Performance evaluation of iterative geometric fitting algorithms

The convergence performance of typical numerical schemes for geometric fitting for computer vision applications is compared. First, the problem and the associated KCR lower bound are stated. Then, three well-known fitting algorithms are described: FNS, HEIV, and renormalization. To these, we add a special variant of Gauss-Newton iterations. For initialization of iterations, random choice, least squares, and Taubin's method are tested. Simulation is conducted for fundamental matrix computation and ellipse fitting, which reveals different characteristics of each method.     

On the Convergence of Fitting Algorithms in Computer Vision

We investigate several numerical schemes for estimating parameters in computer vision problems: HEIV, FNS, renormalization method, and others. We prove mathematically that these algorithms converge rapidly, provided the noise is small. In fact, in just 1-2 iterations they achieve maximum possible statistical accuracy. Our results are supported by a numerical experiment. We also discuss the performance of these algorithms when the noise increases and/or outliers are present. Nikolai Chernov PhD in mathematics from Moscow University in 1984. Researcher in JINR (Dubna, Russia) in 1984–91. Professor of Mathematics at University of Alabama at Birmingham, USA, since 1994.     

FNS,CFNS and HEIV: A Unifying Approach

Estimation of parameters from image tokens is a central problem in computer vision. FNS, CFNS and HEIV are three recently developed methods for solving special but important cases of this problem. The schemes are means for finding unconstrained (FNS, HEIV) and constrained (CFNS) minimisers of cost functions. In earlier work of the authors, FNS, CFNS and a core version of HEIV were applied to a specific cost function. Here we extend the approach to more general cost functions. This allows the FNS, CFNS and HEIV methods to be placed within a common framework. Wojciech Chojnacki is a professor of mathematics in the Department of Mathematics and Natural Sciences at Cardinal Stefan Wyszyski University in Warsaw. He is concurrently a senior research fellow in the School of Computer Science at the University of Adelaide working on a range of problems in computer vision. His research interests include differential equations, mathematical foundations of computer vision, functional analysis, and harmonic analysis. He is author of over 70 articles on pure mathematics and machine vision, and a member of the Polish Mathematical Society. Michael J. Brooks holds the Chair in Artificial Intelligence within the University of Adelaides School of Computer Science, which he heads. He is also leader of the Image Analysis Program within the Cooperative Research Centre for Sensor Signal and Information Processing, based in South Australia. His research interests include structure from motion, self-calibration, metrology, statistical vision-parameter estimation, and video surveillance and analysis. He is author of over 100 articles on vision, actively involved in a variety of commercial applications, an Associate Editor of the International Journal of Computer Vision, and a Fellow of the Australian Computer Society. Anton van den Hengel is a senior lecturer in the School of Computer Science within the University of Adelaide. He is also leader of the Video Surveillance and Analysis Project within the Cooperative Research Centre for Sensor Signal and Information Processing. His research interests include structure from motion, parameter estimation theory, and commercial applications of computer vision. Darren Gawley graduated with first class honours from the School of Computer Science at the University of Adelaide. He holds a temporary lectureship at the same University, and is currently finalising his PhD in the field of computer vision.This revised version was published online in June 2005 with correction to CoverDate     

Estimation of nonlinear errors-in-variables models for computer vision applications

In an errors-in-variables (EIV) model, all the measurements are corrupted by noise. The class of EIV models with constraints separable into the product of two nonlinear functions, one solely in the variables and one solely in the parameters, is general enough to represent most computer vision problems. We show that the estimation of such nonlinear EIV models can be reduced to iteratively estimating a linear model having point dependent, i.e., heteroscedastic, noise process. Particular cases of the proposed heteroscedastic errors-in-variables (HEIV) estimator are related to other techniques described in the vision literature: the Sampson method, renormalization, and the fundamental numerical scheme. In a wide variety of tasks, the HEIV estimator exhibits the same, or superior, performance as these techniques and has a weaker dependence on the quality of the initial solution than the Levenberg-Marquardt method, the standard approach toward estimating nonlinear models.     

结合PCA与HEIV的椭圆目标检测算法

提出结合主元变换与异方差变量含误差模型的椭圆识别与定位方法。根据椭圆长轴对应于椭圆主元方向的特点,利用主元变换法将目标边缘数据变换到主元坐标系,给出新的椭圆轮廓度误差评定方法,将变换后数据点集的椭圆轮廓度误差作为椭圆识别的依据,采用基于异方差变量含误差模型的拟合算法获取椭圆的中心坐标。该方法将任意椭圆转化为标准型椭圆,简化了识别过程,考虑到椭圆数据点的异方差特性,提高了椭圆的定位精度,在噪声方差为0.05情况下,定位精度小于0.04 pixel。     

