Shape Preserving C2 Interpolatory Subdivision Schemes

Stationary interpolatory subdivision schemes which preserve shape properties such as convexity or monotonicity are constructed. The schemes are rational in the data and generate limit functions that are at least C 2. The emphasis is on a class of six-point convexity preserving subdivision schemes that generate C 2 limit functions. In addition, a class of six-point monotonicity preserving schemes that also leads to C 2 limit functions is introduced. As the algebra is far too complicated for an analytical proof of smoothness, validation has been performed by a simple numerical methodology. 1. Introduction Several stationary nonlinear subdivision schemes for the purpose of shape preserving interpolation have been proposed in literature. Many of these schemes use only four points, however, the limit function generated by these schemes is at most C 1 in general. Examples of such schemes are a rational C 1 convexity preserving interpolatory subdivision scheme, see [13,14,10], and a monotonicity * financially supported by the Dutch Technology Foundation STW 2 F. Kuijt and R. van Damme / Shape preserving C 2 subdivision preserving subdivision scheme, see [11]. In this paper shape preserving interpolatory subdivision schemes are constructed that generate limit functions that are at least C 2. These schemes are less local than the subdivision schemes from literature: now six points are used. The main focus is on a class of six-point convexity preserving subdivision schemes that generate C 2 limit functions. In addition, a class of six-point monotonicity preserving schemes is introduced that also leads to C 2 limit functions. The smoothness properties of the subdivision schemes are analysed numerically, as the algebra for an analytical proof of smoothness is far too complicated. Some shape preserving rational spline interpolation methods have been introduced in [8] (monotonicity preservation), and [3], [2] (convexity preservation). Shape preserving subdivision algorithms have been examined in literature, e.g., convexity preserving subdivision is examined in [15] and [6]. However, the proposed methods generate results that are only C 1 in general. The goal of this paper is to examine C 2 interpolatory subdivision schemes. The new subdivision point is defined by making use of a two-point Hermite inter-polant. The derivatives in this Hermite interpolating function however, are estimated by a four-point scheme on suitable derivative data: divided differences in the function values. When we take a cubic two-point Hermite interpolant, and the derivatives are estimated using the well-known linear four-point scheme [5], a linear …