Non-stationary Subdivision Schemes Research Articles

An Efficient Computational Approach for Computing Subdivision Depth of Non-Stationary Binary Subdivision Schemes

Subdivision schemes are equipped with some rules that take a polygon as an input and produce smooth curves or surfaces as an output. This presents the issue of how accurately the polygon approximates the limit curve and surface. What number of iterations/levels would be necessary to achieve the required shape at a user-specified error tolerance? In fact, several methods have been introduced in the case of stationary schemes to address the issue in terms of the error bounds (distance between polygon/polyhedron and limiting shape) and subdivision depth (the number of iterations required to obtain the result at a user-specified error tolerance). However, in the case of non-stationary schemes, this topic needs to be further studied to meet the requirements of new practical applications. This paper highlights a new approach based on a convolution technique to estimate error bounds and subdivision depth for non-stationary schemes. The given technique is independent of any condition on the coefficient of the non-stationary subdivision schemes, and it also produces the best results with the least amount of computational effort. In this paper, we first associated constants with the vectors generated by the given non-stationary schemes, then formulated an expression for the convolution product. This expression gives real values, which monotonically decrease with the increase in the order of the convolution in both the curve and surface cases. This convolution feature plays an important role in obtaining the user-defined error tolerance with fewer iterations. It achieves a trade-off between the number of iterations and user-specified errors. In practice, more iterations are needed to achieve a lower error rate, but we achieved this goal by using fewer iterations.

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 Reverse Non-Stationary Generalized B-Splines Subdivision Scheme

In this paper, two new families of non-stationary subdivision schemes are introduced. The schemes are constructed from uniform generalized B-splines with multiple knots of orders 3 and 4, respectively. Then, we construct a third-order reverse subdivision framework. For that, we derive a generalized multi-resolution mask based on their third-order subdivision filters. For the reverse of the fourth-order scheme, two methods are used; the first one is based on least-squares formulation and the second one is based on solving a linear optimization problem. Numerical examples are given to show the performance of the new schemes in reproducing different shapes of initial control polygons.

Construction of a family of non-stationary combined ternary subdivision schemes reproducing exponential polynomials

Abstract In this paper, we propose a family of non-stationary combined ternary ( 2 m + 3 ) \left(2m+3) -point subdivision schemes, which possesses the property of generating/reproducing high-order exponential polynomials. This scheme is obtained by adding variable parameters on the generalized ternary subdivision scheme of order 4. For such a scheme, we investigate its support and exponential polynomial generation/reproduction and get that it can generate/reproduce certain exponential polynomials with suitable choices of the parameters and reach 2 m + 3 2m+3 approximation order. Moreover, we discuss its smoothness and show that it can produce C 2 m + 2 {C}^{2m+2} limit curves. Several numerical examples are given to show the performance of the schemes.

A Hybrid Non-Stationary Subdivision Scheme Based on Triangulation

A Hybrid Non-Stationary Subdivision Scheme Based on Triangulation

Construction of Trigonometric Box Splines and the Associated Non-Stationary Subdivision Schemes

Construction of Trigonometric Box Splines and the Associated Non-Stationary Subdivision Schemes

Mixed trigonometric and hyperbolic subdivision scheme with two tension and one shape parameters

In this paper, we propose a new family of non-stationary subdivision scheme, based on mixed trigonometric hyperbolic B-spline basis functions with two tension and one shape parameters. We prove that the proposed subdivision scheme of order k produces -continuous limit curves. By assigning appropriate values to the two tension and shape parameters, many known subdivision schemes are special cases of our approach. Several variants of non-stationary subdivision schemes are discussed, and numerical examples illustrate their performances.

Creating a bridge between cardinal B[formula omitted]-spline fundamental functions for interpolation and subdivision

This paper presents innovative contributions to the fields of cardinal spline interpolation and subdivision. In particular, it unifies cardinal Br-spline fundamental functions for interpolation that are made of r=ML+1 (L∈N∪{0}) distinct pieces between each pair of interpolation nodes and are featured by the properties of C2M−2 smoothness, approximation order 2M and support width 2M(r+1)r, with the basic limit functions of a special class of non-stationary subdivision schemes of arity M.After introducing a general result, we focus our attention on the subclass of fourth-order accurate, C2 smooth Br-splines with maximum width of the compact support 6. The binary subdivision scheme yielding these fundamental functions outperforms the existing interpolatory schemes and seems to be the most adequate starting point to obtain compactly supported fundamental (spline) functions for local interpolation over quadrilateral and triangular meshes.

