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

Multi-wave imaging in attenuating media

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.
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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