Sensor-target Geometry Research Articles

Co-registration accuracy between Landsat-8 and Sentinel-2 orthorectified products

Landsat orthorectified products use Ground Control Points (GCPs) and Digital Elevation Models (DEM) to improve the geolocation accuracy and temporal consistency, and to account for the relief displacements due to the sensor-target geometry. In Collection-2, to improve the geometric harmonization between Landsat and Sentinel-2 (S2) orthorectified products, the Landsat GCP's absolute and relative accuracies were improved using the S2 Global Reference Image (GRI) dataset through a continent-level bundle adjustment method. The GRI is a highly accurate global image dataset that was developed by the European Space Agency (ESA) to improve the S2 multi-temporal geolocation accuracy. Since late August 2021, ESA has been using the GRI dataset in the geometric refinement process to generate S2 terrain-corrected (L1C) products. This paper presents the co-registration accuracy between the Landsat-8 (L8) Collection-2 terrain-corrected products and the S2 L1C products that were processed with and without the use of the GRI dataset. The image-to-image registration (I2I) analysis performed between the L8 and S2 data products over a set of globally distributed tiles shows a significant improvement in their co-registration accuracy when GRI is used in the S2 L1C product generation. The co-registration error is estimated to be <6 m circular error at 90% probability (CE90) when GRI is used, and >12 m CE90 when GRI is not used in the S2 product generation process. A similar I2I analysis was conducted between S2 L1C products, L8 L1TP products, and L8 and Landsat 9 (L9) L1TP products. The analysis shows that the S2 L1C products are co-registered with each other temporally to better than 5.1 m CE90 when GRI is used. The L8 L1TP products and L8 versus L9 L1TP products are both co-registered temporally to better than 3 m CE90.

Sensor Selection and Deployment for Range-Only Target Localization Using Optimal Sensor-Target Geometry

Sensor Selection and Deployment for Range-Only Target Localization Using Optimal Sensor-Target Geometry

Optimal Geometry and Motion Coordination for Multisensor Target Tracking with Bearings-Only Measurements.

This paper focuses on the optimal geometry and motion coordination problem of mobile bearings-only sensors for improving target tracking performance. A general optimal sensor-target geometry is derived with uniform sensor-target distance using D-optimality for arbitrary n (n≥2) bearings-only sensors. The optimal geometry is characterized by the partition cases dividing n into the sum of integers no less than two. Then, a motion coordination method is developed to steer the sensors to reach the circular radius orbit (CRO) around the target with a minimum sensor-target distance and move with a circular formation. The sensors are first driven to approach the target directly when outside the CRO. When the sensor reaches the CRO, they are then allocated to different subsets according to the partition cases through matching the optimal geometry. The sensor motion is optimized under constraints to achieve the matched optimal geometry by minimizing the sum of the distance traveled by the sensors. Finally, two illustrative examples are used to demonstrate the effectiveness of the proposed approach.

Optimal Measurement of Drone Swarm in RSS-Based Passive Localization With Region Constraints

Passive geolocation by multiple unmanned aerial vehicles (UAVs) covers a wide range of military and civilian applications including rescue, wild life tracking and electronic warfare. The sensor-target geometry is known to significantly affect the localization precision. The existing optimal sensor placement strategies mainly work on the cases without any constraints on the sensor locations. However, UAVs cannot fly/hover simply in arbitrary region due to realistic constraints, such as the geographical limitations, the security issues, and the max flying speed. In this paper, optimal geometrical configurations of UAVs in received signal strength (RSS)-based localization under region constraints are investigated. Employing the D-optimal criteria, i.e., maximizing the determinant of Fisher information matrix (FIM), such optimal problem is formulated. Based on the rigorous algebra and geometrical derivations, optimal and also closed-form configurations of UAVs under different flying states are proposed. Finally, the effectiveness and practicality of the proposed configurations are demonstrated by simulation examples.

