Sequential B-Spline Surface Construction Using Multiresolution Data Clouds

2012 ◽  
Vol 12 (2) ◽  
Author(s):  
Yuan Yuan ◽  
Shiyu Zhou
Keyword(s):  
Control Points ◽  
Finite Sample ◽  
B Spline ◽  
Spline Surface ◽  
Spline Surfaces ◽  
Filtering Technique ◽  
Engineering Practices ◽  
Asymptotical Convergence ◽  
Surface Construction ◽  
Fitting Error

B-spline surfaces are widely used in engineering practices as a flexible and efficient mathematical model for product design, analysis, and assessment. In this paper, we propose a new sequential B-spline surface construction procedure using multiresolution measurements. At each iterative step of the proposed procedure, we first update knots vectors based on bias and variance decomposition of the fitting error and then incorporate new data into the current surface approximation to fit the control points using Kalman filtering technique. The asymptotical convergence property of the proposed procedure is proved under the framework of sieves method. Using numerical case studies, the effectiveness of the method under finite sample is tested and demonstrated.

Filling N-Sided Holes With Trimmed B-Spline Surfaces Based on Energy-Minimization Method

2015 ◽  
Vol 15 (1) ◽  
Author(s):  
Xiaodong Liu
Keyword(s):  
Energy Minimization ◽  
Computer Aided Design ◽  
Control Points ◽  
Minimization Method ◽  
B Spline ◽  
Spline Surface ◽  
Spline Surfaces ◽  
Key Issues ◽  
Complex Constraints ◽  
Aided Design

Using a trimmed rectangular B-Spline surface to fill an n-sided hole is a much desired operation in computer aided design (CAD), but few papers have addressed this issue. Based on an energy-minimization or variational B-Spline technique, the paper presents the technique of using one single trimmed rectangular B-Spline surface to fill an n-sided hole. The method is efficient and robust, and takes a fraction of a second to fill n-sided holes with high-quality waterproof B-Spline surfaces under complex constraints. As the foundation of filling n-sided holes, the paper also presents the framework and addresses the key issues on variational B-Spline technique. Without any precalculation, the variational B-Spline technique discussed in this paper can solve virtually any B-Spline surface with up to 20,000 control points in real time, which is much more efficient and powerful than previous work in the variational B-Spline field. Moreover, the result is accurate and satisfies CAD systems' high-precision requirements.

Filling N-Sided Holes Using Trimmed B-Spline Surfaces

2012 ◽  
Author(s):  
Xiaodong Liu
Keyword(s):  
Energy Minimization ◽  
Control Points ◽  
Variational Technique ◽  
High Quality ◽  
B Spline ◽  
Cad Systems ◽  
Spline Surface ◽  
Spline Surfaces ◽  
Key Issues ◽  
Complex Constraints

Using one single trimmed B-Spline surface to fill an n-sided hole is a much desired operation in CAD, but few papers have addressed this issue. The paper presents the method of using trimmed B-Spline surfaces to fill n-sided holes based on energy minimization or variational technique. The method is efficient and robust, and takes less than one second to fill n-sided holes with high quality B-Spline surfaces under complex constraints. As the foundation of filling n-sided holes, some key issues on variational B-Spline technique are also discussed. The variational technique discussed is significantly much more efficient and powerful than previous research, and the result is very accurate to satisfy CAD systems’ high-precision requirements. We demonstrate that, without any pre-calculation, the discussed technique is efficient enough to solve a B-Spline surface with up to 20,000 control points in real time while satisfying an arbitrary combination of point and curve constraints.

B2: An Interactive Quality Visualization and Improvement Technique for B-Spline Surfaces

1998 ◽  
Author(s):  
Yifan Chen ◽  
Klaus-Peter Beier
Keyword(s):  
Surface Quality ◽  
Control Points ◽  
Small Surface ◽  
B Spline ◽  
Original Surface ◽  
Interactive Technique ◽  
Spline Surface ◽  
Spline Surfaces ◽  
Surface Irregularities

Abstract A new interactive technique for B-spline surface quality visualization and improvement, called the B2 method, is presented. This method interpolates the control points of a given B-spline surface using a second B-spline surface. If small irregularities exist in the control points of the original surface, they will be magnified through the second B-spline and demonstrated as large distortions in its control points. This facilitates the detection of small surface irregularities. Subsequently, the surface may be improved through direct and interactive adjustment of the second B-spline’s control polyhedron.

