A generalized non-stationary curve subdivision （GNS for short） scheme of arbitrary order k≥3 with a parameter has been proposed by Fang et al. in the paper （Fang Mei-e et al., CAGD, 2010（27）： 720-733）. It has ...A generalized non-stationary curve subdivision （GNS for short） scheme of arbitrary order k≥3 with a parameter has been proposed by Fang et al. in the paper （Fang Mei-e et al., CAGD, 2010（27）： 720-733）. It has been proved that the proposed scheme of order k generates C^k-2 continuous curves for k≥4. But the proof of the smoothness in this paper is uncompleted. Moreover, the Cl-continuity of the third order scheme has not been discussed. For this reason, in this paper, we provide a full corrected proof of the smoothness of the GNS scheme of order k for k≥3.展开更多
Currently, the approximation methods of the Gaussian filter by some other spline filters have been developed. However, thesc methods are only suitable for the study of one-dimensional filtering, when these methods are...Currently, the approximation methods of the Gaussian filter by some other spline filters have been developed. However, thesc methods are only suitable for the study of one-dimensional filtering, when these methods are used for three-dimensional filtering, it is found that a rounding error and quantization error would be passed to the next in every part. In this paper, a new and high-precision implementation approach for Gaussian filter is described, which is suitable for three-dimensional reference filtering. Based on the theory of generalized B-spline function and the variational principle, the transmission characteristics of a digital filter can be changed through the sensitivity of the parameters （t1, t2）, and which can also reduce the rounding error and quantization error by the filter in a parallel form instead of the cascade form, Finally, the approximation filter of Gaussian filter is obtained. In order to verify the feasibility of the new algorithm, the reference extraction of the conventional methods are also used and compared. The experiments are conducted on the measured optical surface, and the results show that the total calculation by the new algorithm only requires 0.07 s for 480×480 data points; the amplitude deviation between the reference of the parallel form filter and the Gaussian filter is smaller; the new method is closer to the characteristic of the Gaussian filter through the analysis of three-dimensional roughness parameters, comparing with the cascade generalized B-spline approximating Gaussian. So the new algorithm is also efficient and accurate for the implementation of Gaussian filter in the application of surface roughness measurement.展开更多
ωB-splines have many optimal properties and can reproduce plentiful commonly-used analytical curves.In this paper,we further propose a non-stationary subdivision method of hierarchically and efficiently generatingωB...ωB-splines have many optimal properties and can reproduce plentiful commonly-used analytical curves.In this paper,we further propose a non-stationary subdivision method of hierarchically and efficiently generatingωB-spline curves of arbitrary order ofωB-spline curves and prove its C^k?2-continuity by two kinds of methods.The first method directly prove that the sequence of control polygons of subdivision of order k converges to a C^k?2-continuousωB-spline curve of order k.The second one is based on the theories upon subdivision masks and asymptotic equivalence etc.,which is more convenient to be further extended to the case of surface subdivision.And the problem of approximation order of this non-stationary subdivision scheme is also discussed.Then a uniform ωB-spline curve has both perfect mathematical representation and efficient generation method,which will benefit the application ofωB-splines.展开更多
A family of Said-Bézier type generalized Ball （SBGB） bases and surfaces with a parameter H over triangular domain is introduced,which unifies Bézier surface and Said-Ball surface and includes several inter...A family of Said-Bézier type generalized Ball （SBGB） bases and surfaces with a parameter H over triangular domain is introduced,which unifies Bézier surface and Said-Ball surface and includes several intermediate surfaces. To convert different bases and surfaces,the dual functionals of bases are presented. As an application of dual functionals,the subdivision formulas for surfaces are established.展开更多
In this work, we have obtained numerical solutions of the generalized Korteweg-de Vries (GKdV) equation by using septic B-spline collocation finite element method. The suggested numerical algorithm is controlled by ap...In this work, we have obtained numerical solutions of the generalized Korteweg-de Vries (GKdV) equation by using septic B-spline collocation finite element method. The suggested numerical algorithm is controlled by applying test problems including;single soliton wave. Our numerical algorithm, attributed to a Crank Nicolson approximation in time, is unconditionally stable. To control the performance of the newly applied method, the error norms, <em>L</em><sub>2</sub> and <em>L</em><sub>∞</sub> and invariants <em>I</em><sub>1</sub>, <em>I</em><sub>2</sub> and <em>I</em><sub>3</sub> have been calculated. Our numerical results are compared with some of those available in the literature.展开更多
