A Self-calibration Bundle Adjustment Algorithm Based on Block Matrix Cholesky Decomposition Technology

In this study,the problem of bundle adjustment was revisited,and a novel algorithm based on block matrix Cholesky decomposition was proposed to solve the thorny problem of self-calibration bundle adjustment.The innovation points are reflected in the following aspects:①The proposed algorithm is not dependent on the Schur complement,and the calculation process is simple and clear;②The complexities of time and space tend to O(n)in the context of world point number is far greater than that of images and cameras,so the calculation magnitude and memory consumption can be reduced significantly;③The proposed algorithm can carry out self-calibration bundle adjustment in single-camera,multi-camera,and variable-camera modes;④Some measures are employed to improve the optimization effects.Experimental tests showed that the proposed algorithm has the ability to achieve state-of-the-art performance in accuracy and robustness,and it has a strong adaptability as well,because the optimized results are accurate and robust even if the initial values have large deviations from the truth.This study could provide theoretical guidance and technical support for the image-based positioning and 3D reconstruction in the fields of photogrammetry,computer vision and robotics.

Fast Inverse Cholesky Decomposition for Rectangular Toeplitz-Block MatriX

A fast Cholesky decomposition and a fast inverse Cholesky decomposition method for A T A are presented,where A is an m×n rectangular Toeplitz block matrix,we give the FCD algorithm for computing R , and the FICD algorithm for computing R -1 ,both allow for an efficient parallel implementation,for solving a least squares problem and require only O(mn) operations.

Matrix decomposition and Lagrangian dual method for discrete portfolio optimization under concave transaction costs

In this paper, the discrete mean-variance model is considered for portfolio selection under concave transaction costs. By using the Cholesky decomposition technique, the convariance matrix to obtain a separable mixed integer nonlinear optimization problem is decomposed. A brand-and-bound algorithm based on Lagrangian relaxation is then proposed. Computational results are reported for test problems with the data randomly generated and those from the US stock market.

Cholesky GAS models for large time-varying covariance matrices

This paper develops a new class of multivariate models for large-dimensional time-varying covariance matrices,called Cholesky generalized autoregressive score(GAS)models,which are based on the Cholesky decomposition of the covariance matrix and assume that the parameters are score-driven.Specifically,two types of score-driven updates are considered:one is closer to the GARCH family,and the other is inspired by the stochastic volatility model.We demonstrate that the models can be estimated equation-wise and are computationally feasible for high-dimensional cases.Moreover,we design an equationwise dynamic model averaging or selection algorithm which simultaneously extracts model and parameter uncertainties,equipped with dynamically estimated model parameters.The simulation results illustrate the superiority of the proposed models.Finally,using a sizeable daily return dataset that includes 124 sectors in the Chinese stock market,two empirical studies with a small sample and a full sample are conducted to verify the advantages of our models.The full sample analysis by a dynamic correlation network documents significant structural changes in the Chinese stock market before and after COVID-19.

An iterative algorithm for solving ill-conditioned linear least squares problems

Linear Least Squares（LLS） problems are particularly difficult to solve because they are frequently ill-conditioned, and involve large quantities of data. Ill-conditioned LLS problems are commonly seen in mathematics and geosciences, where regularization algorithms are employed to seek optimal solutions. For many problems, even with the use of regularization algorithms it may be impossible to obtain an accurate solution. Riley and Golub suggested an iterative scheme for solving LLS problems. For the early iteration algorithm, it is difficult to improve the well-conditioned perturbed matrix and accelerate the convergence at the same time. Aiming at this problem, self-adaptive iteration algorithm（SAIA） is proposed in this paper for solving severe ill-conditioned LLS problems. The algorithm is different from other popular algorithms proposed in recent references. It avoids matrix inverse by using Cholesky decomposition, and tunes the perturbation parameter according to the rate of residual error decline in the iterative process. Example shows that the algorithm can greatly reduce iteration times, accelerate the convergence,and also greatly enhance the computation accuracy.

