Ray transforms on a conformal class of curves

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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

Nicholas Hoell ^1, and Guillaume Bal ^2,

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

Abstract
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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.

Keywords: Beltrami equation, quasiconformal, harmonic, Transport equation, filtered backprojection., attenuated ray transform.

Mathematics Subject Classification: Primary: 35R30, 35F45; Secondary: 53C6.

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.

show all references

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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