Two Numerical Methods for Recovering Small Inclusions from the Scattering Amplitude at a Fixed Frequency

Ammari, Habib; Яковлева, Екатерина; Lesselier, Dominique

doi:10.1137/040612518

Cited by 45 publications

(53 citation statements)

References 22 publications

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“…Earlier studies are limited to point-like scatterers. More recent studies [11,2,12,8,9, 1] approach an application of the MU-SIC algorithm to scatterers of some specified size, relative to the wavelength, and are based on the finite-dimensional multi-static response matrix for point-like scatterers. An anonymous referee brought to our attention [3] dealing with closely related questions for linearized scattering.…”

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confidence: 99%

MUSIC for Extended Scatterers as an Instance of the Factorization Method

Arens¹,

Lechleiter²,

Lüke³

2009

SIAM J. Appl. Math.

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Abstract. Recent work of Luke and Devaney showed that there exists an implementation of a modified linear sampling method that is equivalent to a MUSIC algorithm for scattering from sound soft obstacles. The correspondence is independent of the size of the scatterer or the wavelength of the incident field. As the proof was not constructive, an explicit implementation could not be justified. In the present work, we show that MUSIC is an instance of the factorization method applied to any nonabsorbing scatterer, thus providing a justification for the MUSIC algorithm at arbitrary illuminating frequency for arbitrary nonabsorbing scatterers. These results are also extended to scattering from cracks. With explicit constructions in hand, we are also able to provide error and stability estimates for practical implementations in noisy environments with limited data and to explain a curious behavior of the factorization method in the case of noisy data.Key words. inverse problems, scattering theory, factorization method, image processing, MU-SIC AMS subject classifications. 35R30, 35P25, 94A081. Introduction. The Multiple Signal Classification (MUSIC) algorithm is a well-known method for determining the location of emitters from sensors with arbitrary locations and arbitrary directional characteristics such as gain, phase or polarization in a noise/interference environment [27]. The method is based on the observation that the far field patterns of point sources with centers near the true signal source are nearly orthogonal to the noise subspace of the far field operator, or the multi-static response matrix, as it is known in the signal and image processing literature. The connection between the MUSIC algorithm and other inverse scattering techniques has been explored by several authors [5,16,23]. Earlier studies are limited to point-like scatterers. More recent studies [11,2,12,8,9, 1] approach an application of the MU-SIC algorithm to scatterers of some specified size, relative to the wavelength, and are based on the finite-dimensional multi-static response matrix for point-like scatterers. An anonymous referee brought to our attention [3] dealing with closely related questions for linearized scattering. In [23] Luke and Devaney combine the linear sampling method of Colton and Kirsch [6] with the point source method of Potthast [26] to show the existence of a field that is orthogonal to the far field patterns of point sources centered on the boundary of the scatterers. This extends the MUSIC algorithm to obstacles of arbitrary size, independent of the frequency. Missing from their analysis, however, is a concrete way to construct such a field. Numerical experiments indicate that fields constructed from elements of the noise subspace of the far field operator (that is, the subspace corresponding to small eigenvalues of the far field operator) yield the desired properties. In the present work we use Kirsch's factorization method [14] to prove this to be the case, thus yielding a constructive proof of the MUSIC algorithm...

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confidence: 99%

MUSIC for Extended Scatterers as an Instance of the Factorization Method

Arens¹,

Lechleiter²,

Lüke³

2009

SIAM J. Appl. Math.

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“…This expansion has the same structure as those derived in [1,8], where the authors study the influence of inclusions of small volume embedded in a smooth, continuous, reference medium. In this context, the singularity of the Green's function on the right-hand side has been exploited quite successfully in numerical algorithm for detection [2,6].…”

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confidence: 99%

Asymptotics for the potential in a periodic network with localized defects

Bonnetier¹,

Sista²

2007

Proc Appl Math and Mech

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“…A mathematical study of the properties of the eigenstructure of the response matrix C can be made following the arguments given in [4]. However, the analysis becomes more complicated because of the form of the Green's function G of the unperturbed waveguide.…”

Section: Reconstruction Of Multiple Inclusionsmentioning

confidence: 99%