A New Statistical Approach to the Performance Analysis of Vision-based Localization

TL;DR

This work introduces a stochastic-geometry framework for vision-based localization where landmarks are modeled as a marked Poisson point process and may be visually indistinguishable within the same mark. Range measurements to nearby landmarks yield geometric constraints that enable identification of the observed landmark combination, despite non-uniqueness in appearance. It proves that three noise-free distances suffice to identify the correct landmark combination in 2D, and provides an analytical characterization of localizability under noise via the joint distribution of measurements, marks, and combination size. The paper also develops an algorithm based on geometric constraints (with two observation policies: random and nearest) and derives bounds on localization performance, supported by numerical results that highlight the trade-offs between measurement count, policy choice, and landmark density.

Abstract

Many modern wireless devices with accurate positioning needs also have access to vision sensors, such as a camera, radar, and Light Detection and Ranging (LiDAR). In scenarios where wireless-based positioning is either inaccurate or unavailable, using information from vision sensors becomes highly desirable for determining the precise location of the wireless device. Specifically, vision data can be used to estimate distances between the target (where the sensors are mounted) and nearby landmarks. However, a significant challenge in positioning using these measurements is the inability to uniquely identify which specific landmark is visible in the data. For instance, when the target is located close to a lamppost, it becomes challenging to precisely identify the specific lamppost (among several in the region) that is near the target. This work proposes a new framework for target localization using range measurements to multiple proximate landmarks. The geometric constraints introduced by these measurements are utilized to narrow down candidate landmark combinations corresponding to the range measurements and, consequently, the target's location on a map. By modeling landmarks as a marked Poisson point process (PPP), we show that three noise-free range measurements are sufficient to uniquely determine the correct combination of landmarks in a two-dimensional plane. For noisy measurements, we provide a mathematical characterization of the probability of correctly identifying the observed landmark combination based on a novel joint distribution of key random variables. Our results demonstrate that the landmark combination can be identified using ranges, even when individual landmarks are visually indistinguishable.

Figures (6)

• Figure 1: The illustration of the map shows five types of landmarks: lampposts, bus stops, parking lots, gas stations, and ATMs. Three landmarks are visible at Loc 1, but their unique locations remain uncertain because the same combination of landmarks is also visible from a different location, such as Loc 2. Although the landmarks appear identical, differences in distances, angles, or a combination of both can distinguish them.
• Figure 2: The illustration of possible locations for the second landmark. The first landmark is fixed at the origin, represented as a blue point, and the second landmark is depicted as a green point. The target's location, marked as red points, is at a distance of $d_1$ from the first landmark and $d_2$ from the second landmark. If a target location satisfying these distance constraints exists, the possible locations of the second landmark are restricted to the orange annulus.
• Figure 3: The illustration of possible locations for the third landmark. The blue point represents the first landmark, and the green point represents the second landmark. The red points indicate the locations of the target. The orange annulus shows the possible locations of the third landmark.
• Figure 4: An illustration of the geometric constraints among the target and three landmarks. $r_i$ represents the range measurement of the distance between the target and the $i$-th landmark located at ${\mathbf{x}}_i = (x_i, y_i)$. Blue dashed circles with radii $r_i$ represent the potential locations of the target.
• Figure 5: The performance comparison between two observation policies in terms of (a) the sizes of the solution set ${\mathcal{S}}$, (b) the percentage of landmark combinations removed by the proposed geometric constraints, and (c) the true positive rate and false positive rate evaluated in the stochastic setting.

Theorems & Definitions (11)

• Definition 1: Minkowski Sum
• Lemma 1
• Definition 2
• Lemma 2
• Lemma 3
• Theorem 1
• Lemma 4
• Lemma 5
• Lemma 6
• Theorem 2