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February  2014, 8(1): 103-125. doi: 10.3934/ipi.2014.8.103

## Ray transforms on a conformal class of curves

 1 University of Toronto, Department of Mathematics, Toronto, ON, M5S 2E4, Canada 2 Columbia University, Department of Applied Physics and Applied Mathematics, New York, NY, 10025, United States

Received  May 2010 Revised  October 2011 Published  March 2014

We introduce a technique for recovering a sufficiently smooth function from its ray transforms over rotationally related curves in the unit disc of 2-dimensional Euclidean space. The method is based on a complexification of the underlying vector fields defining the initial transport and inversion formulae are then given in a unified form. The method is used to analyze the attenuated ray transform in the same setting.
Citation: Nicholas Hoell, Guillaume Bal. Ray transforms on a conformal class of curves. Inverse Problems and Imaging, 2014, 8 (1) : 103-125. doi: 10.3934/ipi.2014.8.103
##### References:
 [1] L. Ahlfors, Complex Analysis, McGraw-Hill, 1978. [2] L. V. Ahlfors, Lectures on Quasiconformal Mappings, University Lecture Series, American Mathematical Society, 2006. [3] L. Ahlfors and L. Bers, Riemann's mapping theorem for variable metrics, Ann. of Math, 72 (1960), 385-404. [4] E. Arbuzov, A. Bukhgeim and S. Kazantsev, Two-dimensional tomography problems and the theory of A-analytic functions, Siberian Advances in Mathematics, 8 (1998), 1-20. [5] K. Astala, T. Iwaniec and G. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane, Princeton University Press, 2009. [6] G. Bal, Ray transforms in hyperbolic geometry, J. Math. Pures Appl., 84 (2005), 1362-1392. doi: 10.1016/j.matpur.2005.02.001. [7] G. Bal, On the attenuated Radon transform with full and partial measurements, Inverse Problems, 20 (2004), 399-418. doi: 10.1088/0266-5611/20/2/006. [8] H. Begehr, Complex Analytic Methods for Partial Differential Equations, World Scientific Publishing Co., 1994. [9] C. Berenstein and E. C. Tarabush, Integral geometry in hyperbolic spaces and electrical impedance tomography, SIAM J. Appl. Math., 56 (1996), 755-764. doi: 10.1137/S0036139994277348. [10] P. Colwell, Blaschke Products: Bounded Analytical Functions, University of Michigan Press, 1985. [11] L. Ehrenpreis, The Universality of the Radon Transform, Oxford Mathematical Monographs, Oxford University Press, USA, 2003. doi: 10.1093/acprof:oso/9780198509783.001.0001. [12] L. C. Evans, Partial Differential Equations, 19 of Graduate Studies in Mathematics, American Mathematical Society, 1998. [13] D. Finch, Uniqueness for the X-ray transform in the physical range, Inverse Problems, 2 (1986), 197-203. doi: 10.1088/0266-5611/2/2/010. [14] M. J. Ablowitz and A. S. Fokas, Complex Variables: Introduction and Applications, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2003. doi: 10.1017/CBO9780511791246. [15] J. B. Garnett, Bounded Analytic Functions, Springer New York, 1981. [16] R. E. Greene and S. G. Krantz, Function Theory of One Complex Variable, 40 of Graduate Studies in Mathematics, American Mathematical Society, 2006. [17] D. Griffiths, Introduction to Elementary Particles, Wiley-VCH, 2008. doi: 10.1002/9783527618460. [18] S. Helgason, The Radon Transform, 5 of Progress in Mathematics, Birkhäuser Boston, 1980. [19] ______, Groups and Geometric Analysis (Integral Geometry, Invariant Differential Operators and Spherical Functions), American Mathematical Society, 2000. [20] ______, The inversion of the x-ray transform on a compact symmetric space, Journal of Lie Theory, 17 (2007), 307-315. [21] L. Hormander, Complex Analysis in Several Variables, North Holland, 1990. [22] S. S. Romesh Kumar, Inner functions and substitution operators, Acta Sci. Math. (Szegal), 58 (1993), 509-516. [23] J. M. Lee, Riemannian Manifolds: An Introduction to Curvature, 176 in Graduate Texts in Mathematics, Springer, 1997. [24] F. Natterer, Inversion of the attenuated radon transform, Inverse Problems, 17 (2001), 113-119. doi: 10.1088/0266-5611/17/1/309. [25] F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction, (Monographs on Mathematical Modeling and Computation), Society for Industrial Mathematics, 2007. doi: 10.1118/1.1455744. [26] Z. Nehari, Conformal Mappings, McGraw-Hill Book Company, Inc., 1952. [27] R. Novikov, An inversion formula for the attenuated x-ray transformation, Ark. Math, 40 (2002), 145-167. doi: 10.1007/BF02384507. [28] L. Pestov and G. Uhlmann, On characterization of range and inversion formulas for the geodesic x-ray transform, International Math. Research Notices, 80 (2004), 4331-4347. doi: 10.1155/S1073792804142116. [29] H. Renelt, Elliptic Systems and Quasiconformal Mappings, John Wiley & Sons Inc, 1988. [30] V. Rubakov and S. S. Wilson, Classical Theory of Gauge Fields, Princeton University Press, 2002. [31] B. Rubin, Notes on radon transforms in integral geometry, Fract. Calc. Appl. Anal., 6 (2003), 25-72. [32] M. Salo and G. Uhlmann, The attenuated ray transform on simple surfaces, J. Diff. Geom., 88 (2011), 161-187. [33] D. Sarason, Complex Function Theory, American Mathematical Society, second ed., 2007. [34] V. Sharafudtinov, Integral Geometry of Tensor Fields, VSP, Utrecht, The Netherlands, 1994. doi: 10.1515/9783110900095. [35] G. Uhlmann, Inside Out: Inverse Problems and Applications, Cambridge University Press, 2003. doi: 10.1090/conm/333. [36] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970. [37] M. E. Taylor, Partial Differential Equations, vol. 115-117 of Applied Mathematical Sciences, Springer, 1996.

