Uniqueness and non-uniqueness in inverse radiative transfer
HTML articles powered by AMS MathViewer
- by Plamen Stefanov and Alexandru Tamasan PDF
- Proc. Amer. Math. Soc. 137 (2009), 2335-2344 Request permission
Abstract:
We characterize the non-uniqueness in the inverse problem for the stationary transport model, in which the absorption $a$ and the scattering coefficient $k$ of the media are to be recovered from the albedo operator. We show that “gauge equivalent” pairs $(a,k)$ yield the same albedo operator, and that we can recover uniquely the class of the gauge equivalent pairs. We apply this result to show unique determination of the media when the absorption $a$ depends on the line of travel through each point while the scattering $k$ obeys a symmetry property. Previously known results concerned the directional independent absorption $a$.References
- D. S. Anikonov, The uniqueness of the determination of the coefficient and right-hand side of the transport equation, Differencial′nye Uravnenija 11 (1975), 8–18, 200 (Russian). MR 0371321
- D. S. Anikonov, Multidimensional inverse problems for the transport equation, Differentsial′nye Uravneniya 20 (1984), no. 5, 817–824 (Russian). MR 747361
- Yu. E. Anikonov and B. A. Bubnov, Inverse problems of transport theory, Dokl. Akad. Nauk SSSR 299 (1988), no. 5, 1037–1040 (Russian); English transl., Soviet Math. Dokl. 37 (1988), no. 2, 497–500. MR 947226
- Hans Babovsky, Identification of scattering media from reflected flows, SIAM J. Appl. Math. 51 (1991), no. 6, 1676–1704. MR 1136006, DOI 10.1137/0151086
- Guillaume Bal, Inverse problems for homogeneous transport equations. II. The multidimensional case, Inverse Problems 16 (2000), no. 4, 1013–1028. MR 1776480, DOI 10.1088/0266-5611/16/4/309
- G. Bal and A. Jollivet, Stability estimates in stationary inverse transport, Inverse Probl. Imaging 2(2008), pp. 427–454.
- Guillaume Bal, Ian Langmore, and François Monard, Inverse transport with isotropic sources and angularly averaged measurements, Inverse Probl. Imaging 2 (2008), no. 1, 23–42. MR 2375321, DOI 10.3934/ipi.2008.2.23
- Anatoly N. Bondarenko, The structure of the fundamental solution of the time-independent transport equation, J. Math. Anal. Appl. 221 (1998), no. 2, 430–451. MR 1621726, DOI 10.1006/jmaa.1997.5842
- Kenneth M. Case and Paul F. Zweifel, Linear transport theory, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1967. MR 0225547
- M. Choulli and P. Stefanov, Inverse scattering and inverse boundary value problems for the linear Boltzmann equation, Comm. Partial Differential Equations 21 (1996), no. 5-6, 763–785. MR 1391523, DOI 10.1080/03605309608821207
- Mourad Choulli and Plamen Stefanov, An inverse boundary value problem for the stationary transport equation, Osaka J. Math. 36 (1999), no. 1, 87–104. MR 1670750
- Robert Dautray and Jacques-Louis Lions, Mathematical analysis and numerical methods for science and technology. Vol. 6, Springer-Verlag, Berlin, 1993. Evolution problems. II; With the collaboration of Claude Bardos, Michel Cessenat, Alain Kavenoky, Patrick Lascaux, Bertrand Mercier, Olivier Pironneau, Bruno Scheurer and Rémi Sentis; Translated from the French by Alan Craig. MR 1295030, DOI 10.1007/978-3-642-58004-8
- Y. Kurylev, M. Lassas, and G. Uhlmann, Rigidity of broken geodesic flow and inverse problems, to appear in American Journal of Mathematics.
- Ian Langmore, The stationary transport problem with angularly averaged measurements, Inverse Problems 24 (2008), no. 1, 015024, 23. MR 2384783, DOI 10.1088/0266-5611/24/1/015024
- I. Langmore and S. McDowall, Optical tomography for variable refractive index with angularly averaged measurements, Comm. PDE 33(2008), pp. 2180–2207.
