From approximating subdivision schemes for exponential splines to high-performance interpolating algorithms

Abstract

a b s t r a c tIn this work we construct three novel families of approximating subdivision schemes that generate piecewise exponential polynomials and we show how to convert these into interpolating schemes of great interest in curve design for their ability to reproduce important analytical shapes and to provide highly smooth limit curves with a controllable tension.In particular, throughout this paper we will focus on the derivation of 6-point interpolating schemes that turn out to be unique in combining vital ingredients like C 2 -continuity, simplicity of definition, ease of implementation, user independency, tension control and ability to reproduce salient trigonometric and transcendental curves.

Exponential pseudo-splines: Looking beyond exponential B-splines

“…Such interpolatory 4-and 6-point schemes are new nonstationary variants of the well-known Dubuc-Deslauriers schemes in [22]. They differ from the ones previously proposed in [2,39] for the space of exponential polynomials they reproduce. Figure 2 illustrates the basic limit function of the extreme members of two different families of exponential pseudo-splines, i.e.…”

“…The interest in nonstationary subdivision schemes arose in the last ten years after it was pointed out that they can be equipped with tension parameters that allow us to get as close as desired to the original mesh and to obtain considerable variations of shape (see [16,30,39,40,42,43]). Indeed, differently from stationary subdivision schemes, nonstationary subdivision schemes are capable of reproducing conic sections, spirals or, in general, of generating exponential polynomials x r e θx , x ∈ R, r ∈ N ∪ {0}, θ ∈ C. This generation property is important not only in geometric design (see, e.g., [30,32,37,42,43]), but also in many other applications, e.g., in biomedical imaging (see, e.g., [20,21]) and in Isogeometric Analysis (see, e.g., [19,31]).…”

“…In consideration of the fact that the main goal of this work is the construction of a special class of nonstationary subdivision symbols capable of generating as limit functions exponential polynomials, we continue by recalling the following definitions (see, e.g, [7,17,39]). …”

“…The most popular examples of subdivision schemes are B-spline subdivision schemes and their nonstationary counterparts, namely exponential B-spline subdivision schemes (see, e.g., [16,43]), characterized by the property of representing polynomials and exponential polynomials, respectively. Since in many applicative areas the capability of representing shapes described by exponential polynomial functions is fundamental, interpolating and approximating subdivision schemes based on exponential B-splines and inheriting their generation properties, have been recently introduced (see, e.g., [2,3,5,7,13,14,16,17,28,35,39]). We recall that, while the term generation usually refers to the subdivision scheme capability of providing specific types of limit functions, with reproduction we mean the capability of a subdivision scheme to reproduce in the limit exactly the same function from which the data are sampled.…”

Local refinement for 3D deformable parametric surfaces

“…We define by th α,m rksu kPZ the coefficients of the refinement filter [16,17]. The generator ϕ α is called a refinable function if it verifies the refinement relation given by…”