A unified framework for interpolating and approximating univariate subdivision

In this paper we show that the refinement rules of interpolating and approximating univariate subdivision schemes with odd-width masks of finite support can be derived ones from the others by simple operations on the mask coefficients. These operations are formalized as multiplication/division of the associated generating functions by a proper link polynomial.We then apply the proposed result to some families of stationary and non-stationary subdivision schemes, showing that it also provides a constructive method for the definition of novel refinement algorithms.     

A unified three point approximating subdivision scheme

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.     

A ternary 4-point approximating subdivision scheme

In the implementation of subdivision scheme, three of the most important issues are smoothness, size of support, and approximation order. Our objective is to introduce an improved ternary 4-point approximating subdivision scheme derived from cubic polynomial interpolation, which has smaller support and higher smoothness, comparing to binary 4-point and 6-point schemes, ternary 3-point and 4-point schemes (see Table 2). The method is easily generalized to ternary (2n + 2)-point approximating subdivision schemes. We choose a ternary scheme because a way to get smaller support is to raise arity. And we use polynomial reproduction to get higher approximation order easily.     

From approximating subdivision schemes for exponential splines to high-performance interpolating algorithms

In this work we construct three novel families of approximating subdivision schemes that generate piecewise exponential polynomials and we show how to convert these into interpolating schemes of great interest in curve design for their ability to reproduce important analytical shapes and to provide highly smooth limit curves with a controllable tension.     

A non-linear circle-preserving subdivision scheme

We describe a new method for constructing a sequence of refined polygons, which starts with a sequence of points and associated normals. The newly generated points are sampled from circles which approximate adjacent points and the corresponding normals. By iterating the refinement procedure, we get a limit curve interpolating the data. We show that the limit curve is , and that it reproduces circles. The method is invariant with respect to group of Euclidean similarities (including rigid transformations and scaling). We also discuss an experimental setup for a construction and various possible extensions of the method.      

体模型的一种细分格式

In this paper, a subdivision scheme which generalizes a surface scheme in previous papers to volume meshes is designed. The scheme exhibits significant control over shrink-age/size of volumetric models. It also has the ability to conveniently incorporate boundaries and creases into a smooth limit shape of models. The method presented here is much simpler and easier as compared to MacCracken and Joy‘s. This method makes no restrictions on the local topology of meshes. Particularly, it can be applied without any change to meshes of nonmanifold topology.     

From approximating to interpolatory non-stationary subdivision schemes with the same generation properties

In this paper we describe a general, computationally feasible strategy to deduce a family of interpolatory non-stationary subdivision schemes from a symmetric non-stationary, non-interpolatory one satisfying quite mild assumptions. To achieve this result we extend our previous work (Conti et al., Linear Algebra Appl 431(10):1971?C1987, 2009) to full generality by removing additional assumptions on the input symbols. For the so obtained interpolatory schemes we prove that they are capable of reproducing the same space of exponential polynomials as the one generated by the original approximating scheme. Moreover, we specialize the computational methods for the case of symbols obtained by shifted non-stationary affine combinations of exponential B-splines, that are at the basis of most non-stationary subdivision schemes. In this case we find that the associated family of interpolatory symbols can be determined to satisfy a suitable set of generalized interpolating conditions at the set of the zeros (with reversed signs) of the input symbol. Finally, we discuss some computational examples by showing that the proposed approach can yield novel smooth non-stationary interpolatory subdivision schemes possessing very interesting reproduction properties.     

A new cross subdivision scheme for surface design

In this paper, a new primal approximation -subdivision scheme for surface design, being named the cross scheme or C-scheme for short, is presented. The C²-continuity of the C-scheme is mathematically analyzed. The new scheme can be effectively applied to any 3D mesh of quadrilaterals. Extraordinary vertices and miscellaneous boundary scenarios are handled as well. Extensive implementation shows that it performs better than the classical Catmull-Clark subdivision scheme both at nearby vertices and along the non-vertical and non-horizontal directions.     

Normals of the butterfly subdivision scheme surfaces and their applications

The paper presents explicit formulas for calculating normals to surfaces generated by the butterfly interpolatory subdivision scheme from a general initial triangulation of control points. Two applications of these formulas are presented: building offsets to surfaces generated by the butterfly scheme and Gouraud shading of surfaces generated by this scheme as well as shading of their offsets.     

Monotonicity of the CABARET scheme approximating a hyperbolic equation with a sign-changing characteristic field

The monotonicity of the CABARET scheme approximating a hyperbolic differential equation with a sign-changing characteristic field is analyzed. Monotonicity conditions for this scheme are obtained in domains where the characteristics have a sign-definite propagation velocity and near sonic lines, on which the propagation velocity changes its sign. These properties of the CABARET scheme are illustrated by test computations.     

A new one-step iterative scheme for approximating common fixed points of two multivalued nonexpansive mappings

In this paper, we introduce a new one-step iterative process to approximate the common fixed points of two multivalued nonexpansive mappings. We will also prove a strong convergence theorem in a uniformly convex Banach space under the multivalued version of so-called Condition (A′).     

Fractal range of a 3-point ternary interpolatory subdivision scheme with two parameters

In this paper, we analyze the fractal property of Hassan’s 3-point ternary interpolatory subdivision scheme with two parameters. The fractal range of the scheme is obtained and illustrated. Many examples show that the obtained results suggest a clear direction to generate fractal curves and surfaces by using this scheme.     

On the monotonicity of the CABARET scheme approximating a scalar conservation law with a convex flux

The monotonicity of the CABARET scheme approximating a scalar conservation law with a convex flux is analyzed. Monotonicity conditions for this scheme are obtained assuming that the propagation velocity of characteristics of the approximated conservative equation is positive. Test computations are presented that illustrate these properties of the CABARET scheme.     

Optimal smoothing and interpolating splines with constraints

This paper considers the problem for designing optimal smoothing and interpolating splines with equality and/or inequality constraints. The splines are constituted by employing normalized uniform B-splines as the basis functions, namely as weighted sum of shifted B-splines of degree k. Then a central issue is to determine an optimal vector of the so-called control points. By employing such an approach, it is shown that various types of constraints are formulated as linear function of the control points, and the problems reduce to quadratic programming problems. We demonstrate the effectiveness and usefulness by numerical examples including approximation of probability density functions, approximation of discontinuous functions, and trajectory planning.