Nonlinear Least-squares Solution Research Articles

Star Image Registration Modeling and Parameter Calibration for a 3CMOS Star Sensor.

This paper presents an image registration method specifically designed for a star sensor equipped with three complementary metal oxide semiconductor (CMOS) detectors. Its purpose is to register the red-, green-, and blue-channel star images acquired from three CMOS detectors, assuring the precision of star image fusion and centroid extraction in subsequent stages. This study starts with a theoretical analysis aimed at investigating the effect of inconsistent three-channel imaging parameters on the position of feature points. Based on this analysis, this paper establishes a registration model for transforming the red- and blue-channel star images into the green channel's coordinate system. Subsequently, the method estimates model parameters by finding a nonlinear least-squares solution. The experimental results demonstrate the correctness of the theoretical analysis and the proposed registration method. This method can achieve subpixel alignment accuracy in both the x and y directions, thus effectively ensuring the performance of subsequent operation steps in the 3CMOS star sensor.

Bias Analysis of Parameter Estimator Based on Gauss-Newton Method Applied to Ultra-Wideband Positioning

Ultra-wideband (UWB) is considered as a promising technology in short-distance indoor wireless positioning due to its accurate time resolution and good penetration through objects. Since the functional model of UWB positioning is nonlinear, the optimal solution is generally estimated by the way of continuous iteration. As an iterative descent method of high efficiency, the Gauss–Newton method is widely used to estimate the position. The nonlinear distance equations are linearized, and the solution can be found iteratively. Therefore, the nonlinear least-squares solution is generally biased even if the observations are normally distributed. In outdoor satellite positioning, the ranging distances are long enough so that the bias caused by nonlinearity is very small. However, in UWB positioning, the relative ranging error is large, and the positioning system is prone to become ill-posed, hence the bias due to nonlinearity is not negligible. In this study, both the statistical factor and geometric factor for bias in the nonlinear least-squares estimator of UWB positioning are discussed. In order to assess whether the linearized model is sufficiently approximate for the positioning estimation, a hypothesis test criterion based on Mahalanobis distance is proposed. The simulation and measurement experiments are performed to analyze the factors affecting the bias in UWB positioning. Experimental results are given to demonstrate that the linearization is valid and the bias in UWB positioning estimation can be neglected for the relatively high measurement precision. Moreover, for a positioning configuration, when the anchors are evenly distributed, the amount of nonlinearity is orthogonal to the ranging space of the design matrix, the UWB positioning estimation tends to be unbiased. Meanwhile, the hypothesis test based on Mahalanobis distance is carried out to determine the validity of the linearized model. When the bias is large for UWB positioning, the bias estimate can be used to correct the estimator to guarantee the unbiasedness for UWB positioning. Furthermore, the correction of parameter estimator bias is more effective in the case of relatively low measurement precision or ill-conditioned configuration.

Monte-Carlo-based uncertainty propagation in the context of Gauss–Markov model: a case study in coordinate transformation

One of the main tasks of professionals in the Earth sciences is to convert coordinates from one reference frame into another. The coordinates transformation is widely needed and applied in all branches of modern geospatial activities. To convert from one reference frame to another, it is necessary initially to determine the parameters of a coordinate transformation model. A common way to estimate the transformation parameters is using the least-squares theory within a linearized Gauss-Markov Model (GMM). Another approach arises from a numerical method. Here, a Monte Carlo Method (MCM) is used to infer the uncertainty of estimators. This method is based on: (i) the assignment of probability distributions to the coordinates in the two reference frames, (ii) the determination of a discrete representation of the probability distribution for the transformation parameters, and (iii) the determination of the associated uncertainties from this discrete representation of the estimates of the transformation parameters. In this contribution, we compare the weighted least-squares within the GMM (WLSE-GM) and the proposed method regarding the transformation problem of computing 2D similarity transformation parameters. The results show that, the transformation parameters uncertainties are higher for the LS-MC than the WLSE-GM. This is due to the fact that the WLSE-GM solution does not take into account the uncertainties associated with the system matrix. In future studies the Monte Carlo method should be applied to the nonlinear least-squares solution.

