Least-squares Solution Research Articles

A robust time delay estimation method for ultrasonic echo signals and elastography

Modern medicine cannot ignore the significance of elastography in diagnosis and treatment plans. Despite improvements in accuracy and spatial resolution of elastograms, robustness against noise remains a neglected attribute. A method that can perform in a satisfactory manner under noisy conditions may prove useful for various elastography methods. Here, we propose a method based on eigenvalue decomposition (EVD). In this method, the estimated time delay is defined as the index of the maximum element in the eigenvector that corresponds to the minimum eigenvalue in the covariance matrix of the received signal. Moreover, the implementation of the least-squares (LS) solution and the lower-upper (LU) decomposition contributes to improving the speed of computation and the accuracy of the estimation under low signal-to-noise ratio (SNR) conditions. To assess the performance of the proposed algorithm, it is evaluated along with generalized cross-correlation (GCC) and EVD. The simulation results clearly confirm that the jitter variance achieved in the proposed algorithm outperforms GCC and EVD in the proximity of the Cramer-Rau lower band. Moreover, our algorithm provides satisfactory performance in terms of variance and bias against sub-sample delay at low SNRS. According to the experimental results, the calculated values of the elastographic signal-to-noise ratio (SNRe) and the elastographic contrast-to-noise ratio (CNRe) of the proposed algorithm were 16.7 and 20.09, respectively, clearly better than the values of the other two methods. Furthermore, the proposed algorithm offers less execution time (about 30% of GCC), with a computational complexity equal to GCC and better than EVD.

Vibration error compensation algorithm in the development of laser interference absolute gravimeters

Abstract. Measurement error arising from vibration interference is recognized as the primary obstacle limiting the accuracy and stability of laser interference absolute gravimeters. The present work addresses this issue by proposing a global search optimization algorithm that determines the optimal absolute value of gravity based on the measured time–displacement coordinates of a falling body and the signal obtained from the passive vibration isolation system of the inertial reference corner cube in a laser interference absolute gravimeter. Results of numerical calculations conducted under vibration interference conditions with added white noise resulting in a signal-to-noise ratio of 40 dB demonstrate the following. The accuracy and standard deviation of the gravimeter obtained using the proposed algorithm are −0.04 µGal (1µGal=1×10-8 m s−2) and 0.24 µGal, respectively, while those values obtained by the standard least-squares solution are 10.19 and 154.11 µGal, respectively. The test results indicate that the average response of the reference value of acceleration due to gravity superimposed by a disturbance of 1.00 µGal is 1.01 µGal using the proposed algorithm and 0.87 µGal using the standard least-squares solution.

Trading off 1-norm and sparsity against rank for linear models using mathematical optimization: 1-norm minimizing partially reflexive ah-symmetric generalized inverses

The M-P (Moore-Penrose) pseudoinverse has as a key application the computation of least-squares solutions of inconsistent systems of linear equations. Irrespective of whether a given input matrix is sparse, its M-P pseudoinverse can be dense, potentially leading to high computational burden, especially when we are dealing with high-dimensional matrices. The M-P pseudoinverse is uniquely characterized by four properties, but only two of them need to be satisfied for the computation of least-squares solutions. Fampa and Lee (2018) and Xu, Fampa, Lee, and Ponte (2019) propose local-search procedures to construct sparse block-structured generalized inverses that satisfy the two key M-P properties, plus one more (the so-called reflexive property). That additional M-P property is equivalent to imposing a minimum-rank condition on the generalized inverse. (Vector) 1-norm minimization is used to induce sparsity and, importantly, to keep the magnitudes of entries under control for the generalized-inverses constructed. Here, we investigate the trade-off between low 1-norm and low rank for generalized inverses that can be used in the computation of least-squares solutions. We propose several algorithmic approaches that start from a $1$-norm minimizing generalized inverse that satisfies the two key M-P properties, and gradually decrease its rank, by iteratively imposing the reflexive property. The algorithms iterate until the generalized inverse has the least possible rank. During the iterations, we produce intermediate solutions, trading off low 1-norm (and typically high sparsity) against low rank.

