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doi: 10.3934/ipi.2022013
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## Photoacoustic tomography in attenuating media with partial data

 Department of Mathematics, Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna 4860, Macul, Santiago, Chile

Received  October 2021 Revised  February 2022 Early access March 2022

The attenuation of ultrasound waves in photoacoustic and thermoacoustic imaging presents an important drawback in the applicability of these modalities. This issue has been addressed previously in the applied and theoretical literature, and some advances have been made on the topic. In particular, stability inequalities have been proposed for the inverse problem of initial source recovery with partial observations under the assumption of unique determination of the initial pressure. The main goal of this work is to fill this gap, this is, we prove the uniqueness property for the inverse problem and establish the associated stability estimates as well. The problem of reconstructing the initial condition of acoustic waves in the complete-data setting is revisited and a new Neumann series reconstruction formula is obtained for the case of partial observations in a semi-bounded geometry. A numerical simulation is also included to test the method.

Citation: Benjamin Palacios. Photoacoustic tomography in attenuating media with partial data. Inverse Problems and Imaging, doi: 10.3934/ipi.2022013
##### References:
 [1] S. Acosta and C. Montalto, Multiwave imaging in an enclosure with variable wave speed, Inverse Problems, 31 (2015), 065009, 12 pp. doi: 10.1088/0266-5611/31/6/065009. [2] S. Acosta and B. Palacios, Thermoacoustic tomography for an integro-differential wave equation modeling attenuation, J. Differential Equations, 264 (2018), 1984-2010.  doi: 10.1016/j.jde.2017.10.012. [3] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065.  doi: 10.1137/0330055. [4] O. Chervova and L. Oksanen, Time reversal method with stabilizing boundary conditions for photoacoustic tomography, Inverse Problems, 32 (2016), 125004, 16 pp. doi: 10.1088/0266-5611/32/12/125004. [5] L. C. Evans, Partial Differential Equations, Vol 19, American Mathematical Soc., 1998. [6] D. Finch, S. K. Patch and Ra kesh, Determining a function from its mean values over a family of spheres, SIAM J. Math. Anal., 35 (2004), 1213-1240.  doi: 10.1137/S0036141002417814. [7] X. Fu, Stabilization of hyperbolic equations with mixed boundary conditions, Math. Control Relat. Fields, 5 (2015), 761-780.  doi: 10.3934/mcrf.2015.5.761. [8] M. Haltmeier and L. V. Nguyen, Reconstruction algorithms for photoacoustic tomography in heterogeneous damping media, J. Math. Imaging Vision, 61 (2019), 1007-1021.  doi: 10.1007/s10851-019-00879-y. [9] A. Homan, Multi-wave imaging in attenuating media, Inverse Probl. Imaging, 7 (2013), 1235-1250.  doi: 10.3934/ipi.2013.7.1235. [10] L. Hörmander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, Moscow, 1986. [11] L. Hörmander, The Analysis of Linear Partial Differential Operators Ⅳ: Fourier Integral Operators, Springer, 2009. [12] M. Ikawa, On the mixed problem for hyperbolic equations of second order with the Neumann boundary condition, Osaka Math. J., 7 (1970), 203-223. [13] Q. H. Liu and J. Tao, The perfectly matched layer for acoustic waves in absorptive media, J. Acoust. Soc. Am., 102 (1997), 2072-2082.  doi: 10.1121/1.419657. [14] L. V. Nguyen and L. A. Kunyansky, A dissipative time reversal technique for photoacoustic tomography in a cavity, SIAM J. Imaging Sci., 9 (2016), 748-769.  doi: 10.1137/15M1049683. [15] B. Palacios, Reconstruction for multi-wave imaging in attenuating media with large damping coefficient, Inverse Problems, 32 (2016), 125008, 15 pp. doi: 10.1088/0266-5611/32/12/125008. [16] J. Qian, P. Stefanov, G. Uhlmann and H. Zhao, An efficient Neumann series-based algorithm for thermoacoustic and photoacoustic tomography with variable sound speed, SIAM J. Imaging Sci., 4 (2011), 850-883.  doi: 10.1137/100817280. [17] P. Stefanov, Conditionally stable unique continuation and applications to thermoacoustic tomography, Math. Eng., 1 (2019), 789-799.  doi: 10.3934/mine.2019.4.789. [18] 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. [19] P. Stefanov and G. Uhlmann, Thermoacoustic tomography arising in brain imaging, Inverse Problems, 27 (2011), 045004 26 pp. doi: 10.1088/0266-5611/27/4/045004. [20] P. Stefanov and G. Uhlmann, Multi-wave methods via ultrasound, In Inverse Problems and Applications, Inside Out II, MSRI Publications, 60 (2013), 271–323. [21] P. Stefanov and G. Uhlmann, Recovery of a source term or a speed with one measurement and applications, Trans. Amer. Math. Soc., 365 (2013), 5737-5758.  doi: 10.1090/S0002-9947-2013-05703-0. [22] P. Stefanov and Y. Yang, Multiwave tomography in a closed domain: Averaged sharp time reversal, Inverse Problems, 31 (2015), 065007, 23 pp. doi: 10.1088/0266-5611/31/6/065007. [23] D. Tataru, On the regularity of boundary traces for the wave equation, Ann. Sc. Norm. Super. Pisa Cl. Sci., 26 (1998), 185-206. [24] D. Tataru, Unique continuation for operators with partially analytic coefficients, J. Math. Pures Appl., 78 (1999), 505-521.  doi: 10.1016/S0021-7824(99)00016-1.

