Multidimensional integral inversion, with applications in shape reconstruction. (English) Zbl 1099.65128

Cuyt, Annie; Golub, Gene; Milanfar, Peyman; Verdonk, Brigitte

SIAM J. Sci. Comput. 27, No. 3, 1058-1070 (2005).

The paper concerns a reconstruction of a function or its domain from its moments. The case of multidimensional general compact objects is under consideration. It is shown how a bivariate Stielties transform of a square-integrable function \(f\) with compact support and with finite moments can be approximated by homogeneous Padé approximants on the slices: \(S_\theta\theta=\{(z\cos \theta,z\sin\theta)\); \(z\in R\}\) in the 2-dimensional case. In the proposed shape reconstruction algorithm the given moments are used for computing Padé approximants.
Then the inverse problem of computing the function \(f\) on a discrete set from the approximate Stielties transform is solved by using a truncated singular value decomposition for the regularization of the obtained ill-conditioning system of linear equations. The procedure is extended to three-dimensional case. The new technique is illustrated by several numerical examples of the reconstruction. Two- and three-dimensional objects are reconstructed by approximating their characteristic functions or the function describing the top surface of a cylindrical object.

Reviewer: Teresa Regińska (Warszawa)

MSC:

65R10	Numerical methods for integral transforms
44A60	Moment problems
65F20	Numerical solutions to overdetermined systems, pseudoinverses
65D32	Numerical quadrature and cubature formulas
41A21	Padé approximation
44A15	Special integral transforms (Legendre, Hilbert, etc.)

Keywords:

moment problem; multivariate Stielties transform; Padé approximants; inverse problem; singular value decomposition; numerical examples