Carolina Vittoria Beccari

A non-stationary uniform tension controlled interpolating 4-point scheme reproducing conics

In this paper we propose a non-stationary C^1-continuous interpolating 4-point scheme which provides users with a single tension parameter that can be either arbitrarily increased, to tighten the limit curve towards the piecewise linear interpolant between the data points, or appropriately chosen in order to represent elements of the linear spaces spanned respectively by the functions {1,x,x^2,x^3}, {1,x,e^s^x,e^-^s^x} and {1,x,e^i^s^x,e^-^i^s^x}. As a consequence, for special values of the tension parameter, such a scheme will be capable of reproducing all conic sections exactly. Exploiting the reproduction property of the scheme, we derive an algorithm that automatically provides the initial tension parameter required to exactly reproduce a curve belonging to one of the previously mentioned spaces, whenever the initial data are uniformly sampled on it. The performance of the scheme is illustrated by a number of examples that show the wide variety of effects we can achieve in correspondence of different tension values.

An interpolating 4-point C2 ternary non-stationary subdivision scheme with tension control

In this paper we present a non-stationary 4-point ternary interpolatory subdivision scheme which provides the user with a tension parameter that, when increased within its range of definition, can generate C^2-continuous limit curves showing considerable variations of shape. As a generalization we additionally propose a locally-controlled C^2-continuous subdivision scheme, which allows a different tension value to be assigned to every edge of the original control polygon.

A fast interactive reverse-engineering system

A new method of reverse engineering for fast, simple and interactive acquisition and reconstruction of a virtual three-dimensional (3D) model is presented. We propose an active stereo acquisition system, which makes use of two infrared cameras and a wireless active-pen device, supported by a reconstruction method based on subdivision surfaces. In the 3D interactive hand sketching process the user draws and refines the 3D style-curves, which characterize the shape to be constructed, by simply dragging the active-pen device; then the system automatically produces a low-resolution mesh that is naturally refined through subdivision surfaces. Several examples demonstrate the ability of the proposed advanced design methodology to produce complex 3D geometric models by the interactive and iterative process that provides the user with a real-time visual feedback on the ongoing work.

Shape controlled interpolatory ternary subdivision

Ternary subdivision schemes compare favorably with their binary analogues because they are able to generate limit functions with the same (or higher) smoothness but smaller support. In this work we consider the two issues of local tension control and conics reproduction in univariate interpolating ternary refinements. We show that both these features can be included in a unique interpolating 4-point subdivision method by means of non-stationary insertion rules that do not affect the improved smoothness and locality of ternary schemes. This is realized by exploiting local shape parameters associated with the initial polyline edges.

Construction and characterization of non-uniform local interpolating polynomial splines

This paper presents a general framework for the construction of piecewise-polynomial local interpolants with given smoothness and approximation order, defined on non-uniform knot partitions. We design such splines through a suitable combination of polynomial interpolants with either polynomial or rational, compactly supported blending functions. In particular, when the blending functions are rational, our approach provides spline interpolants having low, and sometimes minimum degree. Thanks to its generality, the proposed framework also allows us to recover uniform local interpolating splines previously proposed in the literature, to generalize them to the non-uniform case, and to complete families of arbitrary support width. Furthermore it provides new local interpolating polynomial splines with prescribed smoothness and polynomial reproduction properties.

Subdivision surfaces integrated in a CAD system

The main roadblock that has limited the usage of subdivision surfaces in computer-aided design (CAD) systems is the lack of quality and precision that a model must achieve for being suitable in the engineering and manufacturing phases of design. The second roadblock concerns the integration into the modeling workflows, that, for engineering purposes, means providing a precise and controlled way of defining and editing models possibly composed of different geometric representations. This paper documents the experience in the context of a European project whose goal was the integration of subdivision surfaces in a CAD system. To this aim, a new CAD system paradigm with an extensible geometric kernel is introduced, where any new shape description can be integrated through the two successive steps of parameterization and evaluation, and a hybrid boundary representation is used to easily model different kinds of shapes. In this way, the newly introduced geometric description automatically inherits any pre-existing CAD tools, and it can interact in a natural way with the other geometric representations supported by the CAD system. To overcome the irregular behavior of subdivision surfaces in the neighborhood of extraordinary points, we locally modify the limit surface of the subdivision scheme so as to tune the analytic properties without affecting its geometric shape. Such a correction is inspired by the polynomial blending approach in Levin (2006) [1] and Zorin (2006) [2], which we extend in some aspects and generalize to multipatch surfaces evaluable at arbitrary parameter values. Some modeling examples will demonstrate the benefits of the proposed integration, and some tests will confirm the effectiveness of the proposed local correction patching method.

A unified framework for interpolating and approximating univariate subdivision

In this paper we show that the refinement rules of interpolating and approximating univariate subdivision schemes with odd-width masks of finite support can be derived ones from the others by simple operations on the mask coefficients. These operations are formalized as multiplication/division of the associated generating functions by a proper link polynomial. We then apply the proposed result to some families of stationary and non-stationary subdivision schemes, showing that it also provides a constructive method for the definition of novel refinement algorithms.

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.

RAGS: Rational geometric splines for surfaces of arbitrary topology

Abstract A construction of spline spaces suitable for representing smooth parametric surfaces of arbitrary topological genus and arbitrary order of continuity is proposed. The obtained splines are a direct generalization of bivariate polynomial splines on planar partitions. They are defined as composite functions consisting of rational functions and are parametrized by a single parameter domain, which is a piecewise planar surface, such as a triangulation of a cloud of 3D points. The idea of the construction is to utilize linear rational transformations (or transition maps) to endow the piecewise planar surface with a particular C ∞ -differentiable structure appropriate for defining rational splines.

Non-uniform non-tensor product local interpolatory subdivision surfaces

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.