Construction and analysis of unified 4-point interpolating nonstationary subdivision surfaces

Subdivision schemes (SSs) have been the heart of computer-aided geometric design almost from its origin, and several unifications of SSs have been established. SSs are commonly used in computer graphics, and several ways were discovered to connect smooth curves/surfaces generated by SSs to applied geometry. To construct the link between nonstationary SSs and applied geometry, in this paper, we unify the interpolating nonstationary subdivision scheme (INSS) with a tension control parameter, which is considered as a generalization of 4-point binary nonstationary SSs. The proposed scheme produces a limit surface having C^{1} smoothness. It generates circular images, spirals, or parts of conics, which are important requirements for practical applications in computer graphics and geometric modeling. We also establish the rules for arbitrary topology for extraordinary vertices (valence ≥3). The well-known subdivision Kobbelt scheme (Kobbelt in Comput. Graph. Forum 15(3):409–420, 1996) is a particular case. We can visualize the performance of the unified scheme by taking different values of the tension parameter. It provides an exact reproduction of parametric surfaces and is used in the processing of free-form surfaces in engineering.

A Nonstationary Ternary 4-Point Shape-Preserving Subdivision Scheme

This paper uses the continued fraction technique to construct a nonstationary 4-point ternary interpolatory subdivision scheme, which provides the user with a tension parameter that effectively handles cusps compared with a stationary 4-point ternary interpolatory subdivision scheme. Then, the continuous nonstationary 4-point ternary scheme is analyzed, and the limit curve is at least C 2 -continuous. Furthermore, the monotonicity preservation and convexity preservation are proved.

Shape-Preservation of the Four-Point Ternary Interpolating Non-stationary Subdivision Scheme

In this paper, we present the shape-preserving properties of the four-point ternary non-stationary interpolating subdivision scheme (the four-point scheme). This scheme involves a tension parameter. We derive the conditions on the tension parameter and initial control polygon that permit the creation of positivity- and monotonicity-preserving curves after a finite number of subdivision steps. In addition, the outcomes are generalized to determine conditions for positivity- and monotonicity-preservation of the limit curves. Convexity-preservation of the limit curve of the four-point scheme is also analyzed. The shape-preserving behavior of the four-point scheme is also shown through several numerical examples.

A Family of Binary Univariate Nonstationary Quasi-Interpolatory Subdivision Reproducing Exponential Polynomials

In this paper, by suitably using the so-called push-back operation, a connection between the approximating and interpolatory subdivision, a new family of nonstationary subdivision schemes is presented. Each scheme of this family is a quasi-interpolatory scheme and reproduces a certain space of exponential polynomials. This new family of schemes unifies and extends quite a number of the existing interpolatory schemes reproducing exponential polynomials and noninterpolatory schemes like the cubic exponential B-spline scheme. For these new schemes, we investigate their convergence, smoothness, and accuracy and show that they can reach higher smoothness orders than the interpolatory schemes with the same reproduction property and better accuracy than the exponential B-spline schemes. Several examples are given to illustrate the performance of these new schemes.

Nonstationary interpolatory subdivision schemes reproducing high-order exponential polynomials

We present a family of nonstationary interpolatory subdivision schemes which reproduces high-order exponential polynomials. First, by extending the classical $\fbox {D-D}$ interpolatory schemes, we present the explicit expression of the symbols that identify a family of the nonstationary interpolatory subdivision schemes. These schemes can allow reproduction of more exponential polynomials, and represent exactly circular shapes, parts of conics which are important analytical shapes in geometric modeling. Furthermore, a rigorous analysis regarding the smoothness of the new nonstationary interpolatory schemes is provided. Next, based on the recursive formulas of the symbols, a repeated local operation is proposed for rapidly computing new points from the old points. Finally, two examples are presented to illustrate the performance of the new schemes.