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##### References:
 [1] L. Ahlfors, Complex Analysis, McGraw-Hill, 1978. [2] L. V. Ahlfors, Lectures on Quasiconformal Mappings, University Lecture Series, American Mathematical Society, 2006. [3] L. Ahlfors and L. Bers, Riemann's mapping theorem for variable metrics, Ann. of Math, 72 (1960), 385-404. [4] E. Arbuzov, A. Bukhgeim and S. Kazantsev, Two-dimensional tomography problems and the theory of A-analytic functions, Siberian Advances in Mathematics, 8 (1998), 1-20. [5] K. Astala, T. Iwaniec and G. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane, Princeton University Press, 2009. [6] G. Bal, Ray transforms in hyperbolic geometry, J. Math. Pures Appl., 84 (2005), 1362-1392. doi: 10.1016/j.matpur.2005.02.001. [7] G. Bal, On the attenuated Radon transform with full and partial measurements, Inverse Problems, 20 (2004), 399-418. doi: 10.1088/0266-5611/20/2/006. [8] H. Begehr, Complex Analytic Methods for Partial Differential Equations, World Scientific Publishing Co., 1994. [9] C. Berenstein and E. C. Tarabush, Integral geometry in hyperbolic spaces and electrical impedance tomography, SIAM J. Appl. Math., 56 (1996), 755-764. doi: 10.1137/S0036139994277348. [10] P. Colwell, Blaschke Products: Bounded Analytical Functions, University of Michigan Press, 1985. [11] L. Ehrenpreis, The Universality of the Radon Transform, Oxford Mathematical Monographs, Oxford University Press, USA, 2003. doi: 10.1093/acprof:oso/9780198509783.001.0001. [12] L. C. Evans, Partial Differential Equations, 19 of Graduate Studies in Mathematics, American Mathematical Society, 1998. [13] D. Finch, Uniqueness for the X-ray transform in the physical range, Inverse Problems, 2 (1986), 197-203. doi: 10.1088/0266-5611/2/2/010. [14] M. J. Ablowitz and A. S. Fokas, Complex Variables: Introduction and Applications, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2003. doi: 10.1017/CBO9780511791246. [15] J. B. Garnett, Bounded Analytic Functions, Springer New York, 1981. [16] R. E. Greene and S. G. Krantz, Function Theory of One Complex Variable, 40 of Graduate Studies in Mathematics, American Mathematical Society, 2006. [17] D. Griffiths, Introduction to Elementary Particles, Wiley-VCH, 2008. doi: 10.1002/9783527618460. [18] S. Helgason, The Radon Transform, 5 of Progress in Mathematics, Birkhäuser Boston, 1980. [19] ______, Groups and Geometric Analysis (Integral Geometry, Invariant Differential Operators and Spherical Functions), American Mathematical Society, 2000. [20] ______, The inversion of the x-ray transform on a compact symmetric space, Journal of Lie Theory, 17 (2007), 307-315. [21] L. Hormander, Complex Analysis in Several Variables, North Holland, 1990. [22] S. S. Romesh Kumar, Inner functions and substitution operators, Acta Sci. Math. (Szegal), 58 (1993), 509-516. [23] J. M. Lee, Riemannian Manifolds: An Introduction to Curvature, 176 in Graduate Texts in Mathematics, Springer, 1997. [24] F. Natterer, Inversion of the attenuated radon transform, Inverse Problems, 17 (2001), 113-119. doi: 10.1088/0266-5611/17/1/309. [25] F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction, (Monographs on Mathematical Modeling and Computation), Society for Industrial Mathematics, 2007. doi: 10.1118/1.1455744. [26] Z. Nehari, Conformal Mappings, McGraw-Hill Book Company, Inc., 1952. [27] R. Novikov, An inversion formula for the attenuated x-ray transformation, Ark. Math, 40 (2002), 145-167. doi: 10.1007/BF02384507. [28] L. Pestov and G. Uhlmann, On characterization of range and inversion formulas for the geodesic x-ray transform, International Math. Research Notices, 80 (2004), 4331-4347. doi: 10.1155/S1073792804142116. [29] H. Renelt, Elliptic Systems and Quasiconformal Mappings, John Wiley & Sons Inc, 1988. [30] V. Rubakov and S. S. Wilson, Classical Theory of Gauge Fields, Princeton University Press, 2002. [31] B. Rubin, Notes on radon transforms in integral geometry, Fract. Calc. Appl. Anal., 6 (2003), 25-72. [32] M. Salo and G. Uhlmann, The attenuated ray transform on simple surfaces, J. Diff. Geom., 88 (2011), 161-187. [33] D. Sarason, Complex Function Theory, American Mathematical Society, second ed., 2007. [34] V. Sharafudtinov, Integral Geometry of Tensor Fields, VSP, Utrecht, The Netherlands, 1994. doi: 10.1515/9783110900095. [35] G. Uhlmann, Inside Out: Inverse Problems and Applications, Cambridge University Press, 2003. doi: 10.1090/conm/333. [36] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970. [37] M. E. Taylor, Partial Differential Equations, vol. 115-117 of Applied Mathematical Sciences, Springer, 1996.
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