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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 & 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, (2006). [3] L. Ahlfors and L. Bers, Riemann's mapping theorem for variable metrics,, Ann. of Math, 72 (1960), 385. [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. [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. 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. 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. 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, (2003). doi: 10.1093/acprof:oso/9780198509783.001.0001. [12] L. C. Evans, Partial Differential Equations,, 19 of Graduate Studies in Mathematics, (1998). [13] D. Finch, Uniqueness for the X-ray transform in the physical range,, Inverse Problems, 2 (1986), 197. 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, (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, (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, (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. [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. [23] J. M. Lee, Riemannian Manifolds: An Introduction to Curvature,, 176 in Graduate Texts in Mathematics, (1997). [24] F. Natterer, Inversion of the attenuated radon transform,, Inverse Problems, 17 (2001), 113. 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), (2007). doi: 10.1118/1.1455744. [26] Z. Nehari, Conformal Mappings,, McGraw-Hill Book Company, (1952). [27] R. Novikov, An inversion formula for the attenuated x-ray transformation,, Ark. Math, 40 (2002), 145. 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. 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. [32] M. Salo and G. Uhlmann, The attenuated ray transform on simple surfaces,, J. Diff. Geom., 88 (2011), 161. [33] D. Sarason, Complex Function Theory,, American Mathematical Society, (2007). [34] V. Sharafudtinov, Integral Geometry of Tensor Fields,, VSP, (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, (1996), 115.

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##### References:
 [1] L. Ahlfors, Complex Analysis,, McGraw-Hill, (1978). [2] L. V. Ahlfors, Lectures on Quasiconformal Mappings,, University Lecture Series, (2006). [3] L. Ahlfors and L. Bers, Riemann's mapping theorem for variable metrics,, Ann. of Math, 72 (1960), 385. [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. [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. 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. 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. 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, (2003). doi: 10.1093/acprof:oso/9780198509783.001.0001. [12] L. C. Evans, Partial Differential Equations,, 19 of Graduate Studies in Mathematics, (1998). [13] D. Finch, Uniqueness for the X-ray transform in the physical range,, Inverse Problems, 2 (1986), 197. 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, (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, (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, (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. [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. [23] J. M. Lee, Riemannian Manifolds: An Introduction to Curvature,, 176 in Graduate Texts in Mathematics, (1997). [24] F. Natterer, Inversion of the attenuated radon transform,, Inverse Problems, 17 (2001), 113. 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), (2007). doi: 10.1118/1.1455744. [26] Z. Nehari, Conformal Mappings,, McGraw-Hill Book Company, (1952). [27] R. Novikov, An inversion formula for the attenuated x-ray transformation,, Ark. Math, 40 (2002), 145. 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. 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. [32] M. Salo and G. Uhlmann, The attenuated ray transform on simple surfaces,, J. Diff. Geom., 88 (2011), 161. [33] D. Sarason, Complex Function Theory,, American Mathematical Society, (2007). [34] V. Sharafudtinov, Integral Geometry of Tensor Fields,, VSP, (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, (1996), 115.
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