A unified subdivision approach for multi-dimensional non-manifold modeling

This paper presents a new unified subdivision scheme that is defined over a k-simplicial complex in n-D space with k≤3. We first present a series of definitions to facilitate topological inquiries during the subdivision process. The scheme is derived from the double (k+1)-directional box splines over k-simplicial domains. Thus, it guarantees a certain level of smoothness in the limit on a regular mesh. The subdivision rules are modified by spatial averaging to guarantee C¹ smoothness near extraordinary cases. Within a single framework, we combine the subdivision rules that can produce 1-, 2-, and 3-manifolds in arbitrary n-D space. Possible solutions for non-manifold regions between the manifolds with different dimensions are suggested as a form of selective subdivision rules according to user preference. We briefly describe the subdivision matrix analysis to ensure a reasonable smoothness across extraordinary topologies, and empirical results support our assumption. In addition, through modifications, we show that the scheme can easily represent objects with singularities, such as cusps, creases, or corners. We further develop local adaptive refinement rules that can achieve level-of-detail control for hierarchical modeling. Our implementation is based on the topological properties of a simplicial domain. Therefore, it is flexible and extendable. We also develop a solid modeling system founded on our subdivision schemes to show potential benefits of our work in industrial design, geometric processing, and other applications.     

Analyzing a generalized Loop subdivision scheme

In this paper a class of subdivision schemes generalizing the algorithm of Loop is presented. The stencils have the same support as those from the algorithm of Loop, but allow a variety of weights. By varying the weights a class of C ¹ regular subdivision schemes is obtained. This class includes the algorithm of Loop and the midpoint schemes of order one and two for triangular nets. The proof of C ¹ regularity of the limit surface for arbitrary triangular nets is provided for any choice of feasible weights. The purpose of this generalization of the subdivision algorithm of Loop is to demonstrate the capabilities of the applied analysis technique. Since this class includes schemes that do not generalize box spline subdivision, the analysis of the characteristic map is done with a technique that does not need an explicit piecewise polynomial representation. This technique is computationally simple and can be used to analyze classes of subdivision schemes. It extends previously presented techniques based on geometric criteria.     

B样条的p-nary细分

有关B样条曲线曲面的binary细分技巧及其应用的研究已经获得了许多成果,建立在B样条binary细分基础上的binary细分法收敛性连续性分析的生成多项式法就是其中之一。该文研究了B样条曲线的p-nary细分问题,给出并证明了B样条基函数的p尺度细分方程中细分系数的计算公式及其性质,讨论了用p-nary细分生成非有理及有理B样条曲线的细分规则。采用该文的方法可方便而快速地在计算机上绘制有理B样条曲线。文章的结果可用于对一般p-nary曲线细分法收敛性及连续性的分析。     

Subdivision models for varying-resolution and generalized perturbations

Subdivision schemes are multi-resolution methods used in computer-aided geometric design to generate smooth curves or surfaces. In this paper, we are interested in both smooth and non-smooth subdivision schemes. We propose two models that generalize the subdivision operation and can yield both smooth and non-smooth schemes in a controllable way:

• (1) The ‘varying-resolution’ model allows a structured access to the various resolutions of the refined data, yielding certain patterns. This model generalizes the standard subdivision iterative operation and has interesting interpretations in the geometrical space and also in creativity-oriented domains, such as music. As an infrastructure for this model, we propose representing a subdivision scheme by two dual rules trees. The dual tree is a permuted rules tree that gives a new operator-oriented view on the subdivision process, from which we derive an ‘adjoint scheme’.

• (2) The ‘generalized perturbed schemes’ model can be viewed as a special multi-resolution representation that allows a more flexible control on adding the details. For this model, we define the terms ‘template mask’ and ‘tension vector parameter’.

