American Institute of Mathematical Sciences

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November  2013, 7(4): 1235-1250. doi: 10.3934/ipi.2013.7.1235

Multi-wave imaging in attenuating media

Andrew Homan 1,

1. 

Department of Mathematics, Purdue University, West Lafayette, IN 47906, United States

Received  January 2013 Revised  August 2013 Published  November 2013

We consider a mathematical model of thermoacoustic tomography and other multi-wave imaging techniques with variable sound speed and attenuation. We find that a Neumann series reconstruction algorithm, previously studied under the assumption of zero attenuation, still converges if attenuation is sufficiently small. With complete boundary data, we show the inverse problem has a unique solution, and modified time reversal provides a stable reconstruction. We also consider partial boundary data, and in this case study those singularities that can be stably recovered.
Keywords: damped wave equation, time reversal., Inverse problems, microlocal analysis, thermo-acoustics.
Mathematics Subject Classification: Primary: 35R30; Secondary: 35A27, 92C5.
Citation: Andrew Homan. Multi-wave imaging in attenuating media. Inverse Problems & Imaging, 2013, 7 (4) : 1235-1250. doi: 10.3934/ipi.2013.7.1235
References:
[1]

H. Ammari, E. Bretin, J. Garnier and A. Wahab, Time reversal in attenuating acoustic media,, Contemp. Math., 548 (2011), 151. doi: 10.1090/conm/548/10841. Google Scholar

[2]

G. Bal, K. Ren, G. Uhlmann and T. Zhou, Quantitative thermo-acoustics and related problems,, Inverse Problems, 27 (2011). doi: 10.1088/0266-5611/27/5/055007. Google Scholar

[3]

P. Burgholzer, F. Camacho-Gonzales, D. Sponseiler, G. Mayer and G. Hendorfer, Information changes and time reversal for diffusion-related periodic fields,, Proc. SPIE, 7177 (2009). doi: 10.1117/12.809074. Google Scholar

[4]

B. T. Cox, J. G. Laufer and P. C. Beard, The challenges for photoacoustic imaging,, Proc. SPIE, 7177 (2009). doi: 10.1117/12.806788. Google Scholar

[5]

X. L. Deán-Ben, D. Razansky and V. Ntziachristos, The effects of attenuation in optoacoustic signals,, Phys. Med. Biol., 56 (2011), 6129. Google Scholar

[6]

D. Finch, S. K. Patch and Rakesh, Determining a function from its mean values over a family of spheres,, SIAM J. Math. Anal., 35 (2004), 1213. doi: 10.1137/S0036141002417814. Google Scholar

[7]

Y. Hristova, Time reversal in thermoacoustic tomography - an error estimate,, Inverse Problems, 25 (2009). doi: 10.1088/0266-5611/25/5/055008. Google Scholar

[8]

Y. Hristova, P. Kuchment and L. Nyugen, On reconstruction and time reversal in thermoacoustic tomography in acoustically homogeneous and inhomogeneous media,, Inverse Problems, 24 (2008). doi: 10.1088/0266-5611/24/5/055006. Google Scholar

[9]

X. Jin, C. Li and L. Wang, Effects of acoustic heterogeneities on transcranial brain imaging with microwave-induced thermoacoustic tomography,, Med. Phys., 35 (2008), 3205. doi: 10.1118/1.2938731. Google Scholar

[10]

K. Kalimeris and O. Scherzer, Photoacoustic imaging in attenuating acoustic media based on strongly causal models,, Math. Meth. Appl. Sci., (). doi: 10.1002/mma.2756. Google Scholar

[11]

R. Kowar, Integral equation models for thermoacoustic imaging of acoustic dissipative tissue,, Inverse Problems, 26 (2010). doi: 10.1088/0266-5611/26/9/095005. Google Scholar

[12]

R. Kowar, O. Scherzer and X. Bonnefond, Causality analysis of frequency-dependent wave attenuation,, Math. Meth. in Appl. Sci., 34 (2011), 108. doi: 10.1002/mma.1344. Google Scholar

[13]

