Extrapolation Algorithm Research Articles

Phase unwrapping using an extrapolation-projection algorithm

We explore an approach to the unwrapping of two-dimensional phase functions using a robust extrapolation-projection algorithm. Phase unwrapping is essential for imaging systems that construct the image from phase information. Unlike some existing methods where unwrapping is performed locally on a pixel-by-pixel basis, this work approaches the unwrapping problem from a global point of view. The unwrapping is done iteratively by a modification of the Gerchberg-Papoulis extrapolation algorithm, and the solution is refined by projecting onto the available global data at each iteration. Robustness of the algorithm is demonstrated through its performance in a noisy environment, and in comparison with a least-squares algorithm well-known in the literature.

An unstructured-grid based volume-of-fluid method for extreme wave and freely-floating structure interactions

Abstract A Volume of Fluid (VOF) technique has been further developed and coupled with an incompressible Euler/Navier Stokes solver operating on adaptive, unstructured grids to simulate the interactions of extreme waves and a LNG carrier with full or partially filled tanks. The present implementation follows the classic VOF implementation for the liquid-gas system, considering only the liquid phase. Extrapolation algorithms are used to obtain velocities and pressures in the gas region near the free surface. An arbitrary Lagrangian-Eulerian (ALE) frame of reference is used. The mesh is moved in such a way as to minimize the distortion of the mesh due to body movement. The incompressible Euler/Navier-Stokes equations are solved using projection schemes and a finite element method on unstructured grids, and the free surface is captured by the VOF method. The computer code developed based on the method described above is used in this study to simulate a numerical seakeeping tank, where the waves are generated by the sinusoidal excitation of a piston paddle, and a freely-floating LNG carrier with full or partially filled tanks moves in response to the waves. Both head sea and oblique sea are considered in the simulation. Highly nonlinear wave-body interactions, such as green water on deck and sloshing, have been modeled successfully.

Simulation of flows with violent free surface motion and moving objects using unstructured grids

AbstractA volume of fluid (VOF) technique has been developed and coupled with an incompressible Euler/Navier–Stokes solver operating on adaptive, unstructured grids to simulate the interactions of extreme waves and three‐dimensional structures. The present implementation follows the classic VOF implementation for the liquid–gas system, considering only the liquid phase. Extrapolation algorithms are used to obtain velocities and pressure in the gas region near the free surface. The VOF technique is validated against the classic dam‐break problem, as well as series of 2D sloshing experiments and results from SPH calculations. These and a series of other examples demonstrate that the ability of the present approach to simulate violent free surface flows with strong nonlinear behaviour. Copyright © 2006 John Wiley & Sons, Ltd.

The extrapolation of Numerov's scheme for nonlinear two-point boundary value problems

This paper is concerned with the extrapolation algorithm of Numerov's scheme for semilinear and strongly nonlinear two-point boundary value problems. The asymptotic error expansion of the solution of Numerov's scheme is obtained. Based on the asymptotic error expansion, Richardson's extrapolation is constructed, and so the accuracy of the numerical solution is greatly increased. Numerical results are presented to demonstrate the efficiency of the extrapolation algorithm.

UV–Vis spectrophotometric and XAFS studies of ferric chloride complexes in hyper-saline LiCl solutions at 25–90 °C

The speciation and thermodynamic properties of ferric chloride complexes in hydrothermal solutions and hypersaline brines are still poorly understood, despite the importance of this element as a micronutrient and ore-component. Available experimental data are limited to room temperature and relatively low chloride concentrations. This paper reports results of UV–Vis spectrophotometric and synchrotron XAFS experiments of ferric chloride complexes in chloride concentrations up to 15 m and at temperatures of 25–90 °C. Qualitative interpretation of the UV–Vis spectra shows that FeCl 2+, FeCl 2 +, FeCl 3(aq) and FeCl 4 − were present in the experimental solutions. As chloride concentrations increase, higher ligand number complexes become important with FeCl 4 − predominating in solutions containing more than 10 m at 25 °C. The predominance fields of FeCl 3(aq) and FeCl 4 − expand to lower Cl concentrations with increasing T. Both XANES and UV–Vis spectra reveal a major change in the geometry of the complex between FeCl 2 + and FeCl 3(aq). EXAFS data confirm that the number of chloride ligands increases with increasing chloride concentration and show that Fe 3+, FeCl 2+ and FeCl 2 + share an octahedral geometry. FeCl 3(aq) could be either tetrahedral or trigonal dipyramidal, while FeCl 4 − is expected to be tetrahedral. EXAFS data support a tetrahedral geometry for FeCl 4 −, especially at 90 °C, but do not allow to distinguish between a tetrahedral or trigonal dipyramidal geometry for FeCl 3(aq) because of similar Fe–Cl distances. At room temperature, EXAFS data suggest that FeCl 3(aq) may be a mixture of octahedral and tetrahedral or trigonal dipyramidal forms. The room temperature formation constants for three ferric chloride complexes (FeCl 2 +, FeCl 3(aq) and FeCl 4 −) determined from the UV data are generally in good agreement with previous studies. Calculations based on the properties extrapolated to 300 °C show that hematite solubility is much higher than previously estimated, and that the high orders complexes FeCl 3(aq) and FeCl 4 − are important at high temperatures even in solutions with low chloride concentrations. The accuracy of these properties is limited by a poor understanding of activity–composition relationships in concentrated electrolytes, and by limitations in the available experimental techniques and extrapolation algorithms; however, the inclusion of higher order complexes in numerical models of ore transport and deposition allows for a more accurate qualitative prediction of Fe behaviour in hydrothermal and hypersaline systems.