Inverse Simulation of Fracture Parameters for Cement-Bonded Corundum Refractories

In order to obtain the real solution of the fracture parameters for the wedge-splitting test, numerical simulation and inverse algorithm have been designed to estimate the maximum tensile stress and fracture energy of cement-bonded corundum refractory. The experimental and simulated curves have been systematically compared to produce the bilinear model of cohesive zone material with the inverse algorithm of nonlinear least-squares solution being the most suitable for the simulation of the wedge-splitting test. Furthermore, the inverse simulation procedure has been applied to specimens of various heating temperatures and cement contents. Consequently, the fracture energy and the maximum tensile stress initially decrease and then increase with the temperature. Furthermore, the fracture energy has the tendency of increasing with the cement content, and the maximum tensile stress has the highest peak at the content of 10 wt.%. Additionally, the cement-bonded corundum refractory presents higher brittleness after high-temperature heating (1400°C) or with the cement content of 10 wt.% at 110°C.

Gaussian Mixture Approximation of Angles-Only Initial Orbit Determination Likelihood Function

A method is developed to approximate the angles-only orbit determination likelihood function using a Gaussian mixture to incorporate information about an admissible region. The resulting probability density function can provide the a priori information for a Gaussian mixture orbit determination filter. This admissible region approach can compensate for the effects of measurement uncertainty over a short track of angles-only data better than deterministic approaches can. The new technique starts with a nonlinear batch least-squares solution. The solution enforces soft constraints on an admissible region defined in terms of minimum periapsis and maximum apoapsis. Although this soft-constrained solution lies in or near the admissible region, it does not characterize that region well. It provides a starting point to develop a Gaussian mixture approximation of the batch least-squares likelihood function as modified through multiplication by a finite-support function that is zero outside the admissible region and 1 inside that region. This Gaussian mixture is optimized to fit the resulting probability density in the two-dimensional subspace of position/velocity space that has the most uncertainty. The procedure starts by approximating the finite-support distribution using a single optimized Gaussian mixand. It automatically increases the number of mixands and reoptimizes until the resulting mixture fits the finite-support distribution to a prespecified precision. This optimal fitting procedure allows the Gaussian mixture to use a low number of mixands while fitting the finite-support probability density function well. By approximating the product of a finite-support function and the original likelihood function, the new method gains the capability to transition smoothly between regimes where the admissibility constraints dominate, i.e., high-altitude/short-measurement-arc cases, and those where they are less important, i.e., low-altitude/long-measurement-arc cases.

An Improved Two-Port Transmission Line Permittivity and Permeability Determination Method With Shorted Sample

A modified measurement technique and nonlinear least-squares solution method is introduced for determining complex permittivity and permeability in transmission lines. In addition to a two-port S-parameter measurement, a one-port measurement of the shorted sample is employed. For low-loss samples, material property determination using the traditional method can be plagued by resonances when the sample thickness is an integer multiple of the guided half-wavelength in the sample medium. The introduction of the shorted reflectivity measurement reduces the large uncertainties inherent in the resonance effect, and increases the likelihood of determining the correct zero in the fitting of the material properties. The short-circuited sample $S_{11}$ measurement has an increased magnitude compared with the standard two-port measurement, which greatly reduces the uncertainty of this term. The resulting complex permeability and permittivity values obtained with this method are smoother improved solutions, and have lower uncertainty. Results are presented comparing the traditional Nicolson–Ross–Weir solution and National Institute of Standards and Technology iterative method with the proposed technique for a cast epoxy and a ferrite-loaded microwave absorber sample in WR42 waveguide. We also investigate finite element simulated S-parameters of a fictional material and compare the derived material parameters obtained with the proposed method to the “true” permittivity and permeability values.

