Spherical Triangular B-splines with Application to Data Fitting

Triangular B-splines surfaces are a tool for representing arbitrary piecewise polynomial surfaces over planar triangulations, while automatically maintaining continuity properties across patch boundaries. Recently, Alfeld et al. [1] introduced the concept of spherical barycentric coordinates which allowed them to formulate Bernstein-Bézier polynomials over the sphere. In this paper we use the concept of spherical barycentric coordinates to develop a similar formulation for triangular B-splines, which we call spherical triangular B-splines. These splines defined over spherical triangulations share the same continuity properties and similar evaluation algorithms with their planar counterparts, but possess none of the annoying degeneracies found when trying to represent closed surfaces using planar parametric surfaces. We also present an example showing the use of these splines for approximating spherical scattered data.     

Artifact analysis on triangular box-splines and subdivision surfaces defined by triangular polyhedra

Surface artifacts are features in a surface which cannot be avoided by movement of control points. They are present in B-splines, box splines and subdivision surfaces. We showed how the subdivision process can be used as a tool to analyse artifacts in surfaces defined by quadrilateral polyhedra ( [Sabin et al., 2005] and [Augsd?rfer et al., 2011]).In this paper we are utilising the subdivision process to develop a generic expression which can be employed to determine the magnitude of artifacts in surfaces defined by any regular triangular polyhedra. We demonstrate the method by analysing box-splines and regular regions of subdivision surfaces based on triangular meshes: Loop subdivision, Butterfly subdivision and a novel interpolating scheme with two smoothing stages. We compare our results for surfaces defined by triangular polyhedra to those for surfaces defined by quadrilateral polyhedra.     

Accelerated Evaluation of Box Splines via a Parallel Inverse FFT

Box splines are a multivariate extension of uniform univariate B-splines. Direct evaluation of a box spline basis function can he difficult but they have a relatively simple Fourier transform and can therefore be evaluated with an inverse FFT. Symmetry recursive evaluation of the coefficients, and parallelization can be used to improve absolute performance. A windowing function can also he used to reduce truncation artifacts. We explore all these options in the context of a high-performance parallel implementation. Our goal is the provision of an empirical touchstone for the inverse FFT evaluation of box spline basis functions, for eventual application to forward projection (splat-based) volume rendering.     

Biquartic C-surface splines over irregular meshes

C¹-surface splines define tangent continuous surfaces from control points in the manner of tensor-product (B-)splines, but allow a wider class of control meshes capable of outlining arbitrary free-form surfaces with or without boundary. In particular, irregular meshes with non-quadrilateral cells and more or fewer than four cells meeting at a point can be input and are treated in the same conceptual frame work as tensor-product B-splines; that is, the mesh points serve as control points of a smooth piecewise polynomial surface representation that is local and evaluates by averaging. Biquartic surface splines extend and complement the definition of C¹-surface splines in a previous paper (Peters, J SLAM J. Numer. Anal. Vol 32 No 2 (1993) 645–666) improving continuity and shape properties in the case where the user chooses to model entirely with four-sided patches. While tangent continuity is guaranteed, it is shown that no polynomial, symmetry-preserving construction with adjustable blends can guarantee its surfaces to lie in the local convex hull of the control mesh for very sharp blends where three patches join. Biquartic C¹-surface splines do as well as possible by guaranteeing the property whenever more than three patches join and whenever the blend exceeds a certain small threshold.     

Discrete box splines and refinement algorithms

We introduce discrete box splines and use them to give a general knotline refinement algorithm for surfaces which are linear combinations of translates of a box spline. The Lane-Riesenfeld algorithm is obtained as a special case. Since this represents a new algorithm for refining certain nontensor product schemes, it is hoped that this will lead to further applications in computer aided geometric design.     

Degree elevation of unified and extended spline curves

Unified and extended splines （UE-splines）, which unifl and extend polynomial, trigonometric, and hyperbolic B-splines, inherit most properties of B-splines and have some advantages over B-splines. The interest of this paper is the degree elevation algorithm of UE-spline curves and its geometric meaning. Our main idea is to elevate the degree of UE-spline curves one knot interval by one knot interval. First, we construct a new class of basis functions, called bi-order UE-spline basis flmctions which are defined by the integral definition of splines. Then some important properties of bi-order UE-splines are given, especially for tile transformation formulae of the basis functions before and after inserting a knot into the knot vector. Finally, we prove that the degree elevation of UE-spline curves can be interpreted as a process of corner cutting on the control polygons, just as in the manner of B-splines. This degree elevation algorithm possesses strong geometric intuition.     

