First-order Taylor Expansion Research Articles

Task-based detectability in CT image reconstruction by filtered backprojection and penalized likelihood estimation.

Nonstationarity is an important aspect of imaging performance in CT and cone-beam CT (CBCT), especially for systems employing iterative reconstruction. This work presents a theoretical framework for both filtered-backprojection (FBP) and penalized-likelihood (PL) reconstruction that includes explicit descriptions of nonstationary noise, spatial resolution, and task-based detectability index. Potential utility of the model was demonstrated in the optimal selection of regularization parameters in PL reconstruction. Analytical models for local modulation transfer function (MTF) and noise-power spectrum (NPS) were investigated for both FBP and PL reconstruction, including explicit dependence on the object and spatial location. For FBP, a cascaded systems analysis framework was adapted to account for nonstationarity by separately calculating fluence and system gains for each ray passing through any given voxel. For PL, the point-spread function and covariance were derived using the implicit function theorem and first-order Taylor expansion according to Fessler ["Mean and variance of implicitly defined biased estimators (such as penalized maximum likelihood): Applications to tomography," IEEE Trans. Image Process. 5(3), 493-506 (1996)]. Detectability index was calculated for a variety of simple tasks. The model for PL was used in selecting the regularization strength parameter to optimize task-based performance, with both a constant and a spatially varying regularization map. Theoretical models of FBP and PL were validated in 2D simulated fan-beam data and found to yield accurate predictions of local MTF and NPS as a function of the object and the spatial location. The NPS for both FBP and PL exhibit similar anisotropic nature depending on the pathlength (and therefore, the object and spatial location within the object) traversed by each ray, with the PL NPS experiencing greater smoothing along directions with higher noise. The MTF of FBP is isotropic and independent of location to a first order approximation, whereas the MTF of PL is anisotropic in a manner complementary to the NPS. Task-based detectability demonstrates dependence on the task, object, spatial location, and smoothing parameters. A spatially varying regularization "map" designed from locally optimal regularization can improve overall detectability beyond that achievable with the commonly used constant regularization parameter. Analytical models for task-based FBP and PL reconstruction are predictive of nonstationary noise and resolution characteristics, providing a valuable framework for understanding and optimizing system performance in CT and CBCT.

Fluid Approximation of Point-queue Model

Point-queue model is widely used in the dynamic user equilibrium (DUE) analysis in discrete-time or continuous-time form. In this paper, a continuous time point queue is proposed. In the former studies, the negativity of the queue length of the original point-queue model is shown and some improvement has been made. Based on the observation that the original point-queue model is actually a queuing model with a server and a buffer with infinite capacity, a fluid approximation (FA) model is proposed to interpret the original point-queue model. Three essential components are a flow balance function, an exit flow function and a time-dependent capacity utilization ratio function, which are all in continuous form. During the analysis, the theoretical proof and numerical study of the non-negativity of queue length are accomplished. With the first-order Taylor expansion, this paper applies the Gronwall's inequality to prove the non-negativity of queue-length. Through numerical testing different specific FA models in the solution scheme, we can show that the negativity of the queue length in the FA model is overcome and some differences between the FA and former studies are discussed. Based on the testing, the capability of our model in approximating the point- queue model is demonstrated.

An interval perturbation method for exterior acoustic field prediction with uncertain-but-bounded parameters

This paper proposes two interval analysis methods, called the first-order interval parameter perturbation method (FIPPM) and the modified interval parameter perturbation method (MIPPM), for use in exterior acoustic field prediction when there are uncertainties in both the material properties and the external load. Interval variables are used to quantitatively describe the uncertain parameters in the face of limited information. The conventional first-order Taylor expansion and perturbation terms are employed in the FIPPM, while the MIPPM introduces modified Taylor series to approximate the non-linear interval matrix and vector. The high-order terms of the Neumann expansion are retained to calculate the interval matrix inverse. A numerical example is given by comparing the results with a Monte Carlo simulation to demonstrate the feasibility and effectiveness of the proposed methods at evaluating the sound pressure ranges in an exterior acoustic field.