The non-smooth schemes are created by the permutations of the ‘varying-resolution’ model or by certain choices of the ‘generalized perturbed schemes’ model. We then present procedures that integrate and demonstrate these models and some enhancements that bear a special meaning in creative contexts, such as music, imaging and texture. We describe two new applications for our models: (a) data and music analysis and synthesis, which also manifests the usefulness of the non-smooth schemes and the approximations proposed, and (b) the acceleration of convergence and smoothness analysis, using the ‘dual rules tree’.     

Curve subdivision schemes for geometric modelling

A new binary four-point approximating subdivision scheme has been presented that generates the limiting curve of C ¹ continuity. A global tension parameter has been introduced to improve the performance of the binary four-point approximating subdivision scheme that generates a family of C ¹ limiting curves. The ternary four-point approximating subdivision scheme has also been introduced that generates a limiting curve of C ² continuity. The proposed schemes are close to being interpolating. The Laurent polynomial method has been used to investigate the order of derivative continuity of the schemes and Hölder exponents of the schemes have also been calculated. Performances of the subdivision schemes have been exposed by considering several examples.     

Smooth spline surface generation over meshes of irregular topology

An efficient method for generating a smooth spline surface over an irregular mesh is presented in this paper. Similar to the methods proposed by [1, 2, 3, 4], this method generates a generalised bi-quadratic B-spline surface and achieves C ¹ smoothness. However, the rules to construct the control points for the proposed spline surfaces are much simpler and easier to follow. The construction process consists of two steps: subdividing the initial mesh once using the Catmull–Clark [5] subdivision rules and generating a collection of smoothly connected surface patches using the resultant mesh. As most of the final mesh is quadrilateral apart from the neighbourhood of the extraordinary points, most of the surface patches are regular quadratic B-splines. The neighbourhood of the extraordinary points is covered by quadratic Zheng–Ball patches [6].     

Creases and boundary conditions for subdivision curves

Our goal is to find subdivision rules at creases in arbitrary degree subdivision for piece-wise polynomial curves, but without introducing new control points e.g. by knot insertion. Crease rules are well understood for low degree (cubic and lower) curves. We compare three main approaches: knot insertion, ghost points, and modifying subdivision rules. While knot insertion and ghost points work for arbitrary degrees for B-splines, these methods introduce unnecessary (ghost) control points.The situation is not so simple in modifying subdivision rules. Based on subdivision and subspace selection matrices, a novel approach to finding boundary and sharp subdivision rules that generalises to any degree is presented. Our approach leads to new higher-degree polynomial subdivision schemes with crease control without introducing new control points.     

A proof of the smoothness of the 6-point interpolatory scheme

In this paper the Laurent polynomial method is used to show that the smoothness of the 6-point Weissman interpolatory subdivision scheme is C ².     

An approximating non-stationary subdivision scheme

We present a 3-point C² approximating non-stationary subdivision scheme for designing curves. The weights of the masks of the scheme are defined in terms of some values of trigonometric B-spline basis functions. The scheme has some interesting properties and it is compared with the 2-point and 3-point schemes generating uniform trigonometric spline curves of order 3 and 5. The comparison and the performance of the scheme are demonstrated by examples.     

Quad/triangle subdivision,nonhomogeneous refinement equation and polynomial reproduction

The quad/triangular subdivision, whose control net and refined meshes consist of both quads and triangles, provides better visual quality of subdivision surfaces. While some theoretical results such as polynomial reproduction and smoothness analysis of quad/triangle schemes have been obtained in the literature, some issues such as the basis functions at quad/triangle vertices and design of interpolatory quad/triangle schemes need further study. In our study of quad/triangle schemes, we observe that a quad/triangle subdivision scheme can be derived from a nonhomogeneous refinement equation. Hence, the basis functions at quad/triangle vertices are shifts of the refinable function associated with a nonhomogeneous refinement equation. In this paper a quad/triangle subdivision surface is expressed analytically as the linear combination of these basis functions and the polynomial reproduction of matrix-valued quad/triangle schemes is studied. The result on polynomial reproduction achieved here is critical for the smoothness analysis and construction of matrix-valued quad/triangle schemes. Several new schemes are also constructed.     