Barycentre method for solving distance equations

When iteratively solving the distance equations, the Newton's method has quadratic convergence but it requires the second-order derivatives. The Gauss–Newton's method needs no information about the second-order derivatives but it may fail without the line search strategy. A simple method called barycentre method is proposed to locally solving the distance equations without the Hessian matrix, the matrix inversion and the line search strategy. The geometrical meaning of the non-linear least-squares solution of the distance equations is revealed that it is the barycentre of a particle system composed of the observational weights at the endpoints of observed distance vectors. By the geometrical meaning of the non-linear least squares, the authors structure an iterative equation to solve the distance equation, and then the convergence and the computational complexity of the method proposed is discussed. It shows that the barycentre method is a conservatively steepest decent method to guarantee the convergence and this method has good performances for solving well-conditioned problems. Ultimately, the method proposed is applied to well-condition and ill-posed positioning equations and the main results are verified.

Enhancing the Symmetry and Proportion of 3D Face Geometry

We present an engine for enhancing the geometry of a 3D face mesh model while making the enhanced version share close similarity with the original. After obtaining the feature points of a given scanned 3D face model, we first perform a local and global symmetrization on the key facial features. We then apply an overall proportion optimization to the frontal face based on Neoclassical Canons and golden ratios. A nonlinear least-squares solution is adopted to adjust the feature points so that the face profile complies with the aesthetic criteria, which are derived from the profile cosmetology. Through the above processes, we obtain the optimized feature points, which will lead to a more attractive face. According to the original feature points and the optimized ones, we perform Laplacian deformation to adjust the remaining points of the face in order to preserve the geometric details. The analysis of user study in this paper validates the effectiveness of our 3D face geometry enhancement engine.

Fingerprinting Mobile User Positions in Sensor Networks: Attacks and Countermeasures

We demonstrate that the network flux over the sensor network provides fingerprint information about the mobile users within the field. Such information is exoteric in the physical space and easy to access through passive sniffing. We present a theoretical model to abstract the network flux according to the statuses of mobile users. We fit the theoretical model with the network flux measurements through Nonlinear Least Squares (NLS) and develop an algorithm that iteratively approaches the NLS solution by Sequential Monte Carlo Estimation. With sparse measurements of the flux information at individual sensor nodes, we show that it is easy to identify the mobile users within the network and instantly track their movements without breaking into the details of the communicational packets. Our study indicates that most of existing systems are vulnerable to such attack against the privacy of mobile users. We further propose a set of countermeasures that redistribute and reshape the network traffic to preserve the location privacy of mobile users. With a trace driven simulation, we demonstrate the substantial threats of the attacks and the effectiveness of the proposed countermeasures.

Parametric Identification of Dynamic Model of Assembled Flexible Manipulator

In order to restrain the vibration of flexible manipulator,friction damping of assembled flexible link can be used to restrain the vibration of flexible manipulator. The contrast experiment with solid flexible link indicates that assembled flexible link has effective vibration-restraining capability. The dynamic model of assembled flexible link is built with an equivalent mass,a nonlinear spring,a damper and a spring-slider. According to the data obtained from FEA simulation of forced vibration of assembled flexible link,its shape function and reaction force hysteresis curve are determined. The dynamic model parameter of assembled flexible link is determined according to kinetic energy equivalence principle,analysis of reaction force hysteresis curve and nonlinear least-squares solution. In addition,S-function block is written in M-file to compute spring-slider. Finally,the simulink model is created according to the dynamic model. The response of free vibration from simulation is consistent with experiment. This method provides an effective way for dynamics modeling and vibration-damping design of assembled flexible link.

