5,651 research outputs found
Orthogonal polynomials for area-type measures and image recovery
Let be a finite union of disjoint and bounded Jordan domains in the
complex plane, let be a compact subset of and consider the
set obtained from by removing ; i.e.,
. We refer to as an archipelago and
as an archipelago with lakes. Denote by
and , the sequences of the Bergman polynomials
associated with and , respectively; that is, the orthonormal
polynomials with respect to the area measure on and . The purpose
of the paper is to show that and have comparable
asymptotic properties, thereby demonstrating that the asymptotic properties of
the Bergman polynomials for are determined by the boundary of . As
a consequence we can analyze certain asymptotic properties of
by using the corresponding results for , which were obtained in a
recent work by B. Gustafsson, M. Putinar, and two of the present authors. The
results lead to a reconstruction algorithm for recovering the shape of an
archipelago with lakes from a partial set of its complex moments.Comment: 24 pages, 9 figure
Numerical hyperinterpolation over nonstandard planar regions
We discuss an algorithm (implemented in Matlab) that computes numerically total-degree bivariate orthogonal polynomials (OPs) given an algebraic cubature formula with positive weights, and constructs the orthogonal projection (hyperinterpolation) of a function sampled at the cubature nodes. The method is applicable to nonstandard regions where OPs are not known analytically, for example convex and concave polygons, or circular sections such as sectors, lenses and lunes
Optimal designs for three-dimensional shape analysis with spherical harmonic descriptors
We determine optimal designs for some regression models which are frequently
used for describing three-dimensional shapes. These models are based on a
Fourier expansion of a function defined on the unit sphere in terms of
spherical harmonic basis functions. In particular, it is demonstrated that the
uniform distribution on the sphere is optimal with respect to all
criteria proposed by Kiefer in 1974 and also optimal with respect to a
criterion which maximizes a mean of the smallest eigenvalues of the
variance--covariance matrix. This criterion is related to principal component
analysis, which is the common tool for analyzing this type of image data.
Moreover, discrete designs on the sphere are derived, which yield the same
information matrix in the spherical harmonic regression model as the uniform
distribution and are therefore directly implementable in practice. It is
demonstrated that the new designs are substantially more efficient than the
commonly used designs in three-dimensional shape analysis.Comment: Published at http://dx.doi.org/10.1214/009053605000000552 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
- …