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
In this article, the objective is to introduce an algorithm to produce the quaternary m-point (for any integer m>1) approximating subdivision schemes, which have smaller support and higher smoothness, comparing to ...In this article, the objective is to introduce an algorithm to produce the quaternary m-point (for any integer m>1) approximating subdivision schemes, which have smaller support and higher smoothness, comparing to binary and ternary schemes. The proposed algorithm has been derived from uniform B-spline basis function using the Cox-de Boor recursion formula. In order to determine the convergence and smoothness of the proposed schemes, the Laurent polynomial method has been used.展开更多
In this paper, we propose and analyze a subdivision scheme which unifies 3-point approximating subdivision schemes of any arity in its compact form and has less support, computational cost and error bounds.? The usefu...In this paper, we propose and analyze a subdivision scheme which unifies 3-point approximating subdivision schemes of any arity in its compact form and has less support, computational cost and error bounds.? The usefulness of the scheme is illustrated by considering different examples along with its comparison with the established subdivision schemes. Moreover, B-splines of degree 4and well known 3-point schemes [1, 2, 3, 4, 6, 11, 12, 14, 15] are special cases of our proposed scheme.展开更多
Based on the butterfly subdivision scheme and the modified butterfly subdivision scheme, an improved butterfly subdivision scheme is proposed. The scheme uses a small stencil of six points to calculate new inserting v...Based on the butterfly subdivision scheme and the modified butterfly subdivision scheme, an improved butterfly subdivision scheme is proposed. The scheme uses a small stencil of six points to calculate new inserting vertex, 2n new vertices are inserted in the 2n triangle faces in each recursion, and the n old vertices are kept, special treatment is given to the boundary, achieving higher smoothness while using small stencils is realized. With the proposed scheme, the number of triangle faces increases only by a factor of 3 in each refinement step. Compared with the butterfly subdivision scheme and the modified butterfly subdivision scheme, the size of triangle faces changes more gradually, which allows one to have greater control over the resolution of a refined mesh.展开更多
This paper offers a general formula for surface subdivision rules for quad meshes by using 2-D Lagrange interpolating polynomial [1]. We also see that the result obtained is equivalent to the tensor product of (2N + 4...This paper offers a general formula for surface subdivision rules for quad meshes by using 2-D Lagrange interpolating polynomial [1]. We also see that the result obtained is equivalent to the tensor product of (2N + 4)-point n-ary interpolating curve scheme for N ≥ 0 and n ≥ 2. The simple interpolatory subdivision scheme for quadrilateral nets with arbitrary topology is presented by L. Kobbelt [2], which can be directly calculated from the proposed formula. Furthermore, some characteristics and applications of the proposed work are also discussed.展开更多
In this paper it is proved that L p solutions of a refinement equation exist if and only if the corresponding subdivision scheme with suitable initial function converges in L p without any assumption on the ...In this paper it is proved that L p solutions of a refinement equation exist if and only if the corresponding subdivision scheme with suitable initial function converges in L p without any assumption on the stability of the solutions of the refinement equation.A characterization for convergence of subdivision scheme is also given in terms of the refinement mask.Thus a complete answer to the relation between the existence of L p solutions of the refinement equation and the convergence of the corresponding subdivision schemes is given.展开更多
In this paper, some characterizations on the convergence rate of both the homogeneous and nonhomogeneous subdivision schemes in Sobolev space are studied and given.