Performance Analysis and Sensor-Target Geometry Optimization for TOA and TDOA-Based Hybrid Source Localization Method

Currently, the performance analysis of positioning algorithms and optimization of ground station deployment schemes are predominantly based on pure TOA or TDOA measurement information, and the relevant theoretical analysis is primarily the geometric analysis of optimal station deployment for fixed point targets, with few placement ranges and amount of station constraints. In practice, however, there are typically several measurements from TOA and TDOA stations, with a focus on positioning precision within a certain region or line trajectory, as well as the necessity for constraints on the ground station placement range. This paper proposes an efficient method for hybrid source localization using TOA and TDOA measurement information, establishes a mathematical model for hybrid source localization based on TOA and TDOA measurement information, derives and simulates the Gauss–Newton iterative localization algorithm with the least squares criterion, and performs a theoretical analysis of the least squares error and CRLB boundary to improve the accuracy of target localization in the aforementioned scenarios. Taking the average CRLB value of target line trajectory positioning error as the objective function, the ground station placement scheme of TOA- and TDOA-receiving sensors is optimized by utilizing a Genetic Algorithm with strong global optimization capability under the constraints of station placement range and station quantities, and a station placement geometry with better performance than typical station placement is obtained. Meanwhile, we summarize the general placement principles for TOA and TDOA hybrid source localization of target line trajectories.

Experimental Study on the Target–Receiver Formation Problem with the Exploitation of Coherent and Non-Coherent Bearing Information

Localization of emitting sources is a fundamental task in sonar applications. One of the most important factors that affect the localization performance is the sensor–target geometry. The sensor formation problem is usually addressed in related work assuming that the target is static and the location is known to a certain degree, but this is not the case for many underwater surveillance problems. In this paper, we deal with the target–receiver formation problem from a different perspective, and propose to investigate the effect of target–receiver geometry on localization performance by exploiting the spatial spectrum of the direct position determination (DPD) methods. For a given multi-array system, the transformation of geometrical patterns can be explicitly demonstrated as the target moves along the track. Meaningful characteristics of the DPD methods are obtained from the experimental results, where coherent and non-coherent bearing information is used and compared. The feasibility of the DPD approaches in the ocean environments is also investigated by comparing with a matched filter processing (MFP)-based multi-array processor in order to validate the credibility of the results in this paper.

Optimal Geometry Analysis for Target Localization With Bayesian Priors

Relative sensor-target geometry is well known to significantly affect the performance of target localization using a sensor network. This article analyzes the optimal sensor-target geometries for the problem of target localization with Bayesian priors. We present a unified geometry optimization framework for different types of sensor networks including bearing-only sensor network, range-only sensor network, received signal strength-only sensor network and mixed network of these sensor types. For geometry optimization of Bayesian target localization with these sensor networks, we establish the equivalence between the A-optimality criterion (i.e., minimizing the estimation mean squared error) and the D-optimality criterion (i.e., minimizing the area of the estimation uncertainty ellipse). Under these optimality criteria, the geometry optimization problem is shown to be mathematically equivalent to the problem of minimizing the modulus of a vector sum. This thus makes the optimal geometry conditions algebraically simple and easy to be computed. We conclude the article with extensive simulation studies to verify the accuracy of the analytical findings.

Closed‐form formula of Cramer–Rao lower bound for 3D TOA target localisation

This Letter focuses on the problem of analytically evaluating the performance metric using the Cramer–Rao lower bound (CRLB) for three-dimensional (3D) time-of-arrival (TOA) target localisation. The necessary and sufficient condition on the invertibility (non-singularity) of the Fisher information matrix (FIM) is presented, and then the existence of the CRLB is analysed. Two types of equivalent closed-form formulas of the CRLB are proposed, and any one of which can be adopted to analytically evaluate the best achievable accuracy of 3D TOA localisation. Moreover, three types of the worst sensor-target geometries described by the unit vectors of line-of-sight of TOA sensors are extensively illustrated, which will lead to a worse localisation accuracy or even result in singularities of the FIM and should be avoided in the deployment of 3D TOA sensors.

Optimal Sensor Placement for 2-D Range-Only Target Localization in Constrained Sensor Geometry

Localization of an emitting or reflecting target is one of the most important issues in a wide range of applications including radar, sonar, wireless communication and sensor networks. Due to significant effect on the positioning accuracy, designing the optimal sensor-target geometry has been considered as an important problem in the localization literature. The existing sensor placement methods mainly solve the problem in the cases without any constraints on the sensors locations. In the realistic scenarios, however, the sensors cannot be placed simply in arbitrary locations due to such constraints as the geographical limitations, communication problems between the sensor pairs and the security issues. In this paper, the optimal sensor-target geometries under sensor location constraints for range-based positioning are determined based on the A-optimality, E-optimality and D-optimality criteria. First, the special case of two-sensor in a circular area is considered as the permissible sensors placement region, and then, the results are extended to more number of sensors in a connected arbitrarily-shaped sensor region. The proposed sensor placement method can be applied to any application that the target is located outside the sensor region.