Influence of bearing ball recirculation on error motions of linear axes

For positioning systems utilizing linear guides and trucks with recirculating balls, a method is presented that uses the measured total error motions and the measured phase of ball loops within trucks to determine the influence of each ball loop on the error motions. The influence of ball recirculation on the error motions is estimated a priori via a least-squares solution based on data collected from a multitude of motion tests in which varying phases were measured by sensors integrated into the trucks. This method enables real-time estimation of performance degradations and identification of their sources.

Functional Principal Component Analysis of Radio–Optical Reference Frame Tie

Abstract The Gaia optical reference frame is intrinsically undefined with respect to global orientation and spin, so it needs to be anchored in the radio-based International Celestial Reference Frame (ICRF) to provide a referenced and quasi-inertial celestial coordinate system. The link between the two fundamental frames is realized through two samples of distant extragalactic sources, mostly active galactic nuclei and quasars, but only the smaller sample of radio-loud ICRF sources with optical counterparts is available to determine the mutual orientation. The robustness of this link can be mathematically formulated in the framework of functional principal component analysis using a set of vector spherical harmonics to represent the differences in celestial positions of the common objects. The weakest eigenvectors are computed, which describe the greatest deficiency of the link. The deficient or poorly determined terms are specific vector fields on the sphere that carry the largest errors of absolute astrometry using Gaia in reference to the ICRF. This analysis provides guidelines for the future development of the ICRF maximizing the accuracy of the link over the entire celestial sphere. A measure of the robustness of a least-squares solution, which can be applied to any linear model fitting problem, is introduced to help discriminate between reference frame tie models of different degrees.

2D zonal integration with unordered data.

Numerical integration of two-dimensional gradient data is an important step for many slope-measuring optical instruments. However, existing methods are limited by low accuracy or data location restrictions. The zonal integration algorithm in this paper is a generalized process that works with unordered data via Taylor series approximations of finite difference calculations. This method does not require iteration, and all significant steps rely on matrix calculations for a least-squares solution. Simultaneous integration and interpolation is achieved with high accuracy and arbitrary data locations.

Approximated least-squares solutions of a generalized Sylvester-transpose matrix equation via gradient-descent iterative algorithm

This paper proposes an effective gradient-descent iterative algorithm for solving a generalized Sylvester-transpose equation with rectangular matrix coefficients. The algorithm is applicable for the equation and its interesting special cases when the associated matrix has full column-rank. The main idea of the algorithm is to have a minimum error at each iteration. The algorithm produces a sequence of approximated solutions converging to either the unique solution, or the unique least-squares solution when the problem has no solution. The convergence analysis points out that the algorithm converges fast for a small condition number of the associated matrix. Numerical examples demonstrate the efficiency and effectiveness of the algorithm compared to renowned and recent iterative methods.

Canonical Correlation Analysis Based on Site-Response Proxies to Predict Site-Specific Amplification Functions in Switzerland

ABSTRACTThe national seismic networks of Switzerland comprise more than 200 stations. At the station sites, the empirical amplification functions (EAFs) are routinely computed after each earthquake using a generalized inversion method based on separation of source, path, and site effects. The seismic stations are also characterized through geophysical measurements aiming to estimate shear-wave velocity profiles and horizontal-to-vertical spectral ratio of ambient vibrations (HVNR). Using this information, the correlation between the HVNR and EAF is assessed through canonical correlation. Once established, the canonical correlation is used to reconstruct the expected EAFpred at each considered station site in the dataset. The prediction is individually made for all seismic stations in the dataset, excluding every time the investigated station is from the calibration dataset; the reconstruction of the EAFpred is performed resorting to two parallel methods. The first method uses a combination of the canonical correlation parameters and Moran index, and the second one solves in a least-squares sense an overdetermined linear equation system including the canonical couples deemed as reliable. After a first round of predictions, a systematic lower EAFpred in soft sediment sites and a higher EAFpred in hard-rock sites is observed. A possible explanation for this behavior is found in the “normalization” to the Swiss standard rock profile in the computation of the EAF at the Swiss stations. Therefore, to reduce this effect, geological and geophysical parameters are considered in addition to the HVNR in the canonical correlation. We observe that the final solution improves when the least-squares solution approach is used with a combination of HVNR, VS30, and thickness of the ice cover at the last glacial maximum. Moreover, a blind test is performed using data not considered in the calibration dataset. The results highlight the ability of the method to provide an estimate of the site amplification over chosen frequency bins.

