Interpolatory Subdivision Scheme Research Articles

Generalization of the incenter subdivision scheme

We introduce a new interpolatory subdivision scheme generalizing the incenter subdivision [8]. The proposed scheme is equipped with a shape controlling tension parameter, is Hermitian, and reproduces circles from non-uniform samples. We prove that for any value of the free parameter the limit curve is G1 continuous. The scheme is shape preserving and avoids undesirable oscillations by producing curves with a finite number of inflection points at the regions where the control polygon suggests a change of convexity. Several examples are presented demonstrating the properties of the scheme.

A Unified Interpolating Subdivision Scheme for Curves/Surfaces by Using Newton Interpolating Polynomial

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.

Interproximate curve subdivision

This paper presents a new curve subdivision algorithm called interproximate subdivision for generating curves that interpolate some given vertices and approximate the other vertices. By the interproximate subdivision, only the vertices specified to be interpolated are fixed and the other vertices are updated at each refinement step. The refinement rules are derived to ensure that the eigenvalues of the refinement matrix satisfy the necessary condition of C2 continuity. The interproximate subdivision also contains tension parameters assigned to vertices or edges for shape adjustment. Compared to the 4-point interpolatory subdivision scheme, the interproximate subdivision does not force the new inserted vertices to be interpolated and is thus expected to have improved behavior; and compared to the cubic B-spline refinement scheme, the interproximate subdivision is able to generate curves interpolating user-specified vertices. In addition, the paper also presents two extensions of the interproximate subdivision: one automatically adapts the tension parameters locally according to the geometry of the control polygon during the refinement to achieve convexity preservation and the other automatically relaxes the interpolating property of some vertices to achieve better shape behavior.

A New 5-Point Ternary Interpolating Subdivision Scheme and Its Differentiability

A new 5-point ternary interpolating scheme with a shape parameter is introduced. The resulting curve is for a certain range of parameters. The differentiable properties of the proposed scheme to extend its application in the generation of smooth curves are explored. Application of the proposed scheme is given to show its visual smoothness. The scheme is also extended to a 5-point tensor product ternary interpolating scheme, and its numerical examples are also included.

A constructive algebraic strategy for interpolatory subdivision schemes induced by bivariate box splines

This paper describes an algebraic construction of bivariate interpolatory subdivision masks induced by three-directional box spline subdivision schemes. Specifically, given a three-directional box spline, we address the problem of defining a corresponding interpolatory subdivision scheme by constructing an appropriate correction mask to convolve with the three-directional box spline mask. The proposed approach is based on the analysis of certain polynomial identities in two variables and leads to interesting new interpolatory bivariate subdivision schemes.

A Generalized Four Point Interpolatory Subdivision Scheme for Curve Design

ßIn this paper we propose a new kind of nonlinear and geometry driven subdivision scheme for curve interpolation. We introduce serval parameters in the new scheme.When the parameter ß is taken as 0, the new scheme presented in this paper regresses to the initial four point subdivision scheme, and when ß→∞ , the new scheme is convexity preserving. With proper choices of the subdßivision parameters,it can overcome the shortcoming of the initial four point subdivision scheme proposed.

Interpolatory blending net subdivision schemes of Dubuc–Deslauriers type

Net subdivision schemes recursively refine nets of univariate continuous functions defined on the lines of planar grids, and generate as limits bivariate continuous functions. In this paper a family of interpolatory net subdivision schemes related to the family of Dubuc–Deslauriers interpolatory subdivision schemes is constructed and analyzed. The construction is based on Gordon blending interpolants to nets of univariate functions, and on a particular class of blending functions with properties related to the Dubuc–Deslauriers schemes. The general analysis tools for net subdivision schemes, developed in a previous paper by the authors, together with the properties of the blending functions, lead to the proof of the convergence of these schemes to limit functions having the same integer smoothness as the limits of the corresponding Dubuc–Deslauriers schemes. These results are proved for net subdivision schemes corresponding to the first 84 members of the Dubuc–Deslauriers family, and conjectured for the rest. A concrete example of a family of piecewise polynomial blending functions is considered, together with the corresponding family of net subdivision schemes. The performance of the first two net subdivision schemes in this family is demonstrated by two examples.

Subdivision Schemes for Positive Definite Matrices

The class of symmetric positive definite matrices is an important class both in theory and application. Although this class is well studied, little is known about how to efficiently interpolate such data within the class. We extend the 4-point interpolatory subdivision scheme, as a method of interpolation, to data consisting of symmetric positive definite matrices. This extension is based on an explicit formula for calculating a binary “geodetic average”. Our method generates a smooth curve of matrices, which retain many important properties of the interpolated matrices. Furthermore, the scheme is robust and easy to implement.