On the simulation of flows with violent free surface motion

A volume of fluid (VOF) technique has been developed and coupled with an incompressible Euler/Navier–Stokes solver operating on adaptive, unstructured grids to simulate the interactions of extreme waves and three-dimensional structures. The present implementation follows the classic VOF implementation for the liquid–gas system, considering only the liquid phase. Extrapolation algorithms to obtain velocities and pressure in the gas region near the free surface have been implemented. The VOF technique is validated against the classic dam-break problem, as well as series of 2D sloshing experiments and results from smoothed particle hydrodynamics (SPH) calculations. These and a series of other examples demonstrate that the present CFD method is capable of simulating violent free surface flows with strong nonlinear behavior.

A General Setting for the Parametric Google Matrix

The spectral and Jordan structures of the web hyperlink matrix _G_(_c_) = _cG_ + (1 - _c_)_ev^T_ have been analyzed when _G_ is the basic (stochastic) Google matrix, _c_ is a real parameter such that 0 < _c_ < 1, _v_ is a nonnegative probability vector, and _e_ is the all-ones vector. Typical studies have relied heavily on special properties of nonnegative, positive, and stochastic matrices. There is a unique nonnegative vector _y_(_c_) such that _y_(_c_)_^TG_(_c_) = _y_(_c_)_^T_ and _y_(_c_)_^T__e_ = 1. This _PageRank_ vector _y_(_c_)can be computed effectively by the power method. We consider a square complex matrix _A_ and nonzero complex vectors _x_ and _v_ such that _Ax_ = λ_x_ and _v\*x_ = 1. We use standard matrix analytic tools to determine the eigenvalues, the Jordan blocks, and a distinguished left λ-eigenvector of _A_(_c_) = _cA_ + (1 - _c_)λ_xv_\* as a function of a complex variable _c_. If λ is a semisimple eigenvalue of _A_, there is a uniquely determined projection _P_ such that lim_{<em>c</em>→1}_y_(_c_) = _Pv_ for every _v_ such that _v\*x_ = 1; this limit may fail to exist for some _v_ if λ is not semisimple. As a special case of our results, we obtain a complex analog of PageRank for the web hyperlink matrix _G_(_c_) with a complex parameter _c_. We study regularity, limits, expansions, and conditioning of _y_(_c_), and we propose a complex extrapolation algorithm that may provide an efficient way to compute PageRank.

Extrapolation algorithm for affine-convex feasibility problems

The convex feasibility problem under consideration is to find a common point of a countable family of closed affine subspaces and convex sets in a Hilbert space. To solve such problems, we propose a general parallel block-iterative algorithmic framework in which the affine subspaces are exploited to introduce extrapolated over-relaxations. This framework encompasses a wide range of projection, subgradient projection, proximal, and fixed point methods encountered in various branches of applied mathematics. The asymptotic behavior of the method is investigated and numerical experiments are provided to illustrate the benefits of the extrapolations.

A novel technique for the approximation of 3-D antenna radiation patterns

In this paper, a novel technique for the approximation of three-dimensional (3-D) antenna radiation patterns is presented. The proposed method combines the two principal cuts in order to acquire an adequate estimate of the 3-D antenna radiation solid. The absolute error from the theoretical solution is analyzed along with other statistical measures for various types of antenna structures, namely omni-directional and directive arrays. The performance of the method is compared against other existing extrapolation algorithms. The proposed technique exhibits low approximation errors and is easily integrated into 3-D radio propagation planning tools (such as ray-tracing algorithms).

