Subdivision Schemes For Surfaces Research Articles

A Surface Subdivision Scheme Based on Four‐Directional S13 Non‐Box Splines

AbstractIn this paper, we propose a novel surface subdivision scheme called non‐box subdivision, which is generalized from four‐directional S13 non‐box splines. The resulting subdivision surfaces achieve C1 continuity with the convex hull property. This scheme can be regarded as either a four‐directional subdivision or a special quadrilateral subdivision. When used as a quadrilateral subdivision, the proposed scheme can control the shape of the limit surface more flexibly than traditional schemes due to the natural introduction of auxiliary face control vertices.

Formula omitted]-B-splines non-stationary subdivision schemes for grids of arbitrary topology design

This paper proposes a new class of B-spline-like functions that blend trigonometric and hyperbolic functions. A new family of non-stationary subdivision schemes of order k≥3 has been built using the refinement of these B-splines. The schemes yield a Ck−2 limit curve that represents the spline. A tensor-product subdivision approach is used to produce a limit surface from an initial mesh with regular vertices. For extraordinary vertices, new rules are proposed. Except at extraordinary points, where they are C1 continuous, the scheme generates Ck−2 continuous limit surfaces. The proposed scheme is a particular case of many well-known curve and surface subdivision schemes. Some examples are given to demonstrate how well the new schemes perform in generating various shapes of limit curves and surfaces.

A Family of Binary Approximating Subdivision Schemes based on Binomial Distribution

A simplest way is introduced to generate a generalized algorithm of univariate and bivariate subdivision schemes. This generalized algorithm is based on the symbol of uniform B-splines subdivision schemes and probability generating function of Binomial distribution. We present a family of binary approximating subdivision schemes which has maximum continuity and less support size. Our proposed family members P4, P5, P6, and P7, have C7, C9, C11 and C13 continuities respectively. In fact, we use Binomial probability distribution to increase the continuity of uniform B-splines subdivision schemes up to more than double. We present the complete analysis of one family member of proposed schemes and give a visual performance to check smoothness graphically. In our analysis, we present continuity, Holder regularity, degree of generation, degree of reproduction and limit stencils analysis of proposed family of subdivision schemes. We also present a survey of high continuity subdivision schemes. Comparison shows that our proposed family of subdivision schemes gives high continuity of subdivision schemes comparative to existing subdivision schemes. We have found that as complexity increases the continuity also increases. In the last, we give the general formula for tensor product surface subdivision schemes and also present the visual performance of proposed tensor product surface subdivision schemes.

Subdivision and multiresolution for PUPs

Partition of Unity Parametrics (PUPs) is a generalization of NURBS that permits the use of arbitrary basis functions to model parametric curves and surfaces. An interesting problem for PUPs is the identification of subdivision, reverse subdivision, and multiresolution schemes for this recently developed and flexible class of parametric curves and surfaces.In this paper, we introduce a systematic approach to derive uniform subdivision schemes for PUPs curves and tensor-product surfaces. Our approach formulates PUPs subdivision as a least squares problem. This allows us to find exact subdivision filters for refinable basis functions and optimal approximate schemes for irrefinable ones. Additionally, we derive PUPs multiresolution masks based on their subdivision filters. We formulate the problem as a constrained least squares optimization, such that the resulting multiresolution schemes are banded and optimal in terms of minimizing multiresolution reconstruction error.Finally, to illustrate our methods, we provide sample subdivision and multiresolution schemes with different properties. These include specific examples targeted towards applications of PUPs multiresolution schemes for compression, feature transfer, and macroscopic editing.

Construction and Analysis of Binary Subdivision Schemes for Curves and Surfaces Originated from Chaikin Points

We present a new variant of Lane-Riesenfeld algorithm for curves and surfaces both. Our refining operator is the modification of Chaikin/Doo-Sabin subdivision operator, while each smoothing operator is the weighted average of the four/sixteen adjacent points. Our refining operator depends on two parameters (shape and smoothing parameters). So we get new families of univariate and bivariate approximating subdivision schemes with two parameters. The bivariate schemes are the nontensor product schemes for quadrilateral meshes. Moreover, we also present analysis of our families of schemes. Furthermore, our schemes give cubic polynomial reproduction for a specific value of the shape parameter. The nonuniform setting of our univariate and bivariate schemes gives better performance than that of the uniform schemes.

