A Variational Message Passing Framework for Multi-Sensor Multi-Object Tracking using Raw Radar Signals

1. Problem Statement

The proliferation of Unmanned Aerial Vehicles (UAVs) poses significant challenges for airspace surveillance. Drones are typically small, have low Radar Cross Sections (RCS), and move slowly, often blending into background clutter.

• Limitations of Conventional Methods: Traditional Detect-Then-Track (DTT) approaches rely on pre-processed measurements (e.g., matched filtering followed by Constant False Alarm Rate detection). These methods often discard low Signal-to-Noise Ratio (SNR) signal components during the detection stage, leading to missed detections and track loss in cluttered or low-SNR environments.
• Limitations of Existing Track-Before-Detect (TBD): While TBD methods operate on raw signals to preserve low-SNR information, many existing approaches rely on simplified likelihood models (e.g., assuming independent samples or point-spread functions). These approximations fail to capture signal correlations caused by closely spaced objects (superposition effects) and often assume known nuisance parameters (like noise variance or reflectivity), which degrades performance in dense scenarios.

2. Methodology

The authors propose a Variational Message Passing (VMP) based Direct Multi-Object Tracking (Direct-MOT) framework that operates directly on raw MIMO radar signals.

A. System and Signal Model

• Architecture: The system utilizes $L$ monostatic MIMO radars, each with $N_T$ transmitters and $N_R$ receivers.
• Superimposed Signal Model: Instead of treating objects as independent measurements, the model employs a superimposed signal model. The received baseband signal at a receiver is modeled as the sum of reflections from all potential objects plus noise. This explicitly captures correlations in the raw data caused by closely spaced objects.
• State Representation: The system tracks $K$ Potential Objects (POs). For each PO $k$, the state vector $\Phi_{k,n}$ includes position and velocity. A binary variable $\xi_{k,n}$ indicates existence (birth/death), and a complex coefficient $\alpha_{k,n,l}$ represents reflectivity.

B. Probabilistic Framework

The authors construct a unified Bayesian network with the following components:

1. Hierarchical Bernoulli-Gamma Model:
   • Bernoulli: Models object existence ($\xi$).
   • Gamma: Models the precision (inverse variance) of object reflectivities ($\gamma$). This allows the algorithm to learn the reliability of individual radar-object links adaptively.
2. State Evolution: Objects follow a linear kinematic model (constant velocity) with process noise.
3. Nuisance Parameter Learning: The framework jointly estimates unknown parameters, including object reflectivities, noise variance, and process noise precision, rather than assuming them to be known.

C. Inference Algorithm (VMP)

Since the exact posterior distribution is intractable, the authors use a Mean-Field Approximation to derive a VMP algorithm:

• Factorization: The joint posterior is approximated by a factorized surrogate distribution $q(\cdot)$.
• Message Passing: The algorithm iteratively updates marginal distributions for states, existence indicators, and nuisance parameters by passing messages between nodes in the factor graph.
• Gaussian Approximation: To ensure computational efficiency and closed-form solutions, messages related to the kinematic state are approximated as Gaussian distributions by minimizing the Kullback-Leibler (KL) divergence against the true message.
• Joint Optimization: To handle the high correlation between existence ($\xi$) and reflectivity ($\alpha$), the algorithm optimizes the Evidence Lower Bound (ELBO) jointly for these factors, accelerating convergence.

D. Practical Implementation

• Track Management: The algorithm includes specific logic for track initialization (searching for new objects when existence probability exceeds a threshold) and pruning (removing tracks when probability drops).
• Multi-Radar Fusion: Data fusion is performed coherently across all radar nodes within the probabilistic model, allowing sensors with poor SNR to be compensated by those with high SNR.
• Complexity Management: The algorithm restricts updates to the estimated number of objects ($\hat{K}_n$) rather than the maximum possible ($K_{max}$), and limits the search space for new objects to a kinematic filter (7 standard deviations) to ensure real-time feasibility.

3. Key Contributions

1. Direct Signal-Level Inference: A novel VMP-based method that performs detection, track formation, and state estimation directly on raw MIMO radar signals, bypassing the intermediate detection step.
2. Coherent Multi-Radar Fusion: Explicit modeling of data fusion across multiple MIMO radars, handling the superposition of signals from closely spaced objects without simplifying assumptions.
3. Adaptive Reliability Modeling: Introduction of a hierarchical Bernoulli-Gamma model that simultaneously estimates object existence and the reliability (precision) of reflectivity for each radar-object link, enabling robust handling of heterogeneous sensor data.
4. Computational Efficiency: Derivation of efficient message updates using mean-field approximation and Gaussian fitting, making the algorithm suitable for real-time applications despite the complexity of raw signal processing.
5. Comprehensive Evaluation: Extensive simulation results demonstrating superiority over state-of-the-art baselines in challenging conditions.

4. Results

The proposed VMP method was evaluated against a conventional DTT pipeline consisting of Super-Resolution Sparse Bayesian Learning (SBL) for detection followed by Belief Propagation (BP) for tracking. Simulations were conducted in three challenging scenarios:

• Random Crossing Tracks: Objects cross paths in low SNR ($\approx -18$ dB).
• Parallel Tracks: Objects move very closely (1.5m separation) for extended periods.
• Sensor Handover: A single object moves through the field of view, observed by only one or two sensors at a time.

Performance Metrics:

• OSPA Error: The VMP method achieved significantly lower Optimal Sub-pattern Assignment (OSPA) errors compared to SBL+BP, particularly in low-SNR and closely-spaced scenarios.
  • In the crossing scenario, SBL+BP suffered from cardinality errors (missed detections) due to signal correlation, leading to track termination. VMP maintained tracks by fusing data coherently.
  • In the parallel track scenario, SBL+BP dropped tracks when objects were below the classical resolution limit. VMP maintained consistent low error.
• Robustness: The VMP algorithm handled amplitude fluctuations (Swerling-III model) better than the DTT approach, which struggled when signal power dropped below detection thresholds.
• Runtime: While the SBL+BP method is dominated by the SBL detection stage, the VMP algorithm demonstrated comparable or better mean runtime (approx. 1.29s vs 1.59s for SBL+BP in the test scenario) on an Apple M1 Pro chip, proving its viability for real-time implementation.

5. Significance

This paper represents a significant advancement in radar signal processing for UAV surveillance. By shifting from a "detect-then-track" paradigm to a "direct-tracking" paradigm using Variational Message Passing, the method:

• Preserves Information: It retains weak signal components that conventional detectors would discard.
• Handles Correlation: It mathematically accounts for the interference and superposition of signals from closely spaced targets, a common failure point for traditional trackers.
• Enables Robust Fusion: It provides a rigorous Bayesian framework for fusing data from multiple heterogeneous sensors, automatically weighting them based on their inferred reliability.

The proposed framework offers a robust solution for next-generation airspace surveillance systems where detecting small, slow, and low-RCS drones in cluttered environments is critical.