Interpolatory Subdivision Scheme Research Articles

Non-uniform interpolatory subdivision surface

This paper presents an interpolatory subdivision scheme with non-uniform parametrization for arbitrary polygon meshes with arbitrary manifold topology. This is the first attempt to generalize the non-uniform four point interpolatory curve subdivision to surface with extraordinary points. The scheme is constructed from the inspiration of the relation between the non-uniform four-point interpolatory subdivision scheme and the non-uniform B-spline refinement rule. Numerical examples and comparisons with the uniform interpolatory subdivision schemes indicate that the quality of the limit surface can be improved by using non-uniform parameter values for non-uniform initial data.

A Family of 2n-Point Ternary Non-Stationary Interpolating Subdivision Scheme

A Family of 2n-Point Ternary Non-Stationary Interpolating Subdivision Scheme

A non-stationary combined subdivision scheme generating exponential polynomials

In this paper, a non-stationary combined subdivision scheme is presented, which can unify several existing non-stationary approximating and interpolatory subdivision schemes. This scheme is obtained by generalizing the connection between the approximating and interpolatory schemes in the stationary case, first formalized by Maillot & Stam using a push-back operator, to the non-stationary case. For such a combined scheme, we investigate its Cl convergence and exponential polynomial generation/reproduction property and get that it can reach C4 degree of smoothness and generate/reproduce certain exponential polynomials with suitable choices of the parameters. Besides, we give a more generalized combined scheme for the purpose of generating and reproducing more general exponential polynomials. The performance of our new schemes is illustrated by several numerical examples.

Smooth interpolation of key frames in a Riemannian shell space

Splines and subdivision curves are flexible tools in the design and manipulation of curves in Euclidean space. In this paper we study generalizations of interpolating splines and subdivision schemes to the Riemannian manifold of shell surfaces in which the associated metric measures both bending and membrane distortion. The shells under consideration are assumed to be represented by Loop subdivision surfaces. This enables the animation of shells via the smooth interpolation of a given set of key frame control meshes. Using a variational time discretization of geodesics efficient numerical implementations can be derived. These are based on discrete geodesic interpolation, a discrete geometric logarithm, a discrete exponential map, and discrete parallel transport. With these building blocks at hand discrete Riemannian cardinal splines and three different types of discrete, interpolatory subdivision schemes are defined. Numerical results for two different subdivision shell models underline the potential of this approach in key frame animation.

A family of non-oscillatory 6-point interpolatory subdivision schemes

In this paper we propose and analyze a new family of nonlinear subdivision schemes which can be considered non-oscillatory versions of the 6-point Deslauries-Dubuc (DD) interpolatory scheme, just as the Power p schemes are considered nonlinear non-oscillatory versions of the 4-point DD interpolatory scheme. Their design principle may be related to that of the Power p schemes and it is based on a weighted analog of the Power p mean. We prove that the new schemes reproduce exactly polynomials of degree three and stay ’close’ to the 6-point DD scheme in smooth regions. In addition, we prove that the first and second difference schemes are well defined for each member of the family, which allows us to give a simple proof of the uniform convergence of these schemes and also to study their stability as in [19, 22]. However our theoretical study of stability is not conclusive and we perform a series of numerical experiments that seem to point out that only a few members of the new family of schemes are stable. On the other hand, extensive numerical testing reveals that, for smooth data, the approximation order and the regularity of the limit function may be similar to that of the 6-point DD scheme and larger than what is obtained with the Power p schemes.

A formula for estimating the deviation of a binary interpolatory subdivision curve from its data polygon

This paper introduces a new formula to evaluate the deviation of a binary interpolatory subdivision curve from its data polygon. We first bound the deviation of the new control points of each subdivision step from its data polygon by accumulating the distances between the new control points and the midpoints of their corresponding edges. Then, by finding the maximum deviation of each subdivision step, a formula for estimating the deviation of the limit curve from its data polygon can be deduced. As the applications of the formula, we evaluate the deviations of the uniform, centripetal and chord parametrization four-point interpolatory subdivision scheme, and find that the bounds derived by our method are sharper than bounds by [3]. Of course, we also deduce the new deviations of the six-point interpolatory subdivision scheme, Dyn et al’s four- and six-point subdivision schemes with tension parameters, and Deslauriers–Dubucs eight- and ten-point subdivision schemes.