In this paper, we propose a three point approximating subdivision scheme, with three shape parameters, that unifies three different existing three point approximating schemes. Some sufficient conditions for subdivisio...In this paper, we propose a three point approximating subdivision scheme, with three shape parameters, that unifies three different existing three point approximating schemes. Some sufficient conditions for subdivision curve C0 to C3 continuity and convergence of the scheme for generating tensor product surfaces for certain ranges of parameters by using Laurent polynomial method are discussed. The systems of curve and surface design based on our scheme have been developed successfully in garment CAD especially for clothes modelling.展开更多
A family of two-order Hermite vector-interpolating subdivision schemes is proposed and its convergence and con- tinuity are analyzed. The iterative level can be estimated for given error. The sufficient conditions of ...A family of two-order Hermite vector-interpolating subdivision schemes is proposed and its convergence and con- tinuity are analyzed. The iterative level can be estimated for given error. The sufficient conditions of C2 continuity are proved. Geometric features of subdivision curves, such as line segments, cusps and inflection points, are obtained by appending some conditions to initial vectorial Hermite sequence. An algorithm is presented for generating geometric features. For an initial se- quence of two-order Hermite elements from unit circle, the numerical error of the 4th subdivided level is O(10?4).展开更多
In this paper we will first prove that the nontrivial L p solutions of the vector refinement equation exist if and only if the corresponding subdivision scheme with a suitable initial function converges in L p...In this paper we will first prove that the nontrivial L p solutions of the vector refinement equation exist if and only if the corresponding subdivision scheme with a suitable initial function converges in L p without assumption of the stability of the solutions. Then we obtain a characterization of the convergence of the subdivision scheme in terms of the mask. This gives a complete answer to the existence of L p solutions of the refinement equation and the convergence of the corresponding subdivision schemes.展开更多
This paper presents a general formula for (2m + 2)-point n-ary interpolating subdivision scheme for curves for any?integer m ≥ 0 and n ≥ 2 by using Newton interpolating polynomial. As a consequence, the proposed wor...This paper presents a general formula for (2m + 2)-point n-ary interpolating subdivision scheme for curves for any?integer m ≥ 0 and n ≥ 2 by using Newton interpolating polynomial. As a consequence, the proposed work is extended for surface case, which is equivalent to the tensor product of above proposed curve case. These formulas merge several notorious curve/surface schemes. Furthermore, visual performance of the subdivision schemes is also presented.展开更多
A new algorithm to compute continuous wavelet transforms at dyadic scales is proposed here. Our approach has a similar implementation with the standard algorithme a trous and can coincide with it in the one dimensiona...A new algorithm to compute continuous wavelet transforms at dyadic scales is proposed here. Our approach has a similar implementation with the standard algorithme a trous and can coincide with it in the one dimensional lower order spline case.Our algorithm can have arbitrary order of approximation and is applicable to the multidimensional case.We present this algorithm in a general case with emphasis on splines anti quast in terpolations.Numerical examples are included to justify our theorerical discussion.展开更多
A general formula for 4-point α-Ary approximating subdivision scheme for curve designing is introduced for any arity α≥2. The new scheme is extension of B-spline of degree 6. Laurent polynomial method is used to in...A general formula for 4-point α-Ary approximating subdivision scheme for curve designing is introduced for any arity α≥2. The new scheme is extension of B-spline of degree 6. Laurent polynomial method is used to investigate the continuity of the scheme. The variety of effects can be achieved in correspondence for different values of parameter. The applications of the proposed scheme are illustrated in comparison with the established subdivision schemes.展开更多
Subdivision schemes provide important techniques for the fast generation of curves and surfaces. A recusive refinement of a given control polygon will lead in the limit to a desired visually smooth object. These metho...Subdivision schemes provide important techniques for the fast generation of curves and surfaces. A recusive refinement of a given control polygon will lead in the limit to a desired visually smooth object. These methods play also an important role in wavelet analysis. In this paper, we use a rather simple way to characterize the convergence of subdivision schemes for multivariate cases. The results will be used to investigate the regularity of the solutions for dilation equations.展开更多
The proper orthogonal decomposition （POD） is a model reduction technique for the simulation Of physical processes governed by partial differential equations （e.g., fluid flows）. It has been successfully used in th...The proper orthogonal decomposition （POD） is a model reduction technique for the simulation Of physical processes governed by partial differential equations （e.g., fluid flows）. It has been successfully used in the reduced-order modeling of complex systems. In this paper, the applications of the POD method are extended, i.e., the POD method is applied to a classical finite difference （FD） scheme for the non-stationary Stokes equation with a real practical applied background. A reduced FD scheme is established with lower dimensions and sufficiently high accuracy, and the error estimates are provided between the reduced and the classical FD solutions. Some numerical examples illustrate that the numerical results are consistent with theoretical conclusions. Moreover, it is shown that the reduced FD scheme based on the POD method is feasible and efficient in solving the FD scheme for the non-stationary Stokes equation.展开更多
We present a general formula to generate the family of odd-point ternary approximating subdivision schemes with a shape parameter for describing curves. The influence of parameter to the limit curves and the sufficien...We present a general formula to generate the family of odd-point ternary approximating subdivision schemes with a shape parameter for describing curves. The influence of parameter to the limit curves and the sufficient conditions of the continuities from C0 to C5 of 3- and 5-point schemes are discussed. Our family of 3-point and 5-point ternary schemes has higher order of derivative continuity than the family of 3-point and 5-point schemes presented by [Jian-ao Lian, On a-ary subdivision for curve design: II. 3-point and 5-point interpolatory schemes, Applications and Applied Mathematics: An International Journal, 3(2), 2008, 176-187]. Moreover, a 3-point ternary cubic B-spline is special case of our family of 3-point ternary scheme. The visual quality of schemes with examples is also demonstrated.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.