Non-linear least-squares solution to the moiré hole method problem in orthotropic materials. Part II: Material elastic constants

This is the second of two papers dealing with inverse problems arising from the hole method problem in orthotropic materials. Two inverse problems are of interest in this second paper: one relates to the use of the hole method as a means of separation of stresses; the other deals with using it for orthotropic material elastic constant identification. The problem of separation of stresses is posed and briefly discussed, but the orthotropic material identification problem is addressed, fully showing how to obtain the five orthotropic elastic constants, four of which are independent, when knowing the applied stresses and examining the isothetics or moire fringes around a through hole in a biaxially loaded thin wood plate.

Non-linear least-squares solution to the moiré hole method problem in orthotropic materials. Part 1: Residual stresses

The hole method problem relates to two inverse problems of interest: the first, most commonly addressed by practitioners, is to obtain residual stresses; the other, generally neglected, inverse problem can be posed as either a stress separation problem or a material elastic properties identification problem. In both this Paper I and Paper II, we pose and solve this dual hole method problem in an orthotropic plate, using computer generated moire isothetics, by means of a non-linear least-squares approach. In Paper I we address the residual stress problem. In Paper II we pose the use of moire isothetics as a means to achieve separation of stresses, but we deal with the determination of the five orthotropic elastic constants, four of which are independent.

Non-linear Least-squares Solution to the Moire Hole Method Problem in Orthotropic Materials. Part II: Material Elastic Constants

Non-linear Least-squares Solution to the Moire Hole Method Problem in Orthotropic Materials. Part II: Material Elastic Constants

Non-linear Least-squares Solution to the Moire Hole Method Problem in Orthotropic Materials. Part I: Residual Stresses

Non-linear Least-squares Solution to the Moire Hole Method Problem in Orthotropic Materials. Part I: Residual Stresses

Statistics of the slope-method estimator

The slope method has customarily been used and is still used for inversion of atmospheric optical parameters, extinction, and backscatter in homogeneous atmospheres from lidar returns. Our aim is to study the underlying statistics of the old slope method and ultimately to compare its inversion performance with that of the present-day nonlinear least-squares solution (the so-called exponential-curve fitting). The contents are twofold: First, an analytical study is conducted to characterize the bias and the mean-square-estimation error of the regression operator, which permits estimation of the optical parameters from the logarithm of the range-compensated lidar return. Second, universal plots for most short- and far-range tropospheric backscatter lidars are presented as a rule of thumb for obtaining the optimum regression interval length that yields unbiased estimates. As a result, the simple graphic basis of the slope method is still maintained, and its inversion performance improves up to that of the present-day computer-oriented exponential-curve fitting, which ends the controversy between these two algorithms.

Receiver array calibration using disparate sources

We present a new array calibration procedure for over-the-horizon (OTH) radar, using disparate sources. Unlike previous array calibration methods, which require a specific type or class of sources for calibrating the array, the method we propose can use combinations of single-mode, multimode, and near-field sources; each source with either known or unknown DOAs (directions-of-arrival). Multidimensional MUSIC is exploited for time-invariant DOA sources, while single-snapshot techniques are used for sources that have time-varying DOAs. A nonlinear separable least-squares solution to the array calibration problem is used to estimate the array coupling matrix and sensor positions. Simulation results indicate that good estimates are obtained for the unknown parameters and further the array sidelobe levels and bearing errors are significantly reduced when these estimated parameters are used in array processing. The algorithm performance was also compared with the Cramer-Rao lower bound and found to be statistically efficient.

Preliminary studies on fitting experimental differential capacitance to theoretical curves

