An inversion formula for the transport equation in ℝ³ using complex analysis in several variables

References

[1] R. Abreu Blaya and J. Bory Reyes, On the Riemann Hilbert type problems in Clifford analysis, Adv. Appl. Clifford Algebr. 11 (2001), no. 1, 15–26. 10.1007/BF03042036Search in Google Scholar

[2] E. V. Arbuzov, A. L. Bukhgeĭm and S. G. Kazantsev, Two-dimensional tomography problems and the theory of A-analytic functions Siberian Adv. Math. 8 (1998), no. 4, 1–20. Search in Google Scholar

[3] G. Bal, Columbia University (2004), Lecture Notes in http://www.colombia.edu/?gb2030/. Search in Google Scholar

[4] H. H. Barrett and W. Swindell, Radiological Imaging. Vol. 1, Academic Press, New York, 1981. 10.1016/B978-0-08-057230-7.50008-1Search in Google Scholar

[5] S. Bernstein, R. Hielscher and H. Schaeben, The generalized totally geodesic Radon transform and its application to texture analysis, Math. Methods Appl. Sci. 32 (2009), no. 4, 379–394. 10.1002/mma.1042Search in Google Scholar

[6] H. Boas, Topics in several complex variables lecture notes, preprint (2005), http://www.math.tamu.edu␣/~boas/courses/685-2005c/. Search in Google Scholar

[7] P. Chandramohan, B. U. Nayak and V. S. Raju, Application of longshore transport equations to the Andhra coast, east coast of India, Coastal Eng. 12 (1988), no. 3, 285–297. 10.1016/0378-3839(88)90009-9Search in Google Scholar

[8] Z. Chen, G. Huan and Y. Ma, Computational Methods for Multiphase Flows in Porous Media, SIAM, Philadelphia, 2006. 10.1137/1.9780898718942Search in Google Scholar

[9] S. R. Deans, The Radon Transform and some of its Applications, John Wiley & Sons, New York, 1983. Search in Google Scholar

[10] D. V. Finch, Cone beam reconstruction with sources on a curve, SIAM J. Appl. Math. 45 (1985), no. 4, 665–673. 10.1137/0145039Search in Google Scholar

[11] D. V. Finch, Approximate reconstruction formulae for the cone beam transform. I, unpulished (1987). Search in Google Scholar

[12] I. M. Gel’fand, M. I. Graev and N. Y. Vilenkin, Generalized Functions. Vol. 5: Integral Geometry and Representation Theory, Academic Press, New York, 1966. Search in Google Scholar

[13] I. M. Gel’fand and G. E. Shilov, Generalized Functions. Vol. I: Properties and Operations, Academic Press, New York, 1964. Search in Google Scholar

[14] F. Golfier, M. Quintard and S. Whitaker S. 2002, Heat and mass transfer in tubes: An analysis using the method of volume averaging, J. Porous Media 5 (2002), no. 2, 169–185. 10.1615/JPorMedia.v5.i3.10Search in Google Scholar

[15] P. Grangeat, Analysis d’un système d’imagerie 3D par´réconstruction à partir de X, en géométrie conique PhD Thesis, Ecole Nationale Supérieure des Té1écommunications, Paris, 1987. Search in Google Scholar

[16] R. C. Gunning, Introduction to Holomorphic Functions of Several Variables. Vol. I: Function Theory, Wadsworth & Brooks/Cole, Pacific Grove, 1990. Search in Google Scholar

[17] C. Hamaker, K. T. Smith, D. C. Solmon and S. L. Wagner, The divergent beam X-ray transform, Rocky Mountain J. Math. 10 (1980), no. 1, 253–283. 10.1216/RMJ-1980-10-1-253Search in Google Scholar

[18] S. Helgason, The Radon Transform, 2nd ed., Progr. Math. 5, Birkhäuser, Boston, 1999. 10.1007/978-1-4757-1463-0Search in Google Scholar

[19] S. M. Hong, A. T. Pham and C. Jungemann, Deterministic Solvers for the Boltzmann Transport Equation, Springer, Wien, 2011. 10.1007/978-3-7091-0778-2Search in Google Scholar

[20] F. John, Plane Waves and Spherical Means Applied to Partial Differential Equations, Interscience, New York, 1955. Search in Google Scholar

[21] A. A. Kirillov, A problem of I. M. Gel’fand, Dokl. Akad. Nauk SSSR 137 (1961), 276–277; translation in Soviet Math. Dokl. 2 (1961), 268–269. Search in Google Scholar

