Subdivision Scheme Research Articles

Hermite B-Splines: n-Refinability and Mask Factorization

This paper deals with polynomial Hermite splines. In the first part, we provide a simple and fast procedure to compute the refinement mask of the Hermite B-splines of any order and in the case of a general scaling factor. Our procedure is solely derived from the polynomial reproduction properties satisfied by Hermite splines and it does not require the explicit construction or evaluation of the basis functions. The second part of the paper discusses the factorization properties of the Hermite B-spline masks in terms of the augmented Taylor operator, which is shown to be the minimal annihilator for the space of discrete monomial Hermite sequences of a fixed degree. All our results can be of use, in particular, in the context of Hermite subdivision schemes and multi-wavelets.

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.

Subdivision Collocation Method for One-Dimensional Bratu’s Problem

The purpose of this article is to employ the subdivision collocation method to resolve Bratu’s boundary value problem by using approximating subdivision scheme. The main purpose of this researcher is to explore the application of subdivision schemes in the field of physical sciences. Our approach converts the problem into a set of algebraic equations. Numerical approximations of the solution of the problem and absolute errors are compared with existing methods. The comparison shows that the proposed method gives a more accurate solution than the existing methods.

A Hybrid Non-Stationary Subdivision Scheme Based on Triangulation

A Hybrid Non-Stationary Subdivision Scheme Based on Triangulation

Computational Design of Lightweight Trusses

Trusses are load-carrying light-weight structures consisting of bars connected at joints ubiquitously applied in a variety of engineering scenarios. Designing optimal trusses that satisfy functional specifications with a minimal amount of material has interested both theoreticians and practitioners for more than a century. In this paper, we introduce two main ideas to improve upon the state of the art. First, we formulate an alternating linear programming problem for geometry optimization. Second, we introduce two sets of complementary topological operations, including a novel subdivision scheme for global topology refinement inspired by Michell’s famed theoretical study. Based on these two ideas, we build an efficient computational framework for the design of lightweight trusses. We show that our method achieves trusses with smaller volumes and is faster compared with recent state-of-the-art approaches.

Hyperbolic interpolatory geometric subdivision schemes

The study of planar and spherical geometric subdivision schemes was done in Dyn and Hormann (2012); Bellaihou and Ikemakhen (2020). In this paper we complete this study by examining the hyperbolic case. We define general interpolatory geometric subdivision schemes generating curves on the hyperbolic plane by using geodesic polygons and the hyperbolic trigonometry. We show that a hyperbolic interpolatory geometric subdivision scheme is convergent if the sequence of maximum edge lengths is summable and the limit curve is G1-continuous if in addition the sequence of maximum angular defects is summable. In particular, we study the case of bisector interpolatory schemes. Some examples are given to demonstrate the properties of these schemes and some fascinating images on Poincaré disk are produced from these schemes.

Clothoid fitting and geometric Hermite subdivision

We consider geometric Hermite subdivision for planar curves, i.e., iteratively refining an input polygon with additional tangent or normal vector information sitting in the vertices. The building block for the (nonlinear) subdivision schemes we propose is based on clothoidal averaging, i.e., averaging w.r.t. locally interpolating clothoids, which are curves of linear curvature. To this end, we derive a new strategy to approximate Hermite interpolating clothoids. We employ the proposed approach to define the geometric Hermite analogues of the well-known Lane-Riesenfeld and four-point schemes. We present numerical results produced by the proposed schemes and discuss their features.

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

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

Analysis of a New Nonlinear Interpolatory Subdivision Scheme on σ Quasi-Uniform Grids

In this paper, we introduce and analyze the behavior of a nonlinear subdivision operator called PPH, which comes from its associated PPH nonlinear reconstruction operator on nonuniform grids. The acronym PPH stands for Piecewise Polynomial Harmonic, since the reconstruction is built by using piecewise polynomials defined by means of an adaption based on the use of the weighted Harmonic mean. The novelty of this work lies in the generalization of the already existing PPH subdivision scheme to the nonuniform case. We define the corresponding subdivision scheme and study some important issues related to subdivision schemes such as convergence, smoothness of the limit function, and preservation of convexity. In order to obtain general results, we consider σ quasi-uniform grids. We also perform some numerical experiments to reinforce the theoretical results.

