May  2015, 9(2): 289-300. doi: 10.3934/ipi.2015.9.289

Recovery of the absorption coefficient in radiative transport from a single measurement

1. 

Department of Pediatrics - Cardiology, Baylor College of Medicine, Houston, TX, United States

Received  December 2013 Revised  January 2015 Published  March 2015

In this paper, we investigate the recovery of the absorption coefficient from boundary data assuming that the region of interest is illuminated at an initial time. We consider a sufficiently strong and isotropic, but otherwise unknown initial state of radiation. This work is part of an effort to reconstruct optical properties using unknown illumination embedded in the unknown medium.
    We break the problem into two steps. First, in a linear framework, we seek the simultaneous recovery of a forcing term of the form $\sigma(t,x,\theta) f(x)$ (with $\sigma$ known) and an isotropic initial condition $u_{0}(x)$ using the single measurement induced by these data. Based on exact boundary controllability, we derive a system of equations for the unknown terms $f$ and $u_{0}$. The system is shown to be Fredholm if $\sigma$ satisfies a certain positivity condition. We show that for generic term $\sigma$ and weakly absorbing media, this linear inverse problem is uniquely solvable with a stability estimate. In the second step, we use the stability results from the linear problem to address the nonlinearity in the recovery of a weak absorbing coefficient. We obtain a locally Lipschitz stability estimate.
Citation: Sebastian Acosta. Recovery of the absorption coefficient in radiative transport from a single measurement. Inverse Problems and Imaging, 2015, 9 (2) : 289-300. doi: 10.3934/ipi.2015.9.289
References:
[1]

S. Acosta, Time reversal for radiative transport with applications to inverse and control problems, Inverse Problems, 29 (2013), 085014, 19pp. doi: 10.1088/0266-5611/29/8/085014.

[2]

V. Agoshkov, Boundary Value Problems for Transport Equations, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 1998. doi: 10.1007/978-1-4612-1994-1.

[3]

D. S. Anikonov, A. E. Kovtanyuk and I. V. Prokhorov, Transport Equation and Tomography, Inverse and Ill-posed Problems Series, VSP, Utrecht, 2002.

[4]

S. Arridge and J. Schotland, Optical tomography: Forward and inverse problems, Inverse Problems, 25 (2009), 123010. doi: 10.1088/0266-5611/25/12/123010.

[5]

S. R. Arridge, Optical tomography in medical imaging, Inverse Problems, 15 (1999), R41-R93. doi: 10.1088/0266-5611/15/2/022.

[6]

S. R. Arridge, Methods in diffuse optical imaging, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 369 (2011), 4558-4576. doi: 10.1098/rsta.2011.0311.

[7]

G. Bal, Inverse transport theory and applications, Inverse Problems, 25 (2009), 053001, 48pp. doi: 10.1088/0266-5611/25/5/053001.

[8]

G. Bal and O. Pinaud, Kinetic models for imaging in random media, Multiscale Model. Simul., 6 (2007), 792-819. doi: 10.1137/060678464.

[9]

G. Bal and O. Pinaud, Imaging using transport models for wave-wave correlations, Math. Models Methods Appl. Sci., 21 (2011), 1071-1093. doi: 10.1142/S0218202511005258.

[10]

G. Bal and K. Ren, Transport-based imaging in random media, SIAM J. Appl. Math., 68 (2008), 1738-1762. doi: 10.1137/070690122.

[11]

K. M. Case and P. F. Zweifel, Linear Transport Theory, Addison-Wesley series in nuclear engineering, Reading, Mass., Addison-Wesley Pub. Co., 1967, URL http://ezproxy.rice.edu/login?url=http://search.ebscohost.com/login.aspx?direct=true&db=cat00402a&AN=rice.652580&site=eds-live&scope=site.

[12]

C. Cercignani and E. Gabetta (eds.), Transport Phenomena and Kinetic Theory. Applications to Gases, Semiconductors, Photons, and Biological Systems, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 2007. doi: 10.1007/978-0-8176-4554-0.

[13]

M. Cessenat, Théorèmes de trace $L^p$ pour des espaces de fonctions de la neutronique, C. R. Acad. Sci. Paris Sér. I Math., 299 (1984), 831-834.

[14]

M. Cessenat, Théorèmes de trace pour des espaces de fonctions de la neutronique, C. R. Acad. Sci. Paris Sér. I Math., 300 (1985), 89-92.

[15]

R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 6. Evolution problems. II, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-58004-8.