A combined approximating and interpolating subdivision scheme with C2 continuity

In this paper a combined approximating and interpolating subdivision scheme is presented. The relationship between approximating subdivision and interpolating subdivision is derived by directly performing operations on geometric rules. The behavior of the limit curve produced by our combined subdivision scheme is analyzed by the Laurent polynomial and attains C 2 degree of smoothness. Furthermore, a non-uniform combined subdivision with shape control parameters is introduced, which allows a different tension value for every edge of the original control polygon.

An alternative method for constructing interpolatory subdivision from approximating subdivision

This paper presents a new perspective for constructing interpolatory subdivision from primal approximating subdivision. The basic idea is constructing the subdivision rule for new inserted vertices of a new interpolatory subdivision scheme based on an approximating subdivision algorithm applied to a local configuration of the mesh with one vertex updated for interpolation of the vertex. This idea is demonstrated by presenting two new interpolatory subdivision schemes based on Catmull–Clark subdivision for an arbitrary polygonal mesh and Loop subdivision for a triangular mesh, respectively. These algorithms are simple and have a small stencil for computing new points. The new perspective also shows a link between those classic approximating and interpolatory subdivision algorithms such as cubic B-spline curve subdivision and the four-point interpolatory subdivision, Catmull–Clark subdivision and Kobbeltʼs interpolatory scheme, and Loop subdivision and the butterfly algorithm.

The Mask of Odd Points n‐Ary Interpolating Subdivision Scheme

We present an explicit formula for the mask of odd points n‐ary, for any odd n⩾3, interpolating subdivision schemes. This formula provides the mask of lower and higher arity schemes. The 3‐point and 5‐point a‐ary schemes introduced by Lian, 2008, and (2m + 1)‐point a‐ary schemes introduced by, Lian, 2009, are special cases of our explicit formula. Moreover, other well‐known existing odd point n‐ary schemes including the schemes introduced by Zheng et al., 2009, can easily be generated by our formula. In addition, error bounds between subdivision curves and control polygons of schemes are computed. It has been noticed that error bounds decrease when the complexity of the scheme decreases and vice versa. Also, as we increase arity of the schemes the error bounds decrease. Furthermore, we present brief comparison of total absolute curvature of subdivision schemes having different arity with different complexity. Convexity preservation property of scheme is also presented.

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---1987, 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.

Approximation order equivalence properties of manifold-valued data subdivision schemes

There has been an emerging interest in developing an approximation theory for manifold-valued functions. In this paper, we address the following fundamental problem: Let M be a manifold with a metric d. For each smoothness factor r > 0 and approximation order R > 0, is there an approximation operator Ah = Ah;r,R that maps samples of any f : R→ M on a grid of size h to an approximant fh = Ah(f |hZ) : R→ M with the properties that a) supx d(fh(x), f(x)) = O(h) whenever f is a bounded C function, and b) fh is C smooth ? The case of M = R is of course well-studied. In the recent paper [14], the authors show that subdivision methods can be used to create arbitrarily smooth interpolants for M -valued data, addressing b) above. In this paper we further show that interpolatory subdivision schemes can be used to solve a) above. So, altogether, we establish the fact that if a linear interpolatory subdivision scheme possesses a smoothness order r and an approximation order R, then a nonlinear interpolatroy subdivision scheme for M -valued data constructed based on this linear scheme has the same smoothness and approximation orders. In other words, subdivision schemes furnish a constructive approximation method for solving the open problem posted above. We discuss the construction of quasi-interpolants of manifold-valued data based on general (not necessarily interpolatory) subdivision schemes. Acknowledgments. The work of this research was partially supported by the National Science Foundation grant DMS 0542237. The empirical observation on approximation equivalence (Figure 1) was first presented in the Sixth International Conference on Curves and Surfaces (Avignon, France, June 29-July 5, 2006.) and further discussed in a mini-symposium in the 12th International Conference in Approximation Theory (San Antonio, Texas, March 4-8, 2007) and the MAIA 2007 conference (Alesund, Norway, August 22-26, 2007.) A preliminary effort on this problem was made in [5]. The second named author thanks Nira Dyn for inviting him to a workshop on “Subdivision and Refinability” (Pontignano, Siena, Italy, May 1-4, 2008), at which the main result of this paper was first presented.