Smoothness of Subdivision Surfaces with Boundary

Subdivision rules for meshes with boundary are essential for practical applications of subdivision surfaces. These rules have to result in piecewise \(C^{\ell }\)-continuous boundary limit curves and ensure \(C^{\ell }\)-continuity of the surface itself. Extending the theory of Zorin (Constr Approx 16(3):359–397, 2000), we present in this paper general necessary and sufficient conditions for \(C^{\ell }\)-continuity of subdivision schemes for surfaces with boundary, and specialize these to practically applicable sufficient conditions for \(C^1\)-continuity. We use these conditions to show that certain boundary rules for Loop and Catmull–Clark are in fact \(C^1\) continuous.

Constrained polynomial approximation of quadric surfaces

Quadric surfaces are one of the geometric elements most commonly used for shape expression and mechanical accessory cartography in three dimensions. Since quadric surfaces cannot be represented exactly using polynomials, it is not suitable to deal with most problems encountered in engineering application. For this reason, we present a new method for polynomial approximation of quadric surfaces. The approximation tensor product surface obtained by the proposed method has no limit on its degree, i.e., arbitrary degree. It has C(k,h) (k,h=0,1,2) continuity at four corners with the original quadric surface. Meanwhile, the method has an optimal approximation result in the L2-norm, and it can also be applied to piecewise continuous quadric surface patches or a quadric surface patch combined with surface subdivision scheme. And the resulting piecewise approximation patches are globally C0 continuous. Finally, error estimation for the approximation method is given and numerical examples are presented to validate the feasibility and effectiveness of our method.

A generalized surface subdivision scheme of arbitrary order with a tension parameter

This article presents a generalized B-spline surface subdivision scheme of arbitrary order with a tension parameter. We first propose a tensor-product subdivision scheme that produces ku×kv order generalized B-spline limit surfaces. Generalized B-spline surface is the unified and extended form of B-splines, trigonometric B-splines and hyperbolic B-splines (Fang et al. 2010). The tensor product subdivision scheme can be used to generate various surfaces of revolution, including those generated by classical analytic curves that can be exactly represented by generalized B-spline curves. By extending a bi-order (say k) tensor-product scheme to meshes of arbitrary topology, we further propose a generalized surface subdivision scheme with a tension parameter. Several well-known subdivision schemes, including Doo–Sabin subdivision, Catmull–Clark subdivision, and two other subdivision schemes proposed by Morin et al. (2001) and Stam (2001), become special cases of the generalized subdivision scheme. The tension parameter can be used to adjust the shape of subdivision surfaces. The scheme produces higher order Ck−2 continuous limit surfaces except at extraordinary points where the continuity is C1. Convenient and hierarchical methods are also presented for embedding sharp features and semi-sharp features on the resulting limit surfaces.

Tight numerical bounds for digital terrain modeling by interpolatory subdivision schemes

Abstract Surface subdivision schemes are used extensively in scientific and practical applications to generate continuous surfaces in an iterative manner, starting from a set of points. The first subdivision step defines the so called control polygon. This initial polygon can be considered as a very coarse approximation to the final surface. Each iterative step is governed by a local averaging rule which is designed to generate new points by taking some weighted averages of the positions of the neighboring vertices from the previous iteration. If the old vertices (i.e. vertices from the previous iteration) are not to be altered, the subdivision scheme is called an interpolatory subdivision scheme. Some of the most popular interpolatory subdivision schemes are the two point, four point and six point subdivision schemes. These subdivision rules define convergent schemes. The limit surface is continuous and, in some cases, the limit is C 1 or C 2 . This paper is devoted to estimate error bounds between the limit surface and the control polygon defined after k subdivision stages. The results are applied to the case that initial data corresponds with real terrains. The explicit and tight numerical bounds make possible to deal with some basic questions in connection with surfaces that are not defined by analytic formulas. Some previous approaches also give numerical bounds but they are too large to be used for some practical purposes. More precisely, from the numerical results, it is possible to analyze the smoothness of real mountains from a quantitative point of view. These kinds of results are valuable in Cartography because common techniques are based on visual perceptions. A second advantage of the given bounds is that they indicate what approximation scheme is more suitable in order to reconstruct the terrain.