On Numerical methods for determination of Earth gravity field model using mass satellite gravity gradiometry data

On the basis of Space-Wise Least Square method, three numerical methods including Cholesky de- composition, pre-conditioned conjugate gradient and Open Multi-Processing parallel algorithm are applied into the determination of gravity field with satellite gravity gradiometry data. The results show that, Cholesky de- composition method has been unable to meet the requirements of computation efficiency when the computer hardware is limited. Pre-conditioned conjugate gradient method can improve the computation efficiency of huge matrix inversion, but it also brings a certain loss of precision. The application of Open Multi-Processing parallel algorithm could achieve a good compromise between accuracy and computation efficiency.

Fast algorithm and numerical simulation for ray-tracing in 3D structure

Beginning with the method of whole path iterative ray-tracing and according to the positive definiteness of the coefficient matrix of the systems of linear equations, a symmetry olock tridiagonal matrix was decomposed into the product of block bidiagonal triangular matrix and its transpose by means of Cholesky decomposition. Then an algorithm for solving systems of block bidiagonal triangular linear equations was given, which is not necessary to treat with the zero elements out of banded systems. A fast algorithm for solving the systems of symmetry block tridiagonal linear equations was deduced, which can quicken the speed of ray-tracing. Finally, the simulation based on this algorithm for ray-tracing in three dimensional media was carried out. Meanwhile, the segmentally-iterative ray-tracing method and banded method for solving the systems of block tridiagonal linear equations were compared in the same model mentioned above. The convergence condition was assumed that the L-2 norm summation for mk, 1 and mk. 2 in the whole ray path was limited in 10-6. And the calculating speeds of these methods were compared. The results show that the calculating speed of this algorithm is faster than that of conventional method and the calculated results are accurate enough. In addition, its precision can be controlled according to the requirement of ray-tracing.

Another Form for LAMBDA Method

The LAMBDA method that was proposed by Teunissen is introduced. Then, on the basis of both the back-sequential conditional LS technique and the upper-triangular Cholesky decomposition, another form for LAMBDA method is proposed. This new form for LAMBDA method has the same principle and calculation speed as the traditional LAMBDA method.

Efficient Simulation of Stationary Multivariate Gaussian Random Fields with Given Cross-Covariance

The present paper introduces a new approach to simulate any stationary multivariate Gaussian random field whose cross-covariances are predefined continuous and integrable functions. Such a field is given by convolution of a vector of univariate random fields and a functional matrix which is derived by Cholesky decomposition of the Fourier transform of the predefined cross-covariance matrix. In contrast to common methods, no restrictive model for the cross-covariance is needed. It is stationary and can also be reduced to the isotropic case. The computational effort is very low since fast Fourier transform can be used for simulation. As will be shown the algorithm is computationally faster than a recently published spectral turning bands model. The applicability is demonstrated using a common numerical example with varied spatial correlation structure. The model was developed to support simulation algorithms for mineral microstructures in geoscience.

An Extension of the Non-central Wishart Distribution with Integer Shape Vector

This research paper deals with an extension of the non-central Wishart introduced in 1944 by Anderson and Girshick,that is the non-central Riesz distribution when the scale parameter is derived from a discrete vector.It is related to the matrix of normal samples with monotonous missing data.We characterize this distribution by means of its Laplace transform and we give an algorithm for generating it.Then we investigate,based on the method of the moment,the estimation of the parameters of the proposed model.The performance of the proposed estimators is evaluated by a numerical study.

