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

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

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

Totally positive refinable functions with general dilation M

We construct a new class of approximating functions that are M-refinable and provide shape preserving approximations. The refinable functions in the class are smooth, compactly supported, centrally symmetric and totally positive. Moreover, their refinable masks are associated with convergent subdivision schemes. The presence of one or more shape parameters gives a great flexibility in the applications. Some examples for dilation M=4and M=5are also given

A UNIQUE COMBINATION OF MASK IN BINARY FOUR-POINT SUBDIVISION SCHEME

A unique binary four-point approximating subdivision scheme has been developed in which one part of binary formula have stationary mask and other part have the non-stationary mask. The resulting curves have the smoothness of C3 continuous for the wider range of shape control parameter. The role of the parameter has been depicted using the square form of discrete control points

Dual Hermite subdivision schemes of de Rham-type

International audienceThough a Hermite subdivision scheme is non-stationary by nature, its non-stationarity can be of two types, making useful the distinction between Inherently Stationary and Inherently Non-Stationary Hermite subdivision schemes. This paper focuses on the class of inherently stationary, dual non-interpolatory Hermite subdivision schemes that can be obtained from known Hermite interpolatory ones, by applying a generalization of the de Rham corner cutting strategy. Exploiting specific tools for the analysis of inherently stationary Hermite subdivision schemes we show that, giving up the interpolation condition, the smoothness of the associated basic limit function can be increased by one, while its support width is only enlarged by one. To accomplish the analysis of de Rham-type Hermite subdivision schemes two new theoretical results are derived and the new notion of HC-convergence is introduced. It allows the construction of Hermite-type subdivision schemes of order d + 1 with the first element of the vector valued limit function having regularity ≥ d

A combined approximating and interpolating subdivision scheme with C2 continuity

AbstractIn 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 C2 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