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 (C 1 or C 2) splines on a uniform triangulation (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 . 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-Bæ#169;zier 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

The dimension of C1C^1 splines of arbitrary degree on a tetrahedral partition

We consider the linear space of piecewise polynomials in three variables which are globally smooth, i.e., trivariate $C^1$ splines. The splines are defined on a uniform tetrahedral partition $\Delta$, which is a natural generalization of the four-directional mesh. By using Bernstein-B{\´e}zier techniques, we establish formulae for the dimension of the $C^1$ splines of arbitrary degree

Bivariate spline interpolation with optimal approximation order

Let be a triangulation of some polygonal domain f c R2 and let S9 (A) denote the space of all bivariate polynomial splines of smoothness r and degree q with respect to A. We develop the first Hermite-type interpolation scheme for S9 (A), q >_ 3r + 2, whose approximation error is bounded above by Kh4+i, where h is the maximal diameter of the triangles in A, and the constant K only depends on the smallest angle of the triangulation and is independent of near-degenerate edges and nearsingular vertices. Moreover, the fundamental functions of our scheme are minimally supported and form a locally linearly independent basis for a superspline subspace of Sr, (A). This shows that the optimal approximation order can be achieved by using minimally supported splines. Our method of proof is completely different from the quasi-interpolation techniques for the study of the approximation power of bivariate splines developed in [71 and [181

HIV infection and HERV expression: a review

The human genome contains multiple copies of retrovirus genomes known as endogenous retroviruses (ERVs) that have entered the germ-line at some point in evolution. Several of these proviruses have retained (partial) coding capacity, so that a number of viral proteins or even virus particles are expressed under various conditions. Human ERVs (HERVs) belong to the beta-, gamma-, or spuma- retrovirus groups. Endogenous delta- and lenti- viruses are notably absent in humans, although endogenous lentivirus genomes have been found in lower primates. Exogenous retroviruses that currently form a health threat to humans intriguingly belong to those absent groups. The best studied of the two infectious human retroviruses is the lentivirus human immunodeficiency virus (HIV) which has an overwhelming influence on its host by infecting cells of the immune system. One HIV-induced change is the induction of HERV transcription, often leading to induced HERV protein expression. This review will discuss the potential HIV-HERV interactions