Knot insertion algorithm is one of the most important technologies of B-spline method. By inserting a knot the local prop- erties of B-spline curve and the control flexibility of its shape can be fiu＇ther improved, a...Knot insertion algorithm is one of the most important technologies of B-spline method. By inserting a knot the local prop- erties of B-spline curve and the control flexibility of its shape can be fiu＇ther improved, also the segmentation of the curve can be rea- lized. ECT spline curve is drew by the multi-knots spline curve with associated matrix in ECT spline space; Muehlbach G and Tang Y and many others have deduced the existence and uniqueness of the ECT spline function and developed many of its important properties .This paper mainly focuses on the knot insertion algorithm of ECT B-spline curve.It is the widest popularization of B-spline Behm algorithm and theory. Inspired by the Behm algorithm, in the ECT spline space, structure of generalized P61ya poly- nomials and generalized de Boor Fix dual functional, expressing new control points which are inserted after the knot by linear com- bination of original control vertex the single knot, and there are two cases, one is the single knot, the other is the double knot. Then finally comes the insertion algorithm of ECT spline curve knot. By application of the knot insertion algorithm, this paper also gives out the knot insertion algorithm of four order geometric continuous piecewise polynomial B-spline and algebraic trigonometric spline B-spline, which is consistent with previous results.展开更多
Abstract Generalized B-splines have been employed as geometric modeling and numerical simu- lation tools for isogeometric analysis （IGA for short）. However, the previous models used in IGA, such as trigonometric gen...Abstract Generalized B-splines have been employed as geometric modeling and numerical simu- lation tools for isogeometric analysis （IGA for short）. However, the previous models used in IGA, such as trigonometric generalized B-splines or hyperbolic generalized B-splines, are not the unified mathematical representation of conics and polynomial parametric curves/surfaces. In this paper, a unified approach to construct the generalized non-uniform B-splines over the space spanned by {α（t）,β（t）,ξ（t）, η（t）, 1, t,……. , tn-4} is proposed, and the corresponding isogeometric analysis framework for PDE solving is also studied. Compared with the NURBS-IGA method, the proposed frameworks have several advantages such as high accuracy, easy-to-compute derivatives and integrals due to the non-rational form. Furthermore, with the proposed spline models, isogeometric analysis can be performed on the computational domain bounded by transcendental curves/surfaces, such as the involute of circle, the helix/helicoid, the catenary/catenoid and the cycloid. Several numerical examples for isogeometrie heat conduction problems are presented to show the effectiveness of the proposed methods.展开更多
Starting from piecewise constant functions, a novel family of generalized symmetric B-splines, with realizable ideal low-pass filters, are constructed. The first order generalized B-spline low-pass filter is closely r...Starting from piecewise constant functions, a novel family of generalized symmetric B-splines, with realizable ideal low-pass filters, are constructed. The first order generalized B-spline low-pass filter is closely related to functions analytic in a neighborhood of the unit disc and the generalized sinc functions. The properties of this kind of low-pass filters are investigated. The behavior of the generalized B-spline low-pass filter related to normalized Gaussian distribution is considered.展开更多
程序化内容生成作为一种数字化内容生产的辅助手段,已经越来越多地应用到了游戏、建筑设计、数字娱乐等领域,极大提高了数字内容的生产效率。作为程序化内容生成算法的一种,波函数坍缩算法(wave function collapse,WFC)提出了一种网格...程序化内容生成作为一种数字化内容生产的辅助手段,已经越来越多地应用到了游戏、建筑设计、数字娱乐等领域,极大提高了数字内容的生产效率。作为程序化内容生成算法的一种,波函数坍缩算法(wave function collapse,WFC)提出了一种网格化的基于关联规则的解决方案。借助Unity3d平台,以WFC算法为基础,提出了插槽多维细分的改进方法,该方法能够在单层次的场景模型生成基础上,基于规则有序进行场景的细化,并辅助以一定的交互功能,最终创建出精细、可信且具有细节可控性的场景环境。展开更多
The new algorithms for finding B-Spline or Bezier curves and surfaces intersections using recursive subdivision techniques are presented, which use extrapolating acceleration technique, and have convergent precision o...The new algorithms for finding B-Spline or Bezier curves and surfaces intersections using recursive subdivision techniques are presented, which use extrapolating acceleration technique, and have convergent precision of order 2. Matrix method is used to subdivide the curves or surfaces which makes the subdivision more concise and intuitive. Dividing depths of Bezier curves and surfaces are used to subdivide the curves or surfaces adaptively Therefore the convergent precision and the computing efficiency of finding the intersections of curves and surfaces have been improved by the methods proposed in the paper.展开更多
基金Supported by National Natural Science Foundation of China(Nos.61272032,60904070)
文摘A generalized non-stationary curve subdivision （GNS for short） scheme of arbitrary order k≥3 with a parameter has been proposed by Fang et al. in the paper （Fang Mei-e et al., CAGD, 2010（27）： 720-733）. It has been proved that the proposed scheme of order k generates C^k-2 continuous curves for k≥4. But the proof of the smoothness in this paper is uncompleted. Moreover, the Cl-continuity of the third order scheme has not been discussed. For this reason, in this paper, we provide a full corrected proof of the smoothness of the GNS scheme of order k for k≥3.