A New Class of Guided C2 Subdivision Surfaces Combining Good Shape with Nested Refinement

Converting quadrilateral meshes to smooth manifolds, guided subdivision offers a way to combine the good highlight line distribution of recent G‐spline constructions with the refinability of subdivision surfaces. This avoids the complex refinement of G‐spline constructions and the poor shape of standard subdivision. Guided subdivision can then be used both to generate the surface and hierarchically compute functions on the surface. Specifically, we present a C² subdivision algorithm of polynomial degree bi‐6 and a curvature bounded algorithm of degree bi‐5. We prove that the common eigenstructure of this class of subdivision algorithms is determined by their guide and demonstrate that their eigenspectrum (speed of contraction) can be adjusted without harming the shape. For practical implementation, a finite number of subdivision steps can be completed by a high‐quality cap. Near irregular points this allows leveraging standard polynomial tools both for rendering of the surface and for approximately integrating functions on the surface.     

A Geometric Approach for Multi-Degree Spline

Multi-degree spline (MD-spline for short) is a generalization of B-spline which comprises of polynomial segments of various degrees.The present paper provides a new definition for MD-spline curves in a geometric intuitive way based on an efficient and simple evaluation algorithm.MD-spline curves maintain various desirable properties of B-spline curves,such as convex hull,local support and variation diminishing properties.They can also be refined exactly with knot insertion.The continuity between two adjacent segments with different degrees is at least C1 and that between two adjacent segments of same degrees d is Cd 1.Benefited by the exact refinement algorithm,we also provide several operators for MD-spline curves,such as converting each curve segment into B′ezier form,an efficient merging algorithm and a new curve subdivision scheme which allows different degrees for each segment.     

A Method for Constructing Interpolatory Subdivision Schemes and Blending Subdivisions

This paper presents a universal method for constructing interpolatory subdivision schemes from known approximatory subdivisions. The method establishes geometric rules of the associated interpolatory subdivision through addition of further weighted averaging operations to the approximatory subdivision. The paper thus provides a novel approach for designing new interpolatory subdivision schemes. In addition, a family of subdivision surfaces varying from the given approximatory scheme to its associated interpolatory scheme, namely the blending subdivisions, can also be established. Based on the proposed method, variants of several known interpolatory subdivision schemes are constructed. A new interpolatory subdivision scheme is also developed using the same technique. Brief analysis of a family of blending subdivisions associated with the Loop subdivision scheme demonstrates that this particular family of subdivisions are globally C¹ continuous while maintaining bounded curvature for regular meshes. As a further extension of the blending subdivisions, a volume‐preserving subdivision strategy is also proposed in the paper.     

Adaptive quasi-interpolating quartic splines

We present an adaptive quasi-interpolating quartic spline construction for regularly sampled surface data. The method is based on a uniform quasi-interpolating scheme, employing quartic triangular patches with C ¹-continuity and optimal approximation order within this class. Our contribution is the adaption of this scheme to surfaces of varying geometric complexity, where the tiling resolution can be locally defined, for example driven by approximation errors. This way, the construction of high-quality spline surfaces is enhanced by the flexibility of adaptive pseudo-regular triangle meshes. Numerical examples illustrate the use of this method for adaptive terrain modeling, where uniform schemes produce huge numbers of patches.     

