Convexity preserving interpolatory subdivision with conic precision

The paper is concerned with the problem of shape preserving interpolatory subdivision. For arbitrarily spaced, planar input data an efficient non-linear subdivision algorithm is presented that results in $G^1$ limit curves, reproduces conic sections and respects the convexity properties of the initial data. Significant numerical examples illustrate the effectiveness of the proposed method

Polynomial-based non-uniform interpolatory subdivision with features control

Starting from a well-known construction of polynomial-based interpolatory 4-point schemes, in this paper we present an original affine combination of quadratic polynomial samples that leads to a non-uniform 4-point scheme with edge parameters. This blending-type formulation is then further generalized to provide a powerful subdivision algorithm that combines the fairing curve of a non-uniform refinement with the advantages of a shape-controlled interpolation method and an arbitrary point insertion rule. The result is a non-uniform interpolatory 4-point scheme that is unique in combining a number of distinctive properties. In fact it generates visually-pleasing limit curves where special features ranging from cusps and flat edges to point/edge tension effects may be included without creating undesired undulations. Moreover such a scheme is capable of inserting new points at any positions of existing intervals, so that the most convenient parameter values may be chosen as well as the intervals for insertion. Such a fully flexible curve scheme is a fundamental step towards the construction of high-quality interpolatory subdivision surfaces with features control

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 [C.Conti, L.Gemignani, L.Romani, Linear Algebra Appl. 431 (2009), no. 10, 1971-1987] 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 exponential polynomial space 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

Exponential Splines and Pseudo-Splines: Generation versus reproduction of exponential polynomials

Subdivision schemes are iterative methods for the design of smooth curves and surfaces. Any linear subdivision scheme can be identified by a sequence of Laurent polynomials, also called subdivision symbols, which describe the linear rules determining successive refinements of coarse initial meshes. One important property of subdivision schemes is their capability of exactly reproducing in the limit specific types of functions from which the data is sampled. Indeed, this property is linked to the approximation order of the scheme and to its regularity. When the capability of reproducing polynomials is required, it is possible to define a family of subdivision schemes that allows to meet various demands for balancing approximation order, regularity and support size. The members of this family are known in the literature with the name of pseudo-splines. In case reproduction of exponential polynomials instead of polynomials is requested, the resulting family turns out to be the non-stationary counterpart of the one of pseudo-splines, that we here call the family of exponential pseudo-splines. The goal of this work is to derive the explicit expressions of the subdivision symbols of exponential pseudo-splines and to study their symmetry properties as well as their convergence and regularity.Comment: 25 page

Ternary Three Point Non-Stationary Subdivision Scheme

Abstract: A ternary three-point approximating non-stationary subdivision scheme is presented that generates the family of C 2 limiting curve. The proposed scheme can be considered as the non-stationary counterpart of the ternary three-point approximating stationary scheme. The comparison of the proposed scheme has been demonstrated using different examples with the existing ternary three-point stationary scheme, which shows that the limit curves of the proposed scheme behave more pleasantly and are very close to generate the conic section

Non-uniform interpolatory subdivision schemes with improved smoothness

Subdivision schemes are used to generate smooth curves or surfaces by iteratively refining an initial control polygon or mesh. We focus on univariate, linear, binary subdivision schemes, where the vertices of the refined polygon are computed as linear combinations of the current neighbouring vertices. In the classical stationary setting, there are just two such subdivision rules, which are used throughout all subdivision steps to construct the new vertices with even and odd indices, respectively. These schemes are well understood and many tools have been developed for deriving their properties, including the smoothness of the limit curves. For non-stationary schemes, the subdivision rules are not fixed and can be different in each subdivision step. Non-uniform schemes are even more general, as they allow the subdivision rules to be different for every new vertex that is generated by the scheme. The properties of non-stationary and non-uniform schemes are usually derived by relating the scheme to a corresponding stationary scheme and then exploiting the fact that the properties of the stationary scheme carry over under certain proximity conditions. In particular, this approach can be used to show that the limit curves of a non-stationary or non-uniform scheme are as smooth as those of a corresponding stationary scheme. In this paper we show that non-uniform subdivision schemes have the potential to generate limit curves that are smoother than those of stationary schemes with the same support size of the subdivision rule. For that, we derive interpolatory 2-point and 4-point schemes that generate C-1 and C-2 limit curves, respectively. These values of smoothness exceed the smoothness of classical interpolating schemes with the same support size by one. (C) 2022 The Author(s). Published by Elsevier B.V

An interpolating curve subdivision scheme based on discrete first derivative

This paper develops a new scheme of four points for interpolating curve subdivision based on the discrete fi rst derivative (DFDS), which reduces the apparition of undesirable oscillations that can be formed on the limit curve when the control points do not follow a uniform parameterization. We used a set of 3000 curves whose control points were randomly generated. Smooth curves were obtained after seven steps of subdivision using fi ve schemes DFDS, Four-Point (4P), New four-point (N4P), Tight four-point (T4P) and the geometrically controlled scheme (GC4P). The tortuosity property was evaluated on every smooth curve. An analysis for the frequency distributions of this property using the Kruskal-Wallis test reveals that DFDS scheme has the lowest values in a close range