Orhan Kurt

Bio: Orhan Kurt is an academic researcher from Kocaeli University. The author has contributed to research in topics: Least squares & Non-linear least squares. The author has an hindex of 2, co-authored 8 publications receiving 12 citations.

An integrated solution for reducing ill-conditioning and testing the results in non-linear 3D similarity transformations

Abstract: To find 3D similarity transformation parameters (a scale, three rotational angles, and three translation elements) between two orthogonal coordinate systems in 3D is an ill-posed non-linear inverse problem by means of common points (their Cartesian components are known in the both systems). The problem can be solved via Linearized Least Squares (LLS) or Direct (non-iterative) Least Squares (DLS). Since the parameters in LLS take different quantities (and units) from each other, the condition problems can arise during the solution of normal equations. In this paper, we propose a combined solution to reducing ill-conditioning and to perform precision analysis and global outlier test in LLS accordingly. The way is based on column norming and uses the normalized unknowns instead of the original ones at the solution stage of the normal equations. While the global outlier test is fulfilled on the normalized unknowns, the original unknowns and their precisions obtained using the normalized matrix with li...

Determination of the Geometric Form of Hierapolis Theater in Pamukkale, Turkey

Abstract: This work addresses the issue of fitting a geometrical shape to the Hierapolis theater near the city of Denizli, Turkey. The aim of the study was to determine the geometrical shape of the theater that would best fit the set of measured site data points. The geometrical form of the theater was surveyed by using an electronic total station instrument. Because theater rings are generally elliptical in shape, the well-known ellipse-fitting algorithm was presented for computing the geometrical parameters. In this study, the authors plan to analyze the fitting algorithm presented, determine the best-fitting geometrical shape (e.g., an ellipse, a circle) to the Hierapolis theater, and characterize its features. Because the fitting accuracies are not sufficient to decide which geometrical shape is appropriate, a statistical testing procedure was applied to identify statistical differences between the related geometries. The most probable geometrical shapes for this purpose were determined with the use of ...

Geometric interpretation and precision analysis of algebraic ellipse fitting using least squares method

Abstract: This paper presents a new approach for precision estimation for algebraic ellipse fitting based on combined least squares method. Our approach is based on coordinate description of the ellipse geometry to determine the error distances of the fitting method. Since it is an effective fitting algorithm the well-known Direct Ellipse Fitting method was selected as an algebraic method for precision estimation. Once an ellipse fitted to the given data points, algebraic distance residuals for each data point and fitting accuracy can be computed. Generally, the adopted approach has revealed geometrical aspect of precision estimation for algebraic ellipse fitting. The experimental results revealed that our approach might be a good choice for precision estimation of the ellipse fitting method.

On ill-conditioning in 3D similarity transformation

Abstract: D similarity transformations are often used for datum transformation in Geomatic Engineering. The transformations are routinely performed between the point coordinates evaluated from GNSS (Global Navigation Satellite Systems) observations and the point coordinates given in national datum. A mathematical model established among two different 3D coordinate systems contains tree type datum parameters which are translations, rotations and a scale. Coefficients of the parameters reflect large variations from each other's. So, the unknown coefficient matrix in normal equations is to be ill-conditioned. In this study, it is indicated how the ill-condition structure is reduced to acceptable level by means of choosing at different unit of unknown parameters. Finally, the transformation from Kocaeli Metropolitan Municipality GNSS network coordinates to Turkish Geodetic Datum is used as a numerical example.