A non-stationary Catmull–Clark subdivision scheme with shape control

In this paper, a non-stationary Catmull–Clark subdivision scheme is presented which enables the user to change the shape of the surface obtained from a given initial mesh. Such a non-stationary subdivision scheme is constructed by taking tensor product of a properly modified cubic exponential B-spline scheme in the regular mesh and giving suitable rules in the neighborhoods of extraordinary points. For this new scheme, we show that it can generate tangent plane continuous surfaces. Besides, we propose a generalization, which could locally control the limit surface for the purpose of generating more flexible surfaces and providing some features control. Some examples are given to show the performance of the new schemes.

A new class of 2m-point binary non-stationary subdivision schemes

A new class of 2m-point non-stationary subdivision schemes (SSs) is presented, including some of their important properties, such as continuity, curvature, torsion monotonicity, and convexity preservation. The multivariate analysis of subdivision schemes is extended to a class of non-stationary schemes which are asymptotically equivalent to converging stationary or non-stationary schemes. A comparison between the proposed schemes, their stationary counterparts and some existing non-stationary schemes has been depicted through examples. It is observed that the proposed SSs give better approximation and more effective results.

A New Class of 2q-Point Nonstationary Subdivision Schemes and Their Applications

The main objective of this study is to introduce a new class of 2 q -point approximating nonstationary subdivision schemes (ANSSs) by applying Lagrange-like interpolant. The theory of asymptotic equivalence is applied to find the continuity of the ANSSs. These schemes can be nicely generalized to contain local shape parameters that allow the user to locally adjust the shape of the limit curve/surface. Moreover, many existing approximating stationary subdivision schemes (ASSSs) can be obtained as nonstationary counterparts of the proposed ANSSs.

Convergence and Normal Continuity Analysis of Nonstationary Subdivision Schemes Near Extraordinary Vertices and Faces

Convergence and normal continuity analysis of a bivariate nonstationary (level-dependent) subdivision scheme for 2-manifold meshes with arbitrary topology is still an open issue. Exploiting ideas from the theory of asymptotically equivalent subdivision schemes, in this paper we derive new sufficient conditions for establishing convergence and normal continuity of any rotationally symmetric, nonstationary subdivision scheme near an extraordinary vertex/face.

Family of odd point non-stationary subdivision schemes and their applications

The (2s-1)-point non-stationary binary subdivision schemes (SSs) for curve design are introduced for any integer sgeq 2. The Lagrange polynomials are used to construct a new family of schemes that can reproduce polynomials of degree (2s-2). The usefulness of the schemes is illustrated in the examples. Moreover, the new schemes are the non-stationary counterparts of the stationary schemes of (Daniel and Shunmugaraj in 3rd International Conference on Geometric Modeling and Imaging, pp. 3–8, 2008; Hassan and Dodgson in Curve and Surface Fitting: Sant-Malo 2002, pp. 199–208, 2003; Hormann and Sabin in Comput. Aided Geom. Des. 25:41–52, 2008; Mustafa et al. in Lobachevskii J. Math. 30(2):138–145, 2009; Siddiqi and Ahmad in Appl. Math. Lett. 20:707–711, 2007; Siddiqi and Rehan in Appl. Math. Comput. 216:970–982, 2010; Siddiqi and Rehan in Eur. J. Sci. Res. 32(4):553–561, 2009). Furthermore, it is concluded that the basic shapes in terms of limiting curves produced by the proposed schemes with fewer initial control points have less tendency to depart from their tangent as well as their osculating plane compared to the limiting curves produced by existing non-stationary subdivision schemes.

Non-stationary versions of fixed-point theory, with applications to fractals and subdivision

Iterated Function Systems (IFSs) have been at the heart of fractal geometry almost from its origin, and several generalizations for the notion of IFS have been suggested. Subdivision schemes are widely used in computer graphics and attempts have been made to link fractals generated by IFSs to limits generated by subdivision schemes. With an eye towards establishing connection between non-stationary subdivision schemes and fractals, this paper investigates “trajectories of maps defined by function systems” which are considered as generalizations of the traditional IFS. The significance of ’forward’ and ’backward’ trajectories of general sequences of maps is studied. The convergence properties of these trajectories constitute a non-stationary version of the classical fixed-point theory. Unlike the ordinary fractals which are self-similar at different scales, the attractors of trajectories of maps defined by function systems may have different structures at different scales.

Non-Stationary Four-Point Binary Blending Subdivision Schemes

Non-Stationary Four-Point Binary Blending Subdivision Schemes