# American Institute of Mathematical Sciences

doi: 10.3934/ipi.2021041

## A mathematical approach towards THz tomography for non-destructive imaging

 1 Johann Radon Institute Linz, Altenbergerstraße 69, A-4040 Linz, Austria 2 Doctoral Program Computational Mathematics, Johannes Kepler University Linz, Altenbergerstraße 69, A-4040 Linz, Austria 3 Industrial Mathematics Institute, Johannes Kepler University Linz, Altenbergerstraße 69, A-4040 Linz, Austria 4 Research Center for Non-Destructive Testing GmbH (RECENDT), Altenbergerstraße 69, A-4040 Linz, Austria

* Corresponding author: Simon Hubmer

Received  October 2020 Revised  April 2021 Published  May 2021

In this paper, we consider the imaging problem of terahertz (THz) tomography, in particular as it appears in non-destructive testing. We derive a nonlinear mathematical model describing a full THz tomography experiment, and consider linear approximations connecting THz tomography with standard computerized tomography and the Radon transform. Based on the derived models we propose different reconstruction approaches for solving the THz tomography problem, which we then compare on experimental data obtained from THz measurements of a plastic sample.

Citation: Simon Hubmer, Alexander Ploier, Ronny Ramlau, Peter Fosodeder, Sandrine van Frank. A mathematical approach towards THz tomography for non-destructive imaging. Inverse Problems & Imaging, doi: 10.3934/ipi.2021041
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Schematic drawing and image of the measurement setup. The THz radiation is generated by a transmitting antenna (Tx). Two Off-Axis Parabolic Mirrors (OPM) are used to create a focussed THz beam. After interacting with an object, the THz beam is guided to the detecting antenna (Rx) by two OPMs again. Exemplarily, a measured reference signal through air and one measured signal through an object is shown
THz beam profile in the focal plane along the parallel scanning direction. The energy density distribution was measured by shifting a rectangular aperture through the focal spot in z-direction
Triangular plastic sample (left), the measured electric field $E_ {i,j}$ corresponding to $( {s_i,\theta_j}) = (35,0)$ (right, blue), and the reference field $E_{\rm{ref}}$ (right, orange)
An example for the presence of multiple peaks in a THz signal. (left) The THz beam partially travels through air and the object. (right) This gives rise to two dominant peaks in the THz signal, one arising from the pulse that travelled through air and a second one that travelled through the object respectively. For our reconstructions only the largest peak (main peak) is used
Pre-processed data $-2\log(\left\vert{P_ {i,j}/ P_{\rm{ref}}}\right\vert)$ (left) and $-\log(I_ {i,j}/ I_{\rm{ref}})$ (right)
Simulated measurement data $\left\vert{P_ {i,j}/ P_{\rm{ref}}}\right\vert$ for the triangular plastic sample depicted in Figure 3 (left), and the resulting reconstruction obtained via the nonlinear Landweber approach introduced in Section 4.1 (right)
Pre-processed data $\left\vert{P_ {i,j}/ P_{\rm{ref}}}\right\vert$ obtained from THz measurements of the triangular plastic sample depicted in Figure 3 (left), and the resulting reconstruction obtained via the nonlinear Landweber approach introduced in Section 4.1 (right)
Reconstructions for Problem 2 obtained from the data $-2\log(\left\vert{P_ {i,j}/ P_{\rm{ref}}}\right\vert)$ depicted in Figure 5 (right) via the application of the following reconstruction methods introduced in Section 4.2: filtered back-projection (top left), contour tomography (top right), Landweber iteration (bottom left), Tikhonov regularization (bottom right)
Reconstructions for Problem 3 obtained from the data $-\log(I_ {i,j}/ I_{\rm{ref}})$ depicted in Figure 5 (left) via the application of the following reconstruction methods introduced in Section 4.2: filtered back-projection (top left), contour tomography (top right), Landweber iteration (bottom left), Tikhonov regularization (bottom right)
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