show all references

##### References:
 [1] S. Acosta and C. Montalto, Multiwave imaging in an enclosure with variable wave speed, Inverse Problems, 31 (2015), 065009, 12 pp. doi: 10.1088/0266-5611/31/6/065009. [2] S. Acosta and B. Palacios, Thermoacoustic tomography for an integro-differential wave equation modeling attenuation, J. Differential Equations, 264 (2018), 1984-2010.  doi: 10.1016/j.jde.2017.10.012. [3] C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065.  doi: 10.1137/0330055. [4] O. Chervova and L. Oksanen, Time reversal method with stabilizing boundary conditions for photoacoustic tomography, Inverse Problems, 32 (2016), 125004, 16 pp. doi: 10.1088/0266-5611/32/12/125004. [5] L. C. Evans, Partial Differential Equations, Vol 19, American Mathematical Soc., 1998. [6] D. Finch, S. K. Patch and Ra kesh, Determining a function from its mean values over a family of spheres, SIAM J. Math. Anal., 35 (2004), 1213-1240.  doi: 10.1137/S0036141002417814. [7] X. Fu, Stabilization of hyperbolic equations with mixed boundary conditions, Math. Control Relat. Fields, 5 (2015), 761-780.  doi: 10.3934/mcrf.2015.5.761. [8] M. Haltmeier and L. V. Nguyen, Reconstruction algorithms for photoacoustic tomography in heterogeneous damping media, J. Math. Imaging Vision, 61 (2019), 1007-1021.  doi: 10.1007/s10851-019-00879-y. [9] A. Homan, Multi-wave imaging in attenuating media, Inverse Probl. Imaging, 7 (2013), 1235-1250.  doi: 10.3934/ipi.2013.7.1235. [10] L. Hörmander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis, Moscow, 1986. [11] L. Hörmander, The Analysis of Linear Partial Differential Operators Ⅳ: Fourier Integral Operators, Springer, 2009. [12] M. Ikawa, On the mixed problem for hyperbolic equations of second order with the Neumann boundary condition, Osaka Math. J., 7 (1970), 203-223. [13] Q. H. Liu and J. Tao, The perfectly matched layer for acoustic waves in absorptive media, J. Acoust. Soc. Am., 102 (1997), 2072-2082.  doi: 10.1121/1.419657. [14] L. V. Nguyen and L. A. Kunyansky, A dissipative time reversal technique for photoacoustic tomography in a cavity, SIAM J. Imaging Sci., 9 (2016), 748-769.  doi: 10.1137/15M1049683. [15] B. Palacios, Reconstruction for multi-wave imaging in attenuating media with large damping coefficient, Inverse Problems, 32 (2016), 125008, 15 pp. doi: 10.1088/0266-5611/32/12/125008. [16] J. Qian, P. Stefanov, G. Uhlmann and H. Zhao, An efficient Neumann series-based algorithm for thermoacoustic and photoacoustic tomography with variable sound speed, SIAM J. Imaging Sci., 4 (2011), 850-883.  doi: 10.1137/100817280. [17] P. Stefanov, Conditionally stable unique continuation and applications to thermoacoustic tomography, Math. Eng., 1 (2019), 789-799.  doi: 10.3934/mine.2019.4.789. [18] 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. [19] P. Stefanov and G. Uhlmann, Thermoacoustic tomography arising in brain imaging, Inverse Problems, 27 (2011), 045004 26 pp. doi: 10.1088/0266-5611/27/4/045004. [20] P. Stefanov and G. Uhlmann, Multi-wave methods via ultrasound, In Inverse Problems and Applications, Inside Out II, MSRI Publications, 60 (2013), 271–323. [21] P. Stefanov and G. Uhlmann, Recovery of a source term or a speed with one measurement and applications, Trans. Amer. Math. Soc., 365 (2013), 5737-5758.  doi: 10.1090/S0002-9947-2013-05703-0. [22] P. Stefanov and Y. Yang, Multiwave tomography in a closed domain: Averaged sharp time reversal, Inverse Problems, 31 (2015), 065007, 23 pp. doi: 10.1088/0266-5611/31/6/065007. [23] D. Tataru, On the regularity of boundary traces for the wave equation, Ann. Sc. Norm. Super. Pisa Cl. Sci., 26 (1998), 185-206. [24] D. Tataru, Unique continuation for operators with partially analytic coefficients, J. Math. Pures Appl., 78 (1999), 505-521.  doi: 10.1016/S0021-7824(99)00016-1.
Sound speed as in (21)
Initial condition $f$ for simulation 1 (left) and 2 (right). The gray line surrounding the domain represents the support of $\lambda$, hence, the observation region
Reconstruction of initial condition for the first simulation: back-projection (top left) and 20 terms of Neumann series (top right), both with pixel values on the interval $[-0.2, 1]$. Bottom: cross section at $y = 0$. The gray line, the black solid line and the gray dash-dotted line correspond, respectively, to the true initial source, the Neumann series approximation and the back-projection approximations
Reconstruction of initial condition for the second simulation: back-projection (left) and 60 terms of the Neumann series (right), both with pixel values on the interval $[-0.2, 1]$. Middle and bottom: cross section at $y = 0.4$ and $y = -0.4$ respectively. The gray line, the black solid line and the gray dash-dotted line correspond, respectively, to the true initial source, the Neumann series approximation and the back-projection approximations
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