S. Das, A. Rafe
Texas State University,
United States
Keywords: Sheaf Theory, Causal Identifiability, Multimodal AI, Algebraic Topology, Sensor Fusion
Summary:
Roadway safety is a national imperative that increasingly relies on the fusion of heterogeneous data streams from cameras, LiDAR, radar, and connected vehicle technologies. While modern Intelligent Transportation Systems generate vast amounts of sensor data, a critical gap remains in the analytical tools used to process it: there is currently no rigorous mathematical framework to determine when multi-sensor data fusion is reliable. Current approaches, including deep learning and statistical surrogates, often function as "black boxes" that prioritize predictive performance over structural validity. These methods struggle to reconcile fragmented local data into a consistent system-wide understanding, particularly when sensor networks are sparse or topologically complex. Without formal guarantees of identifiability, safety-critical decisions risk being made on fragile or biased inferences. This research introduces a novel, mathematically principled framework for multi-sensor identifiability grounded in algebraic topology, specifically sheaf theory. Unlike traditional graph-based models, this approach models the roadway network as a topological space where sensor fields of view act as open sets. We construct two primary sheaves over this topology: a sheaf of observable data (e.g., sensor feeds) and a sheaf of latent safety states (e.g., unobserved crash risks). The relationship between these sheaves is modeled by a "sensing morphism," and the inherent ambiguities of the system are captured by the kernel of this morphism. The core innovation of this work is the use of sheaf cohomology as a computable diagnostic tool for system reliability. We posit that a global safety state is identifiable from local sensor observations if and only if the first sheaf cohomology group of the kernel sheaf vanishes. Intuitively, this topological invariant quantifies obstructions to data fusion; a non-vanishing cohomology group indicates that locally consistent sensor data cannot be glued into a unique, coherent global risk assessment due to sensor gaps or conflicts. This framework enables the development of redundancy certificates, mathematical guarantees that a system’s identifiability remains stable even if specific sensors fail. Furthermore, it transforms sensor placement from a heuristic exercise into an exact optimization problem, identifying minimal sensor configurations required to ensure global consistency. By replacing opaque fusion models with transparent, topological derivations, this research provides the necessary theoretical foundation for trustworthy AI in safety-critical infrastructure, ensuring that automated safety assessments are not only accurate but provably valid.