Iterative Least-squares Method Research Articles

A practical iterative two-view metric reconstruction with uncalibrated cameras

This paper presents a practical iterative algorithm for two-view metric reconstruction without any prior knowledge about the scene and motion in a nonsingular geometry configuration. The principal point is assumed to locate at the image center with zero skew and the same aspect ratio, and the interior parameters are fixed, so the self-calibration becomes focal-length calibration. Existing focal length calibration methods are direct solutions of a quadric composed of fundamental matrix, which are sensitive to noise. A quaternion-based linear iterative Least-Square Method is proposed in this paper, and one-dimensional searching for optimal focal length in a constrained region instead of solving optimization problems with inequality constraints is applied to simplify the computation complexity, then unique rotational matrix and translate vector are recovered. Experiments with simulation data and real images are given to verify the algorithm.

Efficient GOCE satellite gravity field recovery based on least-squares using QR decomposition

We develop and apply an efficient strategy for Earth gravity field recovery from satellite gravity gradiometry data. Our approach is based upon the Paige-Saunders iterative least-squares method using QR decomposition (LSQR). We modify the original algorithm for space-geodetic applications: firstly, we investigate how convergence can be accelerated by means of both subspace and block-diagonal preconditioning. The efficiency of the latter dominates if the design matrix exhibits block-dominant structure. Secondly, we address Tikhonov-Phillips regularization in general. Thirdly, we demonstrate an effective implementation of the algorithm in a high-performance computing environment. In this context, an important issue is to avoid the twofold computation of the design matrix in each iteration. The computational platform is a 64-processor shared-memory supercomputer. The runtime results prove the successful parallelization of the LSQR solver. The numerical examples are chosen in view of the forthcoming satellite mission GOCE (Gravity field and steady-state Ocean Circulation Explorer). The closed-loop scenario covers 1 month of simulated data with 5 s sampling. We focus exclusively on the analysis of radial components of satellite accelerations and gravity gradients. Our extensions to the basic algorithm enable the method to be competitive with well-established inversion strategies in satellite geodesy, such as conjugate gradient methods or the brute-force approach. In its current development stage, the LSQR method appears ready to deal with real-data applications.

Identification of continuous-time systems with multiple unknown time delays by global nonlinear least-squares and instrumental variable methods

This paper considers the identification problem of multiple input single output (MISO) continuous-time systems with unknown time delays of the inputs, from sampled input–output data. An iterative global separable nonlinear least-squares (GSEPNLS) method which estimates the time delays and transfer function parameters separably is derived to significantly reduce the possibility of convergence to a local minimum, by using the stochastic global-optimization techniques. Furthermore, the GSEPNLS method is modified to a novel global separable nonlinear instrumental variable (GSEPNIV) method to remove the biases of the estimates in the presence of high measurement noise. Simulation results show that the proposed algorithms work quite well.

Optimum Adjustment for Distortion in Semiconductor Lithography Equipment

In this paper, we consider the problem of lens distortion adjustment of semiconductor lithography equipment. The objective of adjustment is to minimize the maximum absolute value of distortion. Formerly, an approximate solution method based on the least-squares method has been used. Recently, an approximate solution method based on iterative least-squares method with weight was proposed. However, calculation of that method often takes long time and a better calculation method has been desired. In this paper, we propose an exact solution method based on LP (Linear Programming) to minimize the maximum absolute value of distortion. Now, LP solvers find an optimal solution very fast owing to the progress in linear programming research. Consequently, optimal solutions for 20 instances obtained by our method provided adjustment parameter settings about 29% from 27% better than that by the solution method based on the least-squares method.

Accurate Estimation of Phase Distribution on Reference Plane in Grating Projection Measurement

A binary second order rational polynomial is adopted to simulate and extend the phase distribution on reference plane. In order to get the coefficients of the polynomial accurately, iterative least-square method based on the first order Taylor series expansion is used. The effect of the real reference plane profile error on measuring result is reduced by using extended unwrapped phase to substitute the original unwrapped phase. The effects of the random phase error and the system geometrical parameter error are decreased. The measuring accuracy of the system is improved. The principle of the 3D profile measuring system based on grating projection, the theoretic analysis for accurate estimate of phase distribution on reference plane, and the experimental results are presented.