On four-point penalized Lagrange subdivision schemes

This paper is devoted to the definition and analysis of new subdivision schemes called penalized Lagrange. Their construction is based on an original reformulation for the construction of the coefficients of the mask associated to the classical 4-points Lagrange interpolatory subdivision scheme: these coefficients can be formally interpreted as the solution of a linear system similar to the one resulting from the constrained minimization problem in Kriging theory which is commonly used for reconstruction in geostatistical studies. In such a framework, the introduction in the formulation of a so-called error variance can be viewed as a penalization of the oscillations of the coefficients. Following this idea, we propose to penalize the 4-points Lagrange system. This penalization transforms the interpolatory schemes into approximating ones with specific properties suitable for the subdivision of locally noisy or strongly oscillating data. According to a so-called penalization vector, a family of schemes can be generated. A full theoretical study is first performed to analyze this new type of non stationary subdivision schemes. Then, in the framework of position dependant penalization vector, several numerical tests are provided to point out the efficiency of these schemes compared to standard approaches.

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 B...

Efficient evaluation of subdivision schemes with polynomial reproduction property

In this paper we present an efficient framework for the evaluation of subdivision schemes with polynomial reproduction property. For all interested rational parameters between 0 and 1 with the same denominator, their exact limit positions on the subdivision curve can be obtained by solving a system of linear equations. When the framework is applied to binary and ternary 4-point interpolatory subdivision schemes, we find that the corresponding coefficient matrices are strictly diagonally dominant, and so the evaluation processes are robust. For any individual irrational parameters between 0 and 1, its approximate value is computed by a recursive algorithm which can attain an arbitrary error bound. For surface schemes generalizing univariate subdivision schemes with polynomial reproduction property, exact evaluation methods can also be derived by combining Stam’s method with that of this paper.

Convexity preservation of six point C2 interpolating subdivision scheme

In this paper, the convexity preserving property of six-point ternary interpolating subdivision scheme with the tension parameter ω has been analyzed’. It should be noted that the condition on the tension parameter is determined when initial data is strictly convex. The condition has been determined when 7972<ω<111215, the range of C2 continuous limit curve. Significance of derived condition is elaborated through curvature plots. Some Properties of the scheme have also been explored. Conclusively the whole discussion is followed by examples.

Four point interpolatory-corner cutting subdivision

This paper presents a new curve algorithm called four point interpolatory-corner cutting subdivision for generating curves that interpolate some given vertices and approximate the other vertices. By four-point interpolatory-corner cutting subdivision, only the vertices specified to be interpolated are fixed and the other vertices are updated at each refinement step. Compared to the four-point interpolation subdivision scheme, it is expected to have improved behavior, and compared to the cutting smooth subdivision, it is able to generate curve interpolating some given vertices. It is more suitable for practical application. The refinement rules are derived to ensure that the eigenvalues of the refinement matrix satisfy the necessary condition of C1continuity. Four-point interpolatory-corner cutting smooth subdivision also contains tension parameters, it is conducive to regulate the shape of limit curve.

A Parallel Interpolation Subdivision Scheme for Curve and Surface Design

A Parallel Interpolation Subdivision Scheme for Curve and Surface Design

Generalized and Unified Families of Interpolating Subdivision Schemes

We present generalized and unified families of (2n)-point and (2n − 1)- point p-ary interpolating subdivision schemes originated from Lagrange polynomial for any integers n ≥ 2 and p ≥ 3. Almost all existing even-point and odd-point interpolating schemes of lower and higher arity belong to this family of schemes. We also present tensor product version of generalized and unified families of schemes. Moreover error bounds between limit curves and control polygons of schemes are also calculated. It has been observed that error bounds decrease when complexity of the scheme decrease and vice versa. Furthermore, error bounds decrease with the increase of arity of the schemes. We also observe that in general the continuity of interpolating scheme do not increase by increasing complexity and arity of the scheme.

Fractal behavior of ternary 4-point interpolatory subdivision scheme with tension parameter

In this paper, the fractal curves and surfaces of ternary 4-point interpolatory subdivision scheme are developed and analyzed. The fractal range of the tension parameter has also been determined. The behavior of fractal curves and surfaces corresponding to different values of parameter has been depicted through six examples which offers direct means for an effective generation of subdivision fractals. The real life applications of the proposed fractal subdivision scheme has also been demonstrated.