A new data extrapolation algorithm for high resolution ISAR imaging

A new data extrapolation algorithm for high resolution radar imaging is presented. The backscattered data are modeled as an autoregressive process where the prediction coefficients are computed using 1D least-square lattice filters. Unlike the well-known Burg or modified covariance methods, least square lattice modeling yields different prediction coefficients for forward and backward directions. The proposed method does not need to satisfy Levinson recursion, i.e. does not suffer from the limitations of the Burg method such as spectral splitting or bias in the locations of the scattering centers. Moreover, due to its lattice structure it does not need any matrix inversion like the modified covariance method. Results obtained for an experimental target are included to confirm the proposed algorithm.

Cardinal exponential splines: part I - theory and filtering algorithms

Causal exponentials play a fundamental role in classical system theory. Starting from those elementary building blocks, we propose a complete and self-contained signal processing formulation of exponential splines defined on a uniform grid. We specify the corresponding B-spline basis functions and investigate their reproduction properties (Green function and exponential polynomials); we also characterize their stability (Riesz bounds). We show that the exponential B-spline framework allows an exact implementation of continuous-time signal processing operators including convolution, differential operators, and modulation, by simple processing in the discrete B-spline domain. We derive efficient filtering algorithms for multiresolution signal extrapolation and approximation, extending earlier results for polynomial splines. Finally, we present a new asymptotic error formula that predicts the magnitude and the Nth-order decay of the L/sub 2/-approximation error as a function of the knot spacing T.

Algorithm of Extrapolation of a Nonlinear Random Process on the Basis of Its Canonical Decomposition

Based on the apparatus of canonical decompositions, an algorithm of extrapolation of a nonlinear random process is obtained for an arbitrary number of known values and probabilistic relations used for a prediction.

The human brain's algorithm for extrapolating motion, and its possible gender-dependence

Some athletes accomplish feats requiring a timing accuracy 100 times better than their reaction time. This capability implies that the brain can accurately extrapolate the motion of objects. We hypothesize that the brain uses a simple algorithm to accomplish this extrapolation, and that the algorithm is influenced by the subject's gender and recent observations of motion. We describe an experiment in timing the motion of a dot across a computer screen designed to discover the motion extrapolation algorithm. Different types of motion of the dot were studied. The experiment was conducted with 126 college students (two-thirds female), who each performed 1000 trials. By using as many as 126,000 trials, the random noise inherent in individual trials averages out, allowing the underlying algorithm to be revealed. The results show that motion extrapolation is done using the average velocity of a moving object with no correction for changes in velocity during the motion, but with an ad hoc adjustment based on recently observed motions and on gender—males having on average a smaller error than females. A future controlled experiment will be needed to establish whether the observed gender difference is due to the greater experience of males with such related tasks as video games and sports.

Accelerated Iterative Band-Limited Extrapolation Algorithms

The iterative algorithms of Papoulis-Gerchberg and Cadzow are well-known algorithms for solving the band-limited extrapolation problem. These algorithms, however, are usually slowly convergent. We propose an acceleration scheme for both algorithms, based on finding best approximation in appropriate subspaces. The acceleration scheme can also be applied to other cases of signal restoration, which admit Youla's method of alternating orthogonal projections.

Detection of pseudoperiodic patterns using partial acquisition of magnetic resonance images

Improving the resolution of magnetic resonance imaging (MRI), or, alternatively, reducing the acquisition time, can be quite beneficial for many applications. The main motivation of this work is the assumption that any information that is a priori available on the target image could be used to achieve this goal. In order to demonstrate this approach, we present a novel partial acquisition strategy and reconstruction algorithm, suitable for the special case of detection of pseudoperiodic patterns. Pseudoperiodic patterns are frequently encountered in the cerebral cortex due to its columnar functional organization (best exemplified by orientation columns and ocular dominance columns of the visual cortex). We present a new MRI research methodology, in which we seek an activity pattern, and a pattern-specific experiment is devised to detect it. Such specialized experiments extend the limits of conventional MRI experiments by substantially reducing the scan time. Using the fact that pseudoperiodic patterns are localized in the Fourier domain, we present an optimality criterion for partial acquisition of the MR signal and a strategy for obtaining the optimal discrete Fourier transform (DFT) coefficients. A by-product of this strategy is an optimal linear extrapolation estimate. We also present a nonlinear spectral extrapolation algorithm, based on projections onto convex sets (POCSs), used to perform the actual reconstruction. The proposed strategy was tested and analyzed on simulated signals and in MRI phantom experiments.

