Comparison and Analysis of Non-Linear Least Squares Methods for 3-D Coordinates Transformation

Elhabiby, Mohamed; Gao, Yin; Sideris, M. G.

doi:10.1179/003962608x389988

Cited by 12 publications

(8 citation statements)

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“…(18) namely R is a matrix with 3 rows and 3 columns, usually represented by rotation angles (see e.g. El-Habiby et al 2009;Zeng and Yi, 2011), unit quaternion (see e.g. Shen et al 2006;Zeng and Yi 2011), and Rodrigues matrix (see e.g.…”

Section: Iterative Approach Dependent On Function Valuementioning

confidence: 99%

Iterative approach of 3D datum transformation with a non-isotropic weight

Zeng

2015

Acta Geod Geophys

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The analytical solution of 3D datum transformation with an isotropic weight has been elegantly presented based on Procrustes algorithm (singular value decomposition). But the existence of analytical solution of 3D datum transformation with a nonisotropic weight needs further investigation. Based on the Lagrangian extremum law, the paper derives the analytical formula for translation parameter and scale factor, but because the rotation matrix is unsolved, the analytical solution does not exist. For this reason, the paper presents two kinds of iterative approach of 3D datum transformation with a nonisotropic weight. One is the iterative approach dependent on the objective function value, which uses the Lagrangian minimum function in the variable of rotation matrix as the objective function, and the other is the iterative approach dependent on the derivative of function, which uses the 3D datum transformation model that eliminates the translation parameter. In order to improve the speed and reliability of iterative computation, the form of rotation matrix represented by Rodrigues matrix instead of rotation angles or unit quaternion is adopted for the two iterative approaches. A numerical experiment is demonstrated, and comparison analysis of the two iterative approaches is carried out. The result shows from the view of computing speed and reliability, the iterative approach based on derivatives is preferred.

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Section: Iterative Approach Dependent On Function Valuementioning

confidence: 99%

Iterative approach of 3D datum transformation with a non-isotropic weight

Zeng

2015

Acta Geod Geophys

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“…Once completed, the camera position and attitude will be estimated and consequently the UAV pose can be estimated. There are two general approaches to investigate the nonlinear least squares algorithms (El-Habiby, Gao et al 2009). The first one is linearization of the model.…”

Section: Collinearity Equation and Its Application To Vbnmentioning

confidence: 99%

Comparison and Analysis of Nonlinear Least Squares Methods for Vision Based Navigation (Vbn) Algorithms

Sheta

Elhabiby

Sheimy

2012

Int. Arch. Photogramm. Remote Sens. Spatial Inf. Sci.

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A robust scale and rotation invariant image matching algorithm is vital for the Visual Based Navigation (VBN) of aerial vehicles, where matches between an existing geo-referenced database images and the real-time captured images are used to georeference (i.e. six transformation parameters-three rotation and three translation) the real-time captured image from the UAV through the collinearity equations. The georeferencing information is then used in aiding the INS integration Kalman filter as Coordinate UPdaTe (CUPT). It is critical for the collinearity equations to use the proper optimization algorithm to ensure accurate and fast convergence for georeferencing parameters with the minimum required conjugate points necessary for convergence. Fast convergence to a global minimum will require non-linear approach to overcome the high degree of non-linearity that will exist in case of having large oblique images (i.e. large rotation angles).The main objective of this paper is investigating the estimation of the georeferencing parameters necessary for VBN of aerial vehicles in case of having large values of the rotational angles, which will lead to non-linearity of the estimation model. In this case, traditional least squares approaches will fail to estimate the georeferencing parameters, because of the expected non-linearity of the mathematical model. Five different nonlinear least squares methods are presented for estimating the transformation parameters. Four gradient based nonlinear least squares methods (Trust region, Trust region dogleg algorithm, Levenberg-Marquardt, and Quasi-Newton line search method) and one non-gradient method (Nelder-Mead simplex direct search) is employed for the six transformation parameters estimation process. The research was done on simulated data and the results showed that the Nelder-Mead method has failed because of its dependency on the objective function without any derivative information. Although, the tested gradient methods succeeded in converging to the relative optimal solution of the georeferencing parameters. In trust region methods, the number of iterations was more than Levenberg-Marquardt because of the necessity for evaluating the local minimum to ensure if it is the global one or not in each iteration step. As for the Levenberg-Marquardt method, which is considered as a modified Gauss-Newton algorithm employing the trust region approach where a scalar is introduced to assess the choice of the magnitude and the direction of the descent. This scalar determines whether the Gauss-Newton method direction or the steepest descent method direction will be used as an adaptive approach for both linear and nonlinear mathematical models and it successfully converged and achieved the relative optimum solution. These five methods results are compared explicitly to the linear traditional least-squares approach, with detailed statistical analysis of the results, with emphasis on the UAV (VBN) applications.