文摘In this article, the objective is to introduce an algorithm to produce the quaternary m-point (for any integer m>1) approximating subdivision schemes, which have smaller support and higher smoothness, comparing to binary and ternary schemes. The proposed algorithm has been derived from uniform B-spline basis function using the Cox-de Boor recursion formula. In order to determine the convergence and smoothness of the proposed schemes, the Laurent polynomial method has been used.
文摘In this paper, we propose and analyze a subdivision scheme which unifies 3-point approximating subdivision schemes of any arity in its compact form and has less support, computational cost and error bounds.? The usefulness of the scheme is illustrated by considering different examples along with its comparison with the established subdivision schemes. Moreover, B-splines of degree 4and well known 3-point schemes [1, 2, 3, 4, 6, 11, 12, 14, 15] are special cases of our proposed scheme.
文摘Based on the butterfly subdivision scheme and the modified butterfly subdivision scheme, an improved butterfly subdivision scheme is proposed. The scheme uses a small stencil of six points to calculate new inserting vertex, 2n new vertices are inserted in the 2n triangle faces in each recursion, and the n old vertices are kept, special treatment is given to the boundary, achieving higher smoothness while using small stencils is realized. With the proposed scheme, the number of triangle faces increases only by a factor of 3 in each refinement step. Compared with the butterfly subdivision scheme and the modified butterfly subdivision scheme, the size of triangle faces changes more gradually, which allows one to have greater control over the resolution of a refined mesh.
文摘This paper offers a general formula for surface subdivision rules for quad meshes by using 2-D Lagrange interpolating polynomial [1]. We also see that the result obtained is equivalent to the tensor product of (2N + 4)-point n-ary interpolating curve scheme for N ≥ 0 and n ≥ 2. The simple interpolatory subdivision scheme for quadrilateral nets with arbitrary topology is presented by L. Kobbelt [2], which can be directly calculated from the proposed formula. Furthermore, some characteristics and applications of the proposed work are also discussed.
基金Supported by the National Natural Science Foundation of China(1 0 0 71 0 71 )
文摘In this paper it is proved that L p solutions of a refinement equation exist if and only if the corresponding subdivision scheme with suitable initial function converges in L p without any assumption on the stability of the solutions of the refinement equation.A characterization for convergence of subdivision scheme is also given in terms of the refinement mask.Thus a complete answer to the relation between the existence of L p solutions of the refinement equation and the convergence of the corresponding subdivision schemes is given.
文摘In this paper, some characterizations on the convergence rate of both the homogeneous and nonhomogeneous subdivision schemes in Sobolev space are studied and given.
基金Supported by the Indigenous PhD Scholarship Scheme of Higher Education Commission (HEC) Pakistan
文摘In this paper, we propose a three point approximating subdivision scheme, with three shape parameters, that unifies three different existing three point approximating schemes. Some sufficient conditions for subdivision curve C0 to C3 continuity and convergence of the scheme for generating tensor product surfaces for certain ranges of parameters by using Laurent polynomial method are discussed. The systems of curve and surface design based on our scheme have been developed successfully in garment CAD especially for clothes modelling.
