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.

Diffraction Aided Wireless Positioning

Wireless positioning in Non-Line-of-Sight (NLOS) scenarios is highly challenging due to multipath, which leads to deterioration in the positioning estimate. This study reexamines electromagnetic field principles and applies them to wireless positioning, resulting in new techniques that enhance positioning accuracy in NLOS scenarios. Further, we use the proposed method to analyze a public safety scenario where it is essential to determine the position of at-risk individuals within buildings, emphasizing improving the Z-axis position estimate. Our analysis uses the Geometrical Theory of Diffraction (GTD) to provide important signal propagation insights and develop a new NLOS path model. Next, we use Fisher information to derive necessary and sufficient conditions for 3D positioning using our proposed positioning technique and finally to lower bound the possible 3D and z-axis positioning performance. On applying this positioning technique in a public safety scenario, we show that it is possible to greatly improve both 3D and Z-axis positioning performance by directly estimating NLOS path lengths.

Fundamentals of LEO Based Localization

In this paper, we derive the fundamental limits of low earth orbit (LEO) enabled localization by analyzing the available information in signals from multiple LEOs during different transmission time slots received on a multiple antennas and evaluate the utility of these signals for $9$D localization ($3$D position, $3$D orientation, and $3$D velocity estimation). We start by deriving the Fisher Information Matrix (FIM) for the channel parameters that are present in the signals received from LEOs in the same or multiple constellations during multiple transmission time slots. To accomplish this, we define a system model that captures i) time offset between LEOs caused by having relatively cheap clocks, ii) frequency offset between LEOs, iii) the unknown Doppler rate caused by high mobility LEOs, and iv) multiple transmission time slots from a particular LEO. We transform the FIM for the channel parameters to the FIM for the location parameters and determine the required conditions for localization. To do this, we start with the $3$D localization cases: i) $3$D positioning with known velocity and orientation, ii) $3$D orientation estimation with known position and velocity, and iii) $3$D velocity estimation with known position and orientation. Subsequently, we derive the FIM for the full $9$D localization case ($3$D position, $3$D orientation, and $3$D velocity estimation) in terms of the FIM for the $3$D localization. Using these results, we determine the number of LEOs, the operating frequency, the number of transmission time slots, and the number of receive antennas that allow for different levels of location estimation.

Joint 9D Receiver Localization and Ephemeris Correction with LEO and $5$G Base Stations

In this paper, we use the Fisher information matrix (FIM) to analyze the interaction between low-earth orbit (LEO) satellites and $5$G base stations in providing $9$D receiver localization and correcting LEO ephemeris. First, we give a channel model that captures all the information in the LEO-receiver, LEO-BS, and BS-receiver links. Subsequently, we use FIM to capture the amount of information about the channel parameters in these links. Then, we transform these FIM for channel parameters to the FIM for the $9$D ($3$D position, $3$D orientation, and $3$D velocity estimation) receiver localization parameters and the LEO position and velocity offset. Closed-form expressions for the entries in the FIM for these location parameters are presented. Our results on identifiability utilizing the FIM for the location parameters indicate: i) with one LEO, we need three BSs and three time slots to both estimate the $9$D location parameters and correct the LEO position and velocity, ii) with two LEO, we need three BSs and three time slots to both estimate the $9$D location parameters and correct the LEO position and velocity, and iii) with three LEO, we need three BSs and four-time slots to both estimate the $9$D location parameters and correct the LEO position and velocity. Another key insight is that through the Cramer Rao lower bound we show that with a single LEO, three time slots, and three BSs, the receiver positioning error, velocity estimation error, orientation error, LEO position offset estimation error, and LEO velocity offset estimation error are $0.1 \text{ cm}$, $1 \text{ mm/s}$, $10^{-3} \text{ rad}$, $0.01 \text{ m}$, and $1 \text{ m/s}$, respectively.

3D Positioning using a New Diffraction Path Model

Enhancing 3D and Z-axis positioning accuracy is crucial for effective rescue in indoor emergencies, ensuring safety for emergency responders and at-risk individuals. Additionally, reducing the dependence of a positioning system on fixed infrastructure is crucial, given its vulnerability to power failures and damage during emergencies. Further challenges from a signal propagation perspective include poor indoor signal coverage, multipath effects and the problem of Non-Line-OfSight (NLOS) measurement bias. In this study, we utilize the mobility provided by a rapidly deployable Uncrewed Aerial Vehicle (UAV) based wireless network to address these challenges. We recognize diffraction from window edges as a crucial signal propagation mechanism and employ the Geometrical Theory of Diffraction (GTD) to introduce a novel NLOS path length model. Using this path length model, we propose two different techniques to improve the indoor positioning performance for emergency scenarios.

Stochastic Geometry Analysis of a New GSCM with Dual Visibility Regions

The geometry-based stochastic channel models (GSCM), which can describe realistic channel impulse responses, often rely on the existence of both {\em local} and {\em far} scatterers. However, their visibility from both the base station (BS) and mobile station (MS) depends on their relative heights and positions. For example, the condition of visibility of a scatterer from the perspective of a BS is different from that of an MS and depends on the height of the scatterer. To capture this, we propose a novel GSCM where each scatterer has dual disk visibility regions (VRs) centered on itself for both BS and MS, with their radii being our model parameters. Our model consists of {\em short} and {\em tall} scatterers, which are both modeled using independent inhomogeneous Poisson point processes (IPPPs) having distinct dual VRs. We also introduce a probability parameter to account for the varying visibility of tall scatterers from different MSs, effectively emulating their noncontiguous VRs. Using stochastic geometry, we derive the probability mass function (PMF) of the number of multipath components (MPCs), the marginal and joint distance distributions for an active scatterer, the mean time of arrival (ToA), and the mean received power through non-line-of-sight (NLoS) paths for our proposed model. By selecting appropriate model parameters, the propagation characteristics of our GSCM are demonstrated to closely emulate those of the COST-259 model.