Thermal emissivity of coated glazing—simulation versus measurements

Abstract A large variety of coated glazing products are available on the market today. These are used in energy efficient low emissivity (low-e) or solar control windows. Not only the solar optical properties, but also the thermal emissivity of these coated glazing materials are of importance for the performance of such energy efficient windows. The thermal emissivity is calculated from the IR reflectance. A problem is that for accurate determination of the emissivity according to international standards, the reflectance needs to be known between 2000 and 200 cm −1 , and many FTIR spectrophotometers cannot measure below 400 cm −1 . In this paper some different strategies for the extrapolation to 200 cm −1 are discussed. A sensitivity analysis for different types of materials is presented for a few different extrapolation algorithms. The simplest extrapolation procedure assumes a constant reflectance value throughout the extrapolation interval. This appears to work well for surfaces with high reflectance values. A procedure based on a linear relation between the values at a starting wavelength and at the end point of the extrapolation interval or one using a simple second-degree polynomial function can be used when coatings on glass having medium or low reflectance values are evaluated. A guide on how to extrapolate the spectra, according to the different strategies, is included in the Appendix.

Extrapolation Algorithms for Filtering Series of Functions, and Treating the Gibbs Phenomenon

In this paper, we study the application of some convergence acceleration methods to Fourier series, to orthogonal series, and, more generally, to series of functions. Sometimes, the convergence of these series is slow and, moreover, they exhibit a Gibbs phenomenon, in particular when the solution or its first derivative has discontinuities. It is possible to circumvent, at least partially, these drawbacks by applying a convergence acceleration method (in particular, the e-algorithm) or by approximating the series by a rational function (in particular, a Pade approximant). These issues are discussed and some numerical results are presented. We will see that adding its conjugate series as an imaginary part to a Fourier series greatly improves the efficiency of the algorithms for accelerating the convergence of the series and reducing the Gibbs phenomenon. Conjugacy for series of functions will also be considered.

EVALUATION OF THE FOLD LINE BY ASYMPTOTIC EXTRAPOLATION FOR STRUCTURAL ANALYSIS

A technique for imperfection sensitivity analysis with reference to the geometrically nonlinear analysis of structures is presented. The paper discusses how detailed information on structural behavior can be obtained with less computational cost. The imperfection sensitivity analysis of structures is carried out by detecting the critical states on the equilibrium path relating to the various imperfections. In fact we can investigate the behavior of imperfect structures without considering the imperfect equilibrium curve. In this work we obtained the fold line, the one-dimensional equilibrium subset of limit points relating to different values of imperfection, by asymptotic extrapolation from a known singular point. The evaluation of this singular point on the perfect equilibrium curve is carried out by a path-following algorithm. A procedure that overcomes the ill-conditioning of the systems defined in the critical points is described. This proves to be highly advantageous in terms of computational cost in comparison with classical methods of analysis. The paper investigates the behavior of cylindrical shell. In particular the sensitivity analysis for load imperfections is carried out. The asymptotic extrapolation algorithm is also compared with continuation methods for the analysis of imperfect cylindrical shell.

Elasto-plastic Stress Analysis with Nested Finite Element Method

Nested-finite-element methodology (NFEM) is an improved finite-element method to evaluate local concentrations of stress with nested sub-elements. NFEM was applied only to elastic and explicit inelastic problems, because it is unable to calculate residual forces accurately from stresses at integral points. I thus developed an algorithm that would allow implicit analysis of the elastic-plastic problem on the basis of NFEM. The algorithm has the following steps : (1) Calculate the displacements of nodes with elastic NFEM. (2) Calculate stresses at each integral point. (3) Modify stresses at “yielded” integral points in main-elements and sub-elements. (4) Extrapolate stresses from integral points in sub-elements to integral points in main elements, and (5) Calculate residual forces at nodes from stresses. All steps are repeated until the residual forces converge to an allowable limit. The stress extrapolation scheme is the most important part of the method. I developed the extrapolation algorithm based on the shape functions of an iso-parametric element. To conclude, elastic-plastic NFEM can be used to implicitly analyze the stress of inelastic materials.

Local Polynomial Regression and Simulation–Extrapolation

SummaryThe paper introduces a new local polynomial estimator and develops supporting asymptotic theory for nonparametric regression in the presence of covariate measurement error. We address the measurement error with Cook and Stefanski's simulation–extrapolation (SIMEX) algorithm. Our method improves on previous local polynomial estimators for this problem by using a bandwidth selection procedure that addresses SIMEX's particular estimation method and considers higher degree local polynomial estimators. We illustrate the accuracy of our asymptotic expressions with a Monte Carlo study, compare our method with other estimators with a second set of Monte Carlo simulations and apply our method to a data set from nutritional epidemiology. SIMEX was originally developed for parametric models. Although SIMEX is, in principle, applicable to nonparametric models, a serious problem arises with SIMEX in nonparametric situations. The problem is that smoothing parameter selectors that are developed for data without measurement error are no longer appropriate and can result in considerable undersmoothing. We believe that this is the first paper to address this difficulty.