This article compares the isotropic and anisotropic TV regularizations
used in inverse acoustic scattering. It is observed that compared with the traditional
Tikhonov regularization, isotropic and anisotropic TV regularizations perform
better in the sense of edge preserving. While anisotropic TV regularization
will cause distortions along axes. To minimize the energy function with
isotropic and anisotropic regularization terms, we use split Bregman scheme. We
do several 2D numerical experiments to validate the above arguments.
References
[1]
Colton, D. and Kress, R. (1998) Inverse Acoustic and Electromagnetic Scattering Theory. Springer, Berlin. https://doi.org/10.1007/978-3-662-03537-5
[2]
Author (2002) A Simplified Newton Method in Scattering Theory: The TE Case. IEEE Antennas and Propagation Society International Symposium (IEEE Cat. No. 02CH37313), 2, 102-105.
[3]
Colton, D. and Kress, R. (1983) Integral Equation Methods in Scattering Theory. John Willey and Sons, Inc., New York.
[4]
Gutman, S. and Klibanov, M. (1993) Regularized Quasi-Newton Method for Inverse Scattering Problems. Mathematical and Computer Modelling of Dynamical Systems, 18, 5-31. https://doi.org/10.1016/0895-7177(93)90076-B
[5]
Hohage, T. and Langer, S. (2010) Acceleration Techniques for Regularized Newton Methods Applied to Electromagnetic Inverse Medium Scattering Problems. Inverse Problems, 26, 15-28. https://doi.org/10.1088/0266-5611/26/7/074011
[6]
Zaeytijd, J.D., Franchois, A., Eyraud, C. and Geffrin, J.-M. (2007) Full-Wave Three-Dimensional Microwave Imaging with a Regularized Gauss-Newton Method—Theory and Experiment. IEEE Transactions on Antennas and Propagation, 2 November 2007, 3279-3292. https://doi.org/10.1109/TAP.2007.908824
[7]
Colton, D. and Monk, P. (1985) A Novel Method for Solving the Inverse Scattering Problem for Time-Harmonic Waves in the Resonance Region. SIAM Journal on Applied Mathematics, 45, 1039C1053. https://doi.org/10.1137/0145064
[8]
Ito, K., Jin, B. and Zou, J. (2013) A Direct Sampling Method for Inverse Electromagnetic Medium Scattering. Inverse Problems, 29, 095018. https://doi.org/10.1088/0266-5611/29/9/095018
[9]
Cakoni, F., Colton, D. and Monk, P. (2011) The Linear Sampling Method in Inverse Electromagnetic Scattering. Society for Industrial and Applied Mathematics. https://doi.org/10.1137/1.9780898719406
[10]
Kirsch, A. and Grinberg, N.I. (2008) The Factorization Method for Inverse Problems. Oxford University Press, Oxford. https://doi.org/10.1093/acprof:oso/9780199213535.001.0001
[11]
Kirsch, A. (2002) The MUSIC Algorithm and the Factorization Method in Inverse Scattering Theory for Inhomogeneous Media. Inverse Problems, 18, 1025-1040. https://doi.org/10.1088/0266-5611/18/4/306
[12]
Ammari, H., Iakovleva, E., Lesselier, D. and Perrusson, G. (2007) MUSIC-Type Electromagnetic Imaging of a Collection of Small Three-Dimensional Inclusions. SIAM Journal on Scientific Computing, 29, 674-709. https://doi.org/10.1137/050640655
[13]
Dorn, O. and Lesselier, D. (2009) Level Set Methods for Inverse Scattering: Some Recent Developments. Inverse Problems, 25, 125001. https://doi.org/10.1088/0266-5611/25/12/125001
[14]
Goldstein, T, and Osher, S. (2009) The Split Bregman Method for L1 Regularized Problems. SIAM Journal on Imaging Sciences, 2, 323-343. https://doi.org/10.1137/080725891