Designing B-Spline Surfaces With Haptics Based on Energy Minimization Method

2012 ◽  
Author(s):  
Xiaodong Liu
Keyword(s):  
Real Time ◽  
Energy Minimization ◽  
Haptic Interface ◽  
Control Points ◽  
Minimization Method ◽  
B Spline ◽  
System A ◽  
Spline Surface ◽  
Spline Surfaces ◽  
Energy Minimization Method

Designing B-Spline Surfaces is difficult and cumbersome with traditional 2D based interfaces, e.g., a 2D mouse. Based on energy minimization or variational surfacing technique, the paper presents a technique for using a haptic device to design B-Spline surfaces. In our system, a haptic interface is used to directly manipulate/design key constraints in a natural 3D environment, and the system uses energy minimization method to generate the “smoothest” B-Spline surface that satisfies these constraints in real-time. The discussed technique is significantly much more efficient and powerful than previous research, and without any pre-calculation, it can solve a B-Spline surface with up to 20,000 control points in real-time and, at the same time, produce high quality B-Spline surfaces satisfying an arbitrary combination of point and curve constraints.

scholarly journals N-dimensional B-spline surface estimated by lofting for locally improving IRI

2011 ◽  
Vol 1 (1) ◽  
pp. 41-51 ◽  
Author(s):  
K. Koch ◽  
M. Schmidt
Keyword(s):  
Computer Time ◽  
Simultaneous Estimation ◽  
Electron Content ◽  
Control Points ◽  
Total Electron ◽  
B Spline ◽  
Spline Surface ◽  
Spline Surfaces ◽  
Unknown Control ◽  
Dimensional Surface

N-dimensional B-spline surface estimated by lofting for locally improving IRIN-dimensional surfaces are defined by the tensor product of B-spline basis functions. To estimate the unknown control points of these B-spline surfaces, the lofting method also called skinning method by cross-sectional curve fits is applied. It is shown by an analytical proof and numerically confirmed by the example of a four-dimensional surface that the results of the lofting method agree with the ones of the simultaneous estimation of the unknown control points. The numerical complexity for estimating vn control points by the lofting method is O(vn+1) while it results in O(v3n) for the simultaneous estimation. It is also shown that a B-spline surface estimated by a simultaneous estimation can be extended to higher dimensions by the lofting method, thus saving computer time.An application of this method is the local improvement of the International Reference Ionosphere (IRI), e.g. by the slant total electron content (STEC) obtained by dual-frequency observations of the Global Navigation Satellite System (GNSS). Three-dimensional B-spline surfaces at different time epochs have to be determined by the simultaneous estimation of the control points for this improvement. A four-dimensional representation in space and time of the electron density of the ionosphere is desirable. It can be obtained by the lofting method. This takes less computer time than determining the four-dimensional surface solely by a simultaneous estimation.

scholarly journals Complex Uncertainty of Surface Data Modeling via the Type-2 Fuzzy B-Spline Model

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 1054
Author(s):  
Rozaimi Zakaria ◽  
Abd. Fatah Wahab ◽  
Isfarita Ismail ◽  
Mohammad Izat Emir Zulkifly
Keyword(s):  
Complex Surface ◽  
Complex Data ◽  
Fuzzy Data ◽  
Control Points ◽  
B Spline ◽  
Spline Surface ◽  
Data Control ◽  
Surface Data ◽  
Model Complex

This paper discusses the construction of a type-2 fuzzy B-spline model to model complex uncertainty of surface data. To construct this model, the type-2 fuzzy set theory, which includes type-2 fuzzy number concepts and type-2 fuzzy relation, is used to define the complex uncertainty of surface data in type-2 fuzzy data/control points. These type-2 fuzzy data/control points are blended with the B-spline surface function to produce the proposed model, which can be visualized and analyzed further. Various processes, namely fuzzification, type-reduction and defuzzification are defined to achieve a crisp, type-2 fuzzy B-spline surface, representing uncertainty complex surface data. This paper ends with a numerical example of terrain modeling, which shows the effectiveness of handling the uncertainty complex data.

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.

Choosing the optimal number of B-spline control points (Part 2: Approximation of surfaces and applications)

2017 ◽  
Vol 11 (1) ◽  
Author(s):  
Corinna Harmening ◽  
Hans Neuner
Keyword(s):  
Data Analysis ◽  
Laser Scanner ◽  
Point Clouds ◽  
Optimal Number ◽  
Deformation Analysis ◽  
Control Points ◽  
Trial And Error ◽  
B Spline ◽  
B Splines ◽  
Spline Surfaces

AbstractFreeform surfaces like B-splines have proven to be a suitable tool to model laser scanner point clouds and to form the basis for an areal data analysis, for example an areal deformation analysis.A variety of parameters determine the B-spline's appearance, the B-spline's complexity being mostly determined by the number of control points. Usually, this parameter type is chosen by intuitive trial-and-error-procedures.In [The present paper continues these investigations. If necessary, the methods proposed in [The application of those methods to B-spline surfaces reveals the datum problem of those surfaces, meaning that location and number of control points of two B-splines surfaces are only comparable if they are based on the same parameterization. First investigations to solve this problem are presented.

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