Frank Zeilfelder

Developments in bivariate spline interpolation

The aim of this survey is to describe developments in the field of interpolation by bivariate splines. We summarize results on the dimension and the approximation order of bivariate spline spaces, and describe interpolation methods for these spaces. Moreover, numerical examples are given.

Smooth approximation and rendering of large scattered data sets

Presents an efficient method to automatically compute a smooth approximation of large functional scattered data sets given over arbitrarily shaped planar domains. Our approach is based on the construction of a C/sup 1/-continuous bivariate cubic spline and our method offers optimal approximation order. Both local variation and nonuniform distribution of the data are taken into account by using local polynomial least squares approximations of varying degree. Since we only need to solve small linear systems and no triangulation of the scattered data points is required, the overall complexity of the algorithm is linear in the total number of points. Numerical examples dealing with several real-world scattered data sets with up to millions of points demonstrate the efficiency of our method. The resulting spline surface is of high visual quality and can be efficiently evaluated for rendering and modeling. In our implementation we achieve real-time frame rates for typical fly-through sequences and interactive frame rates for recomputing and rendering a locally modified spline surface.

Scattered data fitting by direct extension of local polynomials to bivariate splines

We present a new scattered data fitting method, where local approximating polynomials are directly extended to smooth (C1 or C2) splines on a uniform triangulation Δ (the four-directional mesh). The method is based on designing appropriate minimal determining sets consisting of whole triangles of domain points for a uniformly distributed subset of Δ. This construction allows to use discrete polynomial least squares approximations to the local portions of the data directly as parts of the approximating spline. The remaining Bernstein–Bézier coefficients are efficiently computed by extension, i.e., using the smoothness conditions. To obtain high quality local polynomial approximations even for difficult point constellations (e.g., with voids, clusters, tracks), we adaptively choose the polynomial degrees by controlling the smallest singular value of the local collocation matrices. The computational complexity of the method grows linearly with the number of data points, which facilitates its application to large data sets. Numerical tests involving standard benchmarks as well as real world scattered data sets illustrate the approximation power of the method, its efficiency and ability to produce surfaces of high visual quality, to deal with noisy data, and to be used for surface compression.

Quasi-interpolation by quadratic piecewise polynomials in three variables

A quasi-interpolation method for quadratic piecewise polynomials in three variables is described which can be used for the efficient reconstruction and visualization of gridded volume data. The Bernstein-Bezier coefficients of the splines are immediately available from the given data values by applying a local averaging, where no prescribed derivatives are required. Since the approach does not make use of a particular basis or a subset spanning the spline spaces, we analyze the smoothness properties of the trivariate splines. We prove that the splines yield nearly optimal approximation order while simultaneously its piecewise derivatives provide optimal approximation of the derivatives for smooth functions. The constants of the corresponding error bounds are given explicitly. Numerical tests confirm the results and the efficiency of the method.

Lagrange Interpolation by C 1 Cubic Splines on Triangulated Quadrangulations.

We describe local Lagrange interpolation methods based on C1 cubic splines on triangulations obtained from arbitrary strictly convex quadrangulations by adding one or two diagonals. Our construction makes use of a fast algorithm for coloring quadrangulations, and the overall algorithm has linear complexity while providing optimal order approximation of smooth functions.

Lagrange Interpolation by Bivariate C1-Splines with Optimal Approximation Order

We develop the first local Lagrange interpolation scheme for C1-splines of degree q≥3 on arbitrary triangulations. For doing this, we use a fast coloring algorithm to subdivide about half of the triangles by a Clough–Tocher split in an appropriate way. Based on this coloring, we choose interpolation points such that the corresponding fundamental splines have local support. The interpolating splines yield optimal approximation order and can be computed with linear complexity. Numerical examples with a large number of interpolation points show that our method works efficiently.

Approximation order of bivariate spline interpolation for arbitrary smoothness

By using the algorithm of Nurnberger and Riessinger (1995), we construct Hermite interpolation sets for spaces of bivariate splines Sqr(Δ1) of arbitrary smoothness defined on the uniform type triangulations. It is shown that our Hermite interpolation method yields optimal approximation order for q ⩾ 3.5r + 1. In order to prove this, we use the concept of weak interpolation and arguments of Birkhoff interpolation.

Interpolation by spline spaces on classes of triangulations

We describe a general method for constructing triangulations Δ which are suitable for interpolation by Srq(Δ), r

Visualization of volume data with quadratic super splines

We develop a new approach to reconstruct non-discrete models from gridded volume samples. As a model, we use quadratic trivariate super splines on a uniform tetrahedral partition /spl Delta/. The approximating splines are determined in a natural and completely symmetric way by averaging local data samples, such that appropriate smoothness conditions are automatically satisfied. On each tetra-hedron of /spl Delta/ , the quasi-interpolating spline is a polynomial of total degree two which provides several advantages including efficient computation, evaluation and visualization of the model. We apply Bernstein-Bezier techniques well-known in CAGD to compute and evaluate the trivariate spline and its gradient. With this approach the volume data can be visualized efficiently e.g., with isosurface ray-casting. Along an arbitrary ray the splines are univariate, piecewise quadratics and thus the exact intersection for a prescribed isovalue can be easily determined in an analytic and exact way. Our results confirm the efficiency of the quasi-interpolating method and demonstrate high visual quality for rendered isosurfaces.

Scattered Data Fitting with Bivariate Splines

We describe scattered data fitting by bivariate splines, i.e., splines defined w.r.t. triangulations in the plane. These spaces are powerful tools for the efficient approximation of large sets of scattered data which appear in many real world problems. Bernstein-Bezier techniques can be used for the efficient computation of bivariate splines and for analysing the complex structure of these spaces. We report on the classical approaches and we describe interpolation and approximation methods for bivariate splines that have been developed recently. For the latter methods, we give illustrative examples treating sets of geodetic data (consisting of up to 106 points).