New computational methods for inverse wave scattering with a new filtering technique

This note is concerned with inverse wave scattering in one and two dimensional domains. It seeks to recover an unknown function based on measurements collected at the boundary of the domain. For one-dimensional problem, only one point of the domain is assumed to be accessible. For the two dimensional domain, the outer boundary is assumed to be accessible. It develops two iterative algorithms, in which an assumed initial guess for the unknown function is updated. The first method uses a set of sampling functions to formulate a moment problem for the correction to the assumed value. This method is applied to both one-dimensional and two dimensional domains. For two dimensional Helmholtz equation, it relies on a new effective filtering technique which is another contribution of the present work. The second method uses a direct formulation to recover the correction term. This method is only developed for the one-dimensional case. For all cases presented here, the correction to the assumed value is obtained by solving an over-determined linear system through the use of least-square minimization. Tikhonov regularization is also used to stabilize the least-square solution. A number of numerical examples are used to show their applicability and robustness to noise.

Asteroid absolute magnitudes and phase curve parameters from Gaia photometry

Aims. We perform light curve inversion for 491 asteroids to retrieve phase curve parameters, rotation periods, pole longitudes and latitudes, and convex and triaxial ellipsoid shapes by using the sparse photometric observations from Gaia Data Release 2 and the dense ground-based observations from the DAMIT database. We develop a method for the derivation of reference absolute magnitudes and phase curves from the Gaia data, allowing for comparative studies involving hundreds of asteroids. Methods. For both general convex shapes and ellipsoid shapes, we computed least-squares solutions using either the Levenberg-Marquardt optimization algorithm or the Nelder-Mead downhill simplex method. Virtual observations were generated by adding Gaussian random errors to the observations, and, later on, a Markov chain Monte Carlo method was applied to sample the spin, shape, and scattering parameters. Absolute magnitude and phase curve retrieval was developed for the reference geometry of equatorial illumination and observations based on model magnitudes averaged over rotational phase. Results. The derived photometric slope values showed wide variations within each assumed Tholen class. The computed Gaia G-band absolute magnitudes matched notably well with the V-band absolute magnitudes retrieved from the Jet Propulsion Laboratory Small-Body Database. Finally, the reference phase curves were well fitted with the H, G1, G2 phase function. The resulting G1, G2 distribution differed, in an intriguing way, from the G1, G2 distribution that is based on the phase curves corresponding to light curve brightness maxima.

Distributed inverse optimal control

This paper develops a distributed approach for inverse optimal control (IOC) in multi-agent systems. Here each agent can only communicate with certain nearby neighbors and only accesses segments of system’s trajectory, which is not sufficient for the agent to solve the IOC problem alone. By introducing the concept of the data effectiveness and bridging the connection between each segment and its contribution to solving IOC, we formulate the IOC problem as a problem of achieving least-square solutions via a distributed algorithm. Simulations are provided to validate the proposed distributed IOC approach.

Relative Pose Estimation With a Single Affine Correspondence.

In this article, we present four cases of minimal solutions for two-view relative pose estimation by exploiting the affine transformation between feature points, and we demonstrate efficient solvers for these cases. It is shown that under the planar motion assumption or with knowledge of a vertical direction, a single affine correspondence is sufficient to recover the relative camera pose. The four cases considered are two-view planar relative motion for calibrated cameras as a closed-form and least-squares solutions, a closed-form solution for unknown focal length, and the case of a known vertical direction. These algorithms can be used efficiently for outlier detection within a RANSAC loop and for initial motion estimation. All the methods are evaluated on both synthetic data and real-world datasets. The experimental results demonstrate that our methods outperform comparable state-of-the-art methods in accuracy with the benefit of a reduced number of needed RANSAC iterations. The source code is released at https://github.com/jizhaox/relative_pose_from_affine.

