Multidimensional Integral Inversion, with Applications in Shape Reconstruction

Abstract

In shape reconstruction, the celebrated Fourier slice theorem plays an essential role. It allows one to reconstruct the shape of a quite general object from the knowledge of its Radon transform [S. Helgason, The Radon Transform, Birkhauser Boston, Boston, 1980]---in other words from the knowledge of projections of the object. In case the object is a polygon [G. H. Golub, P. Milanfar, and J. Varah, SIAM J. Sci. Comput., 21 (1999), pp. 1222-1243], or when it defines a quadrature domain in the complex plane [B. Gustafsson, C. He, P. Milanfar, and M. Putinar, Inverse Problems, 16 (2000), pp. 1053-1070], its shape can also be reconstructed from the knowledge of its moments. Essential tools in the solution of the latter inverse problem are quadrature rules and formal orthogonal polynomials. In this paper we show how shape reconstruction from the knowledge of moments can also be realized in the case of general compact objects, not only in two but also in higher dimensions. To this end we use a less-known homogeneous Pade slice property. Again integral transforms---in our case the multivariate Stieltjes transform and univariate Markov transform---formal orthogonal polynomials in the form of Pade denominators, and multidimensional integration formulas or cubature rules play an essential role. We emphasize that the new technique is applicable in all higher dimensions and illustrate it through the reconstruction of several two- and three-dimensional objects.