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

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

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

Ternary shape-preserving subdivision schemes

We analyze the shape-preserving properties of ternary subdivision schemes generated by bell-shaped masks. We prove that any bell-shaped mask, satisfying the basic sum rules, gives rise to a convergent monotonicity preserving subdivision scheme, but convexity preservation is not guaranteed. We show that to reach convexity preservation the first order divided difference scheme needs to be bell-shaped, too. Finally, we show that ternary subdivision schemes associated with certain refinable functions with dilation 3 have shape-preserving properties of higher order

Interpolating m-refinable functions with compact support: The second generation class

We present an algorithm for the construction of a new class of compactly supported interpolating refinable functions that we call the second generation class since, contrary to the existing class, is associated to subdivision schemes with an even-symmetric mask that does not contain the submask 0\u2026,0,1,0,\u20260. As application examples of the proposed algorithm we present interpolating 4-refinable functions that are generated by parameter-dependent, even-symmetric quaternary schemes never considered in the literature so far

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

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