November  2013, 7(4): 1367-1377. doi: 10.3934/ipi.2013.7.1367

Instability of the linearized problem in multiwave tomography of recovery both the source and the speed

1. 

Department of Mathematics, Purdue University, 150 N University Street, West Lafayette, IN 47907

2. 

Department of Mathematics, University of Washington, Seattle, WA 98195-4350

Received  January 2013 Revised  October 2013 Published  November 2013

In this paper we consider the linearized problem of recovering both the sound speed and the thermal absorption arising in thermoacoustic and photoacoustic tomography. We show that the problem is unstable in any scale of Sobolev spaces.
Citation: Plamen Stefanov, Gunther Uhlmann. Instability of the linearized problem in multiwave tomography of recovery both the source and the speed. Inverse Problems and Imaging, 2013, 7 (4) : 1367-1377. doi: 10.3934/ipi.2013.7.1367
References:
[1]

G. Ambartsoumian, R. Gouia-Zarrad and M. A. Lewis, Inversion of the circular Radon transform on an annulus, Inverse Problems, 26 (2010), 105015, 11pp. doi: 10.1088/0266-5611/26/10/105015.

[2]

M. Agranovsky, P. Kuchment and L. Kunyansky, On Reconstruction Formulas and Algorithms for the Thermoacoustic Tomography, Photoacoustic Imaging and Spectroscopy, CRC Press (2009), 89-101. doi: 10.1201/9781420059922.ch8.

[3]

D. Finch, M. Haltmeier and Rakesh, Inversion of spherical means and the wave equation in even dimensions, SIAM J. Appl. Math., 68 (2007), 392-412. doi: 10.1137/070682137.

[4]

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), 1213-1240 (electronic). doi: 10.1137/S0036141002417814.

[5]

D. Finch and Rakesh, The range of the spherical mean value operator for functions supported in a ball, Inverse Problems, 22 (2006), 923-938. doi: 10.1088/0266-5611/22/3/012.

[6]

D. Finch and Rakesh, Recovering a function from its spherical mean values in two and three dimensions, in Photoacoustic Imaging and Spectroscopy, CRC Press (2009). doi: 10.1201/9781420059922.pt3.

[7]

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

[8]

Y. Hristova, Time reversal in thermoacoustic tomography-an error estimate, Inverse Problems, 25 (2009), 055008, 14pp. doi: 10.1088/0266-5611/25/5/055008.

[9]

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.

[10]

M. Haltmeier, T. Schuster and O. Scherzer, Filtered backprojection for thermoacoustic computed tomography in spherical geometry, Math. Methods Appl. Sci., 28 (2005), 1919-1937. doi: 10.1002/mma.648.

[11]

V. Isakov, Inverse Problems for Partial Differential Equations, second ed., Applied Mathematical Sciences, vol. 127, Springer, New York, 2006.

[12]

J. Jose, Rene G. H. Willemink, W. Steenbergen, C. H. Slump, T. G. van Leeuwen and S. Manohar, Speed-of-sound compensated photoacoustic tomography for accurate imagingarXiv:1206.1191. doi: 10.1118/1.4764911.

[13]

P. Kuchment and L. Kunyansky, Mathematics of thermoacoustic tomography, European J. Appl. Math., 19 (2008), 191-224. doi: 10.1017/S0956792508007353.

[14]

R. A Kruger, W. L. Kiser, 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.

[15]

R. A. Kruger, D. R. Reinecke and G. A. Kruger, Thermoacoustic computed tomography-technical considerations, Med. Phys., 26 (1999), 1832-1837. doi: 10.1118/1.598688.

[16]

I. Lasiecka, J.-L. Lions and R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl., 65 (1986), 149-192.

[17]

S. K. Patch, Thermoacoustic tomography - consistency conditions and the partial scan problem, Physics in Medicine and Biology, 49 (2004), 2305-2315. doi: 10.1201/9781420059922.ch9.

[18]

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 Sciences, (2011), 850-883. doi: 10.1137/100817280.

[19]

V. A. Sharafutdinov, Integral Geometry of Tensor Fields, Inverse and Ill-posed Problems Series, VSP, Utrecht, (1994). doi: 10.1515/9783110900095.