A proportional differential control method for a time-delay system using the Taylor expansion approximation

In this paper, a proportional differential control algorithm is studied. According to the system performance indices, we uses the first-order Taylor expansion to approximate the time-delay, present an algorithm to determine the controller parameters so that the desired dynamic performance can be achieved. The numerical examples are provided to show the advantages of the proposed algorithm for small time-delay.

An improving fringe analysis method based on the accuracy of S-transform profilometry

The S transform, as a simple and popular technique for space\ ime–frequency analysis, has been introduced in optical three-dimensional surface shape measurement in recent years. Based on the S transform, S transform “ridge” method (STR) and S transform filtering method (STF) have been proposed to extract the phase information from the single deformed fringe pattern. This paper focuses on studying the STR in fringe pattern analysis. In previous researches about the STR, a linear constraint, which assumed that the phase was locally expressed as first-order Taylor expansion with respect to x and y directions, was implicitly added. Actually at least the second-order partial derivatives of the phase in each local area should be taken into account because they are related to the local curvature of the height distribution of the tested object. Therefore, the traditional STR has larger phase measuring errors in those areas with the rapid height variation on the tested object. This paper proposes an improved STR method, in which the phase is approximately expressed as a quadric in each local area. The phase extraction formula based on the quadric is derived, and the phase correction is carried out as well. Both the simulations and the experiments verify that a more accurate phase map can be obtained by the improved method compared with that by the traditional STR, especially in the areas where height variation is steep.

Sensitivity Analysis of Deviation Source for Fast Assembly Precision Optimization

Assembly precision optimization of complex product has a huge benefit in improving the quality of our products. Due to the impact of a variety of deviation source coupling phenomena, the goal of assembly precision optimization is difficult to be confirmed accurately. In order to achieve optimization of assembly precision accurately and rapidly, sensitivity analysis of deviation source is proposed. First, deviation source sensitivity is defined as the ratio of assembly dimension variation and deviation source dimension variation. Second, according to assembly constraint relations, assembly sequences and locating, deviation transmission paths are established by locating the joints between the adjacent parts, and establishing each part’s datum reference frame. Third, assembly multidimensional vector loops are created using deviation transmission paths, and the corresponding scalar equations of each dimension are established. Then, assembly deviation source sensitivity is calculated by using a first-order Taylor expansion and matrix transformation method. Finally, taking assembly precision optimization of wing flap rocker as an example, the effectiveness and efficiency of the deviation source sensitivity analysis method are verified.

Analysis of the impacts of horizontal translation and scaling on wavefront approximation coefficients with rectangular pupils for Chebyshev and Legendre polynomials

Chebyshev and Legendre polynomials are frequently used in rectangular pupils for wavefront approximation. Ideally, the dataset completely fits with the polynomial basis, which provides the full-pupil approximation coefficients and the corresponding geometric aberrations. However, if there are horizontal translation and scaling, the terms in the original polynomials will become the linear combinations of the coefficients of the other terms. This paper introduces analytical expressions for two typical situations after translation and scaling. With a small translation, first-order Taylor expansion could be used to simplify the computation. Several representative terms could be selected as inputs to compute the coefficient changes before and after translation and scaling. Results show that the outcomes of the analytical solutions and the approximated values under discrete sampling are consistent. With the computation of a group of randomly generated coefficients, we contrasted the changes under different translation and scaling conditions. The larger ratios correlate the larger deviation from the approximated values to the original ones. Finally, we analyzed the peak-to-valley (PV) and root mean square (RMS) deviations from the uses of the first-order approximation and the direct expansion under different translation values. The results show that when the translation is less than 4%, the most deviated 5th term in the first-order 1D-Legendre expansion has a PV deviation less than 7% and an RMS deviation less than 2%. The analytical expressions and the computed results under discrete sampling given in this paper for the multiple typical function basis during translation and scaling in the rectangular areas could be applied in wavefront approximation and analysis.