文摘A family of two-order Hermite vector-interpolating subdivision schemes is proposed and its convergence and con- tinuity are analyzed. The iterative level can be estimated for given error. The sufficient conditions of C2 continuity are proved. Geometric features of subdivision curves, such as line segments, cusps and inflection points, are obtained by appending some conditions to initial vectorial Hermite sequence. An algorithm is presented for generating geometric features. For an initial se- quence of two-order Hermite elements from unit circle, the numerical error of the 4th subdivided level is O(10?4).
文摘In this paper we will first prove that the nontrivial L p solutions of the vector refinement equation exist if and only if the corresponding subdivision scheme with a suitable initial function converges in L p without assumption of the stability of the solutions. Then we obtain a characterization of the convergence of the subdivision scheme in terms of the mask. This gives a complete answer to the existence of L p solutions of the refinement equation and the convergence of the corresponding subdivision schemes.
文摘This paper presents a general formula for (2m + 2)-point n-ary interpolating subdivision scheme for curves for any?integer m ≥ 0 and n ≥ 2 by using Newton interpolating polynomial. As a consequence, the proposed work is extended for surface case, which is equivalent to the tensor product of above proposed curve case. These formulas merge several notorious curve/surface schemes. Furthermore, visual performance of the subdivision schemes is also presented.
文摘A new algorithm to compute continuous wavelet transforms at dyadic scales is proposed here. Our approach has a similar implementation with the standard algorithme a trous and can coincide with it in the one dimensional lower order spline case.Our algorithm can have arbitrary order of approximation and is applicable to the multidimensional case.We present this algorithm in a general case with emphasis on splines anti quast in terpolations.Numerical examples are included to justify our theorerical discussion.
文摘A general formula for 4-point α-Ary approximating subdivision scheme for curve designing is introduced for any arity α≥2. The new scheme is extension of B-spline of degree 6. Laurent polynomial method is used to investigate the continuity of the scheme. The variety of effects can be achieved in correspondence for different values of parameter. The applications of the proposed scheme are illustrated in comparison with the established subdivision schemes.
文摘Subdivision schemes provide important techniques for the fast generation of curves and surfaces. A recusive refinement of a given control polygon will lead in the limit to a desired visually smooth object. These methods play also an important role in wavelet analysis. In this paper, we use a rather simple way to characterize the convergence of subdivision schemes for multivariate cases. The results will be used to investigate the regularity of the solutions for dilation equations.
基金Project supported by the National Natural Science Foundation of China (Nos. 10871022, 11061009, and 40821092)the National Basic Research Program of China (973 Program) (Nos. 2010CB428403, 2009CB421407, and 2010CB951001)the Natural Science Foundation of Hebei Province of China (No. A2010001663)
文摘The proper orthogonal decomposition （POD） is a model reduction technique for the simulation Of physical processes governed by partial differential equations （e.g., fluid flows）. It has been successfully used in the reduced-order modeling of complex systems. In this paper, the applications of the POD method are extended, i.e., the POD method is applied to a classical finite difference （FD） scheme for the non-stationary Stokes equation with a real practical applied background. A reduced FD scheme is established with lower dimensions and sufficiently high accuracy, and the error estimates are provided between the reduced and the classical FD solutions. Some numerical examples illustrate that the numerical results are consistent with theoretical conclusions. Moreover, it is shown that the reduced FD scheme based on the POD method is feasible and efficient in solving the FD scheme for the non-stationary Stokes equation.
文摘We present a general formula to generate the family of odd-point ternary approximating subdivision schemes with a shape parameter for describing curves. The influence of parameter to the limit curves and the sufficient conditions of the continuities from C0 to C5 of 3- and 5-point schemes are discussed. Our family of 3-point and 5-point ternary schemes has higher order of derivative continuity than the family of 3-point and 5-point schemes presented by [Jian-ao Lian, On a-ary subdivision for curve design: II. 3-point and 5-point interpolatory schemes, Applications and Applied Mathematics: An International Journal, 3(2), 2008, 176-187]. Moreover, a 3-point ternary cubic B-spline is special case of our family of 3-point ternary scheme. The visual quality of schemes with examples is also demonstrated.