A GLOBAL NONLINEAR INSTRUMENTAL VARIABLE METHOD FOR IDENTIFICATION OF CONTINUOUS-TIME SYSTEMS WITH UNKNOWN TIME DELAYS

This paper considers the identification problem of continuous-time systems with unknown time delays from sampled input-output data. An iterative global separable nonlinear least-squares (GSEPNLS) method which estimates the time delays and transfer function parameters separably is derived, by using stochastic global-optimization technique to avoid convergence to a local minimum. Futhermore, the GSEPNLS method is modified to a novel global separable nonlinear instrumental variable (GSEPNIV) method to yield consistent estimates if the algorithm converges to the global minimum. Simulation results show that the proposed method works quite well.

Accurate orbital parameters for the bright southern spectroscopic binary ζ Trianguli Australis – an interesting case of a near-circular orbit

Accurate orbital parameters have been obtained for the bright southern single-lined spectroscopic binary star ζ TrA, with the 1-m telescope and Hercules spectrograph at Mt John University Observatory. More than 200 observations have been collected over five months of observation between 2003 March and August. They cover about 12 complete periods (P 13 d). High-precision relative radial velocities have been measured by the cross-correlation technique with HRSP, a dedicated reduction software package developed in the Department of Physics and Astronomy, University of Canterbury. An iterative linear least-squares method has been used to improve the initial set of orbital parameters estimated from the geometrical properties of the radial-velocity curve. The final rms error of the least-squares fit is only about 14 m s -1 so the error bars on the orbital parameters are extremely small. As a particularly interesting result, a very low eccentricity of e = 0.014 42 ± 0.000 21 has been derived, which proves that the orbit is definitely elliptical and not circular as has been suggested in the literature. The orbital period has also been improved by including some historical observations so that the best value is P = 12.975 780 ± 0.000 046 d. The mass of the companion is in the range 0.09-0.45 M ○. , which covers the spectral types between M1 V and M7 V, on the assumption that the companion is a main-sequence star. In that case the orbital inclination has to be larger than about 14°.

Identification of continuous-time systems with unknown time delays by global nonlinear least-squares method

This paper considers the identification problem of continuous-time systems with unknown time delays from sampled input-output data. By using a digital prefilter, an approximated discrete-time estimation model is first derived, in which the system parameters remain in their original form and the time delays need not be an integral multiple of sampling period. Then an iterative separable nonlinear least-squares (SEPNLS) method which estimates the time delays and transfer function parameters separably is derived. Futhermore, we propose an iterative global SEPNLS method to avoid convergence to a local minimum of the SEPNLS method by using of stochastic global-optimization techniques. Simulational results show that the global SEPNLS method is able to converge to global estimate.

Iterative least squares method for regression analysis with censored data

The iterative least squares method (ILSM) is studied for regression analysis with censored data. Simulated data were used to test the performance of this approach, which was compared with the classical least squares (LS) method applied to data in which the censored observations are ignored. A simple algorithm can make the statistical analysis involving censored data more likely to be used by analytical chemists and pharmacists, as is shown by applying the ILSM to real data sets from chromatography and pharmaceutical technology.

Multicoil Dixon chemical species separation with an iterative least-squares estimation method.

This work describes a new approach to multipoint Dixon fat-water separation that is amenable to pulse sequences that require short echo time (TE) increments, such as steady-state free precession (SSFP) and fast spin-echo (FSE) imaging. Using an iterative linear least-squares method that decomposes water and fat images from source images acquired at short TE increments, images with a high signal-to-noise ratio (SNR) and uniform separation of water and fat are obtained. This algorithm extends to multicoil reconstruction with minimal additional complexity. Examples of single- and multicoil fat-water decompositions are shown from source images acquired at both 1.5T and 3.0T. Examples in the knee, ankle, pelvis, abdomen, and heart are shown, using FSE, SSFP, and spoiled gradient-echo (SPGR) pulse sequences. The algorithm was applied to systems with multiple chemical species, and an example of water-fat-silicone separation is shown. An analysis of the noise performance of this method is described, and methods to improve noise performance through multicoil acquisition and field map smoothing are discussed.