[20]

P. Stefanov and G. Uhlmann, Linearizing non-linear inverse problems and an application to inverse backscattering, J. Funct. Anal., 256 (2009), 2842-2866. doi: 10.1016/j.jfa.2008.10.017.

[21]

______, Thermoacoustic tomography with variable sound speed, Inverse Problems, 25 (2009), 075011, 16pp. doi: 10.1088/0266-5611/25/7/075011.

[22]

______, Thermoacoustic tomography arising in brain imaging, Inverse Problems, 27 (2011), 045004, 26pp. doi: 10.1088/0266-5611/27/4/045004.

[23]

P. Stefanov and G. Uhlmann, Multi-wave methods via ultrasound, MSRI Publications, Inside Out, 60 (2012), 271-324.

[24]

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.

[25]

D. Tataru, Unique continuation problems for partial differential equations, Geometric Methods in Inverse Problems and PDE Control, IMA Vol. Math. Appl., 137, Springer, New York, (2004), 239-255. doi: 10.1007/978-1-4684-9375-7_8.

[26]

B. E. Treeby, T. K. Varslot, E. Z. Zhang, J. G. Laufer and P. C. Beard, Automatic sound speed selection in photoacoustic image reconstruction using an autofocus approach, Journal of Biomedical Optics, 16 (2011). doi: 10.1117/1.3619139.

[27]

Y. Xu, P. Kuchment and G. Ambartsoumian, Reconstructions in limited view thermoacoustic tomography, Medical Physics, 31 (2004), 724-733. doi: 10.1118/1.1644531.

[28]

M. Xu and L. V. Wang, Photoacoustic imaging in biomedicine, Review of Scientific Instruments, 77 (2006), 041101. doi: 10.1063/1.2195024.

[29]

Z. Yuan and H. Jiang, Simultaneous recovery of tissue physiological and acoustic properties and the criteria for wavelength selection in multispectral photoacoustic tomography, Opt. Lett., 34 (2009), 1714-1716. doi: 10.1364/OL.34.001714.

[30]

Z. Yuan, Q. Zhang and H. Jiang, Simultaneous reconstruction of acoustic and optical properties of heterogeneous media by quantitative photoacoustic tomography, Opt. Express, 14 (2006), 6749-6754 (eng). doi: 10.1364/OE.14.006749.

[31]

J. Zhang and M. A. Anastasio, Reconstruction of speed-of-sound and electromagnetic absorption distributions in photoacoustic tomography, Proc. SPIE, 6086 (2006). doi: 10.1117/12.647665.

show all references

References:
[1]

G. Ambartsoumian, R. Gouia-Zarrad and M. A. Lewis, Inversion of the circular Radon transform on an annulus, Inverse Problems, 26 (2010), 105015, 11pp. doi: 10.1088/0266-5611/26/10/105015.

[2]

M. Agranovsky, P. Kuchment and L. Kunyansky, On Reconstruction Formulas and Algorithms for the Thermoacoustic Tomography, Photoacoustic Imaging and Spectroscopy, CRC Press (2009), 89-101. doi: 10.1201/9781420059922.ch8.

[3]

D. Finch, M. Haltmeier and Rakesh, Inversion of spherical means and the wave equation in even dimensions, SIAM J. Appl. Math., 68 (2007), 392-412. doi: 10.1137/070682137.

[4]

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), 1213-1240 (electronic). doi: 10.1137/S0036141002417814.

[5]

D. Finch and Rakesh, The range of the spherical mean value operator for functions supported in a ball, Inverse Problems, 22 (2006), 923-938. doi: 10.1088/0266-5611/22/3/012.

[6]

D. Finch and Rakesh, Recovering a function from its spherical mean values in two and three dimensions, in Photoacoustic Imaging and Spectroscopy, CRC Press (2009). doi: 10.1201/9781420059922.pt3.

[7]

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

[8]

Y. Hristova, Time reversal in thermoacoustic tomography-an error estimate, Inverse Problems, 25 (2009), 055008, 14pp. doi: 10.1088/0266-5611/25/5/055008.

[9]

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.