Optimal Sensor Placement for 3-D Time-of-Arrival Target Localization

This paper focuses on finding the optimal sensor placement strategies for circular time-of-arrival (TOA) localization in the three-dimensional (3D) space. The A-optimality criterion, minimizing the trace of the inverse Fisher information matrix (FIM), is applied to determine optimal sensor placements under the Gaussian measurement noise. A comprehensive analysis of optimal sensor-target geometries is provided in the general circumstance with no restriction on the number of sensors, sensor elevation angle and sensor-target range. The analysis is divided into two detailed sub-cases with uniform sensor measurement noise variance. The reachable minimum trace of Cramer-Rao lower bound (CRLB) $=\frac{9\sigma ^2}{4N}$ is derived. Two new closed-form solutions, i.e., the resistor network and special solution methods, are developed to quickly determine the optimal geometries. Furthermore, the theoretical smallest tr(CRLB) $=\frac{9}{4} (\sum _{i=1}^N\frac{1}{\sigma _{i}^{2}})^{-1}$ of using different noise variances is also presented. The analytical results are verified by simulation examples.

Optimal Sensor-Target Geometries for 3-D Static Target Localization Using Received-Signal-Strength Measurements

This letter investigates how to place the received-signal-strength (RSS) sensors to improve the static target localization accuracy in the three-dimensional (3-D) space. By using the A-optimality criterion, i.e., minimizing the trace of the inverse Fisher information matrix (FIM), a new optimal RSS sensor placement strategy is developed when sensors can be placed freely in the 3-D space. The smallest reachable trace of Cramer–Rao lower bound, i.e., the inverse FIM, is derived with the corresponding optimal sensor-target geometries. Besides, a resistor network method and a special configuration strategy are proposed to quickly determine the optimal geometries. The findings are concluded in three remarks, which are used to evaluate and improve the estimation accuracy. Simulation examples verified these findings.

Optimal sensor placement for multi‐source AOA localisation with distance‐dependent noise model

In this study, the optimal sensor placement problem for multi-source angle of arrival localisation is investigated. The authors adopt the A-optimality criterion, maximising the trace of Fisher information matrix, to determine the optimal sensor-target geometry under distance-dependent Gaussian noise model. A recursive representation of the Cramer-Rao lower bound is derived to recast the sensor placement problem into a sequential method, obtaining the optimal sensor geometries in a step by step manner. Note that the state-of-the-art methods are highly sensitive to the source location changes such that they should be relocated by any later changes in target geometries, which is practically impossible in systems with stationary receivers (such as radars). The proposed sequential method can circumvent this drawback by changing only one flexible sensor position rather than the whole geometry of the sensors. Numerical simulations are included to corroborate the theoretical results and determine the cost of such flexibility and robustness on the localisation performance.

A Novel Target-Height Estimation Approach Using Radar-Wave Multipath Propagation for Automotive Applications

Abstract. This paper introduces a novel target height estimation approach using a Frequency Modulation Continuous Wave (FMCW) automotive radar. The presented algorithm takes advantage of radar wave multipath propagation to measure the height of objects in the vehicle surroundings. A multipath propagation model is presented first, then a target height is formulated using geometry, based on the presented propagation model. It is then shown from Sensor-Target geometry that height estimation of targets is highly dependent on the radar range resolution, target range and target height. The high resolution algorithm RELAX is discussed and applied to collected raw data to enhance the radar range resolution capability. This enables a more accurate height estimation especially for low targets. Finally, the results of a measurement campaign using corner reflectors at different heights are discussed to show that target heights can be very accurately resolved by the proposed algorithm and that for low targets an average mean height estimation error of 0.03 m has been achieved by the proposed height finding algorithm.

Optimal Sensor Placement for 3-D Angle-of-Arrival Target Localization

This paper investigates optimal sensor placement strategies for angle-of-arrival (AOA) localization in three-dimensional space. We adopt the A-optimality criterion to determine optimal sensor placements under the Gaussian measurement noise. A comprehensive analysis of optimal sensor-target geometries is provided with no restriction on the number of AOA sensors, sensor-target range, and noise variances. A resistor network analogy is also presented to enable quick determination of optimal sensor-target geometries. The analytical results are verified by extensive simulation studies.