Improved Contrast in Images of Exoplanets Using Direct Signal-to-noise Ratio Optimization

Abstract Direct imaging of exoplanets is usually limited by quasi-static speckles. These uncorrected aberrations in a star’s point-spread function (PSF) obscure faint companions and limit the sensitivity of high-contrast imaging instruments. Most current approaches to processing differential imaging sequences like angular differential imaging and spectral differential imaging produce a self-calibrating data set that is combined using a linear least-squares solution to minimize the noise. Due to temporal and chromatic evolution of a telescope’s PSF, the best correlated reference images are usually the most contaminated by the planet, leading to self-subtraction and reducing the planet throughput. In this paper, we present an algorithm that directly optimizes the nonlinear equation for planet signal-to-noise ratio (S/N). This new algorithm does not require us to reject adjacent reference images and optimally balances noise reduction with self-subtraction. We then show how this algorithm can be applied to multiple images simultaneously for a further reduction in correlated noise, directly maximizing the S/N of the final combined image. Finally, we demonstrate the technique on an illustrative sequence of HR8799 using the new Julia-based Signal to Noise Analysis Pipeline. We show that S/N optimization can provide up to a 5× improvement in contrast close to the star. Applicable to both new and archival data, this technique will allow for the detection of fainter, lower mass, and closer-in companions, or achieve the same sensitivity with less telescope time.

Alternative formulae for robust Weighted Total Least-Squares solutions for Errors-In-Variables models

Weighted Total Least-Squares (WTLS) can optimally solve the issue of parameter estimation in the Errors-In-Variables (EIV) model; however, this method is relatively sensitive to outliers that may exist in the observation vector and/or the coefficient matrix. Hence, an attempt to identify/suppress those outliers is in progress and will ultimately lead to a novel robust estimation procedure similar to the one used in the Gauss-Markov model. The method can be considered as a follow-up to the WTLS solution formulated with the standard Least-Squares framework. We utilize the standardized total residuals to construct the equivalent weights, and apply the median method to obtain a robust estimator of the variance to provide good robustness in the observation and structure spaces. Moreover, a preliminary analysis for the robustness of related estimators within the EIV model is conducted, which shows that the redescending M-estimates are more robust than the monotonic ones. Finally, the efficacy of the proposed algorithm is demonstrated through two applications, i.e. 2D affine transformation and linear regression on simulated data and on real data with some assumptions. Unfortunately, the proposed algorithm may not be reliable for detecting multiple outliers. Therefore, MM-estimates within the EIV model need to be investigated in further research.

Least-squares solutions of boundary-value problems in hybrid systems

This paper applies the mathematical framework of the Theory of Functional Connections to solve the boundary-value problems arising from hybrid systems. A hybrid system is simply a sequence of different differential equations. The Theory of Functional Connections is a technique to derive constrained expressions which are functionals with embedded constraints. These functionals allow to transform a large class of constrained optimization problems into unconstrained problems. The initial and most useful application of this technique is in the solution of differential equations where the problem can be posed as an unconstrained optimization problem and solved with simple numerical techniques, i.e., least-squares.The approach developed in this work derives an analytical constrained expression for the entire range of a hybrid system, enforcing both the boundary conditions as well as the continuity conditions across the sequence of differential equations. This reduces the searched solution space of the hybrid system to only admissible solutions. The transformation allows for a least-squares solution of the sequence for linear differential equations and an iterative least-squares solution for nonlinear differential equations.This technique is validated by numerical tests for three differential equation sequences: linear–linear, linear–nonlinear, and nonlinear–nonlinear. The accuracy levels obtained are all at machine-error, which is consistent with the accuracy experienced in past studies on the application of the Theory of Functional Connections to solve single ordinary differential equations. In these tests, TFC produces solutions up to 3 orders of magnitude more accurate than the spectral method and up to 4 orders of magnitude more accurate than the shooting method. Additionally, in all cases except the linear–linear sequence, the TFC method is faster. Finally, the proposed framework is used to solve the one-dimensional convection–diffusion equation to further highlight its utility outside of hybrid systems.