[16]

T. Durduran, R. Choe, W. Baker and A. Yodh, Diffuse optics for tissue monitoring and tomography, Rep. Prog. Phys., 73 (2010), 076701.

[17]

H. Egger and M. Schlottbom, An $L^p$ theory for stationary radiative transfer, Appl. Anal., 93 (2014), 1283-1296. doi: 10.1080/00036811.2013.826798.

[18]

K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000.

[19]

A. P. Gibson, J. C. Hebden and S. R. Arridge, Recent advances in diffuse optical imaging, Phys. Med. Biol., 50 (2005), R1-R43. doi: 10.1088/0031-9155/50/4/R01.

[20]

F. Golse, P.-L. Lions, B. Perthame and R. Sentis, Regularity of the moments of the solution of a transport equation, J. Funct. Anal., 76 (1988), 110-125. doi: 10.1016/0022-1236(88)90051-1.

[21]

V. Isakov, Inverse Problems for Partial Differential Equations, 2nd edition, Applied Mathematical Sciences, 127, Springer, New York, 2006.

[22]

T. Kato, Perturbation Theory for Linear Operators, Reprint of the 1980 edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995.

[23]

A. D. Kim and M. Moscoso, Radiative transport theory for optical molecular imaging, Inverse Problems, 22 (2006), 23-42. doi: 10.1088/0266-5611/22/1/002.

[24]

M. V. Klibanov and S. E. Pamyatnykh, Global uniqueness for a coefficient inverse problem for the non-stationary transport equation via Carleman estimate, J. Math. Anal. Appl., 343 (2008), 352-365. doi: 10.1016/j.jmaa.2008.01.071.

[25]

M. V. Klibanov and M. Yamamoto, Exact controllability for the time dependent transport equation, SIAM J. Control Optim., 46 (2007), 2071-2195. doi: 10.1137/060652804.

[26]

M. Machida and M. Yamamoto, Global Lipschitz stability in determining coefficients of the radiative transport equation, Inverse Problems, 30 (2014), 035010, 16pp. doi: 10.1088/0266-5611/30/3/035010.

[27]

M. Mokhtar-Kharroubi, Mathematical Topics in Neutron Transport Theory, vol. 46 of Series on Advances in Mathematics for Applied Sciences, World Scientific Publishing Co., Inc., River Edge, NJ, 1997. doi: 10.1142/9789812819833.

[28]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[29]

K. Ren, Recent developments in numerical techniques for transport-based medical imaging methods, Commun. Comput. Phys., 8 (2010), 1-50. doi: 10.4208/cicp.220509.200110a.

[30]

M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, 2nd edition, Texts in Applied Mathematics, 13, Springer-Verlag, New York, 2004.

[31]

P. Stefanov, Inverse problems in transport theory, in Inside Out: Inverse Problems and Applications, Math. Sci. Res. Inst. Publ., 47, Cambridge Univ. Press, Cambridge, 2003, 111-131.

show all references

References:
[1]

S. Acosta, Time reversal for radiative transport with applications to inverse and control problems, Inverse Problems, 29 (2013), 085014, 19pp. doi: 10.1088/0266-5611/29/8/085014.

[2]

V. Agoshkov, Boundary Value Problems for Transport Equations, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 1998. doi: 10.1007/978-1-4612-1994-1.

[3]

D. S. Anikonov, A. E. Kovtanyuk and I. V. Prokhorov, Transport Equation and Tomography, Inverse and Ill-posed Problems Series, VSP, Utrecht, 2002.

[4]

S. Arridge and J. Schotland, Optical tomography: Forward and inverse problems, Inverse Problems, 25 (2009), 123010. doi: 10.1088/0266-5611/25/12/123010.

[5]

S. R. Arridge, Optical tomography in medical imaging, Inverse Problems, 15 (1999), R41-R93. doi: 10.1088/0266-5611/15/2/022.

[6]

S. R. Arridge, Methods in diffuse optical imaging, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 369 (2011), 4558-4576. doi: 10.1098/rsta.2011.0311.

[7]

G. Bal, Inverse transport theory and applications, Inverse Problems, 25 (2009), 053001, 48pp. doi: 10.1088/0266-5611/25/5/053001.

[8]

G. Bal and O. Pinaud, Kinetic models for imaging in random media, Multiscale Model. Simul., 6 (2007), 792-819. doi: 10.1137/060678464.

