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Full Aperture Reconstruction of the Acoustic Far-Field Pattern from Few Measurements

Published online by Cambridge University Press:  20 August 2015

Hélène Barucq ,
Chokri Bekkey  and
Rabia Djellouli
Hélène Barucq*
Affiliation:
INRIA Bordeaux Sud-Ouest Research Center, Team Project Magique-3D, & LMA/CNRS UMR 5142, Université de Pau et des Pays de l’Adour, France
Chokri Bekkey*
Affiliation:
Faculté des Sciences de Monastir, Tunisia
Rabia Djellouli*
Affiliation:
Department of Mathematics, California State University Northridge & Interdisciplinary Research Institute for the Sciences, IRIS, USA
*
Email address:helene.barucq@univ-pau.fr
Email address:chokri.bekkey@ipeit.rnu.tn
Corresponding author.Email:rabia.djellouli@csun.edu
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Abstract

We propose a numerical procedure to extend to full aperture the acoustic far-field pattern (FFP) when measured in only few observation angles. The reconstruction procedure is a multi-step technique that combines a total variation regularized iterative method with the standard Tikhonov regularized pseudo-inversion. The proposed approach distinguishes itself from existing solution methodologies by using an exact representation of the total variation which is crucial for the stability and robustness of Newton algorithms. We present numerical results in the case of two-dimensional acoustic scattering problems to illustrate the potential of the proposed procedure for reconstructing the full aperture of the FFP from very few noisy data such as backscattering synthetic measurements.

Keywords

35J0578A4674J2065F2265T4035N05Acoustic scattering problemlimited apertureinverse obstacle problemill-posed problemtotal variationTikhonov regularizationNewton method
Type
Research Article
Information
Communications in Computational Physics , Volume 11 , Issue 2 , February 2012 , pp. 647 - 659
Copyright
Copyright © Global Science Press Limited 2012

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References

[1]Vogel, C. R., Computational Methods for Inverse Problems, Frontiers in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002.Google Scholar
[2]Colton, D. and Kress, R., Inverse Acoustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences 93, Springer-Verlag, 1992.Google Scholar
[3]Hadamard, J., Lectures on Cauchy’s Problem in Linear Partial Differential Equations, Yale University Press, 1923.Google Scholar
[4]Djellouli, R., Inverse Acoustic Problems, In: Computational Methods for Acoustics Problems, (Magoules, F., editor), Saxe-Coburg Publications, (2008), pp. 263–294.Google Scholar
[5]Ochs, R. L., The limited aperture problem of inverse acoustic scattering: Dirichlet boundary conditions, SIAM J. Appl. Math. 47, (1987), pp. 1320–1341.Google Scholar
[6]Zinn, A., On an optimization method for full- and limited-aperture problem, In: inverse acoustic scattering for a sound-soft obstacle, Inverse Problems, 5, (1989), pp. 239–253.Google Scholar
[7]Kress, R., Integral equations methods in inverse acoustic and electromagnetic scattering, In: Boundary Integral Formulations for Inverse Analysis, (Ingham, and Wrobel, , eds.), Computational Mechanics Publications, Southampton, (1997), pp. 67–92.Google Scholar
[8]Kress, R., Rundell, W., Inverse obstacle scattering using reduced data, SIAM J. Appl. Math., 59, (1999) pp. 442–454.CrossRefGoogle Scholar
[9]Oukaci, F., Quelques problemes numériques d’identification de forme en diffraction acoustique, Ph. D. Thesis, Université de Technologie de Compiegne, 1999.Google Scholar
[10]Djellouli, R., Tezaur, R., and Farhat, C., On the solution of inverse obstacle acoustic scattering problems with a limited aperture, in: Mathematical and Numerical Aspects of Wave Propagation (Cohen, et al. eds.), Jyväskylä, (2003), pp. 625–630.Google Scholar
[11]Barucq, H., Bekkey, C., and Djellouli, R., A Multi-Step Procedure for Enriching Limited Two-Dimensional Acoustic Far-Field Pattern Measurements, INRIA Research Report, No. 7048 (2009). Available online at: http://hal.archives-ouvertes.fr/inria-00420644/fr/.Google Scholar
[12]Olshansky, Y., Stainvas, I., and Turkel, E., Simultaneous Scatterer Shape Estimation and Far-Field Pattern Denoising, in: Proceedings of the ninth International Conference on Mathematical and Numerical Aspects of Wave Propagation (Barucq, et al. eds.), Pau, (2009), pp. 306–307.Google Scholar
[13]Vogel, C. R., Total Variation regularization for ill-posed problems, Department of Mathematical Sciences Technical Report (1993), Montana State University.Google Scholar
[14]Rudin, L. I., Osher, S. and Fatemi, E., Nonlinear total variation based noise removal algorithms, Proceeding of the 11th Annual International Conference of the center for Nonlinear Studies, Physica D, vol.60 (1992), pp. 259–268.Google Scholar
[15]Acar, R. and Vogel, C. R., Analysis for Bounded Variation Penalty Methods for Ill-Posed Problems, Inverse Problems, Vol. 10, No. 6 (1994), pp. 1217–1229.Google Scholar
[16]Hansen, C., Regularization Tools: a Matlab Package for Analysis and Solution of Ill-Posed Problems, Numerical Algorithms, 6, (1994), pp. 1–35.Google Scholar
[17]Pedersen, J., Modular Algorithms for Large-Scale Total Variation Image Deblurring, Master Thesis, Technical University of Denmark, 2005.Google Scholar
[18]Bowman, J.J., Senior, T.B.A., Uslenghi, P.L.E., Electromagnetic and Acoustic Scattering by Simple Shapes, North-Holland Publishing Company, Amsterdam, 1969.Google Scholar