Novel LoS $β-γ$ THz Channel Unifying Molecular Re-radiation Manifestations

This paper introduces a novel line-of-sight (LoS) $\beta-\gamma$ terahertz (THz) channel model that closely mirrors physical reality by considering radiation trapping. Our channel model provides an exhaustive modeling of the physical phenomena including the amount of re-radiation available at the receiver, parametrized by $\beta$, and the balance between scattering and noise contributions, parametrized by $\gamma$, respectively. Our findings indicate a nontrivial relationship between average limiting received signal-to-noise ratio (SNR) and distance emphasizing the significance of $\gamma$ in THz system design. We further propose new maximum likelihood (ML) thresholds for pulse amplitude modulation (PAM) and quadrature amplitude modulation (QAM) schemes, resulting in analytical symbol error rate (SER) expressions that account for different noise variances across constellation points. The results confirm that the analytical SER closely matches the true simulated SER when using an optimal detector. As expected, under maximum molecular re-radiation, the true SER is shown to be lower than that produced by a suboptimal detector that assumes equal noise variances.

RIS-Aided Kinematic Analysis for Remote Rehabilitation

This paper is the first to introduce the idea of using reconfigurable intelligent surfaces (RISs) as passive devices that measure the position and orientation of certain human body parts over time. In this paper, we investigate the possibility of utilizing the available geometric information provided by on-body RISs that reflect signals from an off-body transmitter to an off-body receiver for stroke rehabilitation. More specifically, we investigate the possibility of using on-body RISs to estimate the location information over time of upper limbs that may have been impaired due to stroke. This location information can help medical professionals to estimate the possibly time varying pose and obtain progress on the rehabilitation of the upper limbs. Our analysis is focused on two scenarios: i) after assessment exercises for stroke rehabilitation when the upper limbs are resting at predefined points in the rehabilitation center, and ii) during the assessment exercises. In the first scenario, we explore the possibility of upper limb orientation estimation by deriving the Fisher information matrix (FIM) under near-field and far-field propagation conditions. It is noteworthy that the FIM quantifies how accurately we can estimate location information from a signal, and any subsequent algorithm is bounded by a function of the FIM. Coming to our propagation assumptions, the difference between the near-field and far-field regimes lies in the curvature of the wavefront. In the near-field, a receiver experiences a spherical wavefront, whereas in the far-field, the wavefront is approximately linear. The threshold to be within the near-field can be on the order of $10 \text{ m}.$

On the Limits of Single Anchor Localization: Near-Field vs Far-Field

It is well known that a single anchor can be used to determine the position and orientation of an agent communicating with it. However, it is not clear what information about the anchor or the agent is necessary to perform this localization, especially when the agent is in the near-field of the anchor. Hence, in this paper, to investigate the limits of localizing an agent with some uncertainty in the anchor location, we consider a wireless link consisting of source and destination nodes. More specifically, we present a Fisher information theoretical investigation of the possibility of estimating different combinations of the source and destination's position and orientation from the signal received at the destination. To present a comprehensive study, we perform this Fisher information theoretic investigation under both the near and far field propagation models. One of the key insights is that while the source or destination's $3$D orientation can be jointly estimated with the source or destination's $3$D position in the near-field propagation regime, only the source or destination's $2$D orientation can be jointly estimated with the source or destination's $2$D position in the far-field propagation regime. Also, a simulation of the FIM indicates that in the near-field, we can estimate the source's $3$D orientation angles with no beamforming, but in the far-field, we can not estimate the source's $2$D orientation angles when no beamforming is employed.

NTN-based 6G Localization: Vision, Role of LEOs, and Open Problems

Since the introduction of 5G Release 18, non-terrestrial networks (NTNs) based positioning has garnered significant interest due to its numerous applications, including emergency services, lawful intercept, and charging and tariff services. This release considers single low-earth-orbit (LEO) positioning explicitly for {\em location verification} purposes, which requires a fairly coarse location estimate. To understand the future trajectory of NTN-based localization in 6G, we first provide a comprehensive overview of the evolution of 3rd Generation Partnership Project (3GPP) localization techniques, with specific emphasis on the current activities in 5G related to NTN location verification. In order to provide support for more accurate positioning in 6G using LEOs, we identify two NTN positioning systems that are likely study items for 6G: (i) multi-LEO positioning, and (ii) augmenting single-LEO based location verification setup with Global Navigation Satellite System (GNSS), especially when an insufficient number of GNSS satellites (such as 2) are visible. We evaluate the accuracy of both systems through a 3GPP-compliant simulation study using a Cram\'{e}r-Rao lower bound (CRLB) analysis. Our findings suggest that NTN technology has significant potential to provide accurate positioning of UEs in scenarios where GNSS signals may be weak or unavailable, but there are technical challenges in accommodating these solutions in 3GPP. We conclude with a discussion on the research landscape and key open problems related to NTN-based positioning.