Hierarchical B-splines on regular triangular partitions

Multivariate splines have a wide range of applications in function approximation, finite element analysis and geometric modeling. They have been extensively studied in the last several decades, and specially the theory on bivariate B-splines over regular triangular partition is well developed. However, the above mentioned splines do not have local refinement property – a property that is very important in adaptive function approximation and level of detailed representation of geometric models. In this paper, we introduce the concept of hierarchial bivariate splines over regular triangular partitions and construct basis functions of such spline space that satisfy some nice properties. We provide some examples of hierarchical splines over triangular partitions in surface fitting and in solving numerical PDEs, and the results turn out to be promising.     

Adaptively refined multilevel spline spaces from generating systems

The truncated basis of adaptively refined multilevel spline spaces was introduced by Giannelli et al., 2012, Giannelli et al., 2014. It possesses a number of advantages, including the partition of unity property, decreased support of the basis functions, preservation of coefficients and strong stability, that may make it highly useful for geometric modeling and numerical simulation. We generalize this construction to hierarchies of spaces that are spanned by generating systems that potentially possess linear dependencies. This generalization requires a modified framework, since the existing approach relied on the linear independence of the functions generating the spaces in the hierarchy. Many results, such as the preservation of coefficients, can be extended to the more general setting. As applications of the modified framework, we introduce a hierarchy of hierarchical B-splines, which enables us to perform local refinement in the presence of features, and we also extend the adaptive multilevel framework to spaces spanned by Zwart–Powell (ZP) elements, which are special box splines defined on the criss-cross grid. In the latter case we show how to identify the linear dependencies that are present in the truncated hierarchical generating system and use this result to perform adaptive surface fitting with multilevel ZP elements.     

Algorithm by hybrid splines

Hybrid splines whose bases are constructed by multi-order B-splines are presented. B-splines of order 2 are induced for representing corners, and those of orders 3 or 4 for representing curved parts. The hybrid splines can represent not only smooth parts (class C^m?22:m≥3) but also corners (class C°): Hybrid splines of order 2 and 4 cannot be represented by conventional splines with multiple knots. Satisfactory approximation results are obtained by hybrid splines of order 2 and 4 determined by a least-squares method.     

Calculating with box splines

Box splines are multivariate splines over regular grids. Two recursion formulas for box splines are developed: (1) a Mansfield-de Boor-like expression of box splines as linear combinations of box splines of lower degree and (2) a deBoor-like reduction of the net of box spline control points. The ideas follow those from the paper by deBoor in 1972. The proofs are geometrical and simple.     

Numerical integration on multivariate scattered data by Lobachevsky splines

In this paper, we investigate the numerical integration problem of a real valued function generally known only on multivariate scattered points using Lobachevsky splines, a pioneering version of cardinal B-splines. Starting from their interpolation properties, we focus on the construction of new integration formulas, which are quite flexible requiring no special distribution of nodes. Numerical results using Lobachevsky splines turn out to be interesting and promising for both accuracy and simplicity in computation. Finally, a comparison with integration by radial basis functions confirms the validity of the proposed approach.     

Subdivision algorithms for the generation of box spline surfaces

This paper proposes a general approach to subdivision algorithms used in interactive computer aided design for splines which are linear combinations of translates of any box splines. We show how these algorithms can be used for efficient generation of the corresponding spline surfaces. Our results extend several known special cases.     

A new basis for PHT-splines

PHT-splines (polynomials splines over hierarchical T-meshes) are a generalization of B-splines over hierarchical T-meshes which possess a very efficient local refinement property. This property makes PHT-splines preferable in geometric processing, adaptive finite elements and isogeometric analysis. In this paper, we first make analysis of the previously constructed basis functions of PHT-splines and observe a decay phenomenon of the basis functions under certain refinement of T-meshes, which is not expected in applications. We then propose a new basis consisting of a set of local tensor product B-splines for PHT-splines which overcomes the decay phenomenon. Some examples are provided for solving numerical PDEs with the new basis, and comparison is made between the new basis and the original basis. Experimental results suggest that the new basis provides better numerical stability in solving numerical PDEs.     

Bernoulli functions and periodic B-splines

Bernoulli polynomials and the related Bernoulli functions are of basic importance in theoretical numerical analysis. It was shown by Golomb and others that the periodic Bernoulli functions serve to construct periodic polynomial splines on uniform meshes. In an unknown paper Wegener investigated remainder formulas for polynomial Lagrange interpolation via Bernoulli functions. We will use Wegener's kernel function to construct periodicB-splines. For uniform meshes we will show that Locher's method of interpolation by translation is applicable to periodicB-splines. This yields an easy and stable algorithm for computing periodic polynomial interpolating splines of arbitrary degree on uniform meshes via discrete Fourier transform.     

