Imaging in a cylindrical acoustic waveguide using the linear sampling method

The widespread occurrence of duct defects necessitates effective obstacle detection within ducts. This study presents an imaging method utilizing acoustic waves in a cylindrical duct. Specifically, the linear sampling method is employed to address the inverse scattering problem in a cylindrical acoustic waveguide. By projecting onto modes, we obtain a modal formulation of the linear sampling method, resulting in a system of linear equations. The matrix of this system depends on the scattering coefficients, while the inhomogeneous term is influenced by the positions of the sampling points. For each sampling point, the norm of the solution to this system serves as an indicator for imaging. The method is validated through several numerical experiments, where its feasibility is thoroughly analyzed and confirmed. Notably, this study demonstrates that the linear sampling method can be applied to imaging in three-dimensional cylindrical waveguides, extending its use beyond previous theoretical analyses and two-dimensional experiments. Also, a prior matrix decomposition is used to improve the algorithm's efficiency.