On extraordinary rules of quad-based interpolatory subdivision schemes

This article deals with interpolatory subdivision schemes generalizing the tensor-product version of the Dubuc–Deslauriers 4-point scheme to quadrilateral meshes of arbitrary manifold topology. In particular, we focus our attention on an extension of the C1 regular stencils that respectively exploits (N+2)-point and (2N+8)-point stencils for the computation of an edge-point and a face-point in the vicinity of an extraordinary vertex of valence N≠4 not lying on a boundary. The aim of our work consists in identifying which constraints are required to be respected by the weights appearing in the above stencils in order to get closed limit surfaces that are C1-continuous at extraordinary points, have both principal curvatures bounded and at least one of them nonzero. The obtained constraints are used to easily check these features in the limit surfaces resulting from the application of special extraordinary rules proposed in the literature by different authors. Moreover, the conditions derived on the stencil weights are exploited to design new extraordinary rules that can produce closed limit surfaces of the same quality as the existing proposals, but at a reduced computational cost.

Building blocks for designing arbitrarily smooth subdivision schemes with conic precision

Since subdivision schemes featured by high smoothness and conic precision are strongly required in many application contexts, in this work we define the building blocks to obtain new families of non-stationary subdivision schemes enjoying such properties. To this purpose, we firstly derive a non-stationary extension of the Lane–Riesenfeld algorithm, and we exploit the resulting class of schemes to design a non-stationary family of alternating primal/dual subdivision schemes, all featured by reproduction of {1,x,etx,e−tx},t∈[0,π)∪iR+. Then, we focus our attention on interpolatory subdivision schemes with conic precision, that can be obtained as a byproduct of the above classes. In particular, we present a novel construction of a family of non-stationary interpolatory 2n-point schemes which generalizes the well-known Dubuc–Deslauriers family in such a way the nth (n≥2) family member reproduces Π2n−3∪{etx,e−tx},t∈[0,π)∪iR+, and keeps the original smoothness of its stationary counterpart unchanged.

Generation of fractal curves and surfaces using ternary 4-point interpolatory subdivision scheme

In this paper, the generation of fractal curves and surfaces along with their properties, using ternary 4-point interpolatory subdivision scheme with one parameter, are analyzed. The relationship between the tension parameter and the fractal behavior of the limiting curve is demonstrated through different examples. The specific range of the tension parameter has also been depicted, which provides a clear way to generate fractal curves. Since the fractal scheme introduces, in the paper, have more number of control points therefore it gives more degree of freedom to control the shape of the fractal curve.

A biarc based subdivision scheme for space curve interpolation

This paper presents a biarc-based subdivision scheme for space curve interpolation. Given a sequence of space points, or a sequence of space points and tangent vectors, the scheme produces a smooth curve interpolating all input points by iteratively inserting new points and computing new tangent vectors. For each step of subdivision, the newly inserted point corresponding to an existing edge is a specified joint point of a biarc curve which interpolates the two end points and the tangents. A provisional tangent is also assigned to the newly inserted point. Each of the tangents for the following subdivision step is further updated as a linear blending of the provisional tangent and the tangent at the respective point of a circle passing through the local point and its two adjacent points. If adjacent four initial points and their initial reference tangent vectors are sampled from a spherical surface, the limit curve segment between the middle two initial points exactly lies on the same spherical surface. The biarc based subdivision scheme is proved to be G1 continuous with a nice convexity preserving property. Numerical examples also show that the limit curves are G2 continuous and fair. Several examples are given to demonstrate the excellent properties of the scheme.

Lagrange Interpolatory Subdivision Schemes in Chebyshev Spaces

We develop an efficient approach to the analysis of Lagrange interpolatory subdivision schemes based on Extended Chebyshev spaces of any even dimension. In general, such schemes are non-uniform and non-stationary. The study confirms and extends some ideas concerning more generally the analysis of non-regular subdivision schemes already presented in earlier papers. One crucial step consists in finding (non-regular) grids naturally adapted to the initial scheme in view of defining its derived schemes, a change of grid being possibly necessary for each additional order of smoothness considered. Surprisingly, it may be the case that the natural grids are non-nested, even though the initial scheme is interpolatory. This is so in particular for Chebyshevian Lagrange interpolatory schemes, for which the natural grids are defined in terms of Chebyshevian divided differences. Comparison of the corresponding successive derived schemes with their polynomial counterparts enables us to show that they have similar behaviours.

A New Method for the Analysis of Univariate Nonuniform Subdivision Schemes

This paper presents a new method for the analysis of convergence and smoothness of univariate nonuniform subdivision schemes. The analysis involves ideas from the theory of asymptotically equivalent subdivision schemes and nonuniform Laurent polynomial representation together with a new perturbation result. Application of the new method is presented for the analysis of interpolatory subdivision schemes based upon extended Chebyshev systems and for a class of smoothly varying schemes.