On the construction of multiresolution analyses associated to general subdivision schemes

Subdivision schemes are widely used in numerical mathematics such as signal/image approximation, analysis and control of data or numerical analysis. However, to develop their full power, subdivision schemes should be incorporated into a multiresolution analysis that, mimicking wavelet analyses, provides a multi-scale decomposition of a function, a curve, or a surface. The ingredients needed to define a multiresolution analysis associated to a subdivision scheme are a decimation scheme and detail operators. Their construction is not straightforward as soon as the subdivision scheme is non-interpolatory. This paper is devoted to the construction of decimation schemes and detail operators compatible with general subdivision schemes, including non-linear ones. Analysis of the performances of the constructed analyses is carried out. Some numerical applications are presented in the framework of image approximation.

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.

Analytic Functions in Local Shift-Invariant Spaces and Analytic Limits of Level Dependent Subdivision

In this paper we characterize all subspaces of analytic functions in finitely generated shift-invariant spaces with compactly supported generators and provide explicit descriptions of their elements. We illustrate the differences between our characterizations and Strang-Fix or zero conditions on several examples. Consequently, we depict the analytic functions generated by scalar or vector subdivision with masks of bounded and unbounded support. In particular, we prove that exponential polynomials are indeed the only analytic limits of level dependent scalar subdivision schemes with finitely supported masks.

On New Means with Interesting Practical Applications: Generalized Power Means

Means of positive numbers appear in many applications and have been a traditional matter of study. In this work, we focus on defining a new mean of two positive values with some properties which are essential in applications, ranging from subdivision and multiresolution schemes to the numerical solution of conservation laws. In particular, three main properties are crucial—in essence, the ideas of these properties are roughly the following: to stay close to the minimum of the two values when the two arguments are far away from each other, to be quite similar to the arithmetic mean of the two values when they are similar and to satisfy a Lipchitz condition. We present new means with these properties and improve upon the results obtained with other means, in the sense that they give sharper theoretical constants that are closer to the results obtained in practical examples. This has an immediate correspondence in several applications, as can be observed in the section devoted to a particular example.

Recursive Process for Constructing the Refinement Rules of New Combined Subdivision Schemes and Its Extended Form

In this article, we present a new method to construct a family of2N+2-point binary subdivision schemes with one tension parameter. The construction of the family of schemes is based on repeated local translation of points by certain displacement vectors. Therefore, refinement rules of the2N+2-point schemes are recursively obtained from refinement rules of the2N-point schemes. Thus, we get a new subdivision scheme at each iteration. Moreover, the complexity, polynomial reproduction, and polynomial generation of the schemes are increased by two at each iteration. Furthermore, a family of interproximate subdivision schemes with tension parameters is also introduced which is the extended form of the proposed family of schemes. This family of schemes allows a different tension value for each edge and vertex of the initial control polygon. These schemes generate curves and surfaces such that some initial control points are interpolated and others are approximated.

A Novel Numerical Method for Computing Subdivision Depth of Quaternary Schemes

In this paper, an advanced computational technique has been presented to compute the error bounds and subdivision depth of quaternary subdivision schemes. First, the estimation is computed of the error bound between quaternary subdivision limit curves/surfaces and their polygons after kth-level subdivision by using l0 order of convolution. Secondly, by using the error bounds, the subdivision depth of the quaternary schemes has been computed. Moreover, this technique needs fewer iterations (subdivision depth) to get the optimal error bounds of quaternary subdivision schemes as compared to the existing techniques.

Subdivision of point-normal pairs with application to smoothing feasible robot path

In a previous paper (Lipovetsky and Dyn in Comput Aided Geom Des 48:36–48, 2016), we introduced a weighted binary average of two 2D point-normal pairs, termed circle average, and investigated subdivision schemes based on it. These schemes refine point-normal pairs in 2D and converge to limit curves and limit normals. Such a scheme has the disadvantage that the limit normals are not the normals of the limit curve. In this paper, we address this problem by proposing a new averaging method and obtaining a new family of algorithms based on it. We demonstrate their new editing capabilities and apply this subdivision technique to smooth a precomputed feasible polygonal point robot path.