[22] M. Landucci, Boundary values of holomorphic functions and Cauchy problem for ∂¯ operator in the polydisc, Rend. Semin. Mat. Univ. Padova 62 (1980), 23–34. Search in Google Scholar

[23] C. Laurent-Thiébaut, Transformation de Bochner–Martinelli dans une variété de Stein, Séminaire d’Analyse P. Lelong–P. Dolbeault–H. Skoda, Années 1985/1986, Lecture Notes in Math. 1295, Springer, Berlin (1987), 96–131. 10.1007/BFb0081979Search in Google Scholar

[24] C. Laurent-Thiébaut, Holomorphic Function Theory in Several Variables, Universitext, Springer, London, 2011. 10.1007/978-0-85729-030-4Search in Google Scholar

[25] P. D. Lax and R. S. Phillips, The Paley–Wiener theorem for the Radon transform, Comm. Pure Appl. Math. 23 (1970), 409–424. 10.1002/cpa.3160230311Search in Google Scholar

[26] B. Meinerzhagen, A. T. Pham, S.-M. Hong and C. Jungemann, Solving Boltzmann transport equation without Monte-Carlo algorithms - New methods for industrial TCAD applications, International Conference on Simulation of Semiconductor Processes and Devices (Bologna 2010), IEEE Press, Piscataway (2010), 10.1109/SISPAD.2010.5604501. 10.1109/SISPAD.2010.5604501Search in Google Scholar

[27] R. G. Novikov, An inversion formula for the attenuated X-ray transformation, Ark. Mat. 40 (2002), no. 1, 145–167. 10.1007/BF02384507Search in Google Scholar

[28] V. P. Palamodov, Inversion formulas for the three-dimensional ray transform, Mathematical Methods in Tomography (Oberwolfach 1990), Lecture Notes in Math. 1497, Springer, Berlin (1991), 53–62. 10.1007/BFb0084507Search in Google Scholar

[29] M. Quintard and M. Whitaker, Convection, dispersion, and interfacial transport of contaminants: Homogeneous porous media, Adv. Water Res. 17 (1994), no. 4, 221–239. 10.1016/0309-1708(94)90002-7Search in Google Scholar

[30] W. Rudin, Function Theory in Polydiscs, W. A. Benjamin, New York, 1969. Search in Google Scholar

[31] W. Rudin, Lumer’s Hardy spaces, Michigan Math. J. 24 (1977), no. 1, 1–5. 10.1307/mmj/1029001813Search in Google Scholar

[32] S. M. Saberi Fathi, Inversion formula for the non-uniformly attenuated x-ray transform for emission imaging in ℝ3 using quaternionic analysis, J. Phys. A 43 (2010), no. 33, Article ID 335202. 10.1088/1751-8113/43/33/335202Search in Google Scholar

[33] S. M. Saberi Fathi and T. T. Truong, On the inverse of the directional derivative operator in ℝN, J. Phys. A 42 (2009), no. 4, Article ID 045203. 10.1088/1751-8113/42/4/045203Search in Google Scholar

[34] S. M. Saberi-Fathi, T. T. Truong and M. K. Nguyen, A quaternionic approach to x-ray transform inversion in ℝ3, J. Phys. A 42 (2009), no. 41, Article ID 415205. 10.1088/1751-8113/42/41/415205Search in Google Scholar

[35] B. D. Smith, Cone-beam tomography: Recent advances and a tutorial review, Optical Eng. 29 (1990), no. 5, 524–534. 10.1117/12.55621Search in Google Scholar

[36] K. T. Smith, D. C. Solmon and S. L. Wagner, Practical and mathematical aspects of the problem of reconstructing objects from radiographs, Bull. Amer. Math. Soc. 83 (1977), no. 6, 1227–1270. 10.1090/S0002-9904-1977-14406-6Search in Google Scholar

[37] D. C. Solmon, The X-ray transform, J. Math. Anal. Appl. 56 (1976), no. 1, 61–83. 10.1016/0022-247X(76)90008-1Search in Google Scholar

[38] H. K. Tuy, An inversion formula for cone-beam reconstruction, SIAM J. Appl. Math. 43 (1983), no. 3, 546–552. 10.1137/0143035Search in Google Scholar

[39] F. Zanotti and R. G. Carbonell, Development of Transport Equations for Multiphase Systems. III: Application to heat transfer in packed beds, Chem. Eng. Sci. 39 (1984), no. 2, 299–311. 10.1016/0009-2509(84)80028-7Search in Google Scholar