SIAM J.S CI.C OMPUT.c 2005Society for Industrial and Applied Mathematics Vol.27,No.3,pp.1058–1070
MULTIDIMENSIONAL INTEGRAL INVERSION,WITH APPLICATIONS IN SHAPE RECONSTRUCTION∗ANNIE CUYT†,GENE GOLUB‡,PEYMAN MILANFAR§,AND BRIGITTE VERDONK†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,Birkh¨a user Boston,Boston,1980]—in other words from the knowledge of projections of the object.In case the object is a polygon[G.H.Golub, http://www.docsj.com/doc/006a99ef4afe04a1b071de2f.htmlanfar,and J.Varah,SIAM http://www.docsj.com/doc/006a99ef4afe04a1b071de2f.htmlput.,21(1999),pp.1222–1243],or when it deﬁnes a quadrature domain in the complex plane[B.Gustafsson,C.He,http://www.docsj.com/doc/006a99ef4afe04a1b071de2f.htmlanfar,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 Pad´e slice property.Again integral transforms—in our case the multivariate Stieltjes transform and univariate Markov transform—formal orthogonal polynomials in the form of Pad´e 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.
Key words.shape,multidimensional,inverse problem,moment problem
AMS subject classiﬁcations.65D32,65F20,41A21,44A60
1.Problem statement.The problem of reconstructing a function and/or its domain given its moments is encountered in many areas.Several applications from diverse areas such as probability and statistics,signal processing,computed tomography[16,17],and inverse potential theory[4,19](magnetic and gravitational anomaly detection)can be cited,to name just a few.We can expound on some of these applications in a bit more detail.Consider the following diverse set of examples:•A region of the plane can be regarded as the domain of a probability density function.In this case,the problem is that of reconstructing this density func-tion and/or approximating its domain from measurements of its moments.
•Tomographic(line integral)measurements of a body can be converted into moments from which an approximation to its density and boundary can be extracted.
•Measurements of exterior gravitationalﬁeld induced by a body of uniform mass can be turned into moment measurement,from which the shape of the region may be reconstructed.
•Measurements of an exterior magneticﬁeld induced by a body of uniform
∗Received by the editors November5,2003;accepted for publication(in revised form)March29, 2005;published electronically December30,2005.
†Department of Mathematics and Computer Science,University of Antwerp,Middelheimlaan1, Antwerp,B2020,Belgium(annie.cuytua.ac.be,email@example.com).
‡Department of Computer Science,Stanford University,Stanford,CA94305(golub@sccm. http://www.docsj.com/doc/006a99ef4afe04a1b071de2f.html).
§Electrical Engineering Department,University of California at Santa Cruz,Santa Cruz,CA 95064(milanfar@http://www.docsj.com/doc/006a99ef4afe04a1b071de2f.html).