and, (b) average run time v/s number of components in a Gaussian mixture.
  • Figure 5: Target and sensor engagements for scenario 1.
  • ...and 11 more figures

Theorems & Definitions (6)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 1
  • Theorem 4
  • Proposition 2