scholarly journals N-D $C^k$ B-Spline Scattered Data Approximation

2005 ◽  
Author(s):  
Nicholas J. Tustison ◽  
James Gee
Keyword(s):  
Function Class ◽  
Data Fitting ◽  
Scattered Data ◽  
Scattered Data Approximation ◽  
Data Approximation ◽  
B Spline ◽  
B Splines ◽  
Scattered Data Fitting ◽  
Spline Surface ◽  
Ill Conditioning

Since the 1970’s B-splines have evolved to become the {} standard for curve and surface representation due to many of their salient properties. Conventional least-squares scattered data fitting techniques for B-splines require the inversion of potentially large matrices. This is time-consuming as well as susceptible to ill-conditioning which leads to undesired results. Lee {} proposed a novel B-spline algorithm for fitting a 2-D cubic B-spline surface to scattered data in . The proposed algorithm utilizes an optional multilevel approach for better fitting results. We generalize this technique to support N-dimensional data fitting as well as arbitrary degree of B-spline. In addition, we generalize the B-spline kernel function class to accommodate this new image filter.

Integration of Moving Least Squares Method and Multilevel B-Spline Method for Scattered Data Approximation

2011 ◽  
Vol 291-294 ◽  
pp. 2245-2249
Author(s):  
Shi Ju Yan ◽  
Bin Ge
Keyword(s):  
Least Squares ◽  
Least Squares Method ◽  
Scattered Data ◽  
Moving Least Squares ◽  
Scattered Data Approximation ◽  
Approximation Accuracy ◽  
Data Approximation ◽  
B Spline ◽  
Moving Least Squares Method ◽  
Method Accuracy

For scattered data approximation with multilevel B-spline(MBS) method, accuracy could be enhanced by densifying control lattice. Nevertheless, when control lattice density reaches to some extent, approximation accuracy could not be enhanced further. A strategy based on integration of moving least squares(MLS) and multilevel B-spline(MBS) is presented. Experimental results demonstrate that the presented strategy has higher approximation accuracy.

Scattered data fitting by minimal surface

2017 ◽  
Vol 25 (3) ◽  
Author(s):  
Yong-Xia Hao ◽  
Dianchen Lu
Keyword(s):  
Tikhonov Regularization ◽  
Numerical Experiments ◽  
Data Fitting ◽  
Scattered Data ◽  
B Spline ◽  
Scattered Data Fitting ◽  
Area Minimization ◽  
Regularization Parameters ◽  
The Given ◽  
Minimal Area

AbstractThe goal of this paper is to develop a computational model for obtaining the fitting surface to the given scattered data with minimal area. The basic idea of the model is to utilize the B-spline and area minimization. The model is turned into a Tikhonov regularization model finally. By choosing the regularization parameters with the L-curve criterion and the GCV method, respectively, numerical experiments indicate that the model can provide an acceptable compromise between the minimization of the data mismatch term and the area of the surface.

scholarly journals Parallel Hierarchical Genetic Algorithm for Scattered Data Fitting through B-Splines

2019 ◽  
Vol 9 (11) ◽  
pp. 2336 ◽  
Author(s):  
Jose Edgar Lara-Ramirez ◽  
Carlos Hugo Garcia-Capulin ◽  
Maria de Jesus Estudillo-Ayala ◽  
Juan Gabriel Avina-Cervantes ◽  
Raul Enrique Sanchez-Yanez ◽  
...  
Keyword(s):  
Genetic Algorithm ◽  
Curve Fitting ◽  
Fitness Function ◽  
Scattered Data ◽  
Model Parameters ◽  
B Spline ◽  
B Splines ◽  
Scattered Data Fitting ◽  
Data Points ◽  
Hierarchical Genetic Algorithm

Curve fitting to unorganized data points is a very challenging problem that arises in a wide variety of scientific and engineering applications. Given a set of scattered and noisy data points, the goal is to construct a curve that corresponds to the best estimate of the unknown underlying relationship between two variables. Although many papers have addressed the problem, this remains very challenging. In this paper we propose to solve the curve fitting problem to noisy scattered data using a parallel hierarchical genetic algorithm and B-splines. We use a novel hierarchical structure to represent both the model structure and the model parameters. The best B-spline model is searched using bi-objective fitness function. As a result, our method determines the number and locations of the knots, and the B-spline coefficients simultaneously and automatically. In addition, to accelerate the estimation of B-spline parameters the algorithm is implemented with two levels of parallelism, taking advantages of the new hardware platforms. Finally, to validate our approach, we fitted curves from scattered noisy points and results were compared through numerical simulations with several methods, which are widely used in fitting tasks. Results show a better performance on the reference methods.

Generation and Modification of Non-Uniform B-Spline Surface Approximations to PDE Surfaces Using the Finite Element Method

1990 ◽  
Author(s):  
Joanna M. Brown ◽  
Malcolm I. G. Bloor ◽  
M. Susan Bloor ◽  
Michael J. Wilson
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Spline Approximation ◽  
The Finite Element Method ◽  
B Spline ◽  
B Splines ◽  
Spline Surface ◽  
Spline Surfaces ◽  
Spline Basis ◽  
Element Method

Abstract A PDE surface is generated by solving partial differential equations subject to boundary conditions. To obtain an approximation of the PDE surface in the form of a B-spline surface the finite element method, with the basis formed from B-spline basis functions, can be used to solve the equations. The procedure is simplest when uniform B-splines are used, but it is also feasible, and in some cases desirable, to use non-uniform B-splines. It will also be shown that it is possible, if required, to modify the non-uniform B-spline approximation in a variety of ways, using the properties of B-spline surfaces.

scholarly journals Hierarchical Genetic Algorithm for B-Spline Surface Approximation of Smooth Explicit Data

2014 ◽  
Vol 2014 ◽  
pp. 1-11 ◽  
Author(s):  
C. H. Garcia-Capulin ◽  
F. J. Cuevas ◽  
G. Trejo-Caballero ◽  
H. Rostro-Gonzalez
Keyword(s):  
Genetic Algorithm ◽  
Geometric Modeling ◽  
Data Set ◽  
Surface Approximation ◽  
B Spline ◽  
B Splines ◽  
Spline Surface ◽  
Spline Surfaces ◽  
Surface Dimension ◽  
Hierarchical Genetic Algorithm

B-spline surface approximation has been widely used in many applications such as CAD, medical imaging, reverse engineering, and geometric modeling. Given a data set of measures, the surface approximation aims to find a surface that optimally fits the data set. One of the main problems associated with surface approximation by B-splines is the adequate selection of the number and location of the knots, as well as the solution of the system of equations generated by tensor product spline surfaces. In this work, we use a hierarchical genetic algorithm (HGA) to tackle the B-spline surface approximation of smooth explicit data. The proposed approach is based on a novel hierarchical gene structure for the chromosomal representation, which allows us to determine the number and location of the knots for each surface dimension and the B-spline coefficients simultaneously. The method is fully based on genetic algorithms and does not require subjective parameters like smooth factor or knot locations to perform the solution. In order to validate the efficacy of the proposed approach, simulation results from several tests on smooth surfaces and comparison with a successful method have been included.

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