Fuzzy Interval TSK Type‐2 Modeling with Parameterized Conjunctors

In this study, a novel approach is described to the design of an interval type‐2 fuzzy neural system (IT2 FNS). It differs from the classical IT2 FNS in its use of parameterized conjunctors. In the optimization of the IT2 FNS, the membership functions are kept fixed and only the parameters of the conjunctors and the parameters in the consequent are tuned. In this study, the gradient based learning algorithm is used. The approach is tested for the modeling of a benchmark nonlinear function and for the wheel slip control of a quarter car model (QCM). In the stated applications, in the absence of any expert knowledge, some knowledge about the system is gained by the use of the interval type‐2 fuzzy c‐means (IT2 FCM) clustering algorithm. Nevertheless, this requires the number of classes to be known beforehand. To alleviate this problem, some validity indices that have been suggested in the literature and a novel validity index that carries less computational burden are considered to determine the number of classes and the number of fuzzy rules. Simulation studies are presented and compared with the results from the literature.     

基于HEIV模型的摄像机一维标定

多摄像机系统广泛应用于文化创意产业,其高精度标定是迫切需要解决的一个关键问题. 新近出现的摄像机一维标定方法能够克服标定物自身遮挡,特别适合标定多摄像机系统. 然而,现有的摄像机一维标定研究主要集中在降低一维标定物的运动约束,而标定精度较低的问题未受到应有的关注. 本文提出一种基于变量含异质噪声 （Heteroscedastic error-in-variables,HEIV）模型的高精度摄像机一维标定方法. 首先,推导出摄像机一维标定的计算模型;其次,利用该计算模型详细分析了一维标定中的噪声,得出摄像机一维标定可以视为一个HEIV问题的结论;最后给出了基于HEIV模型的摄像机一维标定算法. 与现有的算法相比,该方法可以显著改善一维标定的精度,并且受初始值影响小,收敛速度快. 实验结果验证了该方法的正确性和可行性.     

A Bilinear Approach to the Parameter Estimation of a General Heteroscedastic Linear System,with Application to Conic Fitting

In this paper, we employ low-rank matrix approximation to solve a general parameter estimation problem: where a non-linear system is linearized by treating the carrier terms as separate variables, thereby introducing heteroscedastic noise. We extend the bilinear approach to handle cases with heteroscedastic noise, in the framework of low-rank approximation. The ellipse fitting problem is investigated as a specific example of the general theory. Despite the impression given in the literature, the ellipse fitting problem is still unsolved when the data comes from a small section of the ellipse. Although there are already some good approaches to the problem of ellipse fitting, such as FNS and HEIV, convergence in these iterative approaches is not ensured, as pointed out in the literature. Another limitation of these approaches is that they cannot model the correlations among different rows of the “general measurement matrix”. Our method, of employing the bilinear approach to solve the general heteroscedastic parameter estimation problem, overcomes these limitations: it is convergent, at least to a local optimum, and can cope with a general heteroscedastic problem. Experiments show that the proposed bilinear approach performs better than other competing approaches: although it is still far short of a solution when the data comes from a very small arc of the ellipse.

视觉测量中椭圆自动检测与定位方法

针对视觉测量中椭圆目标的检测效率与定位精度较低的问题,提出一种高效的椭圆识别与定位方法。该方法利用回光反射控制点的强反射特性,提取目标的边缘信息,对连续的边缘进行分组,利用椭圆长轴信息对每组边缘数据进行椭圆识别,采用基于变量含误差模型的椭圆拟合方法,实现椭圆目标中心的精确定位。实验证明,该方法具有快速、自动化、定位精度高、鲁棒性好等优点,在视觉测量中具有广泛的应用前景。     

A PSO based integrated functional link net and interval type-2 fuzzy logic system for predicting stock market indices

This paper presents an integrated functional link interval type-2 fuzzy neural system (FLIT2FNS) for predicting the stock market indices. The hybrid model uses a TSK (Takagi-Sugano-Kang) type fuzzy rule base that employs type-2 fuzzy sets in the antecedent parts and the outputs from the Functional Link Artificial Neural Network (FLANN) in the consequent parts. Two other approaches, namely the integrated FLANN and type-1 fuzzy logic system and Local Linear Wavelet Neural Network (LLWNN) are also presented for a comparative study. Backpropagation and particle swarm optimization (PSO) learning algorithms have been used independently to optimize the parameters of all the forecasting models. To test the model performance, three well known stock market indices like the Standard's & Poor's 500 (S&P 500), Bombay stock exchange (BSE), and Dow Jones industrial average (DJIA) are used. The mean absolute percentage error (MAPE) and root mean square error (RMSE) are used to find out the performance of all the three models. Finally, it is observed that out of three methods, FLIT2FNS performs the best irrespective of the time horizons spanning from 1 day to 1 month.