Optimal Geometry Analysis for Multistatic TOA Localization

Multistatic time-of-arrival (TOA) localization has recently attracted great attention due to performance advantages offered by multistatic radar. In target localization, the relative sensor-target geometry is an important factor that can significantly affect the localization performance. In this paper, we analyze the optimal geometries for a two-dimensional TOA localization configuration commonly presented in the literature with a single transmitter and multiple receivers. Our analysis is based on minimizing the area of estimation confidence region, which is equivalent to maximizing the determinant of the Fisher information matrix. If the coordinate system is rotated such that the bearing angle of the transmitter with respect to the target is zero, an optimal geometry is obtained when a subset of the receivers are collinear with the target at a bearing angle of $\pi /3$ rad and the remaining receivers at a bearing angle of $-\pi /3$ rad. The arrangement of the receivers on either branch of the bearing angles, i.e., $\pi /3$ rad or $-\pi /3$ rad, is decided based on the measurement noise variances at the receivers. We conclude the study with numerical solutions generated by the genetic algorithm and simulation examples for UAV path optimization to verify the accuracy of the analytical findings.

Optimum Sensor Array for Passive Localization from Time Differences of Arrival

The problem how to improve the accuracy of passive localization from time differences of arrival received considerable interest. The localization performance of any unbiased estimator can be explicitly characterized by certain measures, for example, by the Cramer-Rao lower bound (CRLB) on the estimator variance. The lower the CRLB, the better localization performance. It is well known that the relative sensor-target geometry can significantly affect the performance of any particular localization algorithm. It demonstrates, when target is surrounded by the sensors, uniform angular array is the optimum sensor placement, in which the CRLB is minimized.

Multi-Bernoulli filter for target tracking with multi-static Doppler only measurement

Multi-static Doppler-shift has re-emerged recently in the target tracking literature along with passive sensing, especially for aircraft tracking. Tracking with multi-static Doppler only measurement requires efficient multi-sensor fusion approach and optimal sensor network configuration if possible. In this paper, we present a solution for multi-target tracking with Doppler only measurements using the multi-Bernoulli filter. To utilize Doppler measurements from multiple sensors, we investigate different multi-sensor fusion schemes and the sensor-target geometry analysis for optimal multi-static Doppler sensor network configuration. Sensor-target geometry analysis is presented to investigate optimal multi-static Doppler sensor network configuration. Numerical results verify that the proposed sequential Monte Carlo (SMC) multi-Bernoulli filter with sequential update scheme and using the carefully chosen network shows good performance.

Optimal Sensor Arrangements in Angle of Arrival (AoA) and Range Based Localization with Linear Sensor Arrays

This paper investigates the linear separation requirements for Angle-of-Arrival (AoA) and range sensors, in order to achieve the optimal performance in estimating the position of a target from multiple and typically noisy sensor measurements. We analyse the sensor-target geometry in terms of the Cramer–Rao inequality and the corresponding Fisher information matrix, in order to characterize localization performance with respect to the linear spatial distribution of sensors. Here in this paper, we consider both fixed and adjustable linear sensor arrays.

Sensor-Target Geometry for Hybrid Bearing/Range Underwater Localization

In this paper, the influence of sensor-target relative geometry on the potential performance of underwater target localization with hybrid bearing/range sensors is investigated. The optimality criterion function is built on the knowledge of Fisher information matrix (FIM), and another analysis on the mean squared error (MSE) is also presented. For a fixed distance between the sensors to the underwater target, the MSE is minimized if and only if the determinant of the FIM is maximized. The main contribution in this paper is the dependence of the range measurement error on the acoustic propagation distance because of the complex underwater environment, which result in a different FIM expression compared to the ideal assumption case. Simulation results are provided to show the effectiveness of the algorithms presented.

Track Quality Based Multitarget Tracking Approach for Global Nearest-Neighbor Association

In multitarget tracking, in addition to the problem of measurement-to-track association, there are decision problems related to track confirmation and termination. In general, such decisions are taken based on the total number of measurement associations, length of no association sequence, and total lifetime of the track in question. For a better utilization of available information, confidence of the tracker on a particular track can be used. This quantity can be computed using the measurement-to-track association likelihoods corresponding to the particular track, target detection probability for the sensor-target geometry, and false alarm density. A track quality measure is proposed here for assignment-based global nearest neighbor (GNN) trackers. It can be noted that to compute track quality measure for assignment-based data association one needs to consider different detection events than those considered for computation of the track quality measures available in the literature, which are designed for probabilistic data association (PDA) based trackers. In addition to the proposed track quality measure, a multitarget tracker based on it is developed, which is particularly suitable in scenarios with temporarily undetectable targets. In this work, tracks are divided into three sets based on their quality and measurement association history: initial tracks, confirmed tracks, and unobservable tracks. Details of the update procedures of the three track sets are provided. The results show that discriminating tracks on the basis of their track quality can lead to longer track life while decreasing the average false track length.