Least Squares Subdivision Surfaces

AbstractThe usual approach to design subdivision schemes for curves and surfaces basically consists in combining proper rules for regular configurations, with some specific heuristics to handle extraordinary vertices. In this paper, we introduce an alternative approach, called Least Squares Subdivision Surfaces (LS), where the key idea is to iteratively project each vertex onto a local approximation of the current polygonal mesh. While the resulting procedure haves the same complexity as simpler subdivision schemes, our method offers much higher visual quality, especially in the vicinity of extraordinary vertices. Moreover, we show it can be easily generalized to support boundaries and creases. The fitting procedure allows for a local control of the surface from the normals, making LS3 very well suited for interactive freeform modeling applications. We demonstrate our approach on diadic triangular and quadrangular refinement schemes, though it can be applied to any splitting strategies.

Formula omitted]-splines

Both the 4-point and the uniform cubic B -spline subdivisions double the number of vertices of a closed-loop polygon k P and produce sequences of vertices f j and b j respectively. We study the J -spline subdivision scheme J s , introduced by Maillot and Stam, which blends these two methods to produce vertices of the form v j = ( 1 − s ) f j + s b j . Iterative applications of J s yield a family of limit curves, the shape of which is parameterized by s. They include four-point subdivision curves ( J 0 ), uniform cubic B -spline curves ( J 1 ), and uniform quintic B -spline curves ( J 1.5 ). We show that the limit curve is at least C 1 when − 1.7 ≤ s ≤ 5.8 , C 2 when 0 < s < 4 , C 3 when 1 < s ≤ 2.8 , and C 4 when s = 3 / 2 , even though a 4-point yields only C 1 curves and a cubic B -spline yields only C 2 curves. We generalize the J s scheme to a two-parameter family J a , b and propose data-dependent and data-independent solutions for computing values of parameters a and b that minimize various objective functions (distance to the control vertices, deviation from the control polygon, change in surface area, and popping when switching levels of subdivision in multi-resolution rendering). We extend the J -spline subdivision to open curves and to a smooth surface subdivision scheme for quad-meshes with arbitrary connectivity.

A new ternary interpolatory subdivision scheme for polyhedral meshes with arbitrary topology

Interpolation using ternary subdivision is an attractive characteristic in geometric modeling due to its advantage over commonly used binary subdivision. In this paper, a new ternary interpolatory subdivision scheme with one parameter based on arbitrarily triangular meshes is proposed. It can be used to any polyhedral meshes with arbitrary topology. The sufficient conditions of the convergence and the tangent plane continuity of the presented subdivision scheme are obtained using geometric methods and algebraic manipulations. It is simple, fast and has a better local property in comparison with the existing ternary interpolatory surface subdivision schemes.

