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Time domain topological gradient and time reversal analogy: an inverse method for ultrasonic target detection. (English) Zbl 1189.74070
Summary: To find the shape and position of one or several obstacles embedded in a 2D or 3D elastic medium given ultrasonic measurements performed with a linear array of sensors, we use a topological gradient method. It consists in minimizing a cost function which evaluates the adequation between the measurements obtained for the inspected medium and measurements performed on a reference medium known as safe. The topological gradient is a function of both the solutions of the forward and adjoint problems. The mathematical solution of such an adjoint problem corresponds to a physical time reversal operation. This inverse method is therefore physically justified and inheritance of the refocusing properties of the time reversal phenomenon is expected.

MSC:
74J25 Inverse problems for waves in solid mechanics
74J20 Wave scattering in solid mechanics
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[1] Prada, C.; Wu, F.; Fink, M., The iterative time reversal mirror: a solution to self-focusing in the pulse echo mode, Jasa, 90, 1119-1129, (1991)
[2] Prada, C.; Fink, M., Eigenmodes of the time reversal operator: a solution to selective focusing in multiple-target media, Wave motion, 20, 151-163, (1994) · Zbl 0925.76571
[3] Prada, C.; Manneville, S.; Spoliansky, D.; Fink, M., Decomposition of the time reversal operator: detection and selective focusing on two scatterers, Jasa, 99, 2067-2076, (1996)
[4] Fink, M.; Prada, C.; Wu, F.; Cassereau, D., Self focusing with time reversal mirror in inhomogeneous media, Proc. IEEE ultrason. symp., 2, 681-686, (1989)
[5] Fink, M., Time reversal of ultrasonic fields. part I. basic principles, IEEE trans. ultrason. ferroelectr. freq. contr., 39, 555-566, (1992)
[6] Borcea, L.; Papanicolaou, G.; Tsogka, C.; Berryman, J., Imaging and time reversal in random media, Inverse probl., 18, 5, 1247-1279, (2002) · Zbl 1047.74032
[7] Blomgren, P.; Papanicolaou, G.; Zhao, H., Super-resolution in time reversal acoustics, Jasa, 111, 1, 230-248, (2002)
[8] A.J. Devaney, Super-resolution processing of multi-static data using time reversal and MUSIC, JASA, in press.
[9] Devaney, A.J., Introduction to inverse scattering theory and diffraction tomography, () · Zbl 0572.92005
[10] Tarantola, A., Theoretical background for the inversion of seismic waveforms, including elasticity and attenuation, Pageoph 128, 1/2, (1988)
[11] A. Schumacher, Topologieoptimierung von Bauteilstrukturen unter Verwendung von Lopchpositionierungkrieterien, PhD Thesis, Universität-Gesamthochschule-Siegen, 1995.
[12] Eschenauer, H.A.; Kobelev, V.V.; Schumacher, A., Bubble method for topology and shape optimization of structures, J. struct. optim., 8, 42-51, (1994)
[13] Sokolowski, J.; Zochowski, A., On the topological derivative in shape optimization, SIAM J. contr. optim., 37, 4, 1251-1272, (1999) · Zbl 0940.49026
[14] M. Masmoudi, The topological asymptotic, in: R. Glowinski, H. Kawarada, J. Periaux (Eds.), Computational Methods for Control Applications, Gakuto International Series: Mathematical Sciences and Applications, vol. 16, 2002. · Zbl 1082.93584
[15] Garreau, S.; Guillaume, Ph.; Masmoudi, M., The topological asymptotic for PDE systems: the elasticity case, SIAM J. contr. optim., 39, 6, 1756-1778, (2001) · Zbl 0990.49028
[16] Natterer, F.; Wubbeling, F., A propagation – backpropagation method for ultrasound tomography, Inverse probl., 11, 1225-1232, (1995) · Zbl 0839.35146
[17] Oberai, A.A.; Feijoo, G.R.; Pinsky, P.M., Prediction of elastic material properties via an adjoint formulation, Jasa, 110, 5, 2759, (2001)
[18] Oberai, A.A.; Gokhale, N.H.; Feijoo, G.R., Solution of inverse problems in elasticity imaging using the adjoint method, Inverse probl., 19, 297-313, (2003) · Zbl 1171.35490
[19] J. Pommier, L’Asymptotique Topologique en électromagnétisme, PhD Thesis, Université de Toulouse 3, France, 2002.
[20] S. Amstutz, Topological sensitivity for some nonlinear systems, J. Math. Pure Appl., in press. · Zbl 1090.35053
[21] Lions, J.L., Optimal control of systems governed by partial differential equations, (1971), Springer · Zbl 0203.09001
[22] Virieux, J., P-SV wave propagation in heterogeneous media: velocity-stress finite-difference method, Geophysics, 51, 4, 889-901, (1986)
[23] M. Tanter, Application of time reversal focusing to ultrasonic brain hyperthermia, PhD Thesis, University Paris VII, France, 1999. Software URL: http://www.loa.espci.fr/>∼michael/fr/acel/aceltest.htm.
[24] Collino, F.; Tsogka, C., Application of the perfectly matched layer model to the linear elastodynamic problem in anisotropic heterogeneous media, Geophysics, 66, 1, 294-307, (2001)
[25] Cea, J.; Garreau, S.; Guillaume, Ph.; Masmoudi, M., The shape and topological optimizations connection, Comput. meth. appl. mech. eng., 188, 713-726, (2000) · Zbl 0972.74057
[26] Cassereau, D.; Fink, M., Time reversal of ultrasonic fields. part III. basic principles, IEEE trans. ultrason. ferroelectr. freq. contr., 39, 579-592, (1992)
[27] Tanter, M.; Aubry, J.F.; Gerber, J.; Thomas, J.L.; Fink, M., Optimal focusing by spatio-temporal inverse filter. part I. basic principles, Jasa, 110, 1, 37-47, (2001)
[28] Aubry, J.F.; Tanter, M.; Gerber, J.; Thomas, J.L.; Fink, M., Optimal focusing by spatio-temporal inverse filter. part II. experiments, Jasa, 110, 1, 48-58, (2001)
[29] Montaldo, G.; Tanter, M.; Fink, M., Real time inverse filter focusing through iterative time reversal, Jasa, 115, 2, 768-775, (2004)
[30] Montaldo, G.; Tanter, M.; Fink, M., Revisiting iterative time reversal: real time detection of multiple targets, Jasa, 115, 2, 776-784, (2004)
[31] A. Tourin, Diffusion multiple et renversement du temps des ondes ultrasonores, PhD Thesis, Université de Paris VII, 1999.
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