On Algorithms and Heuristics for Constrained Least-Squares Fitting of Circles and Spheres to Support Standards

2018 ◽  
Author(s):  
Craig M. Shakarji ◽  
Vijay Srinivasan
Keyword(s):  
Least Squares ◽  
Theoretical Work ◽  
Two Dimensions ◽  
International Standards ◽  
Three Dimensions ◽  
Constrained Least Squares ◽  
Least Squares Fitting ◽  
Practical Algorithms ◽  
Coordinate Metrology ◽  
Set Of Points

Constrained least-squares fitting has gained considerable popularity among national and international standards committees as the default method for establishing datums on manufactured parts. This has resulted in the emergence of several interesting and urgent problems in computational coordinate metrology. Among them is the problem of fitting inscribing and circumscribing circles (in two-dimensions) and spheres (in three-dimensions) using constrained least-squares criterion to a set of points that are usually described as a ‘point-cloud.’ This paper builds on earlier theoretical work, and provides practical algorithms and heuristics to compute such circles and spheres. Representative codes that implement these algorithms and heuristics are also given to encourage industrial use and rapid adoption of the emerging standards.

On Algorithms and Heuristics for Constrained Least-Squares Fitting of Circles and Spheres to Support Standards

2019 ◽  
Vol 19 (3) ◽  
Author(s):  
Craig M. Shakarji ◽  
Vijay Srinivasan
Keyword(s):  
Least Squares ◽  
Theoretical Work ◽  
Two Dimensions ◽  
International Standards ◽  
Three Dimensions ◽  
Constrained Least Squares ◽  
Least Squares Fitting ◽  
Practical Algorithms ◽  
Coordinate Metrology ◽  
Set Of Points

Constrained least-squares fitting has gained considerable popularity among national and international standards committees as the default method for establishing datums on manufactured parts. This has resulted in the emergence of several interesting and urgent problems in computational coordinate metrology. Among them is the problem of fitting inscribing and circumscribing circles (in two dimensions) and spheres (in three dimensions) using constrained least-squares criterion to a set of points that are usually described as a “point-cloud.” This paper builds on earlier theoretical work, and provides practical algorithms and heuristics to compute such circles and spheres. Representative codes that implement these algorithms and heuristics are also given to encourage industrial use and rapid adoption of the emerging standards.

Convexity and Optimality Conditions for Constrained Least-Squares Fitting of Planes and Parallel Planes to Establish Datums

2018 ◽  
Vol 19 (1) ◽  
Author(s):  
Craig M. Shakarji ◽  
Vijay Srinivasan
Keyword(s):  
Least Squares ◽  
Optimality Conditions ◽  
Global Optimum ◽  
Objective Functions ◽  
Constrained Least Squares ◽  
Least Squares Fitting ◽  
Optimum Solution ◽  
Set Of Points ◽  
Theoretical Issues ◽  
Global Optimum Solution

This paper addresses some important theoretical issues for constrained least-squares fitting of planes and parallel planes to a set of points. In particular, it addresses the convexity of the objective function and the combinatorial characterizations of the optimality conditions. These problems arise in establishing planar datums and systems of planar datums in digital manufacturing. It is shown that even when the set of points (i.e., the input points) are in general position, (1) a primary planar datum can contact 1, 2, or 3 input points, (2) a secondary planar datum can contact 1 or 2 input points, and (3) two parallel planes can each contact 1, 2, or 3 input points, but there are some constraints to these combinatorial counts. In addition, it is shown that the objective functions are convex over the domains of interest. The optimality conditions and convexity of objective functions proved in this paper will enable one to verify whether a given solution is a feasible solution, and to design efficient algorithms to find the global optimum solution.

scholarly journals PARALLEL DELAUNAY REFINEMENT: ALGORITHMS AND ANALYSES

2007 ◽  
Vol 17 (01) ◽  
pp. 1-30 ◽  
Author(s):  
DANIEL A. SPIELMAN ◽  
SHANG-HUA TENG ◽  
ALPER ÜNGÖR
Keyword(s):  
Mesh Size ◽  
Constant Factor ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Delaunay Refinement ◽  
Steiner Points ◽  
Input Domain ◽  
Set Of Points ◽  
Refinement Algorithm

We present a parallel Delaunay refinement algorithm for generating well-shaped meshes in both two and three dimensions. Like its sequential counterparts, the parallel algorithm iteratively improves the quality of a mesh by inserting new points, the Steiner points, into the input domain while maintaining the Delaunay triangulation. The Steiner points are carefully chosen from a set of candidates that includes the circumcenters of poorly-shaped triangular elements. We introduce a notion of independence among possible Steiner points that can be inserted simultaneously during Delaunay refinements and show that such a set of independent points can be constructed efficiently and that the number of parallel iterations is O( log 2Δ), where Δ is the spread of the input — the ratio of the longest to the shortest pairwise distances among input features. In addition, we show that the parallel insertion of these set of points can be realized by sequential Delaunay refinement algorithms such as by Ruppert's algorithm in two dimensions and Shewchuk's algorithm in three dimensions. Therefore, our parallel Delaunay refinement algorithm provides the same shape quality and mesh-size guarantees as these sequential algorithms. For generating quasi-uniform meshes, such as those produced by Chew's algorithms, the number of parallel iterations is in fact O( log Δ). To the best of our knowledge, our algorithm is the first provably polylog(Δ) time parallel Delaunay-refinement algorithm that generates well-shaped meshes of size within a constant factor of the best possible.