[9]

G. Bal and O. Pinaud, Imaging using transport models for wave-wave correlations, Math. Models Methods Appl. Sci., 21 (2011), 1071-1093. doi: 10.1142/S0218202511005258.

[10]

G. Bal and K. Ren, Transport-based imaging in random media, SIAM J. Appl. Math., 68 (2008), 1738-1762. doi: 10.1137/070690122.

[11]

K. M. Case and P. F. Zweifel, Linear Transport Theory, Addison-Wesley series in nuclear engineering, Reading, Mass., Addison-Wesley Pub. Co., 1967, URL http://ezproxy.rice.edu/login?url=http://search.ebscohost.com/login.aspx?direct=true&db=cat00402a&AN=rice.652580&site=eds-live&scope=site.

[12]

C. Cercignani and E. Gabetta (eds.), Transport Phenomena and Kinetic Theory. Applications to Gases, Semiconductors, Photons, and Biological Systems, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 2007. doi: 10.1007/978-0-8176-4554-0.

[13]

M. Cessenat, Théorèmes de trace $L^p$ pour des espaces de fonctions de la neutronique, C. R. Acad. Sci. Paris Sér. I Math., 299 (1984), 831-834.

[14]

M. Cessenat, Théorèmes de trace pour des espaces de fonctions de la neutronique, C. R. Acad. Sci. Paris Sér. I Math., 300 (1985), 89-92.

[15]

R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology. Vol. 6. Evolution problems. II, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-58004-8.

[16]

T. Durduran, R. Choe, W. Baker and A. Yodh, Diffuse optics for tissue monitoring and tomography, Rep. Prog. Phys., 73 (2010), 076701.

[17]

H. Egger and M. Schlottbom, An $L^p$ theory for stationary radiative transfer, Appl. Anal., 93 (2014), 1283-1296. doi: 10.1080/00036811.2013.826798.

[18]

K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000.

[19]

A. P. Gibson, J. C. Hebden and S. R. Arridge, Recent advances in diffuse optical imaging, Phys. Med. Biol., 50 (2005), R1-R43. doi: 10.1088/0031-9155/50/4/R01.

[20]

F. Golse, P.-L. Lions, B. Perthame and R. Sentis, Regularity of the moments of the solution of a transport equation, J. Funct. Anal., 76 (1988), 110-125. doi: 10.1016/0022-1236(88)90051-1.

[21]

V. Isakov, Inverse Problems for Partial Differential Equations, 2nd edition, Applied Mathematical Sciences, 127, Springer, New York, 2006.

[22]

T. Kato, Perturbation Theory for Linear Operators, Reprint of the 1980 edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995.

[23]

A. D. Kim and M. Moscoso, Radiative transport theory for optical molecular imaging, Inverse Problems, 22 (2006), 23-42. doi: 10.1088/0266-5611/22/1/002.

[24]

M. V. Klibanov and S. E. Pamyatnykh, Global uniqueness for a coefficient inverse problem for the non-stationary transport equation via Carleman estimate, J. Math. Anal. Appl., 343 (2008), 352-365. doi: 10.1016/j.jmaa.2008.01.071.

[25]

M. V. Klibanov and M. Yamamoto, Exact controllability for the time dependent transport equation, SIAM J. Control Optim., 46 (2007), 2071-2195. doi: 10.1137/060652804.

[26]

M. Machida and M. Yamamoto, Global Lipschitz stability in determining coefficients of the radiative transport equation, Inverse Problems, 30 (2014), 035010, 16pp. doi: 10.1088/0266-5611/30/3/035010.

[27]

M. Mokhtar-Kharroubi, Mathematical Topics in Neutron Transport Theory, vol. 46 of Series on Advances in Mathematics for Applied Sciences, World Scientific Publishing Co., Inc., River Edge, NJ, 1997. doi: 10.1142/9789812819833.

[28]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[29]

K. Ren, Recent developments in numerical techniques for transport-based medical imaging methods, Commun. Comput. Phys., 8 (2010), 1-50. doi: 10.4208/cicp.220509.200110a.

[30]

M. Renardy and R. C. Rogers, An Introduction to Partial Differential Equations, 2nd edition, Texts in Applied Mathematics, 13, Springer-Verlag, New York, 2004.

[31]

P. Stefanov, Inverse problems in transport theory, in Inside Out: Inverse Problems and Applications, Math. Sci. Res. Inst. Publ., 47, Cambridge Univ. Press, Cambridge, 2003, 111-131.

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