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The Factorization Method for Inverse Problems$
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Andreas Kirsch and Natalia Grinberg

Print publication date: 2007

Print ISBN-13: 9780199213535

Published to Oxford Scholarship Online: September 2008

DOI: 10.1093/acprof:oso/9780199213535.001.0001

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The MUSIC algorithm and scattering by an inhomogeneous medium

The MUSIC algorithm and scattering by an inhomogeneous medium

(p.86) 4 The MUSIC algorithm and scattering by an inhomogeneous medium
The Factorization Method for Inverse Problems

Andreas Kirsch

Natalia Grinberg

Oxford University Press

This chapter examines the case of a penetrable scatterer with an index of refraction that can be space-dependent and is assumed to be different from the constant background index. The inverse scattering problem is to determine the support D of the contrast from far field measurements. The chapter begins with a simple scattering model where the scatterers consists of a finite number of point scatterers. The inverse problem is to determine the locations of these point scatterers from the multistatic response matrix F, which is the discrete analog of the far field operator. In this situation, the Factorization Method is nothing else but the MUSIC-algorithm which is well known in signal processing. The chapter then formulates direct and inverse scattering problem for the scattering by an inhomogeneous medium, reformulates the direct problem as the Lippmann-Schwinger integral equation, and justifies the popular Born approximation. The chapter formulizes the far field operator and proves a characterization of D by the convergence of a Picard series which involves only known data derived from the far field operator. This characterization holds only if the frequency is not an eigenvalue of an unconventional eigenvalue problem of transmission type. The last section shows that there exist at most a quantifiable number of these values.

Keywords:   Helmholtz equation, discrete scattering model, Born approximation, Lippmann-Schwinger integral equation, far field operator, Factorization Method, interior transmission eigenvalues, inverse scattering problem, point scatterers

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