From conics to NURBS: A tutorial and survey

The main geometric features of the nonuniform rational B-splines (NURBS) curve and surface representations are described. It is shown that most of these features are already exhibited by conics, which are a special case of NURBS. The properties typical of NURBS are discussed without dwelling on properties already present in polynomial curves. Conic sections and their representations using rational Bezier curves are reviewed. Cubic NURB curves, geometrical rational splines, rational and B-spline surfaces, and rational Bezier triangles are discussed     

ωB-splines

A new kind of spline with variable frequencies, called ωB-spline, is presented. It not only unifies B-splines, trigonometric and hyperbolic polynomial B-splines, but also produces more new types of splines, ωB-spline bases are defined in the space spanned by {coso） t, sino）t, ], t, ..., t^n, ...} with the sequence of frequencies m where n is an arbitrary nonnegative integer, ωB-splines persist all desirable properties of B-splines. Furthermore, they have some special properties advantageous for modeling free form curves and surfaces.     

Retailoring Box Splines to Lattices for Highly Isotropic Volume Representations

3D box splines are defined by convolving a 1D box function with itself along different directions. In volume visualization, box splines are mainly used as reconstruction kernels that are easy to adapt to various sampling lattices, such as the Cartesian Cubic (CC), Body‐Centered Cubic (BCC), and Face‐Centered Cubic (FCC) lattices. The usual way of tailoring a box spline to a specific lattice is to span the box spline by exactly those principal directions that span the lattice itself. However, in this case, the preferred directions of the box spline and the lattice are the same, amplifying the anisotropic effects of each other. This leads to an anisotropic volume representation with strongly preferred directions. Therefore, in this paper, we retailor box splines to lattices such that the sets of vectors that span the box spline and the lattice are disjoint sets. As the preferred directions of the box spline and the lattice compensate each other, a more isotropic volume representation can be achieved. We demonstrate this by comparing different combinations of box splines and lattices concerning their anisotropic behavior in tomographic reconstruction and volume visualization.     

B-splines of Blaschke product type

In this paper, we construct a class of new splines related to a Blaschke product. They emerge naturally when studying the filter functions of a class of linear time-invariant systems which are related to the boundary values of a Blaschke product for the purpose of sampling non-bandlimited signals using nonlinear Fourier atoms. The new splines generalize the well-known symmetric B-splines. We establish their properties such as integral representation property, a partition of unity property, a recurrence relation and difference property. We also investigate their random behaviour. Finally, our numerical experiments confirm our theories.     

Subdivision surfaces for CAD—an overview

Subdivision surfaces refer to a class of modelling schemes that define an object through recursive subdivision starting from an initial control mesh. Similar to B-splines, the final surface is defined by the vertices of the initial control mesh. These surfaces were initially conceived as an extension of splines in modelling objects with a control mesh of arbitrary topology. They exhibit a number of advantages over traditional splines. Today one can find a variety of subdivision schemes for geometric design and graphics applications. This paper provides an overview of subdivision surfaces with a particular emphasis on schemes generalizing splines. Some common issues on subdivision surface modelling are addressed. Several key topics, such as scheme construction, property analysis, parametric evaluation and subdivision surface fitting, are discussed. Some other important topics are also summarized for potential future research and development. Several examples are provided to highlight the modelling capability of subdivision surfaces for CAD applications.     

Use of box splines in computer tomography

Box splines are attractive for practical multivariate approximation, since they possess good approximation power and can be evaluated very efficiently. We want to give an idea of how their qualities can be made to come into play in the field of image reconstruction in computerized tomography (CT). To keep the exposition simple, we will concentrate on a special situation: our tomograph will be characterized by the bivariate standard scanning geometry and our reconstructions will always lie in scales of the linear space spanned by the integer translates of a fixed piecewise quadratic box spline. On the other hand we give details of an algorithm based on Fourier reconstruction, which produces approximations of optimal order for the box splines used, whilst the amount of computational work required is of no higher order than for classical Fourier reconstruction. We present another reconstruction procedure based on quasi-interpolation, which compares to filtered backprojection in computational complexity. Along with our exposition, we give a generalization of a certain Theorem due to Nievergelt which may be of interest for practical applications.