Foundations of Vision-Based Localization: A New Approach to Localizability Analysis Using Stochastic Geometry

Despite significant algorithmic advances in vision-based positioning, a comprehensive probabilistic framework to study its performance has remained unexplored. The main objective of this paper is to develop such a framework using ideas from stochastic geometry. Due to limitations in sensor resolution, the level of detail in prior information, and computational resources, we may not be able to differentiate between landmarks with similar appearances in the vision data, such as trees, lampposts, and bus stops. While one cannot accurately determine the absolute target position using a single indistinguishable landmark, obtaining an approximate position fix is possible if the target can see multiple landmarks whose geometric placement on the map is unique. Modeling the locations of these indistinguishable landmarks as a Poisson point process (PPP) $\Phi$ on $\mathbb{R}^2$, we develop a new approach to analyze the localizability in this setting. From the target location $\mathbb{x}$, the measurements are obtained from landmarks within the visibility region. These measurements, including ranges and angles to the landmarks, denoted as $f(\mathbb{x})$, can be treated as mappings from the target location. We are interested in understanding the probability that the measurements $f(\mathbb{x})$ are sufficiently distinct from the measurement $f(\mathbb{x}_0)$ at the given location, which we term localizability. Expressions of localizability probability are derived for specific vision-inspired measurements, such as ranges to landmarks and snapshots of their locations. Our analysis reveals that the localizability probability approaches one when the landmark intensity tends to infinity, which means that error-free localization is achievable in this limiting regime.