P. Kuchment and L. Kunyansky, Mathematics of thermoacoustic tomography,, Euro. J. Appl. Math., 19 (2008), 191. doi: 10.1017/S0956792508007353. Google Scholar

[14]

I. Lasiecka, J. L. Lions and R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators,, J. Math. Pures Appl., 65 (1986), 149. Google Scholar

[15]

D. Modgil, M. Anastasio and P. J. La Rivière, Photoacoustic image reconstruction in an attenuating medium using singular value decomposition,, Proc. SPIE, 7177 (2009). Google Scholar

[16]

S. K. Patch and M. Haltmeier, Thermoacoustic tomography - ultrasound attenuation effects,, IEEE Nucl. Sci. Symp. Conf. Rec., 4 (2006), 2604. Google Scholar

[17]

J. Qian, P. Stefanov, G. Uhlmann and H. Zhao, An effecient Neumann-series based algorithm for thermoacoustic and photoacoustic tomography with variable sound speed,, SIAM J. Imaging Sciences, 4 (2011), 850. doi: 10.1137/100817280. Google Scholar

[18]

M. Reed and B. Simon, Methods of Modern Mathematical Physics,, volume 2, (1975). Google Scholar

[19]

P. J. La Rivière, J. Zhang and M. Anastasio, Image reconstruction in optoacoustic tomography for dispersive acoustic media,, Optics Letters, 31 (2006), 781. Google Scholar

[20]

H. Roitner and P. Burgholzer, Effecient modeling and compensation of ultrasound attenuation losses in photoacoustic imaging,, Inverse Problems, 27 (2011). doi: 10.1088/0266-5611/27/1/015003. Google Scholar

[21]

P. Stefanov and G. Uhlmann, Thermoacoustic tomography with variable sound speed,, Inverse Problems, 25 (2009). doi: 10.1088/0266-5611/25/7/075011. Google Scholar

[22]

D. Tataru, Unqiue continuation for operators with partially analytic coefficients,, J. Math. Pures Appl., 78 (1999), 505. doi: 10.1016/S0021-7824(99)00016-1. Google Scholar

[23]

M. Taylor, Pseudodifferential Operators,, Princeton University Press, (1981). Google Scholar

[24]

J. Tittlefitz, Thermoacoustic tomography in elastic media,, Inverse Problems, 28 (2012). doi: 10.1088/0266-5611/28/5/055004. Google Scholar

[25]

B. Treeby, E. Zhang and B. T. Cox, Photoacoustic tomography in absorbing acoustic media using time reversal,, Inverse Problems, 26 (2010). doi: 10.1088/0266-5611/26/11/115003. Google Scholar

show all references

References:
[1]

H. Ammari, E. Bretin, J. Garnier and A. Wahab, Time reversal in attenuating acoustic media,, Contemp. Math., 548 (2011), 151. doi: 10.1090/conm/548/10841. Google Scholar

[2]

G. Bal, K. Ren, G. Uhlmann and T. Zhou, Quantitative thermo-acoustics and related problems,, Inverse Problems, 27 (2011). doi: 10.1088/0266-5611/27/5/055007. Google Scholar

[3]

P. Burgholzer, F. Camacho-Gonzales, D. Sponseiler, G. Mayer and G. Hendorfer, Information changes and time reversal for diffusion-related periodic fields,, Proc. SPIE, 7177 (2009). doi: 10.1117/12.809074. Google Scholar

[4]

B. T. Cox, J. G. Laufer and P. C. Beard, The challenges for photoacoustic imaging,, Proc. SPIE, 7177 (2009). doi: 10.1117/12.806788. Google Scholar

[5]

X. L. Deán-Ben, D. Razansky and V. Ntziachristos, The effects of attenuation in optoacoustic signals,, Phys. Med. Biol., 56 (2011), 6129. Google Scholar

[6]

D. Finch, S. K. Patch and Rakesh, Determining a function from its mean values over a family of spheres,, SIAM J. Math. Anal., 35 (2004), 1213. doi: 10.1137/S0036141002417814. Google Scholar

[7]

