5 research outputs found
Polynomial-based non-uniform interpolatory subdivision with features control
Starting 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 blending-type 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
In this paper, we construct wavelet tight frames with n vanishing moments for
Dubuc-Deslauriers 2npoint semi-regular interpolatory subdivision schemes. Our
motivation for this construction is its practical use for further regularity
analysis of wide classes of semi-regular subdivision. Our constructive tools
are local eigenvalue convergence analysis for semi-regular Dubuc-Deslauriers
subdivision, the Unitary Extension Principle and the generalization of the
Oblique Extension Principle to the irregular setting by Chui, He and
St\"ockler. This group of authors derives suitable approximation of the inverse
Gramian for irregular Bspline subdivision. Our main contribution is the
derivation of the appropriate approximation of the inverse Gramian for the
semi-regular Dubuc-Deslauriers scaling functions ensuring n vanishing moments
of the corresponding framelets
New strategies for curve and arbitrary-topology surface constructions for design
This dissertation presents some novel constructions for curves and surfaces with arbitrary topology in the context of geometric modeling.
In particular, it deals mainly with three intimately connected topics that are of interest in both theoretical and applied research: subdivision surfaces, non-uniform local interpolation (in both univariate and bivariate cases), and spaces of generalized splines.
Specifically, we describe a strategy for the integration of subdivision surfaces in computer-aided design systems and provide examples to show the effectiveness of its implementation.
Moreover, we present a construction of locally supported, non-uniform, piecewise polynomial univariate interpolants of minimum degree with respect to other prescribed design parameters (such as support width, order of continuity and order of approximation).
Still in the setting of non-uniform local interpolation, but in the case of surfaces, we devise a novel parameterization strategy that, together with a suitable patching technique, allows us to define composite surfaces that interpolate given arbitrary-topology meshes or curve networks and satisfy both requirements of regularity and aesthetic shape quality usually needed in the CAD modeling framework.
Finally, in the context of generalized splines, we propose an approach for the construction of the optimal normalized totally positive (B-spline) basis, acknowledged as the best basis of representation for design purposes, as well as a numerical procedure for checking the existence of such a basis in a given generalized spline space.
All the constructions presented here have been devised keeping in mind also the importance of application and implementation, and of the related requirements that numerical procedures must satisfy, in particular in the CAD context