A new fast normal-based interpolating subdivision scheme by cubic Bézier curves

In this paper, we propose a new fast normal-based interpolating subdivision scheme for curve and surface design. Different from the 4-points interpolating subdivision scheme, it is based on cubic Bezier curves and the normal vectors are used to generate a circle. Both a convex edge and an inflexion edge can be subdivided into convex sub-edges and then generate smooth curves. Under proper angle conditions, this subdivision scheme converges and the limit curve will be \(\hbox {G}^{1}\) smoothness. When applying it to subdivide surface on triangle/quadrilateral meshes, we use the normal vectors and have no need to consider the meshes neighboring to the current surface elements. Such advantage leads to that the subdivision scheme has fast rendering speed without changing the topology of the meshes. Subdivision examples and results by our scheme are illustrated and meantime is compared with those generated by other well-known schemes. It shows that this scheme can generate a more smooth curve based on both a convex edge and an inflexion edge, and the limit surface has better smoothness than those of other interpolating schemes.     

A Unified Subdivision Scheme for Polygonal Modeling

Subdivision rules have traditionally been designed to generate smooth surfaces from polygonal meshes. In this paper we propose to employ subdivision rules as a polygonal modeling tool, specifically to add additional level of detail to meshes. However, existing subdivision schemes have several undesirable properties making them ill suited for polygonal modeling. In this paper we propose a general set of subdivision rules which provides users with more control over the subdivision process. Most existing subdivision schemes are special cases. In particular, we provide subdivision rules which blend approximating spline based schemes with interpolatory ones. Also, we generalize subdivision to allow any number of refinements to be performed in a single step.     

三向四次箱样条曲面与Bezier曲面的光滑拼接

以二元四次多项式在三角域和矩形域上的Bezier形式的Blossom为工具,给出了当给定一张三向四次箱样条曲面时,能与之C^0、C^1、C^2拼接的三边或矩形Bezier曲面的控制顶点所要满足的一个显式表示的充分条件。这一结果在使用三向四次箱样条曲面或Loop细分曲面造型,而又需要构造Bezier曲面与之拼接或补洞时,具有理论和实际应用价值。     

Pseudo‐Spline Subdivision Surfaces

Pseudo‐splines provide a rich family of subdivision schemes with a wide range of choices that meet various demands for balancing the approximation power, the length of the support, and the regularity of the limit functions. Special cases of pseudo‐splines include uniform odd‐degree B‐splines and the interpolatory 2n‐point subdivision schemes, and the other pseudo‐splines fill the gap between these two families. In this paper we show how the refinement step of a pseudo‐spline subdivision scheme can be implemented efficiently using repeated local operations, which require only the data in the direct neighbourhood of each vertex, and how to generalize this concept to quadrilateral meshes with arbitrary topology. The resulting pseudo‐spline surfaces can be arbitrarily smooth in regular mesh regions and C¹ at extraordinary vertices as our numerical analysis reveals.     

非静态混合细分法

提出了一种含参数b 的非静态Binary 混合细分法,当参数取0、1 时,分别对应已 有的非静态四点C1 插值细分法及C-B 样条细分法。用渐进等价定理证明了对任意 (0,1]区间的 参数其极限曲线为C2 连续的。从理论上证明了细分法对特殊函数的再生性,及其对圆和椭圆等 特殊曲线的再生性,并通过实验对比说明了对任意的[0,1]区间的参数,该细分法都能再生圆和 椭圆等特殊曲线,而与其渐进等价的静态细分法则不具备该性质。将该细分法推广为含局部控 制参数的广义混合细分法,从而可以达到局部调整极限曲线的目的。     

Local and singularity-free G 1 triangular spline surfaces using a minimum degree scheme

We develop a scheme for constructing G ¹ triangular spline surfaces of arbitrary topological type. To assure that the scheme is local and singularity-free, we analyze the selection of scalar weight functions and the construction of the boundary curve network in detail. With the further requirements of interpolating positions, normals, and surface curvatures, we show that the minimum degree of such a triangular spline surface is 6. And we present a method for constructing boundary curves network, which consists of cubic Bézier curves. To deal with certain singular cases, the base mesh must be locally subdivided and we proposed an adaptive subdivision strategy for it. An application of our G ¹ triangular spline surfaces to the approximation of implicit surfaces is described. The visual quality of this scheme is demonstrated by some examples.