Tomas Sauer

Regularity of multiwavelets

The motivation for this paper is an interesting observation made by Plonka concerning the factorization of the matrix symbol associated with the refinement equation for B-splines with equally spaced multiple knots at integers and subsequent developments which relate this factorization to regularity of refinable vector fields over the real line. Our intention is to contribute to this train of ideas which is partially driven by the importance of refinable vector fields in the construction of multiwavelets.The use of subdivision methods will allow us to consider the problem almost entirely in the spatial domain and leads to exact characterizations of differentiability and Hölder regularity in arbitrary Lp spaces. We first study the close relationship between vector subdivision schemes and a generalized notion of scalar subdivision schemes based on bi-infinite matrices with certain periodicity properties. For the latter type of subdivision scheme we will derive criteria for convergence and Hölder regularity of the limit function, which mainly depend on the spectral radius of a bi-infinite matrix induced by the subdivision operator, and we will show that differentiability of the limit functions can be characterized by factorization properties of the subdivision operator. By switching back to vector subdivision we will transfer these results to refinable vectors fields and obtain characterizations of regularity by factorization and spectral radius properties of the symbol associated to the refinable vector field. Finally, we point out how multiwavelets can be generated from orthonormal refinable bi-infinite vector fields.

Construction of orthogonal bases for polynomials in Bernstein form on triangular and simplex domains

A scheme for constructing orthogonal systems of bivariate polynomials in the Bernstein-Bezier form over triangular domains is formulated. The orthogonal basis functions have a hierarchical ordering by degree, facilitating computation of least-squares approximations of increasing degree (with permanence of coefficients) until the approximation error is subdued below a prescribed tolerance. The orthogonal polynomials reduce to the usual Legendre polynomials along one edge of the domain triangle, and within each fixed degree are characterized by vanishing Bernstein coefficients on successive rows parallel to that edge. Closed-form expressions and recursive algorithms for computing the Bernstein coefficients of these orthogonal bivariate polynomials are derived, and their application to surface smoothing problems is sketched. Finally, an extension of the scheme to the construction of orthogonal bases for polynomials over higher-dimensional simplexes is also presented.

Polynomial Interpolation in Several Variables: Lattices, Differences, and Ideals

Abstract When passing from one to several variables, the nature and structure of polynomial interpolation changes completely: the solvability of the interpolation problem with respect to a given finite dimensional polynomial space, like all polynomials of at most a certain total degree, depends not only on the number, but significantly on the geometry of the nodes. Thus the construction of interpolation sites suitable for a given space of polynomials or of appropriate interpolation spaces for a given set of nodes become challenging and nontrivial problems. The paper will review some of the basic constructions of interpolation lattices which emerge from the geometric characterization due to Chung and Yao. Depending on the structure of the interpolation problem, there are different representations of the interpolation polynomial and several formulas for the error of interpolation, reflecting the underlying point geometry and employing different types of differences. In addition, we point out the close relationship with constructive ideal theory and degree reducing interpolation, whose most prominent representer is the least interpolant, introduced by de Boor et al.

Approximate varieties, approximate ideals and dimension reduction

We consider a method to determine, for a given finite set of points, an algebraic variety of small degree which contains these points. In contrast to most other algorithms in Computer Algebra, this one is adapted to numerical, inexact computations.

From Hermite to stationary subdivision schemes in one and several variables

Vector and Hermite subdivision schemes both act on vector data, but since the latter one interprets the vectors as function values and consecutive derivatives they differ by the “renormalization” of the Hermite scheme in any step. In this paper we give an algebraic factorization method in one and several variables to relate any Hermite subdivision scheme that satisfies the so–called spectral condition to a vector subdivision scheme. These factorizations are natural extensions of the “zero at π” condition known for the masks of refinable functions. Moreover, we show how this factorization can be used to investigate different forms of convergence of the Hermite scheme and why the multivariate situation is conceptionally more intricate than the univariate one. Finally, we give some examples of such factorizations.

Tomography-based customized IOL calculation model.

Purpose: To provide a mathematical calculation scheme for customized intraocular lens (IOL) design based on high resolution anterior segment optical coherence tomography (AS-OCT) of anterior eye segment and axial length data. Material and Methods: We use the corneal and anterior segment data from the high resolution AS-OCT and the axial length data from the IOLMaster to create a pseudophakic eye model. An inverse calculation algorithm for the IOL back surface optimization is introduced. We employ free form surface representation (bi-cubic spline) for the corneal and IOL surface. The merit of this strategy is demonstrated by comparing with a standard spherical model and quadratic function. Four models are calculated: (1) quadratic cornea + quadratic IOL; (2) spline cornea + quadratic IOL; (3) spline cornea + spline IOL; and (4) spherical cornea + spherical IOL. The IOL optimization process for the pseudophakic eye is performed by numerical ray-tracing method within a 6-mm zone. The spot diagram on the fovea (forward ray-tracing) and wavefront at the spectacle plane (backward ray-tracing) are compared for different models respectively. Results: The models with quadratic (1) or spline (3) surface representation showed superior image performance than the spherical model 4. The residual wavefront errors (peak to valley) of models 1, 2, and 3 are below one micron scale. Model 4 showed max wavefront error of about 15 µm peak to valley. However, the combination of quadratic best fit IOL with the free form cornea (model 2) showed one magnitude smaller wavefront error than the spherical representation of both surfaces (model 3). This results from higher order terms in cornea height profile. Conclusions: A four-surface eye model using a numerical ray-tracing method is proposed for customized IOL calculation. High resolution OCT data can be used as a sufficient base for a customized IOL characterization.

Matrix-based calculation scheme for toric intraocular lenses.

Background and purpose:  While a number of intraocular lens power prediction formulas are well established for determination of spherical lenses, no common strategy is published for the computation of toric intraocular lenses. The purpose of this study is to describe a paraxial computing scheme using 4 × 4 system matrices to describe the ‘optical system eye’ containing astigmatic refractive surfaces with their axes at random.

Full rank interpolatory subdivision schemes: Kronecker, filters and multiresolution

In this extension of earlier work, we point out several ways how a multiresolution analysis can be derived from a finitely supported interpolatory matrix mask which has a positive definite symbol on the unit circle except at -1. A major tool in this investigation will be subdivision schemes that are obtained by using convolution or correlation operations based on replacing the usual matrix multiplications by Kronecker products.

On the multivariate Horner scheme II: running error analysis

Abstract As an extension of our previous paper which gave forward and backward error estimates, we perform a running error analysis of the multivariate Horner scheme. This leads to a modified algorithm which computes the value of the polynomial together with an error estimate.

Prony's method in several variables

The paper gives an extension of Prony’s method to the multivariate case which is based on the relationship between polynomial interpolation, normal forms modulo ideals and H-bases. Though the approach is mainly of algebraic nature, we also give an algorithm using techniques from Numerical Linear Algebra to solve the problem in a fast and efficient way.