From extension of Loop's approximation scheme to interpolatory subdivisions

The minimum-supported bivariate C 2 -cubic spline on a 6-directional mesh constructed in our previous work [Chui, C.K., Jiang, Q.T., 2003. Surface subdivision schemes generated by refinable bivariate spline function vectors. Appl. Comput. Harmonic Anal. 15, 147–162] can be used to extend Loop's approximation subdivision scheme to introduce some parameter for controlling surface geometric shapes. This extension is achieved by considering matrix-valued subdivisions, resulting in subdivision templates of the same 1-ring template size as Loop's scheme, but with 2-dimensional matrix-valued weights. Another feature accomplished by considering such an extension is that the two components of the refinable vector-valued spline function can be reformulated, by taking certain linear combinations, to convert the approximation scheme to an interpolatory scheme, but at the expense of an increase in template size for the edge vertices. To maintain the 1-ring template size with guarantee of C 2 smoothness for interpolatory surface subdivisions, a non-spline solution is needed, by applying some constructive scheme such as the procedure discussed in our recent work [Chui, C.K., Jiang, Q.T., 2005b. Matrix-valued symmetric templates for interpolatory surface subdivisions I. Regular vertices. Appl. Comput. Harmonic Anal. 19, 303–339]. The main objective of this paper is to develop the corresponding matrix-valued 1-ring templates for the extraordinary vertices of arbitrary valences, for all of the three schemes mentioned above: the extended Loop approximation scheme, its conversion to an interpolatory scheme, and the non-spline 1-ring interpolatory scheme. The discrete Fourier transform (DFT) is applied to analyze the spectral properties of the corresponding subdivision matrices, assuring that the eigenvalues of the subdivision matrices satisfy certain conditions for C 1 smoothness at the extraordinary vertices for all of the three considerations in this paper.

Meshless geometric subdivision

Point-based surface processing has developed into an attractive alternative to mesh-based processing tools for a number of geometric modeling applications. By working with point clouds directly, processing is based on the raw data and its underlying geometry rather than any arbitrary intermediate representations and generally artificial connectivity relations. We extend this principle into the area of subdivision surfaces by introducing the notion of meshless geometric subdivision. Our approach replaces the role of mesh connectivity with intrinsic point proximity thereby avoiding a number of limitations of mesh-based surface subdivision schemes. Apart from introducing this idea of meshless subdivision, we put forward a first intrinsic meshless subdivision scheme and present a new method for the computation of intrinsic means on Euclidean manifolds.

Refinable bivariate quartic and quintic [formula omitted]-splines for quadrilateral subdivisions

Refinable compactly supported bivariate C 2 quartic and quintic spline function vectors on the four-directional mesh are introduced in this paper to generate matrix-valued templates for approximation and Hermite interpolatory surface subdivision schemes, respectively, for both the 2 and 1-to-4 split quadrilateral topological rules. These splines have their full local polynomial preservation orders. In addition, we extend our study to parametric approach and use the symmetric properties of our refinable quintic spline components as a guideline to reduce the number of free parameters in constructing second order C 2 Hermite interpolatory quadrilateral subdivision schemes with precisely six components.

Matrix-valued symmetric templates for interpolatory surface subdivisions: I. Regular vertices

The objective of this paper is to introduce a general procedure for deriving interpolatory surface subdivision schemes with “symmetric subdivision templates” (SSTs) for regular vertices. While the precise definition of “symmetry” will be clarified in the paper, the property of SSTs is instrumental to facilitate application of the standard procedure for finding symmetric weights for taking weighted averages to accommodate extraordinary (or irregular) vertices in surface subdivisions, a topic to be studied in a continuation paper. By allowing the use of matrices as weights, the SSTs introduced in this paper may be constructed to overcome the size barrier limited to scalar-valued interpolatory subdivision templates, and thus avoiding the unnecessary surface oscillation artifacts. On the other hand, while the old vertices in a (scalar) interpolatory subdivision scheme do not require a subdivision template, we will see that this is not the case for the matrix-valued setting. Here, we employ the same definition of interpolation subdivisions as in the usual scalar consideration, simply by requiring the old vertices to be stationary in the definition of matrix-valued interpolatory subdivisions. Hence, there would be another complication when the templates are extended to accommodate extraordinary vertices if the template sizes are not small. In this paper, we show that even for C 2 interpolatory subdivisions, only one “ring” is sufficient in general, for both old and new vertices. For example, for 1-to-4 split C 2 interpolatory surface subdivisions, we obtain matrix-valued symmetric interpolatory subdivision templates (SISTs) for both triangular and quadrilateral meshes with sizes that agree with those of the Loop and Catmull–Clark schemes, respectively. Matrix-valued SISTs of similar sizes are also constructed for C 2 interpolatory 3 and 2 subdivision schemes in this paper. In addition to small template sizes, an obvious feature of matrix-valued weights is the flexibility for introducing shape-control parameters. Another significance is that, in contrast to the usual scalar setting, matrix-valued SISTs can be formulated in terms of the coefficient sequence of some vector refinement equation of interpolating bivariate C 2 splines with small support. For example, by modifying the spline function vectors introduced in our previous work [C.K. Chui, Q.T. Jiang, Surface subdivision schemes generated by refinable bivariate spline function vectors, Appl. Comput. Harmon. Anal. 15 (2003) 147–162; C.K. Chui, Q.T. Jiang, Refinable bivariate quartic and quintic C 2 -splines for quadrilateral subdivisions, Preprint, 2004], C 2 symmetric interpolatory subdivision schemes associated with refinement equations of C 2 cubic and quartic splines on the 6-directional and 4-directional meshes, respectively, are also constructed in this paper.

