Reconstruction of Small Inhomogeneities from Boundary Measurements
This is the first book to provide a systematic exposition of promising techniques for the reconstruction of small inhomogeneities from boundary measurements. In particular, theoretical results and numerical procedures for the inverse problems for the
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Habib Ammari Hyeonbae Kang
Reconstruction of Small Inhomogeneities from Boundary Measurements
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Authors Habib Ammari Centre de Math´ematiques Appliqu´ees E´ cole Polytechnique UMR CNRS - 7641 Palaiseau 91128 France e-mail: [email protected] Hyeonbae Kang School of Mathematical Sciences Seoul National University Seoul 151-747 Korea e-mail: [email protected]
Library of Congress Control Number: 2004108535
Mathematics Subject Classification (2000): 35R30, 35B30 ISSN 0075-8434 ISBN 3-540-22483-1 Springer Berlin Heidelberg New York DOI: 10.1007/b98245 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specif ically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microf ilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science + Business Media http://www.springeronline.com c Springer-Verlag Berlin Heidelberg 2004 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specif ic statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: Camera-ready TEX output by the authors Printed on acid-free paper 41/3141/du - 5 4 3 2 1 0
Preface
Electrical impedance tomography (EIT) seeks to recover the electrical conductivity distribution inside a body from measurements of current flows and voltages on its surface. The vast and growing literature reflects many possible applications of EIT techniques, e.g., for medical diagnosis or nondestructive evaluation of materials. Since the underlying inverse problem is nonlinear and severely ill-posed, general purpose EIT reconstruction techniques are likely to fail. Therefore it is generally advisable to incorporate a-priori knowledge about the unknown conductivity. One such type of knowledge could be that the body consists of a smooth background containing a number of unknown small inclusions with a significantly different conductivity. This situation arises for example in breast cancer imaging or mine detection. In this case EIT seeks to recover the unknown inclusions. Due to the smallness of the inclusions the associated voltage potentials measured on the surface of the body are very close to the potentials corresponding to the medium without inclusion. So unless one knows exactly what patterns to look for, noise will largely dominate the information contained in the measured data. Furthermore, in applications it is often not necessary to reconstruct the precise values of the conductivity or geometry of the inclusions. The information of re
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