A trust region method for implicit orthogonal distance regression

102

PrefaceLeast-squares is a term which implies that the model parameters are determined by minimizing the square sum of some predefined error measures. If a data set is a set of measurement points in 2-D/3-D space, the most natural error measure is the shortest distance between the model feature and the measurement point. This is because the measurement point is a probable observation of the nearest object point to the measurement point. The least-squares model fitting with this error measure is called orthogonal distance fitting (ODF). The square sum of the error distances can be described in two ways: the distance-based cost function and the coordinate-based cost function. The numerical methods minimizing these two cost functions are respectively called the distance-based algorithm and the coordinate-based algorithm.The ODF task has two sub-problems: not only do the model parameters have to be found but also the shortest distance points on a model feature from each given point. In general, the shortest distance point is by nature nonlinear to the model parameters. Thus, the ODF problem is inherently a nonlinear minimization problem solved through iteration. There are two minimization strategies for solving the ODF problems through iteration. The total method simultaneously finds both the model parameters and the shortest distance points, whilst the variable-separation method finds them alternately using a nested iteration scheme. Four combinations of the two numerical methods and the two minimization strategies can be applied. The combination of the distance-based algorithm with the total method results in an underdetermined linear equation system for iteration and is thus impractical for solving ODF problems. As a result, three algorithmic approaches for solving ODF problems are realistic:-Algorithm I: combination of the coordinate-based algorithm with the total method -Algorithm II: combination of the distance-based algorithm with the variableseparation method -Algorithm III: combination of the coordinate-based algorithm with the variableseparation method.This book presents the ODF Algorithms I-III for implicit and parametric curves/ surfaces in 2-D/3-D space possessing the following algorithmic features:-Estimation of the model parameters minimizing the square sum of the shortest distances between a model feature and the given points in 2-D/3-D space -General application to parametric and implicit curves/surfaces VIII Preface -Estimation of the model parameters in terms of form, position, and rotation -General implementation of the distance-based and coordinate-based algorithms -General implementation of the total and variable-separation methods -Solving ODF problems with additional constraints -Provision of parameter-testing possibilities -Robust and fast convergence -Low computing costs and memory space usage -Low implementation costs for a new model feature -Automatic determination of the initial parameter values for iteration.Chapter 1 first reviews the mathematical descriptions of curves and surfaces in s...

Section: Algebraic Fitting Vs Geometric Fittingmentioning

confidence: 99%

Section: State-of-the-art Orthogonal Distance Fittingmentioning

confidence: 99%

Least Squares Orthogonal Distance Fitting of Curves and Surfaces in Space

Ahn¹

2004

102

“…In comparison with Sullivan et al's algorithm [39], our algorithm, including (24) and (27), simultaneously estimates the parameters a in terms of form, position, and rotation parameters. And, Sullivan et al's algorithm obtains its Jacobian matrix unnecessarily expensively from (2).…”

Section: Pumentioning

confidence: 99%

“…And, as we will show in Section 3.1, they have unnecessarily expensively obtained the partial derivatives @db; X i =@b from (2). Helfrich and Zwick described a similar algorithm in [27].…”

mentioning

confidence: 99%

3. Orthogonal Distance Fitting of Implicit Curves and Surfaces

Ahn¹

2004

AbstractÐDimensional model fitting finds its applications in various fields of science and engineering and is a relevant subject in computer/machine vision and coordinate metrology. In this paper, we present two new fitting algorithms, distance-based and coordinate-based algorithm, for implicit surfaces and plane curves, which minimize the square sum of the orthogonal error distances between the model feature and the given data points. Each of the two algorithms has its own advantages and is to be purposefully applied to a specific fitting task, considering the implementation and memory space cost, and possibilities of observation weighting. By the new algorithms, the model feature parameters are grouped and simultaneously estimated in terms of form, position, and rotation parameters. The form parameters determine the shape of the model feature and the position/rotation parameters describe the rigid body motion of the model feature. The proposed algorithms are applicable to any kind of implicit surface and plane curve. In this paper, we also describe algorithm implementation and show various examples of orthogonal distance fit.

“…For fitting of curves and surfaces, orthogonal distance fitting is of primary concern because of the applied error definition, namely the shortest distance from the given point to the model feature [5,9]. While there are orthogonal distance fitting algorithms for explicit [3], and implicit models [2,7] in the literature, we are considering in this paper fitting algorithms for parametric models [4,6,8,10,11] (Fig. 1).…”

Section: Introductionmentioning

confidence: 99%

4. Orthogonal Distance Fitting of Parametric Curves and Surfaces

Ahn¹

2004

Fitting of parametric curves and surfaces to a set of given data points is a relevant subject in various fields of science and engineering. In this paper, we review the current orthogonal distance fitting algorithms for parametric models in a well organized and easily understandable manner, and present a new algorithm. Each of these algorithms estimates the model parameters minimizing the square sum of the error distances between the model feature and the given data points. The model parameters are grouped and simultaneously estimated in terms of form, position, and rotation parameters. The form parameters determine the shape of the model feature, and the position/rotation parameters describe the rigid body motion of the model feature. The new algorithm is applicable to any kind of parametric curve and surface. We give fitting examples for circle, cylinder, and helix in space.