A Novel Approach to Preserving Geometric Constraints in Curve/Surface MRA Modeling

Author(s):  
Jinxiang Yin ◽  
Guanlong Chen

MRA editing destroys the geometric constraints of curves/surfaces, a high efficient algorithm called wavelet signal separation to preserve geometric constraints during wavelet transformation in MRA modeling is proposed in this paper. The algorithm divides the B-spline control points into those associated and those unassociated with the geometric constraints. Through preserving the signal information associated with the constraint control points, the geometric constraints can be directly preserved after wavelet transformation. The detailed mathematics for this approach is presented. Different examples are included to demonstrate the power of the approach.

2013 ◽  
Vol 411-414 ◽  
pp. 523-526
Author(s):  
Xiao Bing Chen ◽  
Kun Yu

In order to obtain B-spline curve with fewer control points and higher precision, an efficient algorithm for B-spline curve fitting by using feature data points is proposed. During iterations of the proposed algorithm, the projected points, which are the nearest points on fitting curve to discrete data points, are calculated first, then maximal deviations between B-spline curve and connection lines of the data points are controlled, finally new feature points are determined and parameters of feature points are adjusted by parameters of projected points. According to these, B-spline curve with fewer control points and higher precision are obtained rapidly. Experimental result indicates that the proposed algorithm is feasible and effective.


Author(s):  
Wei Zhao ◽  
Shuming Gao ◽  
Yusheng Liu

Due to the complexity, reuse of freeform features represented by B-spline is still an open issue. Based on Poisson equation, a novel approach to the reuse of freeform features represented by uniform rational B-spline (URBS) is proposed in this paper. In order to effectively support reuse, a new representation of freeform features is put forwarded, which is based on a new property about the odd URBS and consists of the principal control points, geometry context and basic surface of the feature. Based on the new representation, reuse of freeform features is achieved by updating the principal control points on the pasted area of the target surface using Poisson equation. The approach is implemented and some examples are given.


Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 1054
Author(s):  
Rozaimi Zakaria ◽  
Abd. Fatah Wahab ◽  
Isfarita Ismail ◽  
Mohammad Izat Emir Zulkifly

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.


2020 ◽  
Vol 10 (1) ◽  
pp. 110-123
Author(s):  
Gaël Kermarrec ◽  
Hamza Alkhatib

Abstract B-spline curves are a linear combination of control points (CP) and B-spline basis functions. They satisfy the strong convex hull property and have a fine and local shape control as changing one CP affects the curve locally, whereas the total number of CP has a more general effect on the control polygon of the spline. Information criteria (IC), such as Akaike IC (AIC) and Bayesian IC (BIC), provide a way to determine an optimal number of CP so that the B-spline approximation fits optimally in a least-squares (LS) sense with scattered and noisy observations. These criteria are based on the log-likelihood of the models and assume often that the error term is independent and identically distributed. This assumption is strong and accounts neither for heteroscedasticity nor for correlations. Thus, such effects have to be considered to avoid under-or overfitting of the observations in the LS adjustment, i.e. bad approximation or noise approximation, respectively. In this contribution, we introduce generalized versions of the BIC derived using the concept of quasi- likelihood estimator (QLE). Our own extensions of the generalized BIC criteria account (i) explicitly for model misspecifications and complexity (ii) and additionally for the correlations of the residuals. To that aim, the correlation model of the residuals is assumed to correspond to a first order autoregressive process AR(1). We apply our general derivations to the specific case of B-spline approximations of curves and surfaces, and couple the information given by the different IC together. Consecutively, a didactical yet simple procedure to interpret the results given by the IC is provided in order to identify an optimal number of parameters to estimate in case of correlated observations. A concrete case study using observations from a bridge scanned with a Terrestrial Laser Scanner (TLS) highlights the proposed procedure.


Author(s):  
Yuan Yuan ◽  
Shiyu Zhou

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.


2009 ◽  
Vol 8 (2) ◽  
pp. 101-107 ◽  
Author(s):  
Zhongke Wu ◽  
Mingquan Zhou ◽  
Xingce Wang

A novel approach to modeling realistic tree easily through interactive methods based on ball B-Spline Curves (BBSCs) and an efficient graph based data structure of tree model is proposed in the paper. As BBSCs are flexible for modifying, deforming and editing, these methods provide intuitive interaction and more freedom for users to model trees. If conjuncted with other methods like generating tree models through L-systems or iterated function systems (IFS), the models are more realistic and natural through modifying and editing. The method can be applied to the design of bonsai tree models.


1994 ◽  
Vol 6 (6) ◽  
pp. 491-498 ◽  
Author(s):  
Hiroaki Ozaki ◽  
◽  
Hua Chiu ◽  

A basic optimization algorithm is presented in this paper, in order to obtain the optimum solution of a two-point boundary value variational problem without constraints. The solution is given by a parallel and iterative computation and described as a set of control points of a uniform B-spline. This algorithm can also be applied to solving problems with some constraints, if we introduce an additional component, namely the potential function, corresponding to constraints in the original objective function. The algorithm is very simple and easily applicable to various engineering problems. As an application, trajectory planning of a manipulator with redundant degrees of freedom is considered under the conditions that the end effector path, the smoothness of movement, and the constraints of the control or the state variables are specified. The validity of the algorithm is well confirmed by numerical examples.


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