Marian Neamtu

Fitting scattered data on sphere-like surfaces using spherical splines

Abstract Spaces of polynomial splines defined on planar traingulations are very useful tools for fitting scattered data in the plane. Recently, [4, 5], using homogeneous polynomials, we have developed analogous spline spaces defined on triangulations on the sphere and on sphere-like surfaces. Using these spaces, it is possible to construct analogs of many of the classical interpolation and fitting methods. Here we examine some of the more interesting ones is detail. For interpolation, we discuss macro-element and minimal energy splines, and for fitting, we consider discrete least squares and penalized least squares.

Bernstein-Be´zier polynomials on spheres and sphere-like surfaces

Abstract In this paper we discuss a natural way to define barycentric coordinates on general sphere-like surfaces. This leads to a theory of Bernstein-Bezier polynomials which parallels the familiar planar case. Our constructions are based on a study of homogeneous polynomials on trihedra in R 3. The special case of Bernstein-Bezier polynomials on a sphere is considered in detail.

Control curves and knot insertion for trigonometric splines

We introduce control curves for trigonometric splines and show that they have properties similar to those for classical polynomial splines. In particular, we discuss knot insertion algorithms, and show that as more and more knots are inserted into a trigonometric spline, the associated control curves converge to the spline. In addition, we establish a convex-hull property and a variation-diminishing result.

Error Bounds for Solving Pseudodifferential Equations on Spheres by Collocation with Zonal Kernels

The problem of solving pseudodifferential equations on spheres by collocation with zonal kernels is considered and bounds for the approximation error are established. The bounds are given in terms of the maximum separation distance of the collocation points, the order of the pseudodifferential operator, and the smoothness of the employed zonal kernel. A by-product of the results is an improvement on the previously known convergence order estimates for Lagrange interpolation.

Dimension and local bases of homogeneous spline spaces

Recently, we have introduced spaces of splines defined on triangulations lying on the sphere or on sphere-like surfaces. These spaces arose out of a new kind of Bernstein–Bezier theory on such surfaces. The purpose of this paper is to contribute to the development of a constructive theory for such spline spaces analogous to the well-known theory of polynomial splines on planar triangulations. Rather than working with splines on sphere-like surfaces directly, we instead investigate more general spaces of homogeneous splines in

Degenerate polynomial patches of degree 4 and 5 used for geometrically smooth interpolation in R 3

\mathbb{R}^3

On the Approximation Order of Splines on Spherical Triangulations

. In particular, we present formulas for the dimensions of such spline spaces, and construct locally supported bases for them.

The Kernel Common Vector Method: A Novel Nonlinear Subspace Classifier for Pattern Recognition

The problem of interpolating scattered 3D data by a geometrically smooth surface is considered. A completely local method is proposed, based on employing degenerate triangular Bernstein-Bezier patches. An analysis of these patches is given and some numerical experiments with quartic and quintic patches are presented.

Designing NURBS Cam Profiles Using Trigonometric Splines

Bounds are provided on how well functions in Sobolev spaces on the sphere can be approximated by spherical splines, where a spherical spline of degree d is a Cr function whose pieces are the restrictions of homogeneous polynomials of degree d to the sphere. The bounds are expressed in terms of appropriate seminorms defined with the help of radial projection, and are obtained using appropriate quasi-interpolation operators.

Approximation and geometric modeling with simplex B-splines associated with irregular triangles

The common vector (CV) method is a linear subspace classifier method which allows one to discriminate between classes of data sets, such as those arising in image and word recognition. This method utilizes subspaces that represent classes during classification. Each subspace is modeled such that common features of all samples in the corresponding class are extracted. To accomplish this goal, the method eliminates features that are in the direction of the eigenvectors corresponding to the nonzero eigenvalues of the covariance matrix of each class. In this paper, we introduce a variation of the CV method, which will be referred to as the modified CV (MCV) method. Then, a novel approach is proposed to apply the MCV method in a nonlinearly mapped higher dimensional feature space. In this approach, all samples are mapped into a higher dimensional feature space using a kernel mapping function, and then, the MCV method is applied in the mapped space. Under certain conditions, each class gives rise to a unique CV, and the method guarantees a 100% recognition rate with respect to the training set data. Moreover, experiments with several test cases also show that the generalization performance of the proposed kernel method is comparable to the generalization performances of other linear subspace classifier methods as well as the kernel-based nonlinear subspace method. While both the MCV method and its kernel counterpart did not outperform the support vector machine (SVM) classifier in most of the reported experiments, the application of our proposed methods is simpler than that of the multiclass SVM classifier. In addition, it is not necessary to adjust any parameters in our approach.