volume | PIER Journals

Abstract

We consider a topological derivative based imaging technique for non-iterative imaging of small and extended perfectly conducting cracks with Dirichlet boundary condition. For this purpose, we introduce topological derivative imaging function based on the asymptotic formula in the existence of narrow crack. We then mathematically analyze its structure in order to investigate why it yields the shape of crack(s). Analyzed structure gives us an optimal condition to get a better image of them. Various numerical experiments support our analysis.

1. Álvarez, D., O. Dorn, N. Irishina, and M. Moscoso, "Crack reconstruction using a level-set strategy," J. Comput. Phys., Vol. 228, 5710-5721, 2009.
doi:10.1016/j.jcp.2009.04.038 Google Scholar

2. Ammari, H., An Introduction to Mathematics of Emerging Biomedical Imaging, Mathematics and Applications Series, Vol. 62, Springer-Verlag, Berlin, 2008.

3. Ammari, H., E. Bonnetier, and Y. Capdeboscq, "Enhanced resolution in structured media SIAM J. Appl. Math.,", Vol. 70, 1428-1452, 2009. Google Scholar

4. Ammari, H. and H. Kang, Reconstruction of Small Inhomo-geneities from Boundary Measurements, Vol. 1846, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2004.

5. Ammari, H., H. Kang, H. Lee, and W.-K. Park, "Asymptotic imaging of perfectly conducting cracks," SIAM J. Sci. Comput., Vol. 32, 894-922, 2010.
doi:10.1137/090749013 Google Scholar

6. Amstutz, S., I. Horchani, and M. Masmoudi, "Crack detection by the topological gradient method," Control and Cybernetics, Vol. 34, 81-101, 2005. Google Scholar

7. Auroux, D. and M. Masmoudi, "Image processing by topological asymptotic analysis," ESAIM: Proc., Vol. 26, 24-44, 2009.
doi:10.1051/proc/2009003 Google Scholar

8. Bonnet, M., "Fast identification of cracks using higher-order topological sensitivity for 2-D potential problems," Eng. Anal. Bound. Elem., Vol. 35, 223-235, 2011.
doi:10.1016/j.enganabound.2010.08.007 Google Scholar

9. Byrne, D., M. O'Halloran, M. Glavin, and E. Jones, "Data independent radar beamforming algorithms for breast cancer detection," Progress In Electromagnetic Research, Vol. 107, 331-348, 2010.
doi:10.2528/PIER10061001 Google Scholar

10. Carpio, A. and M.-L. Rapun, "Solving inhomogeneous inverse problems by topological derivative methods," Inverse Problems, Vol. 24, 045014, 2008.
doi:10.1088/0266-5611/24/4/045014 Google Scholar

11. Chen, G. P. and Z. Q. Zhao, "Ultrasound tomograohy-guide TRM technique for breast tumor detecting in MITAT system," Journal of Electromagnetic Waves and Applications, Vol. 24, No. 11-12, 1459-1471, 2010.
doi:10.1163/156939310792149650 Google Scholar

12. Chen, X., "Subspace-based optimization method in electric impedance tomography," Journal of Electromagnetic Waves and Applications, Vol. 23, No. 11-12, 1397-1406, 2009.
doi:10.1163/156939309789476301 Google Scholar

13. Colton, K. and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences, Vol. 93, Springer, Berlin, 1998.

14. Conceição, R. C., M. O'Halloran, M. Glavin, and E. Jones, "Numerical modelling for ultra wideband radar breast cancer detection and classification," Progress In Electromagnetic Research B, Vol. 34, 145-171, 2011. Google Scholar

15. Donelli, M., "A rescue radar system for the detection of victims trapped under rubble based on the independent component analysis algorithm," Progress In Electromagnetic Research M, Vol. 19, 173-181, 2011.
doi:10.2528/PIERM11061206 Google Scholar

16. Donelli, M., I. J. Craddock, D. Gibbins, and M. Sarafianou, "A three-dimensional time domain microwave imaging method for breast cancer detection based on an evolutionary algorithm," Progress In Electromagnetic Research M, Vol. 18, 179-195, 2011. Google Scholar

17. Dorn, O. and D. Lesselier, "Level set methods for inverse scattering," Inverse Problems, Vol. 22, R67-R131, 2006.
doi:10.1088/0266-5611/22/4/R01 Google Scholar

18. Eschenauer, H. A., V. V. Kobelev, and A. Schumacher, "Bubble method for topology and shape optimization of structures," Struct. Optim., Vol. 8, 42-51, 1994.
doi:10.1007/BF01742933 Google Scholar

19. Kress, R., "Inverse scattering from an open arc," Math. Methods Appl. Sci., Vol. 18, 267-293, 1995.
doi:10.1002/mma.1670180403 Google Scholar

20. Kuo, W.-C., C.-Y. Chuang, M.-Y. Chou, W.-H. Huang, and S.-T. Cheng, "Phase detection with sub-nanometer sensitivity using polarization quadrature encoding method in optical coherence tomography ," Progress In Electromagnetic Research, Vol. 104, 297-311, 2010.
doi:10.2528/PIER10050305 Google Scholar

21. Lesselier, D. and B. Duchene, "Buried, 2-D penetrable objects illuminated by line sources: FFT-based iterative computations of the anomalous field," Progress In Electromagnetic Research, Vol. 5, 351-389, 1991. Google Scholar

22. O'Halloran, M., M. Glavin, and E. Jones, "Rotating antenna microwave imaging system for breast cancer detection," Progress In Electromagnetic Research, Vol. 107, 203-217, 2010.
doi:10.2528/PIER10071002 Google Scholar

23. Park, W.-K., "On the imaging of thin dielectric inclusions via topological derivative concept," Progress In Electromagnetic Research, Vol. 110, 237-252, 2010.
doi:10.2528/PIER10101305 Google Scholar

24. Park, W.-K. and D. Lesselier, "Electromagnetic MUSIC-type imaging of perfectly conducting, arc-like cracks at single frequency," J. Comput. Phys., Vol. 228, 8093-8111, 2009.
doi:10.1016/j.jcp.2009.07.026 Google Scholar

25. Park, W.-K. and D. Lesselier, "Reconstruction of thin electromagnetic inclusions by a level set method," Inverse Problems, Vol. 25, 085010, 2009.
doi:10.1088/0266-5611/25/8/085010 Google Scholar

26. Sokolowski, J. and A. Zochowski, "On the topological derivative in shape optimization," SIAM J. Control Optim., Vol. 37, No. 4, 1251-1272, 1999.
doi:10.1137/S0363012997323230 Google Scholar

27. Zhou, Y., "Microwave imaging based on wideband range profiles," Progress In Electromagnetics Research Letters, Vol. 19, 57-65, 2010. Google Scholar

28. Zhu, G. K. and M. Popovic, "Comparison of radar and thermoacoustic technique in microwave breast imaging," Progress In Electromagnetics Research B, Vol. 35, 1-14, 2011.
doi:10.2528/PIERB11080204 Google Scholar

Dr. Y.-K. Ma

Dr. Pok-Son Kin

Prof. Won-Kwang Park