Identification of Continuous-Time Systems with Unknown Time Delay by Nonlinear Least-Squares and Instrumental Variable Methods

This paper considers the identification problem of continuous-time systems with unknown time delay from sampled input-output data. By using a digital prefilter, an approximated discrete-time estimation model is first derived, in which the system parameters remain in their original form and the time delay need not be an integral multiple of the sampling period. Then an iterative separable nonlinear least-squares (SEPNLS) method which estimates the time delay and transfer function parameters separably is derived. Furthermore, through investigation of the properties of the SEPNLS estimate, a novel separable nonlinear instrumental variable (SEPNIV) method is also proposed. Simulational results show that the SEPNIV method yields consistent estimates.

Multivariate curve resolution in the analysis of vibrational spectroscopy data files.

T he ability of chemometric techniques to reduce and resolve large data matrices into responses and corresponding weights of these responses has been shown with a variety of data sets, and there exist a variety of approaches capable of assessing such bilinear data sets measured with vibrational spectroscopy.1–4 We have focused on using a form of principal factor analysis, which uses matrix decomposition as the initial step to generate abstract factors and scores. This initial estimate is then reŽ ned in a second step to produce spectra and intensity matrices. The second step uses an iterative constrained least-squares method referred to as alternating least squares. The technique, which is particularly useful in extracting estimates of pure analyte spectra and concentration proŽ les from spectral data matrices, is generally referred to

A note on an iterative least-squares estimation method for ARMA and VARMA models

In this note we suggest a new iterative least-squares method for estimating scalar and vector ARMA models. A Monte Carlo study shows that the method has better small sample properties than existing least-squares methods and compares favourably with maximum likelihood estimation as well.

Magnetic equivalence of terminal nuclei in the azide anion broken by nuclear spin relaxation

NMR spectra of water solution of sodium azide selectively 15N labelled in the central position were studied using an iterative least-squares method. In agreement with predictions based on Bloch-Wangsness-Redfield nuclear spin relaxation theory, it is demonstrated that quadrupolar relaxation of the magnetically equivalent terminal 14N (spin-1) nuclei in the azide anion renders the J coupling between these nuclei an observable quantity. In isotropic fluids, this seems to be the first experimental evidence of relaxation-broken magnetic equivalence symmetry.

Fast Panoramic Image Mosaicing Using One-dimensional Flow Estimation

In this paper, we propose a fast method for panoramic image mosaicing using one-dimensional (1D) flow estimation. First, the 1D flow vectors are estimated at each pixel in x and y directions. Then the global affine parameters are estimated from 1D flows using the iterative least-squares method. The accurate parameters are obtained by iteratively repeated optimization in x and y directions. We have implemented the proposed method on a conventional low-cost PC. The experimental results show that the image mosaicing is possible in near-real time.

Parameter estimation for non-linear environmental models using below-detection data

Problems requiring regression analysis of below-detection or censored data frequently arise in the field of environmental engineering. Detection limits of analyzing instruments are the main reason for censored observations of pollutant concentrations. An iterative least-squares method for regression analysis is developed to suit the doubly censored data for non-linear models commonly used in environmental engineering. The algorithm developed, non-linear iterative least squares (NILS), utilizes the values expected for censored observations estimated from probability density functions of doubly censored data in the regression process. It has been shown through extensive simulation that the NILS method proposed returns estimates of model parameters close to those estimated with complete information, even for probability of censoring (POC) equal to 0.7 (i.e. 70% of the data are censored). Simulation results of 500 runs (sample size 20, 40 and 80 data points with POC 0.2, 0.4 and 0.7, respectively) from the algorithm developed are also compared to results obtained using some obvious simplifications on censored data, such as: (i) ignoring censored data; (ii) taking censored data as the detection limit; and (iii) taking censored data as equal to half of the difference between the lower limit (e.g. background level) and the detection limit. These simplifications have no scientific basis. It is shown that these simplifications on censored data cause substantial bias in the parameters estimated, and thus model predictions become uncertain. In addition to simulation analysis, the methodology is also established for actual observations for environmental models taken from the literature.

