Non-uniform non-tensor product local interpolatory subdivision surfaces

Abstract

In this paper we exploit a class of univariate, C 1 interpolating four-point subdivision schemes featured by a piecewise uniform parameterization, to define non-tensor product subdivision schemes interpolating regular grids of control points and generating C 1 limit surfaces with a better behavior than the well-established tensor product subdivision and spline surfaces. As a result, it is emphasized that subdivision methods can be more effective than splines, not only, as widely acknowledged, for the representation of surfaces of arbitrary topology, but also for the generation of smooth interpolants of regular grids of points. To our aim, the piecewise uniform parameterization of the univariate case is generalized to an augmented parameterization, where the knot intervals of the kth level grid of points are computed from the initial ones by an updating relation that keeps the subdivision algorithm linear. The particular parameters configuration, together with the structure of the subdivision rules, turn out to be crucial to prove the continuity and smoothness of the limit surface.

High quality local interpolation by composite parametric surfaces

“…Comparing the expressions in (17) and (20) we can see that the two vectors d(v) andd(v) are one a "shifted" version of the other and therefore "overlap" almost everywhere, with the exception of the first element ind and the last one in d ( Figure 5(a) schematizes the situation for a class of fundamental functions having support width w = 4). These two different intervals are uninfluential to the value and derivatives of the fundamental functions at a boundary point (see Figure 5(b)).…”

“…In fact, the term "augmented" was firstly introduced in [18] with a similar meaning to the one we use here. More recently, an interpolatory subdivision scheme with augmented parametrization has been discussed in [20]. The scheme allows for interpolating each section polyline of the mesh at independent parameter values and therefore the most appropriate parametrization is used to construct each section curve of the limit surface.…”

A general framework for the construction of piecewise-polynomial local interpolants of minimum degree

“…Besides this, non-uniform locally supported splines of B1 and B2 type have recently shown their importance in connection with the study of subdivision schemes that arise by upsampling fundamental spline bases [1,2,13], because the continuity and polynomial reproduction properties of the schemes are related to the corresponding properties of the bases. In the bivariate setting, these subdivision schemes can be generalized to define non-uniform non-tensor-product interpolation methods [4], which have already proven to be effective in some applications [5] and have significant potential to gain even more interest, due to the limited computational cost and high quality of interpolation.…”