# American Institute of Mathematical Sciences

August  2020, 14(4): 665-682. doi: 10.3934/ipi.2020030

## Thermoacoustic Tomography with circular integrating detectors and variable wave speed

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

Received  July 2019 Revised  February 2020 Published  May 2020

Fund Project: Author partly supported by NSF Grant DMS-1600327

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.

Citation: Chase Mathison. Thermoacoustic Tomography with circular integrating detectors and variable wave speed. Inverse Problems and Imaging, 2020, 14 (4) : 665-682. doi: 10.3934/ipi.2020030
##### References:
 [1] M. J. Grote and I. Sim, Efficient PML for the wave equation, preprint, arXiv: 1001.0319. [2] M. Haltmeier, O. Scherzer, P. Burgholzer and G. Paltauf, Thermoacoustic computed tomography with large planar receivers, Inverse Problems, 20 (2004), 1663-1673.  doi: 10.1088/0266-5611/20/5/021. [3] L. Hörmander, Fourier integral operators. Ⅰ., Acta Math., 127 (1971), 79-183.  doi: 10.1007/BF02392052. [4] L. Hörmander, The Analysis of Linear Partial Differential Operators. IV. Fourier Integral Operators, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-642-00136-9. [5] F. John, Partial Differential Equations, Applied Mathematical Sciences, 1, 1$^st$ edition, Springer-Verlag, New York-Berlin, 1971. [6] R. A. Kruger, W. L. Kiser Jr., D. R. Reinecke and G. A. Kruger, Thermoacoustic computed tomography using a conventional linear transducer array, Med. Phys., 30 (2003), 856-860.  doi: 10.1118/1.1565340. [7] P. Kuchment and L. Kunyansky, Mathematics of thermoacoustic tomography, European J. Appl. Math., 19 (2008), 191-224.  doi: 10.1137/1.9781611973297. [8] A. A. Oraevsky, S. L. Jacques, R. O. Esenaliev and F. K. Tittel, Laser-based optoacoustic imaging in biological tissues, in Laser-Tissue Interaction V; and Ultraviolet Radiation Hazards, 2134, Proc. SPIE, 1994,122–129. doi: 10.1117/12.182927. [9] E. Quinto, Radon transforms on curves in the plane, Lect. Appl. Math., 30 (1994), 231-244. [10] P. Stefanov and G. Uhlmann, Thermoacoustic tomography with variable sound speed, Inverse Problems, 25 (2009), 075011, 16 pp. doi: 10.1088/0266-5611/25/7/075011. [11] P. Stefanov and Y. Yang, Thermo- and photoacoustic tomography with variable speed and planar detectors, SIAM J. Math. Anal., 49 (2017), 297-310.  doi: 10.1137/16M1073716. [12] M. E. Taylor, Pseudodifferential Operators, in Princeton Mathematical Series, 34, Princeton University Press, Princeton, NJ, 1981,146–191. [13] M. E. Taylor, Partial differential equations. I. Basic theory, in Applied Mathematical Sciences, 115, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4419-7055-8. [14] F. Trèves, Introduction to pseudodifferential and Fourier integral operators, in The University Series in Mathematics, 2, Plenum Press, New York-London, 1980. [15] M. Xu and L. V. Wang, Photoacoustic imaging in biomedicine, Review of Scientific Instruments, 77 (2006), 041101. doi: 10.1063/1.2195024. [16] G. Zangerl, O. Scherzer and M. Haltmeier, Circular integrating detectors in photo and thermoacoustic tomography, Inverse Probl. Sci. Eng., 17 (2009), 133-142.  doi: 10.1080/17415970802166782. [17] G. Zangerl, O. Scherzer and M. Haltmeier, Exact series reconstruction in photoacoustic tomography with circular integrating detectors, Commun. Math. Sci., 7 (2009), 665-678.  doi: 10.4310/CMS.2009.v7.n3.a8.

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##### References:
 [1] M. J. Grote and I. Sim, Efficient PML for the wave equation, preprint, arXiv: 1001.0319. [2] M. Haltmeier, O. Scherzer, P. Burgholzer and G. Paltauf, Thermoacoustic computed tomography with large planar receivers, Inverse Problems, 20 (2004), 1663-1673.  doi: 10.1088/0266-5611/20/5/021. [3] L. Hörmander, Fourier integral operators. Ⅰ., Acta Math., 127 (1971), 79-183.  doi: 10.1007/BF02392052. [4] L. Hörmander, The Analysis of Linear Partial Differential Operators. IV. Fourier Integral Operators, Springer-Verlag, Berlin, 2009. doi: 10.1007/978-3-642-00136-9. [5] F. John, Partial Differential Equations, Applied Mathematical Sciences, 1, 1$^st$ edition, Springer-Verlag, New York-Berlin, 1971. [6] R. A. Kruger, W. L. Kiser Jr., D. R. Reinecke and G. A. Kruger, Thermoacoustic computed tomography using a conventional linear transducer array, Med. Phys., 30 (2003), 856-860.  doi: 10.1118/1.1565340. [7] P. Kuchment and L. Kunyansky, Mathematics of thermoacoustic tomography, European J. Appl. Math., 19 (2008), 191-224.  doi: 10.1137/1.9781611973297. [8] A. A. Oraevsky, S. L. Jacques, R. O. Esenaliev and F. K. Tittel, Laser-based optoacoustic imaging in biological tissues, in Laser-Tissue Interaction V; and Ultraviolet Radiation Hazards, 2134, Proc. SPIE, 1994,122–129. doi: 10.1117/12.182927. [9] E. Quinto, Radon transforms on curves in the plane, Lect. Appl. Math., 30 (1994), 231-244. [10] P. Stefanov and G. Uhlmann, Thermoacoustic tomography with variable sound speed, Inverse Problems, 25 (2009), 075011, 16 pp. doi: 10.1088/0266-5611/25/7/075011. [11] P. Stefanov and Y. Yang, Thermo- and photoacoustic tomography with variable speed and planar detectors, SIAM J. Math. Anal., 49 (2017), 297-310.  doi: 10.1137/16M1073716. [12] M. E. Taylor, Pseudodifferential Operators, in Princeton Mathematical Series, 34, Princeton University Press, Princeton, NJ, 1981,146–191. [13] M. E. Taylor, Partial differential equations. I. Basic theory, in Applied Mathematical Sciences, 115, Springer-Verlag, New York, 1996. doi: 10.1007/978-1-4419-7055-8. [14] F. Trèves, Introduction to pseudodifferential and Fourier integral operators, in The University Series in Mathematics, 2, Plenum Press, New York-London, 1980. [15] M. Xu and L. V. Wang, Photoacoustic imaging in biomedicine, Review of Scientific Instruments, 77 (2006), 041101. doi: 10.1063/1.2195024. [16] G. Zangerl, O. Scherzer and M. Haltmeier, Circular integrating detectors in photo and thermoacoustic tomography, Inverse Probl. Sci. Eng., 17 (2009), 133-142.  doi: 10.1080/17415970802166782. [17] G. Zangerl, O. Scherzer and M. Haltmeier, Exact series reconstruction in photoacoustic tomography with circular integrating detectors, Commun. Math. Sci., 7 (2009), 665-678.  doi: 10.4310/CMS.2009.v7.n3.a8.
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
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
Variable wave speed of $1+0.3\sin(8x)\cos(5y)\eta(x,y)$, where $\eta(x,y)\in C_0^\infty(B_1(0))$
Results of reconstruction using $R = 1$ and $r = 2$ model (Large radius detector model). This reconstruction was made using full data
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
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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