Fast identification of small perfectly conducting cracks without diagonal elements of multi-static response matrix

ABSTRACT

We explore mathematical structures of indicator functions used in subspace migration to identify the location of a set of perfectly conducting cracks with small lengths when the diagonal elements of a Multi-Static Response (MSR) matrix cannot be obtained. By using the asymptotic formula for the presence of cracks and the structure of the singular vectors linked to the nonzero singular values of the MSR matrix, we prove that the indicator function can be represented by a Bessel function of order zero of the first kind and the total number of incident and observation directions in a full-view inverse scattering problem. We also present experimental results of numerical simulations with noisy data to validate our findings.

KEYWORDS:

• Subspace migration
• multi-Static response matrix
• inverse scattering problem
• bessel function
• numerical simulations

Acknowledgements

The author wishes to thank the anonymous reviewer for his/her comments, which helped to improve the quality of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Won-Kwang Park http://orcid.org/0000-0001-9418-2455

Additional information

Funding

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. NRF-2017R1D1A1A09000547).

Notes on contributors

Won-Kwang Park

Won-Kwang Park received his BS degree in Mathematical Education from Kookmin University, Seoul, Korea, in 2000, MS degree in Mathematics from Yonsei University, Seoul, Korea, in 2004, and PhD degree in Applied Mathematics from Ecole Polytechnique, Palaiseau, France, in 2009. After graduation, he joined the Institute for Mathematics and Scientific Computing, Karl Franzens University of Graz, Austria, as a postdoctoral researcher in 2009. From 2010, he joined the Kookmin University to work first as a full-time lecturer and as an assistant professor at the Department of Mathematics. Currently, he is an associate professor in the Department of Information Security, Cryptology, and Mathematics at Kookmin University. His main research areas include inverse problems, microwave imaging, non-destructive evaluations, scientific computing, and deep learning.