TERNARY INTERPOLATORY SUBDIVISION SCHEMES ORIGINATED FROM SPLINES

A generic technique for construction of ternary interpolatory subdivision schemes, which is based on polynomial and discrete splines, is presented. These schemes have rational symbols. The symbols are explicitly presented in the paper. This is accompanied by a detailed description of the design of the refinement masks and by algorithms that verify the convergence of these schemes. In addition, the smoothness of the limit functions is investigated. The ternary subdivision schemes, whose construction is based on continuous splines, become tools for fast computation ofıory s of arbitrary order at triadic rational points.

A piecewise polynomial approach to analyzing interpolatory subdivision

The four-point interpolatory subdivision scheme of Dubuc and its generalizations to irregularly spaced data studied by Warren and by Daubechies, Guskov, and Sweldens are based on fitting cubic polynomials locally. In this paper, we analyze the convergence of the scheme by viewing the limit function as the limit of piecewise cubic functions arising from the scheme. This allows us to recover the regularity results of Daubechies et al. in a simpler way and to obtain the approximation order of the scheme and its first derivative.

Exact evaluation of limits and tangents for interpolatory subdivision surfaces at rational points

This paper presents a new method for exact evaluation of a limit surface generated by stationary interpolatory subdivision schemes and its associated tangent vectors at arbitrary rational points. The algorithm is designed on the basis of the parametric m -ary expansion and construction of the associated matrix sequence. The evaluation stencil of the control points on the initial mesh is obtained, through computation, by multiplying the finite matrices in a sequence corresponding to the expansion sequence and eigendecomposition of the contractive matrix related to the period of rational numbers. The method proposed in this paper works for other non-polynomial subdivision schemes as well.

Non-uniform interpolatory curve subdivision with edge parameters built upon compactly supported fundamental splines

In this paper we present a family of Non-Uniform Local Interpolatory (NULI) subdivision schemes, derived from compactly supported interpolatory fundamental splines with non-uniform knots (NULIFS). For this spline family, the knot-partition is defined by a sequence of break points and by one additional knot, arbitrarily placed along each knot-interval. The resulting refinement algorithms are linear and turn out to contain a set of edge parameters that, when fixed to a value in the range [0,1], allow us to achieve special shape features by simply moving each auxiliary knot between the break points. Among all the members of this new family of schemes, we will then especially analyze the NULI 4-point refinement. This subdivision scheme has all the fundamental features of the quadratic fundamental spline basis it is originated from, namely compact support, C1 smoothness, second order polynomials reproduction and approximation order 3. In addition the NULI 4-point subdivision algorithm has the possibility of setting consecutive edge parameters to simulate double and triple knots—that are not considered by the authors of the corresponding spline basis—thus allowing for limit curves with crease vertices, without using an ad hoc mask. Numerical examples and comparisons with other methods will be given to the aim of illustrating the performance of the NULI 4-point scheme in the case of highly non-uniform initial data.

Reconstruction of 3D objects from 2D cross-sections with the 4-point subdivision scheme adapted to sets

Reconstruction of 3D objects from 2D cross-sections is an intriguing problem with many potential applications. We approach this problem through a novel multi-resolution method based on iterative refinement of the sets representing the cross-sections. To that end, we introduce a new geometric weighted average of two sets, defined for positive weights (corresponding to interpolation) and when one weight is negative (corresponding to extrapolation). This new average can be used to interpolate between cross-sections of a 3D object in a piecewise way. To obtain a smoother reconstruction of the 3D object, we adapt to sets the 4-point interpolatory subdivision scheme using the new average with both positive and negative weights. The effectiveness of the new method is demonstrated by several examples.

Mixed Loop Scheme and it’s Industry Application

This paper presents a unified and mixed interpolatory and approximation subdivision scheme for triangular meshes. It can be used to solve the "popping effect" problem when switching between meshes at different levels of resolution. The scheme generates surfaces coincidence with the Loop subdivision scheme in the limit condition having the coefficient k equal 0. When k equal 1, it will be changed into a new interpolatory subdivision scheme. A new set of rules is designed for computing the newly inserted vertices around extraordinary masks. Eigen-structure analysis demonstrates that subdivision surfaces generated using the new schemes are C1 continuous. The method is a completely simple one without constructing and solving equations. The example of British Museum shell design show the great flexibility advantage of mixed subdivision design method in shell architecture, even in other industry design field.

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.