基金Supported by National Natural Science Foundation of China(Grant Nos51175085,51375094)Fujian Provincial Education Department Foundation of China(Grant No.JA13059)+1 种基金Open Fund of State Key Laboratory of Tribology of Tsinghua University,China(Grant No.SKLTKF13B02)Fuzhou Science and Technology plan Fund of China(Grant No.2014-G-74)
文摘Currently, the approximation methods of the Gaussian filter by some other spline filters have been developed. However, thesc methods are only suitable for the study of one-dimensional filtering, when these methods are used for three-dimensional filtering, it is found that a rounding error and quantization error would be passed to the next in every part. In this paper, a new and high-precision implementation approach for Gaussian filter is described, which is suitable for three-dimensional reference filtering. Based on the theory of generalized B-spline function and the variational principle, the transmission characteristics of a digital filter can be changed through the sensitivity of the parameters （t1, t2）, and which can also reduce the rounding error and quantization error by the filter in a parallel form instead of the cascade form, Finally, the approximation filter of Gaussian filter is obtained. In order to verify the feasibility of the new algorithm, the reference extraction of the conventional methods are also used and compared. The experiments are conducted on the measured optical surface, and the results show that the total calculation by the new algorithm only requires 0.07 s for 480×480 data points; the amplitude deviation between the reference of the parallel form filter and the Gaussian filter is smaller; the new method is closer to the characteristic of the Gaussian filter through the analysis of three-dimensional roughness parameters, comparing with the cascade generalized B-spline approximating Gaussian. So the new algorithm is also efficient and accurate for the implementation of Gaussian filter in the application of surface roughness measurement.
基金the National Natural Science Foundation of China(61772164,61761136010)the Natural Science Foundation of Zhejiang Province(LY17F020025).
文摘ωB-splines have many optimal properties and can reproduce plentiful commonly-used analytical curves.In this paper,we further propose a non-stationary subdivision method of hierarchically and efficiently generatingωB-spline curves of arbitrary order ofωB-spline curves and prove its C^k?2-continuity by two kinds of methods.The first method directly prove that the sequence of control polygons of subdivision of order k converges to a C^k?2-continuousωB-spline curve of order k.The second one is based on the theories upon subdivision masks and asymptotic equivalence etc.,which is more convenient to be further extended to the case of surface subdivision.And the problem of approximation order of this non-stationary subdivision scheme is also discussed.Then a uniform ωB-spline curve has both perfect mathematical representation and efficient generation method,which will benefit the application ofωB-splines.
文摘A family of Said-Bézier type generalized Ball （SBGB） bases and surfaces with a parameter H over triangular domain is introduced,which unifies Bézier surface and Said-Ball surface and includes several intermediate surfaces. To convert different bases and surfaces,the dual functionals of bases are presented. As an application of dual functionals,the subdivision formulas for surfaces are established.
文摘In this work, we have obtained numerical solutions of the generalized Korteweg-de Vries (GKdV) equation by using septic B-spline collocation finite element method. The suggested numerical algorithm is controlled by applying test problems including;single soliton wave. Our numerical algorithm, attributed to a Crank Nicolson approximation in time, is unconditionally stable. To control the performance of the newly applied method, the error norms, <em>L</em><sub>2</sub> and <em>L</em><sub>∞</sub> and invariants <em>I</em><sub>1</sub>, <em>I</em><sub>2</sub> and <em>I</em><sub>3</sub> have been calculated. Our numerical results are compared with some of those available in the literature.
