StaR Maps: Unveiling Uncertainty in Geospatial Relations

TL;DR

The paper tackles representing and reasoning about uncertainty in geospatial maps for intelligent transportation systems. It introduces Statistical Relational Maps (StaR Maps), a hybrid framework that couples Uncertainty Annotated Maps with spatial-relational queries via first-order logic and probabilistic inference. StaR Maps propagate uncertainty by generating $N$ map realizations from an error model and computing spatial relation statistics (e.g., means and variances) through sampling or moment matching, stored as scalar fields. Experiments on crowd-sourced OpenStreetMap data with synthetic Gaussian translation errors demonstrate accurate probabilistic estimates and show that Gaussian Process-guided refinement improves sample efficiency. The open-source ProMis implementation demonstrates practical applicability for planning and safety-critical tasks in urban environments.

Abstract

The growing complexity of intelligent transportation systems and their applications in public spaces has increased the demand for expressive and versatile knowledge representation. While various mapping efforts have achieved widespread coverage, including detailed annotation of features with semantic labels, it is essential to understand their inherent uncertainties, which are commonly underrepresented by the respective geographic information systems. Hence, it is critical to develop a representation that combines a statistical, probabilistic perspective with the relational nature of geospatial data. Further, such a representation should facilitate an honest view of the data's accuracy and provide an environment for high-level reasoning to obtain novel insights from task-dependent queries. Our work addresses this gap in two ways. First, we present Statistical Relational Maps (StaR Maps) as a representation of uncertain, semantic map data. Second, we demonstrate efficient computation of StaR Maps to scale the approach to wide urban spaces. Through experiments on real-world, crowd-sourced data, we underpin the application and utility of StaR Maps in terms of representing uncertain knowledge and reasoning for complex geospatial information.

Figures (11)

• Figure 1: Statistical Relational Maps (StaR Maps) capture uncertain environments: StaR Maps provide a unified interface to heterogeneous background knowledge. Given symbolic, geographic data, and neural perception of the navigation space, they offer semantic, probabilistic answers to first-order logic based spatial queries.
• Figure 2: The Statistical Relational Maps (StaR Maps) architecture: Uncertain maps, annotated with translational and transformational statistics, are passed into a density estimator. Given a set of sample locations and the map sampler, generating variations of the annotated data according to its uncertainty, spatial relations are estimated. Using, e.g., moment matching, for each sample request the spatial relations are represented as categorical or continuous distributions. Finally, a scalar field for each relation is interpolated or approximated from the samples, providing a basis for spatial reasoning.
• Figure 3: Sampling from UAMs: Each map element carries an expectation of its true spatial occupancy, annotated with uncertainties that result from, e.g., the employed sensors or measurement methodology. Once a set of maps has been sampled, we probe the environment from a selection of points to fit StaR Maps' spatial relations. Here, sampling the distance of a point to the closest uncertain road with different error models (a - c) is illustrated.
• Figure 4: Scalar and vector fields in StaR Maps express uncertainties in spatial relations: Queried road network for which random maps have been generated. (a) and (b) show parameters of a normal distribution that model the distance to the closest road, while (c) models the probability of keeping a distance over 30 m. (d) shows the probability of a location of an agent's navigation space being occupied by buildings. Note a color range from red (low) to blue (high) for (a)-(c) and an inverse color range due to visual clarity for (d).
• Figure 5: Parameter estimation of the distance relation: Here, a histogram of the sampling process of the distance to the closest road is shown for a single point. From the set of samples we compute mean and standard deviation in order to parameterize the Gaussian that will model the distribution.