- Edward W. Larsen, Solution of multidimensional inverse transport problems, J. Math. Phys. 25 (1984), no. 1, 131–135. MR 728896, DOI 10.1063/1.526007
- Edward W. Larsen, Solution of three-dimensional inverse transport problems, Transport Theory Statist. Phys. 17 (1988), no. 2-3, 147–167. MR 963049, DOI 10.1080/00411458808230860
- N. J. McCormick, Inverse radiative transfer problems: A review, Nucl. Sci. Eng. 112(1992), pp. 185–198.
- Stephen R. McDowall, An inverse problem for the transport equation in the presence of a Riemannian metric, Pacific J. Math. 216 (2004), no. 2, 303–326. MR 2094548, DOI 10.2140/pjm.2004.216.303
- Stephen R. McDowall, Optical tomography on simple Riemannian surfaces, Comm. Partial Differential Equations 30 (2005), no. 7-9, 1379–1400. MR 2180309, DOI 10.1080/03605300500258923
- M. Mokhtar-Kharroubi, Mathematical topics in neutron transport theory, Series on Advances in Mathematics for Applied Sciences, vol. 46, World Scientific Publishing Co., Inc., River Edge, NJ, 1997. New aspects; With a chapter by M. Choulli and P. Stefanov. MR 1612403, DOI 10.1142/9789812819833
- Michael Reed and Barry Simon, Methods of modern mathematical physics. III, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1979. Scattering theory. MR 529429
- M. Ribarič and I. Vidav, Analytic properties of the inverse $A(z)^{-1}$ of an analytic linear operator valued function $A(z)$, Arch. Rational Mech. Anal. 32 (1969), 298–310. MR 236741, DOI 10.1007/BF00281506
- V. G. Romanov, Stability estimates in problems of recovering the attenuation coefficient and the scattering indicatrix for the transport equation, J. Inverse Ill-Posed Probl. 4 (1996), no. 4, 297–305. MR 1403885, DOI 10.1515/jiip.1996.4.4.297
- Plamen Stefanov and Gunther Uhlmann, Optical tomography in two dimensions, Methods Appl. Anal. 10 (2003), no. 1, 1–9. MR 2014159, DOI 10.4310/MAA.2003.v10.n1.a1
- Plamen Stefanov and Gunther Uhlmann, An inverse source problem in optical molecular imaging, Anal. PDE 1 (2008), no. 1, 115–126. MR 2444095, DOI 10.2140/apde.2008.1.115
- Alexandru Tamasan, An inverse boundary value problem in two-dimensional transport, Inverse Problems 18 (2002), no. 1, 209–219. MR 1893591, DOI 10.1088/0266-5611/18/1/314
- Alexandru Tamasan, Optical tomography in weakly anisotropic scattering media, Inverse problems: theory and applications (Cortona/Pisa, 2002) Contemp. Math., vol. 333, Amer. Math. Soc., Providence, RI, 2003, pp. 199–207. MR 2032017, DOI 10.1090/conm/333/05964
- Jenn-Nan Wang, Stability estimates of an inverse problem for the stationary transport equation, Ann. Inst. H. Poincaré Phys. Théor. 70 (1999), no. 5, 473–495 (English, with English and French summaries). MR 1697917
Additional Information
- Plamen Stefanov
- Affiliation: Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, Indiana 47907-2067
- MR Author ID: 166695
- Email: stefanov@math.purdue.edu
- Alexandru Tamasan
- Affiliation: Department of Mathematics, University of Central Florida, 4000 Central Florida Boulevard, Orlando, Florida 32816
- MR Author ID: 363173
- Email: tamasan@math.ucf.edu
- Received by editor(s): September 15, 2008
- Published electronically: February 17, 2009
- Additional Notes: The first author was partly supported by NSF FRG Grant No. 0554065
- Communicated by: Walter Craig
- © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 137 (2009), 2335-2344
- MSC (2000): Primary 35R30, 78A46
- DOI: https://doi.org/10.1090/S0002-9939-09-09839-6
- MathSciNet review: 2495267