Joint spectral radius and ternary hermite subdivision

In this paper we construct a family of ternary interpolatory Hermite subdivision schemes of order 1 with small support and {mathscr{H}}mathcal {C}^{2}-smoothness. Indeed, leaving the binary domain, it is possible to derive interpolatory Hermite subdivision schemes with higher regularity than the existing binary examples. The family of schemes we construct is a two-parameter family whose {mathscr{H}}mathcal {C}^{2}-smoothness is guaranteed whenever the parameters are chosen from a certain polygonal region. The construction of this new family is inspired by the geometric insight into the ternary interpolatory scalar three-point subdivision scheme by Hassan and Dodgson. The smoothness of our new family of Hermite schemes is proven by means of joint spectral radius techniques.

A Hermite interpolatory subdivision scheme constructed from quadratic rational Bernstein–Bezier spline

In this paper, a new nonlinear Hermite interpolatory subdivision scheme for curve interpolation is introduced. The scheme is constructed from the rational Bernstein–Bezier (RBB) spline. The limit function of the scheme interpolates both the function values and the derivatives. The work provides convergence analysis, polynomial reproduction, and shape preserving properties of the scheme. In particular, it is shown that the limit functions are globally C1 and the scheme also reproduces quadratic polynomials. Moreover, the scheme preserves monotonicity and convexity. Several examples are provided to justify our claims.

On the Application of the Generalized Means to Construct Multiresolution Schemes Satisfying Certain Inequalities Proving Stability

Multiresolution representations of data are known to be powerful tools in data analysis and processing, and they are particularly interesting for data compression. In order to obtain a proper definition of the edges, a good option is to use nonlinear reconstructions. These nonlinear reconstruction are the heart of the prediction processes which appear in the definition of the nonlinear subdivision and multiresolution schemes. We define and study some nonlinear reconstructions based on the use of nonlinear means, more in concrete the so-called Generalized means. These means have two interesting properties that will allow us to get associated reconstruction operators adapted to the presence of discontinuities, and having the maximum possible order of approximation in smooth areas. Once we have these nonlinear reconstruction operators defined, we can build the related nonlinear subdivision and multiresolution schemes and prove more accurate inequalities regarding the contractivity of the scheme for the first differences and in turn the results about stability. In this paper, we also define a new nonlinear two-dimensional multiresolution scheme as non-separable, i.e., not based on tensor product. We then present the study of the stability issues for the scheme and numerical experiments reinforcing the proven theoretical results and showing the usefulness of the algorithm.

Network analysis of internal migration in Croatia

Migration, and urbanization as its consequence, is among the most intricate political and scientific topics, predicted to have huge effects on human lives in the near future. Thus being said, previous works have mainly focused on international migration, and the research on internal migration outside of the US is scarce, and in the case of Europe—the ubiquitous center of migration affairs—only in its infancy. Observing migration between settlements, especially using network analysis indicators and models, can help to explain and predict migration, as well as urbanization originating from internal migration. We therefore conducted a network analysis of internal migration in Croatia, providing insights into the size of internal migration in population, and relative sizes between intra-settlement migration, inter-settlement migration and population. Through centrality analysis, we provide insights into hierarchy of importance, especially, in terms of the overall flow and overall attractiveness of particular settlements in the network. The analysis of the network structure reveals high presence of reciprocity and thus the importance of internal migration to urbanization, as well as the systematic abandonment of large cities in the east of the country. The application of three different community detection algorithms provides insights for the policy domain in terms of the compatibility of the current country administrative subdivision schemes and the subdivision implied by migration patterns. For network scholars, the analysis at hand reveals the status quo in applied network analysis to migration, the works published, the measures used, and potential metrics outside those applied which may be used to better explain and predict the intricate phenomenon of human migration.