An adaptive robust fuzzy beamformer for steering vector mismatch and reducing interference and noise

This paper presents an efficient beamformer design that accounts for both signal steering vector mismatches and the trade-off between interference and noise reduction. The empirical results show that substantial performance degradation occurs when the exact signal steering vector for the desired signal is not known. Furthermore, total rejection of interference may increase noise, and vice versa. Therefore, an adaptive robust fuzzy beamformer was designed by adopting the method of fuzzy systems to modify both the value of the signal steering vector and the interference-to-noise ratio. This method uses the first-order Taylor expansion of the object function to modify the mismatches of the signal steering vector, and uses the signal covariance matrix eigendecomposition to adjust the ratio of interference reduction to noise reduction. Simulations confirm that the proposed scheme performance is substantially improved and more robust if the effects of the signal steering vector mismatches and the interference to noise ratio are considered in the beamformer design which is based on expert knowledge.

Generalized Euler–Lagrange equation for nonsmooth calculus of variations

In this paper, we first introduce a novel generalized derivative and obtain the generalized first-order Taylor expansion of the nonsmooth functions. Then we derive the generalized Euler–Lagrange equation for the nonsmooth calculus of variations and solve this equation by using Chebyshev pseudospectral method, approximately. Finally, the optimal solutions of some problems in the nonsmooth calculus of variations are approximated.

An iterative calculation method for suspension bridge’s cable system based on exact catenary theory

In this paper, a flexible iterative method capable of considering the effects of slip between the main cable and saddles is presented for the analysis of the cable system in the suspension bridge. In the proposed procedure, nonlinear governing equations were first linearized based on the first-order Taylor expansion, then the tangent stiffness matrix was derived using appropriate numerical methods. Using the proposed flexible iterative procedure which is built upon the framework of Newton-Raphson method, the main cable’s unstrained length and equilibrium forces which satisfy the configuration and mechanical property under bridge’s completion state is obtained according to the main cable’s initial geometry parameters, saddles parameters and hangers arrangement. Based on form-finding analysis, the method is also proposed to calculate the main cable’s internal forces and displacements during the erection of stiffening girder; the reliability and efficiency of the method is demonstrated by two typical numerical examples. Furthermore, the proposed method is used as a pro-processing tool in the finite element analyses of a cable structure. Finally, a numerical example (Yingwuzhou Yangtze River Bridge) is reported to illustrate the advantages of the proposed method, including the accurate predictions of the main cable’s unstrained length and the excursion of the saddles, which is crucial for choosing appropriate saddles parameters.

Iterative linear focal-plane wavefront correction

We propose an efficient approximation to the nonlinear phase diversity (PD) method for wavefront reconstruction and correction from intensity measurements with potential of being used in real-time applications. The new iterative linear phase diversity (ILPD) method assumes that the residual phase aberration is small and makes use of a first-order Taylor expansion of the point spread function (PSF), which allows for arbitrary (large) diversities in order to optimize the phase retrieval. For static disturbances, at each step, the residual phase aberration is estimated based on one defocused image by solving a linear least squares problem, and compensated for with a deformable mirror. Due to the fact that the linear approximation does not have to be updated with each correction step, the computational complexity of the method is reduced to that of a matrix-vector multiplication. The convergence of the ILPD correction steps has been investigated and numerically verified. The comparative study that we make demonstrates the improved performance in computational time with no decrease in accuracy with respect to existing methods that also linearize the PSF.

Perturbation Analysis with Approximate Integration for Propagation Mode in Two-Dimensional Two-Slab Waveguides

On the basis of perturbation expansion from a gapless system, we calculate the propagation constant and propagation mode wave function in two-dimensional two-slab waveguides with a core gap small enough that there is only one propagation mode. We also perform calculations without the approximation for comparison. Our result shows that first-order perturbation contains the first-order Taylor expansion of (core gap)/(core width), and when the integration of the perturbation is suitably approximated, the result of the first-order perturbation is the same as that of the first-order Taylor expansion of (core gap)/(core width).

