Giulio Casciola

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.

Shape preserving surface reconstruction using locally anisotropic radial basis function interpolants

In this paper we deal with the problem of reconstructing surfaces from unorganized sets of points, while capturing the significant geometry details of the modelled surface, such as edges, flat regions, and corners. This is obtained by exploiting the good approximation capabilities of the radial basis functions (RBF), the local nature of the method proposed in [1], and introducing information on shape features and data anisotropies detected from the given surface points. The result is a shape-preserving reconstruction, given by a weighted combination of locally aniso tropic interpolants. For each local interpolant the anisotropy is obtained by replacing the Euclidean norm with a suitable metric which takes into account the local distribution of the points. Thus hyperellipsoid basis functions, named anisotropic RBFs, are defined. Results from the application of the method to the reconstruction of object surfaces in @?^3 are presented, confirming the effectiveness of the approach.

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.

Fast surface reconstruction and hole filling using positive definite radial basis functions

Abstract Surface reconstruction from large unorganized data sets is very challenging, especially if the data present undesired holes. This is usually the case when the data come from laser scanner 3D acquisitions or if they represent damaged objects to be restored. An attractive field of research focuses on situations in which these holes are too geometrically and topologically complex to fill using triangulation algorithms. In this work a local approach to surface reconstruction from point-clouds based on positive definite Radial Basis Functions (RBF) is presented that progressively fills the holes by expanding the neighbouring information. The method is based on the algorithm introduced in [7] which has been successfully tested for the smooth multivariate interpolation of large scattered data sets. The local nature of the algorithm allows for real time handling of large amounts of data, since the computation is limited to suitable small areas, thus avoiding the critical efficiency problem involved in RBF multivariate interpolation. Several tests on simulated and real data sets demonstrate the efficiency and the quality of the reconstructions obtained using the proposed algorithm.

The regularizing properties of anisotropic Radial Basis Functions

In the present work we consider the problem of interpolating scattered data using radial basis functions (RBF). In general, it is well known that this leads to a discrete linear inverse problem that needs to be regularized in order to provide a meaningful solution. The work focuses on a metric-regularization approach, based on a new class of RBF, called anisotropic RBF. The work provides theoretical justifications for the regularization approach and it considers a suitable proposal for the metric, supporting it by numerical examples.

REPARAMETRIZATION OF NURBS CURVES

In geometric design, it is often useful to be able to give an arc length reparametrization for NURBS curves, that keeps the curve a NURBS too. Since parametric rational curves, except for straight lines, cannot be parametrized by arc length, we developed a numerical method of approximating the arc length parametrization function. In this way it was possible to obtain a good parametrization of a NURBS curve with respect to arc length. Numerical results show a good behaviour of the proposed method on several test curves.

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.