[10]

M. Haltmeier, T. Schuster and O. Scherzer, Filtered backprojection for thermoacoustic computed tomography in spherical geometry, Math. Methods Appl. Sci., 28 (2005), 1919-1937. doi: 10.1002/mma.648.

[11]

V. Isakov, Inverse Problems for Partial Differential Equations, second ed., Applied Mathematical Sciences, vol. 127, Springer, New York, 2006.

[12]

J. Jose, Rene G. H. Willemink, W. Steenbergen, C. H. Slump, T. G. van Leeuwen and S. Manohar, Speed-of-sound compensated photoacoustic tomography for accurate imagingarXiv:1206.1191. doi: 10.1118/1.4764911.

[13]

P. Kuchment and L. Kunyansky, Mathematics of thermoacoustic tomography, European J. Appl. Math., 19 (2008), 191-224. doi: 10.1017/S0956792508007353.

[14]

R. A Kruger, W. L. Kiser, 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.

[15]

R. A. Kruger, D. R. Reinecke and G. A. Kruger, Thermoacoustic computed tomography-technical considerations, Med. Phys., 26 (1999), 1832-1837. doi: 10.1118/1.598688.

[16]

I. Lasiecka, J.-L. Lions and R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl., 65 (1986), 149-192.

[17]

S. K. Patch, Thermoacoustic tomography - consistency conditions and the partial scan problem, Physics in Medicine and Biology, 49 (2004), 2305-2315. doi: 10.1201/9781420059922.ch9.

[18]

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 Sciences, (2011), 850-883. doi: 10.1137/100817280.

[19]

V. A. Sharafutdinov, Integral Geometry of Tensor Fields, Inverse and Ill-posed Problems Series, VSP, Utrecht, (1994). doi: 10.1515/9783110900095.

[20]

P. Stefanov and G. Uhlmann, Linearizing non-linear inverse problems and an application to inverse backscattering, J. Funct. Anal., 256 (2009), 2842-2866. doi: 10.1016/j.jfa.2008.10.017.

[21]

______, Thermoacoustic tomography with variable sound speed, Inverse Problems, 25 (2009), 075011, 16pp. doi: 10.1088/0266-5611/25/7/075011.

[22]

______, Thermoacoustic tomography arising in brain imaging, Inverse Problems, 27 (2011), 045004, 26pp. doi: 10.1088/0266-5611/27/4/045004.

[23]

P. Stefanov and G. Uhlmann, Multi-wave methods via ultrasound, MSRI Publications, Inside Out, 60 (2012), 271-324.

[24]

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.

[25]

D. Tataru, Unique continuation problems for partial differential equations, Geometric Methods in Inverse Problems and PDE Control, IMA Vol. Math. Appl., 137, Springer, New York, (2004), 239-255. doi: 10.1007/978-1-4684-9375-7_8.

[26]

B. E. Treeby, T. K. Varslot, E. Z. Zhang, J. G. Laufer and P. C. Beard, Automatic sound speed selection in photoacoustic image reconstruction using an autofocus approach, Journal of Biomedical Optics, 16 (2011). doi: 10.1117/1.3619139.

[27]

Y. Xu, P. Kuchment and G. Ambartsoumian, Reconstructions in limited view thermoacoustic tomography, Medical Physics, 31 (2004), 724-733. doi: 10.1118/1.1644531.

[28]

M. Xu and L. V. Wang, Photoacoustic imaging in biomedicine, Review of Scientific Instruments, 77 (2006), 041101. doi: 10.1063/1.2195024.

[29]

Z. Yuan and H. Jiang, Simultaneous recovery of tissue physiological and acoustic properties and the criteria for wavelength selection in multispectral photoacoustic tomography, Opt. Lett., 34 (2009), 1714-1716. doi: 10.1364/OL.34.001714.

[30]

Z. Yuan, Q. Zhang and H. Jiang, Simultaneous reconstruction of acoustic and optical properties of heterogeneous media by quantitative photoacoustic tomography, Opt. Express, 14 (2006), 6749-6754 (eng). doi: 10.1364/OE.14.006749.

[31]

J. Zhang and M. A. Anastasio, Reconstruction of speed-of-sound and electromagnetic absorption distributions in photoacoustic tomography, Proc. SPIE, 6086 (2006). doi: 10.1117/12.647665.

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