Non-uniform interpolatory curve subdivision with edge parameters built upon compactly supported fundamental splines

Abstract

In this paper we present a family of Non-Uniform Local Interpolatory (NULI) subdivision schemes, derived from compactly supported interpolatory fundamental splines with non-uniform knots (NULIFS). For this spline family, the knot-partition is defined by a sequence of break points and by one additional knot, arbitrarily placed along each knot-interval. The resulting refinement algorithms are linear and turn out to contain a set of edge parameters that, when fixed to a value in the range [0,1], allow us to achieve special shape features by simply moving each auxiliary knot between the break points. Among all the members of this new family of schemes, we will then especially analyze the NULI 4-point refinement. This subdivision scheme has all the fundamental features of the quadratic fundamental spline basis it is originated from, namely compact support, C 1 smoothness, second order polynomials reproduction and approximation order 3. In addition the NULI 4-point subdivision algorithm has the possibility of setting consecutive edge parameters to simulate double and triple knots -that are not considered by the authors of the corresponding spline basis -thus allowing for limit curves with crease vertices, without using an ad hoc mask. Numerical examples and comparisons with other methods will be given to the aim of illustrating the performance of the NULI 4-point scheme in the case of highly non-uniform initial data.

Smooth shapes with spherical topology: Beyond traditional modeling, efficient deformation, and interaction

“…Variants of this scheme have been proposed in Ref. [44] which have also been used to construct conic sections [45].…”

Non-uniform non-tensor product local interpolatory subdivision surfaces

“…Thanks to this updating rule, the subdivision algorithm is linear and, after a few rounds of subdivision, the knot intervals assume a piecewise uniform configuration, namely the parameterization is uniform everywhere except at isolated points corresponding to the initial polyline vertices. Schemes of this kind have been the topic of several papers including [6,7,16,27]. In this case the insertion rule in (1) can be written as…”

“…, which was proposed in [6]. By construction, the two reference schemes with coefficients in (5) and (6) reproduce polynomials respectively in Π 3 and Π 2 ; moreover, in the referenced papers, it was proven that they are C 1 when the parameterization is piecewise uniform.…”

“…In this respect, uniform interpolatory schemes have been investigated in detail, both for the curve and surface cases, and by now fully understood. Interpolatory methods with non-uniform parameterization, were introduced in the seminal work by Daubechies et al [16] and more recently they have been the topic of several papers (see [6,7,17]). This renewed interest has arisen since it was observed that the non-uniform parameterization may significantly reduce interpolation artifacts (like unwanted undulation, cusps and self-intersections) with respect to the uniform.…”

“…In the interpolatory context, this paper is the first addressing a non-tensor product setting. The non-uniform, nontensor product surface subdivision scheme is defined as a generalization of a class of piecewise uniform, univariate, interpolatory 4-point schemes derived by upsampling fundamental bases for interpolation, as those in [6,8,16]. As a consequence, the coefficients of the scheme depend on the non-uniform, local knot intervals upon which the underlying cardinal spline basis is defined.…”