BANDED M-MATRIX SPLITTING PRECONDITIONER FOR RIESZ SPACE FRACTIONAL REACTION-DISPERSION EQUATION

Based on the Crank-Nicolson and the weighted and shifted Grunwald operators,we present an implicit difference scheme for the Riesz space fractional reaction-dispersion equations and also analyze the stability and the convergence of this implicit difference scheme.However,after estimating the condition number of the coefficient matrix of the discretized scheme,we find that this coefficient matrix is ill-conditioned when the spatial mesh-size is sufficiently small.To overcome this deficiency,we further develop an effective banded M-matrix splitting preconditioner for the coefficient matrix.Some properties of this preconditioner together with its preconditioning effect are discussed.Finally,Numerical examples are employed to test the robustness and the effectiveness of the proposed preconditioner.

Calculation Method of Optimal Speed Profile for Hot Plate During Controlled Cooling Process

The healthy and rapid development of the controlled cooling technology was hampered by the uneven cooling phenomenon. During the process of hot plate production,the homogeneous cooling along the length direction of plate was constrained by lots of factors. And because the speed was a flexible control parameter,the calculation method of optimal speed profile was developed based on the measured start cooling temperature and its matrix equation was solved by the Cholesky decomposition method. The optimal speed profile was used in online control system. As a result,the temperature distribution along the plate length direction was relatively uniform,and 95% of measured final cooling temperature difference from the target temperature 700 ℃ was controlled within ±20 ℃.

Joint semiparametric mean-covariance model in longitudinal study

Semiparametric regression models and estimating covariance functions are very useful for longitudinal study. To heed the positive-definiteness constraint, we adopt the modified Cholesky decomposition approach to decompose the covariance structure. Then the covariance structure is fitted by a semiparametric model by imposing parametric within-subject correlation while allowing the nonparametric variation function. We estimate regression functions by using the local linear technique and propose generalized estimating equations for the mean and correlation parameter. Kernel estimators are developed for the estimation of the nonparametric variation function. Asymptotic normality of the the resulting estimators is established. Finally, the simulation study and the real data analysis are used to illustrate the proposed approach.

Efficient image representation for object recognition via pivots selection

Patch-level features are essential for achieving good performance in computer vision tasks. Besides well- known pre-defined patch-level descriptors such as scalein- variant feature transform （SIFT） and histogram of oriented gradient （HOG）, the kernel descriptor （KD） method [1] of- fers a new way to ＂grow-up＂ features from a match-kernel defined over image patch pairs using kernel principal compo- nent analysis （KPCA） and yields impressive results. In this paper, we present efficient kernel descriptor （EKD） and efficient hierarchical kernel descriptor （EHKD）, which are built upon incomplete Cholesky decomposition. EKD au- tomatically selects a small number of pivot features for gener- ating patch-level features to achieve better computational effi- ciency. EHKD recursively applies EKD to form image-level features layer-by-layer. Perhaps due to parsimony, we find surprisingly that the EKD and EHKD approaches achieved competitive results on several public datasets compared with other state-of-the-art methods, at an improved efficiency over KD.

An On-line Finite Element Temperature Field Model for Plate Ultra Fast Cooling Process

Taking the element specific-heat interpolation function into account, a one-dimensional （l-D） finite ele- ment temperature field model for the on-line control of the ultra fast cooling process was developed based on the heat transfer theory. This 1-D model was successfully implemented in one 4 300 mm plate production line. To improve the calculation accuracy of this model, the temperature-dependent material properties inside an element were considered during the modeling process. Furthermore, in order to satisfy the real-time requirements of the on-line model, the variable bandwidth storage method and the Cholesky decomposition method were used in the programming to storage the data and carry out the numerical solution. The on-line application of the proposed model indicated that the devia- tion between the calculated cooling stop temperature and the measured one was less than ± 15 ℃.

A High Performance Multifrontal Code for Linear Solution of Structures Using Multi-Core Microprocessors

A multifrontal code is introduced for the efficient solution of the linear system of equations arising from the analysis of structures. The factorization phase is reduced into a series of interleaved element assembly and dense matrix operations for which the BLAS3 kernels are used. A similar approach is generalized for the forward and back substitution phases for the efficient solution of structures having multiple load conditions. The program performs all assembly and solution steps in parallel. Examples are presented which demonstrate the code’s performance on single and dual core processor computers.

