We study the problem of signal source localization using differential received signal strength (DRSS) measurements. We begin by presenting verifiable geometric conditions for sensor deployment that ensure the model’s asymptotic localizability. Then we establish the consistency and asymptotic efficiency of the maximum likelihood (ML) estimator. However, computing the ML estimator is challenging due to relying on solving a non-convex optimization problem. To address this, we propose an asymptotically efficient two-step estimator that matches the ML estimator’s asymptotic properties while achieving low computational complexity (linear in the number of measurements). The primary challenge lies in obtaining a preliminary consistent estimate in the first step. To achieve this, we construct a linear least-squares (LS) problem through algebraic operations on the nonlinear measurement model to obtain a closed-form solution. This solution is biased due to the correlation between the regressors and the noise of the linear model.We eliminate the bias using the data to yield an asymptotically unbiased and consistent estimate. In the second step, we perform a single Gauss-Newton (GN) iteration using the preliminary consistent estimate as the initial value, achieving the same asymptotic properties as the ML estimator. Finally, simulation results demonstrate the superior performance of the proposed two-step estimator over existing methods for large sample sizes.

Given 2D point correspondences between an image pair, inferring the camera motion is a fundamental issue in the computer vision community. The existing works generally set out from the epipolar constraint and estimate the essential matrix, which is not optimal in the maximum likelihood (ML) sense. In this paper, we dive into the original measurement model with respect to the rotation matrix and normalized translation vector and formulate the ML problem. We then propose an optimal two-step algorithm to solve it: In the first step, we estimate the variance of measurement noises and devise a consistent estimator based on bias elimination; In the second step, we execute a one-step Gauss-Newton iteration on manifold to refine the consistent estimator. We prove that the proposed estimator achieves the same asymptotic statistical properties as the ML estimator: The first is consistency, i.e., the estimator converges to the ground truth as the point number increases; The second is asymptotic efficiency, i.e., the mean squared error of the estimator converges to the theoretical lower bound - Cramer-Rao bound. In addition, we show that our algorithm has linear time complexity. These appealing characteristics endow our estimator with a great advantage in the case of dense point correspondences. Experiments on both synthetic data and real images demonstrate that when the point number reaches the order of hundreds, our estimator outperforms the state-of-the-art ones in terms of estimation accuracy and CPU time.

We consider signal source localization from range-difference measurements. First, we give some readily-checked conditions on measurement noises and sensor deployment to guarantee the asymptotic identifiability of the model and show the consistency and asymptotic normality of the maximum likelihood (ML) estimator. Then, we devise an estimator that owns the same asymptotic property as the ML one. Specifically, we prove that the negative log-likelihood function converges to a function, which has a unique minimum and positive definite Hessian at the true source’s position. Hence, it is promising to execute local iterations, e.g., the Gauss-Newton (GN) algorithm, following a consistent estimate. The main issue involved is obtaining a preliminary consistent estimate. To this aim, we construct a linear least-squares problem via algebraic operation and constraint relaxation and obtain a closed-form solution. We then focus on deriving and eliminating the bias of the linear least-squares estimator, which yields an asymptotically unbiased and further consistent estimate. Noting that the bias is a function of the noise variance, we further devise a consistent noise variance estimator that involves a 3-order polynomial rooting. Based on the preliminary consistent location estimate, a one-step GN iteration suffices to achieve the same asymptotic property as the ML estimator. Simulation results demonstrate the superiority of our proposed algorithm in the large sample case.

State estimation is an essential part of autonomous systems. Integrating the Ultra-Wideband (UWB) technique has been shown to correct the long-term estimation drift and bypass the complexity of loop closure detection. However, few works on robotics treat UWB as a stand-alone state estimation solution. The primary purpose of this work is to investigate planar pose estimation using only UWB range measurements. We prove the excellent property of a two-step scheme, which says we can refine a consistent estimator to be asymptotically efficient by one step of Gauss-Newton iteration. Grounded on this result, we design the GN-ULS estimator, which reduces the computation time significantly compared to previous methods and presents the possibility of using only UWB for real-time state estimation.

The Perspective-n-Point (PnP) problem has been widely studied in both computer vision and photogrammetry societies. With the development of feature extraction techniques, a large number of feature points might be available in a single shot. It is promising to devise a consistent estimator, i.e., the estimate can converge to the true camera pose as the number of points increases. To this end, we propose a consistent PnP solver, named CPnP, with bias elimination. Specifically, linear equations are constructed from the original projection model via measurement model modification and variable elimination, based on which a closed-form least-squares solution is obtained. We then analyze and subtract the asymptotic bias of this solution, resulting in a consistent estimate. Additionally, Gauss-Newton (GN) iterations are executed to refine the consistent solution. Our proposed estimator is efficient in terms of computations—it has$O(n)$time complexity. Simulations and real dataset tests show that our proposed estimator is superior to some well-known ones for images with dense visual features, in terms of estimation precision and computing time.

This paper studies the localization problem using noisy range-difference measurements, or equivalently time difference of arrival (TDOA) measurements. There is a reference sensor, and for each other sensor, the TDOA measurement is obtained with respect to the reference one. By minimizing the sum of squared errors, a nonconvex constrained least squares (CLS) problem is formulated. In this work, we focus on devising an algorithm to seek the global minimizer of the CLS problem, hoping that the numerical solution meets some precision requirement in terms of relative error. Based on the Lagrange multiplier method, we first branch the feasible Lagrange multiplier set into several subsets and develop a workflow in terms of if-then-else control structure to seek the global minimizer by searching for the optimal Lagrange multiplier. The execution order is carefully organized so that it is in line with the general principle of putting the flow that one normally understands to be executed first. We then dive into detailed searching methods in different cases and conduct computational error analysis, giving the error bound on the Lagrange multiplier, when we search for it, to meet the precision requirement on an approximate solution. Based on the above achievements, a programmable global minimizer seeking algorithm is proposed for the CLS problem. Simulations and experimental tests on a public dataset demonstrate the effectiveness of the proposed algorithm.