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Localization, Stability, and Resolution of Topological Derivative Based Imaging Functionals in Elasticity

Habib Ammari Department of Mathematics and Applications, Ecole Normale Sup ́erieure, 75005 Paris, France Elie Bretin Institut Camille Jordan, INSA de Lyon, Villeurbanne Cedex 69621, France Josselin Garnier Wenjia Jing Department of Mathematics and Applications, Ecole Normale Sup ́erieure, 75005 Paris, France Hyeonbae Kang Department of Mathematics, Inha University, Incheon 402-751, Korea Abdul Wahab Department of Mathematics, COMSATS Institute of Information Technology, Wah Cantt. 47040, Pakistan

Analysis of PDEs Optimization and Control mathscidoc:2206.03010

SIAM Journal on Imaging Sciences, 6, (4), 2174-2212, 2013.11
The focus of this work is on rigorous mathematical analysis of the topological derivative based detection algorithms for the localization of an elastic inclusion of vanishing characteristic size. A filtered quadratic misfit is considered, and the performance of the topological derivative imaging functional resulting therefrom is analyzed. Our analysis reveals that the imaging functional may not attain its maximum at the location of the inclusion. Moreover, the resolution of the image is below the diffraction limit. Both phenomena are due to the coupling of pressure and shear waves propagating with different wave speeds and polarization directions. A novel imaging functional based on the weighted Helmholtz decomposition of the topological derivative is, therefore, introduced. It is thereby substantiated that the maximum of the imaging functional is attained at the location of the inclusion and the resolution is enhanced and proves to be the diffraction limit. Finally, we investigate the stability of the proposed imaging functionals with respect to measurement and medium noises.
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@inproceedings{habib2013localization,,
  title={Localization, Stability, and Resolution of Topological Derivative Based Imaging Functionals in Elasticity},
  author={Habib Ammari, Elie Bretin, Josselin Garnier, Wenjia Jing, Hyeonbae Kang, and Abdul Wahab},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220626163712105585460},
  booktitle={SIAM Journal on Imaging Sciences},
  volume={6},
  number={4},
  pages={2174-2212},
  year={2013},
}
Habib Ammari, Elie Bretin, Josselin Garnier, Wenjia Jing, Hyeonbae Kang, and Abdul Wahab. Localization, Stability, and Resolution of Topological Derivative Based Imaging Functionals in Elasticity. 2013. Vol. 6. In SIAM Journal on Imaging Sciences. pp.2174-2212. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20220626163712105585460.
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