# American Institute of Mathematical Sciences

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 & 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.  Google Scholar [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.  Google Scholar [3] D. S. Anikonov, A. E. Kovtanyuk and I. V. Prokhorov, Transport Equation and Tomography, Inverse and Ill-posed Problems Series, VSP, Utrecht, 2002.  Google Scholar [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.  Google Scholar [5] S. R. Arridge, Optical tomography in medical imaging, Inverse Problems, 15 (1999), R41-R93. doi: 10.1088/0266-5611/15/2/022.  Google Scholar [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.  Google Scholar [7] G. Bal, Inverse transport theory and applications, Inverse Problems, 25 (2009), 053001, 48pp. doi: 10.1088/0266-5611/25/5/053001.  Google Scholar [8] G. Bal and O. Pinaud, Kinetic models for imaging in random media, Multiscale Model. Simul., 6 (2007), 792-819. doi: 10.1137/060678464.  Google Scholar [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.  Google Scholar [10] G. Bal and K. Ren, Transport-based imaging in random media, SIAM J. Appl. Math., 68 (2008), 1738-1762. doi: 10.1137/070690122.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] T. Durduran, R. Choe, W. Baker and A. Yodh, Diffuse optics for tissue monitoring and tomography, Rep. Prog. Phys., 73 (2010), 076701. Google Scholar [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.  Google Scholar [18] K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000.  Google Scholar [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.  Google Scholar [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.  Google Scholar [21] V. Isakov, Inverse Problems for Partial Differential Equations, 2nd edition, Applied Mathematical Sciences, 127, Springer, New York, 2006.  Google Scholar [22] T. Kato, Perturbation Theory for Linear Operators, Reprint of the 1980 edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [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.  Google Scholar [3] D. S. Anikonov, A. E. Kovtanyuk and I. V. Prokhorov, Transport Equation and Tomography, Inverse and Ill-posed Problems Series, VSP, Utrecht, 2002.  Google Scholar [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.  Google Scholar [5] S. R. Arridge, Optical tomography in medical imaging, Inverse Problems, 15 (1999), R41-R93. doi: 10.1088/0266-5611/15/2/022.  Google Scholar [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.  Google Scholar [7] G. Bal, Inverse transport theory and applications, Inverse Problems, 25 (2009), 053001, 48pp. doi: 10.1088/0266-5611/25/5/053001.  Google Scholar [8] G. Bal and O. Pinaud, Kinetic models for imaging in random media, Multiscale Model. Simul., 6 (2007), 792-819. doi: 10.1137/060678464.  Google Scholar [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.  Google Scholar [10] G. Bal and K. Ren, Transport-based imaging in random media, SIAM J. Appl. Math., 68 (2008), 1738-1762. doi: 10.1137/070690122.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] T. Durduran, R. Choe, W. Baker and A. Yodh, Diffuse optics for tissue monitoring and tomography, Rep. Prog. Phys., 73 (2010), 076701. Google Scholar [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.  Google Scholar [18] K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000.  Google Scholar [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.  Google Scholar [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.  Google Scholar [21] V. Isakov, Inverse Problems for Partial Differential Equations, 2nd edition, Applied Mathematical Sciences, 127, Springer, New York, 2006.  Google Scholar [22] T. Kato, Perturbation Theory for Linear Operators, Reprint of the 1980 edition, Classics in Mathematics, Springer-Verlag, Berlin, 1995.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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