Association of Percolation Theory with Princen’s Approach To Model the Storage Modulus of Highly Concentrated Emulsions

An approach based on percolation theory is associated to the rigorousness of the two-dimensional (2D) model of close-packed, monodisperse, cylindrical emulsions established by Princen, to obtain an equation to model the dispersed-phase volume fraction (ϕ) dependence of the storage modulus (G′) of highly concentrated emulsions. A first-order Taylor expansion of this general percolation model leads to an expression similar to the three-dimensional (3D) model proposed by Princen and Kiss for real polydisperse emulsions, with a more satisfactory derivation of the function E(ϕ). Finally, the robustness of this model is tested against different experimental data collected from the literature.

On the convergence rate of the unscented transformation

Nonlinear state-space models driven by differential equations have been widely used in science. Their statistical inference generally requires computing the mean and covariance matrix of some nonlinear function of the state variables, which can be done in several ways. For example, such computations may be approximately done by Monte Carlo, which is rather computationally expensive. Linear approximation by the first-order Taylor expansion is a fast alternative. However, the approximation error becomes non-negligible with strongly nonlinear functions. Unscented transformation was proposed to overcome these difficulties, but it lacks theoretical justification. In this paper, we derive some theoretical properties of the unscented transformation and contrast it with the method of linear approximation. Particularly, we derive the convergence rate of the unscented transformation.

A Derivative-Free Kalman Filtering Approach to State Estimation-Based Control of Nonlinear Systems

For nonlinear systems, subject to Gaussian noise, the extended Kalman filter (EKF) is frequently applied for estimating the system's state vector from output measurements. The EFK is based on linearization of the systems' dynamics using a first-order Taylor expansion. Although EKF is efficient in several problems, it is characterized by cumulative errors due to the gradient-based linearization it performs, and this may affect the accuracy of the state estimation or even risk the stability of the state estimation-based control loop. To overcome the flaws of EKF, it has been proposed to use the unscented Kalman filter (UKF) as a method for nonlinear state estimation, which does not introduce linearization errors. Aiming also at finding more efficient implementations of nonlinear Kalman filtering, this paper introduces a derivative-free Kalman filtering approach, which is suitable for state estimation-based control of a class of nonlinear systems. The considered systems are first subject to a linearization transformation, and next state estimation is performed by applying the standard Kalman filter to the linearized model. Unlike EKF, the proposed method provides estimates of the state vector of the nonlinear system without the need for derivatives and Jacobians calculation and without using linearization approximations. The proposed derivative-free Kalman filtering approach has been compared to EKF and UKF in the case of state estimation-based control for a nonlinear DC motor model.

The potential of discs from a ‘mean Green function’

By using various properties of the complete elliptic integrals, we have derived an alternative expression for the gravitational potential of axially symmetric bodies, which is free of singular kernel in contrast with the classical form. This is mainly a radial integral of the local surface density weighted by a regular "mean Green function" which depends explicitly on the body's vertical thickness. Rigorously, this result stands for a wide variety of configurations, as soon as the density structure is vertically homogeneous. Nevertheless, the sensitivity to vertical stratification | the Gaussian profile has been considered | appears weak provided that the surface density is conserved. For bodies with small aspect ratio (i.e. geometrically thin discs), a first-order Taylor expansion furnishes an excellent approximation for this mean Green function, the absolute error being of the fourth order in the aspect ratio. This formula is therefore well suited to studying the structure of self-gravitating discs and rings in the spirit of the "standard model of thin discs" where the vertical structure is often ignored, but it remains accurate for discs and tori of finite thickness. This approximation which perfectly saves the properties of Newton's law everywhere (in particular at large separations), is also very useful for dynamical studies where the body is just a source of gravity acting on external test particles.

