Thermoacoustic Tomography with circular integrating detectors and variable wave speed

Chase Mathison

doi:10.3934/ipi.2020030

Article Contents

2020, Volume 14, Issue 4: 665-682. Doi: 10.3934/ipi.2020030

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Thermoacoustic Tomography with circular integrating detectors and variable wave speed

Chase Mathison

Department of Mathematics, Purdue University, 150 N University St., West Lafayette, IN 47907, USA

Received: June 30, 2019

Revised: January 31, 2020

Published: May 2020

Author partly supported by NSF Grant DMS-1600327

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Abstract

We explore Thermoacoustic Tomography with circular integrating detectors assuming variable, smooth wave speed. We show that the measurement operator in this case is a Fourier Integral Operator and examine how the singularities in initial data and measured data are related through the canonical relation of this operator. We prove which of those singularities in the initial data are visible from a fixed open subset of the set on which measurements are taken. In addition, numerical results are shown for both full and partial data.

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Mathematics Subject Classification: Primary: 35R30, 35S30; Secondary: 92C55.

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Figure 1. Two different experimental setups shown depending on the radius of the integrating detector. On the left is the small radius case, and on the right is the large radius case

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Figure 2. Singularities that may be visible from $ \theta_0 \in \Gamma $ in both the cases (left) $ R-r > 1 $ and (right) $ R = 1, r>2 $ will lie on the geodesics issued from the integrating detectors

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Figure 3. Variable wave speed of $ 1+0.3\sin(8x)\cos(5y)\eta(x,y) $, where $ \eta(x,y)\in C_0^\infty(B_1(0)) $

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Figure 4. Results of reconstruction using $ R = 1 $ and $ r = 2 $ model (Large radius detector model). This reconstruction was made using full data

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Figure 5. Result of reconstruction with partial data using $ R = 2 $, and $ r = 0.8 $ (Small radius detector model). This reconstruction was for $ \theta \in (-\pi/2, 0) $. Shown in the figure are the set on which data is collected as well as some representative circular integrating detectors

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Figure 6. Result of reconstruction with partial data using $ R = 1 $, and $ r = 2 $ (Large radius detector model). This reconstruction was for $ \theta \in (-\pi/2,0) $. Shown in the figure are the set on which data is collected as well as some representative circular integrating detectors

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