Optimal recovery of a radiating source with multiple frequencies along one line

Tommi Brander; Joonas Ilmavirta; Petteri Piiroinen; Teemu Tyni

doi:10.3934/ipi.2020044

Article Contents

2020, Volume 14, Issue 6: 967-983. Doi: 10.3934/ipi.2020044

This issue Previous Article Next Article Two-dimensional seismic data reconstruction using patch tensor completion

Optimal recovery of a radiating source with multiple frequencies along one line

1.
Department of Mathematical Sciences, Faculty of Information Technology and Electrical Engineering
2.
NTNU – Norwegian University of Science and Technology, Trondheim, Norway
3.
Technical University of Denmark, Department of Applied Mathematics and Computer Science, Lyngby, Denmark
4.
University of Jyväskylä, Department of Mathematics and Statistics, P.O. Box 35 (MaD), FI-40014 University of Jyväskylä, Finland
5.
Department of Mathematics and Statistics, P.O 68 (Pietari Kalmin katu 5), 00014 University of Helsinki, Finland
6.
Department of Mathematical Sciences, P.O. Box 3000, FI-90014 University of Oulu, Finland

^* Corresponding author: Tommi Brander
^* Corresponding author: Tommi Brander

Received: May 2019

Revised: May 2020

Early access: August 2020

Published: December 2020

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Abstract

We study an inverse problem where an unknown radiating source is observed with collimated detectors along a single line and the medium has a known attenuation. The research is motivated by applications in SPECT and beam hardening. If measurements are carried out with frequencies ranging in an open set, we show that the source density is uniquely determined by these measurements up to averaging over levelsets of the integrated attenuation. This leads to a generalized Laplace transform. We also discuss some numerical approaches and demonstrate the results with several examples.

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Mathematics Subject Classification: Primary: 44A10, 65R32; Secondary: 44A60, 46N40, 65Z05.

Citation:

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Figure 1. Unknown $ \rho_0 $ (dashed red line) and the numerical solution $ \rho $ (solid blue line) with 0.5% noise level. Example 1 (above) and example 2 (below) with Tikhonov-solution (left), TV-solution (middle) and CGLS-solution (right)

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Figure 2. Unknown $ \rho_0 $ (dashed red line), $ \rho_0 $ averaged over regions where $ p $ is constant (black dot-dash line) and the numerical solution $ \rho $ (solid blue line) with 0.5% noise level. Tikhonov-solution (left), TV-solution (middle) and CGLS-solution (right)

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Figure 3. Unknown $ \rho_0 $ (dashed red line) and the Tikhonov-solution $ \rho $ (solid blue line) with 0.5% noise level and smaller measurement intervals

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Table 1. Averaged relative errors and variances of one hundred solutions on smaller intervals with noise level 0.5%

Intervals	$ (0,1) $	$ (0.2,0.8) $	$ (0.3,0.6) $	$ (0.4,0.5) $
$ \overline{\epsilon}_\mathrm{rel} $	0.117	0.170	0.186	0.320
$ \mathrm{var} $	$ 1.88\cdot 10^{-3} $	$ 1.27\cdot 10^{-2} $	$ 1.27\cdot 10^{-2} $	$ 6.65\cdot 10^{-3} $

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