Polynomial-based non-uniform interpolatory subdivision with features control

Abstract

a b s t r a c tStarting 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 blendingtype 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.

Semi-regular Dubuc–Deslauriers wavelet tight frames

“…To show the second identity in (34), first note that, by the construction of P in Section 3, we have that Pt α = 2 −α t α for t in (1). Thus, due to (16), we get…”

“…In this section, we recall some basic facts about subdivision and wavelet tight frames. A stationary subdivision scheme with the bi-infinite subdivision matrix P : (Z) → (Z) maps recursively the initial data f 0 = [f 0 (k) : k ∈ Z] ∈ (Z) parametrized by the starting mesh t (see (1)) to finer sequences f j = [f j (k) : k ∈ Z] ∈ (Z) parametrized by the meshes 2 −j t, j ∈ N by…”

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…”

“…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.…”

A smoothness criterion for monotonicity-preserving subdivision

“…The use of subdivision to achieve this [15,16] has recently gained greater focus because monotonicity is required for the parameter values used when interpolating points with a parametric curve. If a subdivision method is used to construct the curve, such as in [1,2,7,17], the parameter values will themselves satisfy a subdivision scheme which should preserve monotonicity. As suggested in [17], it might be desirable that these parameter values become 'smoothed out' in some sense in the limit, which we can interpret as saying that the scheme for the parameter values should have at least C 1 smoothness.…”