基金Supported by Financially Supported by the NUAA Fundamental Research Funds(No.NZ2013201)
文摘Knot insertion algorithm is one of the most important technologies of B-spline method. By inserting a knot the local prop- erties of B-spline curve and the control flexibility of its shape can be fiu＇ther improved, also the segmentation of the curve can be rea- lized. ECT spline curve is drew by the multi-knots spline curve with associated matrix in ECT spline space; Muehlbach G and Tang Y and many others have deduced the existence and uniqueness of the ECT spline function and developed many of its important properties .This paper mainly focuses on the knot insertion algorithm of ECT B-spline curve.It is the widest popularization of B-spline Behm algorithm and theory. Inspired by the Behm algorithm, in the ECT spline space, structure of generalized P61ya poly- nomials and generalized de Boor Fix dual functional, expressing new control points which are inserted after the knot by linear com- bination of original control vertex the single knot, and there are two cases, one is the single knot, the other is the double knot. Then finally comes the insertion algorithm of ECT spline curve knot. By application of the knot insertion algorithm, this paper also gives out the knot insertion algorithm of four order geometric continuous piecewise polynomial B-spline and algebraic trigonometric spline B-spline, which is consistent with previous results.
基金supported by Zhejiang Provincial Natural Science Foundation of China under Grant No.LR16F020003the National Nature Science Foundation of China under Grant Nos.61472111,61602138+1 种基金the Open Project Program of the State Key Lab of CAD&CG(A1703)Zhejiang University
文摘Abstract Generalized B-splines have been employed as geometric modeling and numerical simu- lation tools for isogeometric analysis （IGA for short）. However, the previous models used in IGA, such as trigonometric generalized B-splines or hyperbolic generalized B-splines, are not the unified mathematical representation of conics and polynomial parametric curves/surfaces. In this paper, a unified approach to construct the generalized non-uniform B-splines over the space spanned by {α（t）,β（t）,ξ（t）, η（t）, 1, t,……. , tn-4} is proposed, and the corresponding isogeometric analysis framework for PDE solving is also studied. Compared with the NURBS-IGA method, the proposed frameworks have several advantages such as high accuracy, easy-to-compute derivatives and integrals due to the non-rational form. Furthermore, with the proposed spline models, isogeometric analysis can be performed on the computational domain bounded by transcendental curves/surfaces, such as the involute of circle, the helix/helicoid, the catenary/catenoid and the cycloid. Several numerical examples for isogeometrie heat conduction problems are presented to show the effectiveness of the proposed methods.
基金supported by National Natural Science Foundation of China (Grant Nos.61072126 and 11071058)Natural Science Foundation of Guangdong Province (Grant No. S2011010004986)
文摘Starting from piecewise constant functions, a novel family of generalized symmetric B-splines, with realizable ideal low-pass filters, are constructed. The first order generalized B-spline low-pass filter is closely related to functions analytic in a neighborhood of the unit disc and the generalized sinc functions. The properties of this kind of low-pass filters are investigated. The behavior of the generalized B-spline low-pass filter related to normalized Gaussian distribution is considered.
文摘程序化内容生成作为一种数字化内容生产的辅助手段,已经越来越多地应用到了游戏、建筑设计、数字娱乐等领域,极大提高了数字内容的生产效率。作为程序化内容生成算法的一种,波函数坍缩算法(wave function collapse,WFC)提出了一种网格化的基于关联规则的解决方案。借助Unity3d平台,以WFC算法为基础,提出了插槽多维细分的改进方法,该方法能够在单层次的场景模型生成基础上,基于规则有序进行场景的细化,并辅助以一定的交互功能,最终创建出精细、可信且具有细节可控性的场景环境。
文摘The new algorithms for finding B-Spline or Bezier curves and surfaces intersections using recursive subdivision techniques are presented, which use extrapolating acceleration technique, and have convergent precision of order 2. Matrix method is used to subdivide the curves or surfaces which makes the subdivision more concise and intuitive. Dividing depths of Bezier curves and surfaces are used to subdivide the curves or surfaces adaptively Therefore the convergent precision and the computing efficiency of finding the intersections of curves and surfaces have been improved by the methods proposed in the paper.