Least squares evaluations for form and profile errors of ellipse using coordinate data

Abstract: To improve the measurement and evaluation of form error of an elliptic section, an evaluation method based on least squares fitting is investigated to analyze the form and profile errors of an ellipse using coordinate data. Two error indicators for defining ellipticity are discussed, namely the form error and the profile error, and the difference between both is considered as the main parameter for evaluating machining quality of surface and profile. Because the form error and the profile error rely on different evaluation benchmarks, the major axis and the foci rather than the centre of an ellipse are used as the evaluation benchmarks and can accurately evaluate a tolerance range with the separated form error and profile error of workpiece. Additionally, an evaluation program based on the LS model is developed to extract the form error and the profile error of the elliptic section, which is well suited for separating the two errors by a standard program. Finally, the evaluation method about the form and profile errors of the ellipse is applied to the measurement of skirt line of the piston, and results indicate the effectiveness of the evaluation. This approach provides the new evaluation indicators for the measurement of form and profile errors of ellipse, which is found to have better accuracy and can thus be used to solve the difficult of the measurement and evaluation of the piston in industrial production.

Iterative solution of Helmert transformation based on a unit dual quaternion

Abstract: The rigid motion involving both rotation and translation in the 3D space can be simultaneously described by a unit dual quaternion. Considering this excellent property, the paper constructs the Helmert transformation (seven-parameter similarity transformation) model based on a unit dual quaternion and then presents a rigid iterative algorithm of Helmert transformation using a unit dual quaternion. Because of the singularity of the coefficient matrix of the normal equation, the nine parameter (including one scale factor and eight parameters of a dual quaternion) Helmert transformation model is reduced into five parameter (including one scale factor and four parameters of a unit quaternion which can represent the rotation matrix) Helmert transformation one. Besides, a good start estimate of parameter is required for the iterative algorithm, hence another algorithm employed to compute the initial value of parameter is put forward. The numerical experiments involving a case of small rotation angles i.e. geodetic coordinate transformation and a case of big rotation angles i.e. the registration of LIDAR points are studied. The results show the presented algorithms in this paper are correct and valid for the two cases, disregarding the rotation angles are big or small. And the accuracy of computed parameter is comparable to the classic Procrustes algorithm from Grafarend and Awange (J Geod 77:66–76, 2003), the orthonormal matrix algorithm from Zeng (Earth Planets Space 67:105, 2015), and the algorithm from Wang et al. (J Photogramm Remote Sens 94:63–69, 2014).

WTLS iterative algorithm of 3D similarity coordinate transformation based on Gibbs vectors

Abstract: The 3D similarity coordinate transformation is fundamental and frequently encountered in many areas of work such as geodesy, engineering surveying, LIDAR, terrestrial laser scanning, photogrammetry, machine vision, etc. The algorithms of 3D similarity transformation are divided into two categories. One is a closed-form algorithm that is straightforward and fast. However, it cannot provide the accuracy information for the transformation parameters. The other category of algorithm is iterative, and this can offer the accuracy information for the transformation parameters. However, the latter usually needs a good initial value of the unknown. Considering the accuracy information for transformation parameters is essential or indispensable from the viewpoint of uncertainty, this contribution proposes a weighted total least squares (WTLS) iterative algorithm of the 3D similarity coordinate transformation based on Gibbs vectors. It is fast in terms of fewer iterations, reliable and does not need good initial values of transformation parameters. Two cases including the registration of LIDAR points with big rotation angles and a geodetic datum transformation with small rotation angles are demonstrated to validate the new algorithm.

An integrated solution for reducing ill-conditioning and testing the results in non-linear 3D similarity transformations

Abstract: To find 3D similarity transformation parameters (a scale, three rotational angles, and three translation elements) between two orthogonal coordinate systems in 3D is an ill-posed non-linear inverse problem by means of common points (their Cartesian components are known in the both systems). The problem can be solved via Linearized Least Squares (LLS) or Direct (non-iterative) Least Squares (DLS). Since the parameters in LLS take different quantities (and units) from each other, the condition problems can arise during the solution of normal equations. In this paper, we propose a combined solution to reducing ill-conditioning and to perform precision analysis and global outlier test in LLS accordingly. The way is based on column norming and uses the normalized unknowns instead of the original ones at the solution stage of the normal equations. While the global outlier test is fulfilled on the normalized unknowns, the original unknowns and their precisions obtained using the normalized matrix with li...