Efficient Estimation of Longitudinal Data Additive Varying Coefficient Regression Models

We consider a longitudinal data additive varying coefficient regression model, in which the coef- ficients of some factors （covariates） are additive functions of other factors, so that the interactions between different factors can be taken into account effectively. By considering within-subject correlation among repeated measurements over time and additive structure, we propose a feasible weighted two-stage local quasi-likelihood estimation. In the first stage, we construct initial estimators of the additive component functions by B-spline se- ries approximation. With the initial estimators, we transform the additive varying coefficients regression model into a varying coefficients regression model and further apply the local weighted quasi-likelihood method to estimate the varying coefficient functions in the second stage. The resulting second stage estimators are com- putationally expedient and intuitively appealing. They also have the advantages of higher asymptotic efficiency than those neglecting the correlation structure, and an oracle property in the sense that the asymptotic property of each additive component is the same as if the other components were known with certainty. Simulation studies are conducted to demonstrate finite sample behaviors of the proposed estimators, and a real data example is given to illustrate the usefulness of the proposed methodology.

Parsimonious Mean-Covariance Modeling for Longitudinal Data with ARMA Errors

Based on the generalized estimating equation approach,the authors propose a parsimonious mean-covariance model for longitudinal data with autoregressive and moving average error process,which not only unites the existing autoregressive Cholesky factor model and moving average Cholesky factor model but also provides a wide variety of structures of covariance matrix.The resulting estimators for the regression coefficients in both the mean and the covariance are shown to be consistent and asymptotically normally distributed under mild conditions.The authors demonstrate the effectiveness,parsimoniousness and desirable performance of the proposed approach by analyzing the CD4-I-cell counts data set and conducting extensive simulations.

Bayesian Joint Semiparametric Mean–Covariance Modeling for Longitudinal Data

Joint parsimonious modeling the mean and covariance is important for analyzing longitudinal data,because it accounts for the efficiency of parameter estimation and easy interpretation of variability.The main potential risk is that it may lead to inefficient or biased estimators of parameters while misspecification occurs.A good alternative is the semiparametric model.In this paper,a Bayesian approach is proposed for modeling the mean and covariance simultaneously by using semiparametric models and the modified Cholesky decomposition.We use a generalized prior to avoid the knots selection while using B-spline to approximate the nonlinear part and propose a Markov Chain Monte Carlo scheme based on Metropolis–Hastings algorithm for computations.Simulation studies and real data analysis show that the proposed approach yields highly efficient estimators for the parameters and nonparametric parts in the mean,meanwhile providing parsimonious estimation for the covariance structure.

Fast frequency-domain equalization for single-carrier V-BLAST systems over multipath channel

A low complex minimum mean-square error frequency-domain decision feedback （MMSE-FDDF） equalization algorithm is proposed in this paper for the single-carrier V-BLAST systems. Exploiting the factor that the discrete Fourier transform （DFT） is unitary, the proposed receiver can equalize the signals by the MMSE detecting to the spectrums in the frequency domain instead of the waveforms in the time domain. In order to obtain the right decisions, the detector must be able to equalize the overall spectrum with regard to each layer. This work can be performed very efficiently since the system matrix has been designed as a special block-circulant-block matrix. Similar to other V-BLAST-like systems, the detecting order has strong impact on the performance of MMSE-FDDF. Therefore, we further give a fast optimally sorting scheme for the MMSE-FDDF architecture. By using the newly constructed matrix, the coefficients computation and the sorting can be combined into one process, and then we employ the modified Gram-Schmidt （MGS） to simplify the process. The simulation results and the computational complexity analysis show that the proposed MMSE-FDDF has better tradeoff between the performance and the complexity than the existing algorithms. In addition, MMSE-FDDF can avoid the performance floor caused by the overlap-and-save technique in the severe dispersive channel.