Several theoretical forms of the dependence of the electrode differential capacitance on the electrode potential (capacitance curves) have been examined in the context of their effectiveness in fitting experimental data. Each curve has been derived from a rigorously defined physical model. The link between the capacitance curve and the model has been clearly indicated. The goodness of fit and the curvature measures of non-linearity (maximum intrinsic and maximum parameter effects non-linearity, defined in the paper) served as a criterion for the acceptance or rejection of a particular curve and the related model. A new computer program that facilitates the fitting is briefly described. This user-interactive program enables non-linear fitting of an arbitrary capacitance curve, provides fitted data, together with residuals and the dependence of surface coverage on the electrode potential, and calculates the standard deviations for the parameter estimates, their Student t values and the matrix of parameter correlation coefficients. Additionally, the program provides Box's bias of the parameters and both maximum intrinsic and maximum parameter effects curvature measures. If required, the program output can be arranged to deliver simultaneously the dependence of surface charge density on electrode potential or surface coverage, the theoretical electrocapillary curves and other related information. The two-dimensional grids of the non-linear least-squares solution locus for some parameters are presented. The quality of fit is demonstrated for a few examples. The two-, three- and four-parameter models considered in this paper were unable to fit the experimental data within accepted experimental errors although they did not show any other extraordinary deficiencies. No definite conclusions were reached regarding the six-parameter models considered. Some of them can fit the data within experimental error but more work is required on reparametrization. Some six-parameter models behaved equally well from statistical point of view.

A nonlinear least-squares solution with causality constraints applied to transmission line permittivity and permeability determination

A technique for the solution of one-port and two-port scattering equations for complex permittivity and permeability determination is presented. Using a nonlinear regression procedure, the model determines parameters for the specification of the spectral functional form of complex permittivity and permeability. The method is based on a nonlinear regression technique and uses the fact that a causal, analytic function can be represented by poles and zeros. The technique allows the accurate determination of many low- and high-permittivity dielectric and magnetic materials in either the low- or high-loss range. The model allows for small adjustments, consistent with the physics of the problem, to independent variable data such as angular frequency, sample length, sample position, and cut-off wavelength. The model can determine permittivity and permeability for samples where sample length, sample position, and sample holder length are not known precisely. The problem of local minima is discussed. >

Three-dimensional finite strain from crinoid ossicles

Randomly oriented crinoid ossicles are useful markers for the determination of three-dimensional finite strain. Two techniques are presented. Both make use of established methods to measure the two-dimensional strain ellipses on three surfaces, which are then combined to calculate the shape and orientation of the strain ellipsoid. First, ossicles positioned such that a thin section cuts across the cylinder appear as randomly oriented ellipses prior to deformation, and can be analyzed using standard Rfφ methods. Second, ossicles positioned such that the section cuts lengthwise through the cylinder appear as rectangular or sub-rectangular shapes with orthogonal geometries in the undeformed state. Measurements of angular shear strain on two or more such markers are analyzed using a non-linear least-squares solution to the Breddin graph, allowing determination of the best-fit strain ellipse.Both methods are applied to an echinoderm grainstone from the central Helvetic nappes of Switzerland. The results are internally consistent, and compatible with those from other strain analysis techniques.

Ternary mutual diffusion coefficients of ZnCl2?KCl?H2O at 25�C by Rayleigh interferometry

Mutual diffusion coefficients and densities were measured for aqueous ZnCl2−KCl mixtures at 25° by using free-diffusion Rayleigh interferometry and pycnometry, respectively. The ZnCl2 concentrations were fixed at 1.5 mol-dm−3, whereas those of KCl were 0.5, 1.25, 2.0, or 4.0 mol-dm−3. This corresponds to a half charged zinc-chlorine storage battery at various suporting electrolyte concentrations. The main-term coefficient of ZnCl2 only varies by 10% with KCl concentration, whereas that of KCl varies by about 22%. The ZnCl2 cross-term coefficient remains small and positive; in contrast the KCl cross-term coefficient goes through a maximum and is negative at high and low KCl concentrations. At KCl concentrations of 0.5 and 4.0 mol-dm−3, solutions with the KCl Δc≈0 are statically and dynamically (diffusively) unstable at the top and bottom of the boundary. Evaluation of the parameters of the non-linear least-squares solution to the diffusion equation is difficult for the 1.25 mol-dm−3 KCl case, since this system has nearly equal eigenvalues in its diffusion coefficient matrix.