A Simple Linear Response Closure Approximation for Slow Dynamics of a Multiscale System with Linear Coupling

Many applications of contemporary science involve multiscale dynamics, which are typically characterized by the time and space scale separation of patterns of motion, with fewer slowly evolving variables and much larger set of faster evolving variables. This time-space scale separation causes direct numerical simulation of the evolution of the dynamics to be computationally expensive, due both to the large number of variables and the necessity to choose a small discretization time step in order to resolve the fast components of dynamics. In this work we propose a simple method of determining the closed model for slow variables alone, which requires only a single computation of appropriate statistics for the fast dynamics with a certain fixed state of the slow variables. The method is based on the first-order Taylor expansion of the averaged coupling term with respect to the slow variables, which can be computed using the linear fluctuation-dissipation theorem. We show that, with simple linear coupling in both slow and fast variables, this method produces quite comparable statistics to what is exhibited by a complete two-scale model. The main advantage of the method is that it applies even when the statistics of the full multiscale model cannot be simulated due to computational complexity, which makes it practical for real-world large scale applications.

AN EFFICIENT OPTIMIZATION METHOD FOR UNCERTAIN PROBLEMS BASED ON NON-PROBABILISTIC INTERVAL MODEL

In this paper, an efficient optimization method named NIO-SLP is suggested for uncertain structures by using the non-probabilistic interval model to quantify the uncertainty. A general nonlinear interval optimization (NIO) problem is investigated, in which both of the objective function and constraints are nonlinear and uncertain. A sequence of approximate optimization sub-problems are created based on the first-order Taylor expansion and each one degenerates into a simple linear interval optimization (LIO) problem. An iterative mechanism is proposed to update the design space and whereby make the optimization sequence converge at the optimum. Two numerical examples are investigated to demonstrate the effectiveness of the present method.

DYNAMIC LOAD IDENTIFICATION FOR UNCERTAIN STRUCTURES BASED ON INTERVAL ANALYSIS AND REGULARIZATION METHOD

In this paper, an inverse method that combines the interval analysis with regularization is presented to stably identify the bounds of dynamic load acting on the uncertain structures. The uncertain parameters of the structure are treated as intervals and hence only their bounds are needed. Using the first-order Taylor expansion, the identified load can be approximated as a linear function of the uncertain parameters. In this function, it is assumed that the load at the midpoint of the uncertain parameters can be expressed as a series of impulse kernels. The finite element method (FEM) is used to obtain the response function of the impulse kernel and the response to the midpoint load is expressed in a form of convolution. In order to deal with the ill-posedness arising from the deconvolution, two regularization methods are adopted to provide the numerically efficient and stable solution of the desired unknown midpoint load. Then, a sensitivity analysis is suggested to calculate the first derivative of the identified load with respect to each uncertain parameter. Applying the interval extension in interval mathematics, the lower and upper bounds of identified load caused by the uncertainty can be finally determined. Numerical simulation demonstrates that the present method is effective and robust to stably determine the range of the load on the uncertain structures from the noisy measured response in time domain.

Robust Control of Uncertain Piezoelectric Laminated Plates Based on Model Reduction

The reduction method and robust H1 control for uncertain piezoelectric laminated plates with small interval parameters are investigated in this paper. The electromechanical coupling finite element equations of piezoelectric laminated plates are developed based on Hamilton principle. To reduce the high-order uncertain system with small interval parameters, a reduction method is presented by using the balanced model reduction, first-order Taylor expansion, and interval algorithm. Based on the reduction method, a new strategy of robust H1 control design is studied,which includesthe robustH1 controller designofareduceduncertain generalized plantandthe robustness analysisoftheclosedsystemcomposedofthecontrollerandtheoriginaluncertainsystem.Numericalexamplesshow that the reduction method is suitable for a high-order uncertain system with small interval parameters and that the robust H1 control strategy for the uncertain system is feasible and practical.