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“…(see, e.g., Aktuğ (2009) ;Aktuğ 2012;Akyilmaz 2007;Arun et al 1987;Burša 1967;Chang 2015;Chang et al 2017;El-Habiby et al 2009;El-Mowafy et al 2009;Grafarend and Awange 2003;Han 2010;Horn 1986Horn , 1987Han and Van Gelder 2006;Horn et al 1988;Jaw and Chuang 2008;Jitka 2011;Kashani 2006;Krarup 1985;Leick 2004;Leick and van Gelder 1975;Mikhail et al 2001;Neitzel 2010;Soler 1998;Soler and Snay 2004;Soycan and Soycan 2008;Teunissen 1986;Teunissen 1988;Wang et al 2014;Závoti and Kalmár 2016;Zeng 2014;Zeng 2015;Zeng et al 2016;Zeng and Yi 2011). Helmert transformation problem is to determine the seven transformation parameters including three rotation angles, three translation parameters and one scale factor using a set of control points (the number of control points should be equal to or more than three because three equations can be constructed for one point).…”

Section: Introductionmentioning

confidence: 99%

“…Numerous algorithms of Helmert transformation have been presented. On the one hand, the algorithms can be classified to a numerical iterative algorithm, e.g., El-Habiby et al (2009), Zeng and Yi (2011), Paláncz et al 2013, Zeng et al (2016, etc., and an analytical algorithm, e.g., Grafarend and Awange (2003), Shen et al (2006a, b), Zeng (2015), etc. For the numerical iterative algorithm, initial values of transformation parameters are usually required.…”

Section: Introductionmentioning

confidence: 99%

A dual quaternion algorithm of the Helmert transformation problem

Zeng

Fang

Chang

et al. 2018

Earth Planets Space

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Rigid transformation including rotation and translation can be elegantly represented by a unit dual quaternion. Thus, a non-differential model of the Helmert transformation (3D seven-parameter similarity transformation) is established based on unit dual quaternion. This paper presents a rigid iterative algorithm of the Helmert transformation using dual quaternion. One small rotation angle Helmert transformation (actual case) and one big rotation angle Helmert transformation (simulative case) are studied. The investigation indicates the presented dual quaternion algorithm (QDA) has an excellent or fast convergence property. If an accurate initial value of scale is provided, e.g., by the solutions no. 2 and 3 of Závoti and Kalmár (Acta Geod Geophys 51: [245][246][247][248][249][250][251][252][253][254][255][256] 2016) in the case that the weights are identical, QDA needs one iteration to obtain the correct result of transformation parameters; in other words, it can be regarded as an analytical algorithm. For other situations, QDA requires two iterations to recover the transformation parameters no matter how big the rotation angles are and how biased the initial value of scale is. Additionally, QDA is capable to deal with point-wise weight transformation which is more rational than those algorithms which simply take identical weights into account or do not consider the weight difference among control points. From the perspective of transformation accuracy, QDA is comparable to the classic Procrustes algorithm (Grafarend and Awange in J Geod 77: [66][67][68][69][70][71][72][73][74][75][76] 2003) and orthonormal matrix algorithm from Zeng (Earth Planets Space 67:105, 2015. https://doi.

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Comparison and Analysis of Non-Linear Least Squares Methods for 3-D Coordinates Transformation

Cited by 12 publications

References 5 publications

Iterative approach of 3D datum transformation with a non-isotropic weight

Iterative approach of 3D datum transformation with a non-isotropic weight

Comparison and Analysis of Nonlinear Least Squares Methods for Vision Based Navigation (Vbn) Algorithms

A dual quaternion algorithm of the Helmert transformation problem

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