Irina Georgieva

Interpolation of harmonic functions based on Radon projections

We consider an algebraic method for reconstruction of a harmonic function in the unit disk via a finite number of values of its Radon projections. The approach is to seek a harmonic polynomial which matches given values of Radon projections along some chords of the unit circle. We prove an analogue of the famous Marr’s formula for computing the Radon projection of the basis orthogonal polynomials in our setting of harmonic polynomials. Using this result, we show unique solvability for a family of schemes where all chords are chosen at equal distance to the origin. For the special case of chords forming a regular convex polygon, we prove error estimates on the unit circle and in the unit disk. We present an efficient reconstruction algorithm which is robust with respect to noise in the input data and provide numerical examples.

Surface Reconstruction and Lagrange Basis Polynomials

Surface reconstruction, based on line integrals along segments of the unit disk is studied. Various methods concerning with this problem are known. We consider here interpolation over regular schemes of chords by polynomials. We find the interpolant in Lagrange form and investigate some properties of Lagrange basis polynomials. Numerical experiments for both surface and image reconstruction are presented.

Harmonic interpolation based on Radon projections along the sides of regular polygons

Given information about a harmonic function in two variables, consisting of a finite number of values of its Radon projections, i.e., integrals along some chords of the unit circle, we study the problem of interpolating these data by a harmonic polynomial. With the help of symbolic summation techniques we show that this interpolation problem has a unique solution in the case when the chords form a regular polygon. Numerical experiments for this and more general cases are presented.

New results on regularity and errors of harmonic interpolation using Radon projections

We study interpolation of harmonic functions in the unit disk with a finite number of values of the Radon projection along prescribed chords as the input data. We seek the interpolant in the space of harmonic polynomials in such a way that it matches the given projection values exactly. In this setting, we investigate schemes where all chords are divided into two sets of parallel chords. We give necessary and sufficient conditions for a scheme of this type to result in a uniquely solvable interpolation problem. As a second new result, we generalize the previously known error estimates for schemes with equispaced chord angles, both to allow for a larger class of chord choices and to obtain new error estimates in fractional Sobolev norms.

Least Squares Fitting of Harmonic Functions Based on Radon Projections

Given the line integrals of a harmonic function on a finite set of chords of the unit circle, we consider the problem of fitting these Radon projections type of data by a harmonic polynomial in the unit disk. In particular, we focus on the overdetermined case where the amount of given data is greater than the dimension of the polynomial space. We prove sufficient conditions for existence and uniqueness of a harmonic polynomial fitting the data by using least squares method. Combining with recent results on interpolation with harmonic polynomials, we obtain an algorithm of practical application. We extend our results to fitting of more general mixed data consisting of both Radon projections and function values. We perform a comparative numerical study of the least-squares approach with two other reconstruction methods for the case of noisy data.

On interpolation in the unit disk based on both radon projections and function values

We consider interpolation of function in two variables on the unit disk with bivariate polynomials, based on its Radon projections and function values This is closely related to surface and image reconstruction Due to the practical importance of these problems, recently a lot of mathematicians deal with interpolation and smoothing of bivariate functions with a data consisting of prescribed Radon projections or mixed type of data – Radon projections and function values Here we present some new results and numerical experiments in this field.

An algebraic method for reconstruction of harmonic functions via radon projections

We consider an algebraic method for reconstruction of a harmonic function via a finite number of values of its Radon projections. More precisely, for given values of some Radon projections, we seek a harmonic polynomial which matches these data exactly. In the present work, we focus mostly on the case where these measurements are taken along equally spaced chords of the unit circle. We present an efficient reconstruction algorithm which is robust with respect to noise in the input data and provide numerical examples.