Application of a novel resolution approach combining soft- and hard-modelling features to investigate temperature-dependent kinetic processes

A new resolution method based on the combination of hard- and soft-modelling is applied to the analytical study of spectrometrically monitored kinetic processes. This method results from the modification of the iterative soft-modelling multivariate curve resolution-alternating least-squares method (MCR-ALS) by the inclusion of a new hard constraint that forces some or all the concentration profiles to fulfil a kinetic model. In this way, strengths of both soft- and hard-modelling data analysis methods are combined and limitations linked to the application of only one of these pure approaches are overcome. The new method is applied to a real example of cyclometallation reaction. This is a challenging case where the extent of the process changes with temperature and the spectra of the different species formed do not differ extremely from each other. Besides, due to the lability of the initial reagent, the nature and the concentrations of all the reacting components present at the beginning of the process are not always known. The example presented is representative of many other complex kinetic problems, which could not be solved by the use of traditional methods based exclusively on the straightforward fitting of kinetic models.

Full 3-D inversion of electromagnetic data on PC

Three-dimensional (3-D) electromagnetic (EM) inversion might be believed to require high-performance computers. However, with the rapid progress of recent computer technology, running 3-D inversions on personal computers (PC) is becoming a rational choice. This paper describes an attempt to carry out full 3-D inversions of synthetic frequency-domain EM data on a PC. In the inversion, a staggered-grid finite difference scheme is used to solve for the secondary electric field. The system of equations is solved using the incomplete Cholesky biconjugate gradient (ICBCG) method. By applying the static divergence correction proposed by Smith [Geophysics 61 (1996) 1319–1324], the rates of convergence are dramatically improved, and the forward calculation per source location on a medium-size grid takes only a few minutes on a PC. The sensitivities of the EM responses to subsurface resistivity changes are calculated from forward solutions using the reciprocity relation. An inverse problem is formulated so that a model is found that has a smooth structure and at the same time is close to an initial model, and is solved with an iterative least-squares method. A synthetic example for an airborne EM survey shows that the lateral extents of 3-D bodies are well resolved, and the vertical coil coaxial system gives a better lateral resolution than the horizontal coil coplanar system. An example for a ground EM survey shows that expanding the survey coverage or aperture is needed to improve the resolution at depth.

Analysis of extreme hydrologic events with Gumbel distributions: marginal and additive cases

The importance of the Gumbel probability distribution for the description of extreme hydrologic events is examined in this article. The key findings of this work are: (1) an iterative method of least squares was developed and found to be well-suited for the efficient fitting of the two-parameter Gumbel distribution to hydrologic extremes; (2) negative truncation is necessary to adequately describe hydrologic minima (non-negative) data, while the standard Gumbel distribution for maxima is well-suited for modeling extreme (large) hydrologic events; (3) the distribution function of the sum of two independent Gumbel variables, of importance in hydrology, has been derived and successfully applied to spring flow data. Several examples that involve the modeling of hydrologic extremes are presented and analyzed.

Optical tomography of the aurora and EISCAT

Abstract. Tomographic reconstruction of the three-dimensional auroral arc emission is used to obtain vertical and horizontal distributions of the optical auroral emission. Under the given experimental conditions with a very limited angular range and a small number of observers, algebraic reconstruction methods generally yield better results than transform techniques. Different algebraic reconstruction methods are tested with an auroral arc model and the best results are obtained with an iterative least-square method adapted from emission-computed tomography. The observation geometry used during a campaign in Norway in 1995 is tested with the arc model and root-mean-square errors, to be expected under the given geometrical conditions, are calculated. Although optimum geometry was not used, root-mean-square errors of less than 2% for the images and of the order of 30% for the distribution could be obtained. The method is applied to images from real observations. The correspondence of original pictures and projections of the reconstructed volume is discussed, and emission profiles along magnetic field lines through the three-dimensionally reconstructed arc are calibrated into electron density profiles with additional EISCAT measurements. Including a background profile and the temporal changes of the electron density due to recombination, good agreement can be obtained between measured profiles and the time-sequence of calculated profiles. These profiles are used to estimate the conductivity distribution in the vicinity of the EISCAT site. While the radar can only probe the ionosphere along the radar beam, the three-dimensional tomography enables conductivity estimates in a large area around the radar site.Key words. Tomography · Aurora · EISCAT · Ionosphere · Conductivity