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August  2021, 15(4): 745-762. doi: 10.3934/ipi.2021012

## A sampling type method in an electromagnetic waveguide

 Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

Received  July 2020 Revised  November 2020 Published  August 2021 Early access  January 2021

We propose a sampling type method to image scatterer in an electromagnetic waveguide. The waveguide terminates at one end and the measurements are on the other end and in the far field. The imaging function is based on integrating the measurements and a known function over the measurement surface directly. The design and analysis of such imaging function are based on a factorization of a data operator given by the measurements. We show by analysis that the imaging function peaks inside the scatterer, where the coercivity of the factorized operator and the design of the known function play a central role. Finally, numerical examples are provided to demonstrate the performance of the imaging method.

Citation: Shixu Meng. A sampling type method in an electromagnetic waveguide. Inverse Problems & Imaging, 2021, 15 (4) : 745-762. doi: 10.3934/ipi.2021012
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##### References:
Plot of $|\mathcal{H} \Psi ({\boldsymbol{x}}_*;{\boldsymbol{z}}) {\boldsymbol{e}}_j|^2$ as a function of the sampling point ${\boldsymbol{z}}$ for a rectangular waveguide with cross section $(0,10)\times(0,10)$ where ${\boldsymbol{x}}_* = (5,5,-5)$ and $k = 3$. Top left: Iso-surface plot with iso-value $0.6$. Top right: $x_1x_2$-cross section image. Bottom left: $x_2x_3$-cross section image. Bottom right: $x_1x_3$-cross section image
Three dimensional view of the generated mesh
Image of a cuboid. Top left: exact. Top middle: three dimensional image using iso-surface plot with iso-value $0.6$. Top right: three dimensional image using iso-surface plot with iso-value $0.4$. At the bottom, we plot the cross section images where the exact geometry is indicated by the dashed line. Bottom left: $x_1x_2$-cross section image. Bottom middle: $x_2x_3$-cross section image. Bottom right: $x_1x_3$-cross section image
Image of a L-shape scatterer. Top left: exact. Top middle: three dimensional image using iso-surface plot with iso-value $0.5$. Top right: three dimensional image using iso-surface plot with iso-value $0.4$. At the bottom, we plot the cross section images where the exact geometry is indicated by the dashed line. Bottom left: $x_1x_2$-cross section image. Bottom middle: $x_2x_3$-cross section image. Bottom right: $x_1x_3$-cross section image
Image of a cylindrical scatterer. Top left: exact. Top middle: $k = 3$; three dimensional image using iso-surface plot with iso-value $0.4$. Top right: $k = 5$; three dimensional image using iso-surface plot with iso-value $0.3$. In the middle ($k = 3$) and bottom ($k = 5$) row, we plot the cross section images where the exact geometry is indicated by the dashed line. Middle/bottom row left: $x_1x_2$-cross section image. Middle/bottom row middle: $x_2x_3$-cross section image. Middle/bottom row right: $x_1x_3$-cross section image
Top: image of two balls. Bottom: image of three balls. Left: exact. Middle: three dimensional image using iso-surface plot (with iso-value 0.2). Right: $x_1x_2$-cross section image
Image of a large ball. Left: exact. Middle: three dimensional image when $k = 1$. Right: three dimensional image when $k = 3$
Iso-surface (with the same iso-value 0.4) image of the L-shape scatterer. Left: exact. Middle: $10\%$ noise. Right: $30\%$ noise
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