Research on Classification Mining of Tank Driving Simulation Training Data

Training with tank driving simulator is an important method to improve equipment operation skills. In view of the deficiency that it is difficult to find knowledge and rules from complex training data by statistical analysis method in the past driving simulation training, this paper proposes CSAGA-LSSVM algorithm to analyze tank driving simulation training data. Selecting key points to quickly generate shapelets, reducing the number of candidate shapelets; combining shapelets according to distance and time interval to enhance the ability of feature identification; designing adaptive genetic algorithm to dynamically adjust the probability of crossover and mutation to find the optimal parameter solution of least squares support vector machine. The algorithm is applied to the classification mining of shift operation data from a certain tank driving simulator to extract the operation characteristics of personnel.

Multi-Source Image Registration Based on Slic Superpixel Segmentation and Autoencoder

A new Harris-AE feature matching algorithm based on image edge information combined with SLIC superpixel segmentation method is proposed to achieve high-precision registration of visible and infrared images. First, perform edge enhancement and histogram equalization processing on visible light and infrared images, superpixel segmentation technology is used to eliminate sub-images with low information entropy, and then by constructing a multi-scale Gaussian pyramid to detect the Harris corners of the image edge information, using autoencoder neural network to generate feature descriptors corresponding to feature points, and match the feature points of two images through the fast nearest-neighbor algorithm with bidirectional matching strategy. Finally, the RANSAC algorithm is used to purify the matching points, and the spatial geometric change parameters between the two images are estimated by the least-squares solution to complete the registration of visible light and infrared images. Experiments have proved that this algorithm improves the accuracy of matching while shortening the registration time.

Statistical inference on uncertain nonparametric regression model

Nonparametric regression analysis is a useful method to explore the relationships among the variables when a parametric form is not known. Assuming the observations of the model are imprecise and modeling the observed data via uncertain variables, this paper proposes least squares estimation of uncertain nonparametric regression model to explore the functional relationships between response variable and explanatory variable. In particular, we employ B-Splines and local polynomials to approximate the nonparametric function, respectively. Estimation of unknown function can be obtained as a solution of least squares and quadratic programming algorithm can be used to compute efficiently the estimator. Numerical examples are given to illustrate the proposed methods.

Efficient recursive least squares solver for rank-deficient matrices

Updating a linear least-squares solution can be critical for near real-time signal-processing applications. The Greville algorithm proposes a simple formula for updating the pseudoinverse of a matrix A∈Rn×m with rank r. In this paper, we explicitly derive a similar formula by maintaining a general rank factorization, which we call rank-Greville. Based on this formula, we implemented a recursive least-squares algorithm exploiting the rank-deficiency of A, achieving the update of the minimum-norm least-squares solution in O(mr) operations and, therefore, solving the linear least-squares problem from scratch in O(nmr) operations. We empirically confirmed that this algorithm displays a better asymptotic time complexity than LAPACK solvers for rank-deficient matrices. The numerical stability of rank-Greville was found to be comparable to Cholesky-based solvers. Nonetheless, our implementation supports exact numerical representations of rationals, due to its remarkable algebraic simplicity.

On greedy randomized block Kaczmarz method for consistent linear systems

The randomized block Kaczmarz method aims to solve linear system Ax=b by iteratively projecting the current estimate to the solution space of a subset of the constraints. Recent works analyzed the method for the overdetermined least-squares problem, showing expected linear rate of convergence to the ordinary least squares solution with the use of a randomized control scheme to choose the subset at each step. This paper considers the natural follow-up to the randomized control scheme—greedy strategies like the greedy probability criterion and the almost-maximal residual control, and show convergence to a least-squares least-norm solution. Numerical results show that our proposed methods are feasible and have faster convergence rate than the randomized block Kaczmarz method.