Abstract
Consider the reconstruction of the complex refraction index of an object, which is immersed in a known homogeneous background, from the knowledge of scattered waves of the point sources outside of the object. We firstly establish the uniqueness for this inverse problem, which provides the theoretical basis for the reconstruction scheme. Then based on the contrast source inversion (CSI) method, we propose an algorithm determining the refraction index and the artificial wave sources alternately by a dynamic iterative scheme. The algorithm defines the iterates by solving a series of minimization problems with uniformly convex penalty terms, which are allowed to be non-smooth to include L1 and total variation like functionals, ensuring the reconstruction quality when the unknown refraction index has the special features such as sparsity and discontinuity. By choosing the regularizing parameter automatically, the algorithm is terminated in terms of discrepancy principle. The convergence property of the iterative sequence is rigorously proven. Numerical implementations demonstrate the validity of the proposed algorithm.
Similar content being viewed by others
References
Bao G, Liu H Y. Nearly cloaking the electromagnetic fields. SIAM J Appl Math, 2014, 74: 724–742
Blasten E, Imanuvilov O Y, Yamamoto M. Stability and uniqueness for a two-dimensional inverse boundary value problem for less regular potentials. Inverse Probl Imaging, 2015, 9: 709–723
Chen Y. A fast, direct algorithm for the Lippmann-Schwinger integral equation in two dimensions. Adv Comput Math, 2002, 16: 175–190
Colton D, Kress R. Inverse Acoustic and Electromagnetic Scattering Theory. Berlin: Springer, 1992
Greenleaf J F, Johnson S A, Bahn R C. Quantitative cross-sectional imaging of ultrasound parameters. In: Proceedings of IEEE Ultrasonics Symposium. New York: IEEE, 1977, 989–995
Hanke M, Groetsch C W. Nonstationary iterated Tikhonov regularization. J Optim Theory Appl, 1998, 97: 37–53
Hu G H, Liu H Y. Recovering complex elastic scatterers by a single far-field pattern. J Differential Equations, 2014, 257: 469–489
Imanuvilov O Y, Yamamoto M. Uniqueness for inverse boundary value problems by Dirichlet-to-Neumann map on subboundaries. Milan J Math, 2013, 81: 187–258
Jin Q. On a regularized Levenberg-Marquardt method for solving nonlinear inverse problems. Numer Math, 2010, 115: 229–259
Jin Q. A general convergence analysis of some Newton-type methods for nonlinear inverse problems. SIAM J Numer Anal, 2011, 49: 549–573
Jin Q, Stals L. Nonstationary iterated Tikhonov regularization for ill-posed problems in Banach spaces. Inverse Problems, 2012, 28: 104011
Jin Q, Zhong M. Nonstationary iterated Tikhonov regularization in Banach spaces with uniformly convex penalty terms. Numer Math, 2014, 127: 485–513
Kleinman R E, Van den Berg P M. A modified gradient method for two-dimensional problems in tomography. J Comput Appl Math, 1992, 42: 17–35
Kleinman R E, Van den Berg P M. An extended range modified gradient technique for profile inversion. Radio Sci, 1993, 28: 877–884
Kleinman R E, Van den Berg P M. Two-dimensional location and shape reconstruction. Radio Sci, 1994, 29: 1157–1169
Lin H, Azuma T, Qu X, et al. Robust contrast source inversion method with automatic choice rule for regularization parameters for ultrasound waveform tomography. Japan J Appl Phys, 2016, 55: 07KB08
Liu H Y, Zhao H K, Zou C J. Determining scattering support of anisotropic acoustic mediums and obstacles. Commun Math Sci, 2015, 13: 987–1000
Liu J J, Nakamura G, Sini M. Reconstruction of the shape and surface impedance from acoustic scattering data for arbitrary cylinder. SIAM J Appl Math, 2007, 67: 1124–1146
Nachman A I. Global uniqueness for a two-dimensional inverse boundary value problem. Ann of Math (2), 1996, 143: 71–96
Nesterov Y. A method of solving a convex programming problem with convergence rate O(1=k2). Dokl Akad Nauk, 1983, 27: 372–376
Rudin L, Osher S, Fatemi C. Nonlinear total variation based noise removal algorithm. Phys D, 1992, 60: 259–268
Tamano S, Azuma T, Imoto H, et al. Compensation of transducer element positions in a ring array ultrasonic computer tomography system. Japan J Appl Phys, 2015, 54: 07HF24
Van den Berg P M, Abubakar A. Contrast source inversion method: State of art. J Electromagnetic Waves Appl, 2001, 15: 1503–1505
Van den Berg P M, Kleinman R E. A total variation enhanced modified gradient algorithm for profile reconstruction. Inverse Problems, 1995, 11: L5–L10
Van den Berg P M, Kleinman R E. A contrast source inversion method. Inverse Problems, 1997, 13: 1607–1620
Van den Berg P M, Van B, Abubakar A. Extended contrast source inversion. Inverse Problems, 1999, 15: 1325–1344
Wang H B, Liu J J. The two-dimensional direct and inverse scattering problems with generalized oblique derivative boundary condition. SIAM J Appl Math, 2015, 75: 313–334
Zălinscu C. Convex Analysis in General Vector Spaces. River Edge: World Scientific, 2002
Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant Nos. 1421110002, 11531005 and 11501102) and National Science Foundation of Jiangsu Province (Grant No. BK20150594). The authors are grateful to Prof. Yamamoto M. and Dr. Lin H. X. for valuable and helpful comments and discussions.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhong, M., Liu, J. On the reconstruction of media inhomogeneity by inverse wave scattering model. Sci. China Math. 60, 1825–1836 (2017). https://doi.org/10.1007/s11425-016-9054-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11425-016-9054-6