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Summary: Inverse Problems 16 (2000) 10531070. Printed in the UK PII: S0266-5611(00)08372-6
Reconstructing planar domains from their moments
Bj¨orn Gustafsson, Chiyu He, Peyman Milanfar§ and Mihai Putinar
Mathematics Department, The Royal Institute of Technology, Stockholm, S-10044 Sweden
Mathematics Department, University of California, Santa Barbara, CA 93106, USA
§ Electrical Engineering Department, University of California, Santa Cruz, CA 95064, USA
E-mail: gbjorn@math.kth.se, he@math.ucsb.edu, milanfar@cse.ucsc.edu and
mputinar@math.ucsb.edu
Received 1 October 1999, in final form 16 May 2000
Abstract. In many areas of science and engineering it is of interest to find the shape of an
object or region from indirect measurements which can actually be distilled into moments of the
underlying shapes we seek to reconstruct. In this paper, we describe a theoretical framework for the
reconstruction of a class of planar semi-analytic domains from their moments. A part of this class,
known as quadrature domains, can approximate, arbitrarily closely, any bounded domain in the
complex plane, and is therefore of great practical importance. We provide an exact reconstruction
algorithm of quadrature domains. Some numerical demonstrations of the proposed algorithms will
be presented. In addition, relations of the present theory to computer-assisted tomography and a
geophysical inverse problem will be briefly discussed.
1. Introduction
The theoretical subject of this paper is the truncated L problem of moments in two variables
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