scholarly journals Meshless Local Petrov–Galerkin Formulation of Inverse Stefan Problem via Moving Least Squares Approximation

2019 ◽  
Vol 24 (4) ◽  
pp. 101
Author(s):  
A. Karami ◽  
Saeid Abbasbandy ◽  
E. Shivanian
Keyword(s):  
Free Boundary ◽  
Least Squares ◽  
Boundary Problem ◽  
Moving Boundary ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Moving Least Squares ◽  
Heaviside Step Function ◽  
Mlpg Method ◽  
Mls Approximation

In this paper, we study the meshless local Petrov–Galerkin (MLPG) method based on the moving least squares (MLS) approximation for finding a numerical solution to the Stefan free boundary problem. Approximation of this problem, due to the moving boundary, is difficult. To overcome this difficulty, the problem is converted to a fixed boundary problem in which it consists of an inverse and nonlinear problem. In other words, the aim is to determine the temperature distribution and free boundary. The MLPG method using the MLS approximation is formulated to produce the shape functions. The MLS approximation plays an important role in the convergence and stability of the method. Heaviside step function is used as the test function in each local quadrature. For the interior nodes, a meshless Galerkin weak form is used while the meshless collocation method is applied to the the boundary nodes. Since MLPG is a truly meshless method, it does not require any background integration cells. In fact, all integrations are performed locally over small sub-domains (local quadrature domains) of regular shapes, such as intervals in one dimension, circles or squares in two dimensions and spheres or cubes in three dimensions. A two-step time discretization method is used to deal with the time derivatives. It is shown that the proposed method is accurate and stable even under a large measurement noise through several numerical experiments.

On the Enduring Appeal of Least-Squares Fitting in Computational Coordinate Metrology

2011 ◽  
Vol 12 (1) ◽  
Author(s):  
Vijay Srinivasan ◽  
Craig M. Shakarji ◽  
Edward P. Morse
Keyword(s):  
Least Squares ◽  
Geometric Features ◽  
Coordinate Measuring Machines ◽  
Historical Survey ◽  
Least Squares Fitting ◽  
Coordinate Metrology ◽  
International Standardization ◽  
Manufacturing Applications ◽  
Fitting In ◽  
Current Practices

The vast majority of points collected with coordinate measuring machines are not used in isolation; rather, collections of these points are associated with geometric features through fitting routines. In manufacturing applications, there are two fundamental questions that persist about the efficacy of this fitting—first, do the points collected adequately represent the surface under inspection; and second, does the association of substitute (fitted) geometry with the points meet criteria consistent with the standardized geometric specification of the product. This paper addresses the second question for least-squares fitting both as a historical survey of past and current practices, and as a harbinger of the influence of new specification criteria under consideration for international standardization. It also touches upon a set of new issues posed by the international standardization on the first question as related to sampling and least-squares fitting.

scholarly journals On parameter estimation in the bass model by nonlinear least squares fitting the adoption curve

2013 ◽  
Vol 23 (1) ◽  
pp. 145-155 ◽  
Author(s):  
Darija Marković ◽  
Dragan Jukić
Keyword(s):  
Parameter Estimation ◽  
Least Squares ◽  
Weighted Least Squares ◽  
Sufficient Conditions ◽  
Marketing Research ◽  
Nonlinear Least Squares ◽  
Theoretical Work ◽  
Bass Model ◽  
Least Squares Fitting ◽  
Least Squares Estimate

The Bass model is one of the most well-known and widely used first-purchase diffusion models in marketing research. Estimation of its parameters has been approached in the literature by various techniques. In this paper, we consider the parameter estimation approach for the Bass model based on nonlinear weighted least squares fitting of its derivative known as the adoption curve. We show that it is possible that the least squares estimate does not exist. As a main result, two theorems on the existence of the least squares estimate are obtained, as well as their generalization in the ls norm (1 ≤ s < ∞). One of them gives necessary and sufficient conditions which guarantee the existence of the least squares estimate. Several illustrative numerical examples are given to support the theoretical work.

Rejecting outliers and estimating errors in an orthogonal-regression framework

1995 ◽  
Vol 350 (1694) ◽  
pp. 407-439 ◽  
Keyword(s):  
Least Squares ◽  
Ordinary Least Squares ◽  
Two Dimensions ◽  
Three Dimensions ◽  
Matrix Perturbation ◽  
Data Set ◽  
Orthogonal Regression ◽  
Matrix Perturbation Theory ◽  
Alternative Techniques ◽  
Regression Diagnostic

Least squares minimization is by nature global and, hence, vulnerable to distortion by outliers. We present a novel technique to reject outliers from an m -dimensional data set when the underlying model is a hyperplane (a line in two dimensions, a plane in three dimensions). The technique has a sound statistical basis and assumes that Gaussian noise corrupts the otherwise valid data. The majority of alternative techniques available in the literature focus on ordinary least squares , where a single variable is designated to be dependent on all others - a model that is often unsuitable in practice. The method presented here operates in the more general framework of orthogonal regression , and uses a new regression diagnostic based on eigendecomposition. It subsumes the traditional residuals scheme and, using matrix perturbation theory, provides an error model for the solution once the contaminants have been removed.

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