Subdivision schemes for smooth contact surfaces of arbitrary mesh topology in 3D

AbstractThis paper presents a strategy to parameterize contact surfaces of arbitrary mesh topology in 3D with at least C1‐continuity for both quadrilateral and triangular meshes. In the regular mesh domain, four quadrilaterals or six triangles meet in one node, even C2‐continuity is attained. Therefore, we use subdivision surfaces, for which non‐physical pressure jumps are avoided for contact interactions. They are usually present when the contact kinematics is based on facet elements discretizing the interacting bodies. The properties of subdivision surfaces give rise to basically four different implementation strategies. Each strategy has specific features and requires more or less efforts for an implementation in a finite element program. One strategy is superior with respect to the others in the sense that it does not use nodal degrees of freedom of the finite element mesh at the contact surface. Instead, it directly uses the degrees of freedom of the smooth surface. Thereby, remarkably, it does not require an interpolation. We show how the proposed method can be used to parameterize adaptively refined meshes with hanging nodes. This is essential when dealing with finite element models whose geometry is generated by means of subdivision techniques. Three numerical 3D problems demonstrate the improved accuracy, robustness and performance of the proposed method over facet‐based contact surfaces. In particular, the third problem, adopted from biomechanics, shows the advantages when designing complex contact surfaces by means of subdivision techniques. Copyright © 2004 John Wiley &amp; Sons, Ltd.

Interpolatory convexity-preserving subdivision schemes for curves and surfaces

Interpolatory convexity-preserving subdivision shcemes for curves and surfaces are introduced, and a convergence analysis is presented. The schemes are defined by geometric constructions, and they are nonlinear in the control points. It is shown, by geometry-based proofs, that the limit curves and surfaces are C 1.

Using parameters to increase smoothness of curves and surfaces generated by subdivision

Some recent subdivision schemes for curves and surfaces are viewed as perturbations of classical schemes for B-splines, and their convergence analysis is presented. It is shown that, under certain mild conditions, a small perturbation of a convergent subdivision scheme does not impair the smoothness of the limit function. Moreover, a proper perturbation can even increase the smoothness and thus provide a one-parameter class of schemes with better continuity properties.

Test of the Capabilities of CHIEF

The computer program CHIEF was designed by Schenck [J. Acoust. Soc. Amer. 44, 41–58 (1968)] to obtain approximate solutions to exterior steady-state acoustic radiation problems for arbitrary shape surfaces whose normal velocity is specified. To test its capabilities as a research tool, CHIEF was used to obtain the acoustic radiation impedance of caps and rings on oblate spheroids. The radiation impedance was also obtained using a harmonic expansion in oblate spheroidal functions. Improved FORTRAN computer programs were developed to evaluate the necessary spheroidal functions accurately. The accuracy of the CHIEF results are discussed in terms of acoustic size, the choice of surface subdivision schemes, and execution time on the CDC 3800 computer at NRL. New input subroutines to CHIEF, which provide numerous geometrical options, are also discussed.