Y. Hristova, Time reversal in thermoacoustic tomography - an error estimate,, Inverse Problems, 25 (2009). doi: 10.1088/0266-5611/25/5/055008. Google Scholar

[8]

Y. Hristova, P. Kuchment and L. Nyugen, On reconstruction and time reversal in thermoacoustic tomography in acoustically homogeneous and inhomogeneous media,, Inverse Problems, 24 (2008). doi: 10.1088/0266-5611/24/5/055006. Google Scholar

[9]

X. Jin, C. Li and L. Wang, Effects of acoustic heterogeneities on transcranial brain imaging with microwave-induced thermoacoustic tomography,, Med. Phys., 35 (2008), 3205. doi: 10.1118/1.2938731. Google Scholar

[10]

K. Kalimeris and O. Scherzer, Photoacoustic imaging in attenuating acoustic media based on strongly causal models,, Math. Meth. Appl. Sci., (). doi: 10.1002/mma.2756. Google Scholar

[11]

R. Kowar, Integral equation models for thermoacoustic imaging of acoustic dissipative tissue,, Inverse Problems, 26 (2010). doi: 10.1088/0266-5611/26/9/095005. Google Scholar

[12]

R. Kowar, O. Scherzer and X. Bonnefond, Causality analysis of frequency-dependent wave attenuation,, Math. Meth. in Appl. Sci., 34 (2011), 108. doi: 10.1002/mma.1344. Google Scholar

[13]

P. Kuchment and L. Kunyansky, Mathematics of thermoacoustic tomography,, Euro. J. Appl. Math., 19 (2008), 191. doi: 10.1017/S0956792508007353. Google Scholar

[14]

I. Lasiecka, J. L. Lions and R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators,, J. Math. Pures Appl., 65 (1986), 149. Google Scholar

[15]

D. Modgil, M. Anastasio and P. J. La Rivière, Photoacoustic image reconstruction in an attenuating medium using singular value decomposition,, Proc. SPIE, 7177 (2009). Google Scholar

[16]

S. K. Patch and M. Haltmeier, Thermoacoustic tomography - ultrasound attenuation effects,, IEEE Nucl. Sci. Symp. Conf. Rec., 4 (2006), 2604. Google Scholar

[17]

J. Qian, P. Stefanov, G. Uhlmann and H. Zhao, An effecient Neumann-series based algorithm for thermoacoustic and photoacoustic tomography with variable sound speed,, SIAM J. Imaging Sciences, 4 (2011), 850. doi: 10.1137/100817280. Google Scholar

[18]

M. Reed and B. Simon, Methods of Modern Mathematical Physics,, volume 2, (1975). Google Scholar

[19]

P. J. La Rivière, J. Zhang and M. Anastasio, Image reconstruction in optoacoustic tomography for dispersive acoustic media,, Optics Letters, 31 (2006), 781. Google Scholar

[20]

H. Roitner and P. Burgholzer, Effecient modeling and compensation of ultrasound attenuation losses in photoacoustic imaging,, Inverse Problems, 27 (2011). doi: 10.1088/0266-5611/27/1/015003. Google Scholar

[21]

P. Stefanov and G. Uhlmann, Thermoacoustic tomography with variable sound speed,, Inverse Problems, 25 (2009). doi: 10.1088/0266-5611/25/7/075011. Google Scholar

[22]

D. Tataru, Unqiue continuation for operators with partially analytic coefficients,, J. Math. Pures Appl., 78 (1999), 505. doi: 10.1016/S0021-7824(99)00016-1. Google Scholar

[23]

M. Taylor, Pseudodifferential Operators,, Princeton University Press, (1981). Google Scholar

[24]

J. Tittlefitz, Thermoacoustic tomography in elastic media,, Inverse Problems, 28 (2012). doi: 10.1088/0266-5611/28/5/055004. Google Scholar

[25]

B. Treeby, E. Zhang and B. T. Cox, Photoacoustic tomography in absorbing acoustic media using time reversal,, Inverse Problems, 26 (2010). doi: 10.1088/0266-5611/26/11/115003. Google Scholar

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