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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

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 & Imaging, 2014, 8 (1) : 103-125. doi: 10.3934/ipi.2014.8.103
References:
[1]

L. Ahlfors, Complex Analysis,, McGraw-Hill, (1978).   Google Scholar

[2]

L. V. Ahlfors, Lectures on Quasiconformal Mappings,, University Lecture Series, (2006).   Google Scholar

[3]

L. Ahlfors and L. Bers, Riemann's mapping theorem for variable metrics,, Ann. of Math, 72 (1960), 385.   Google Scholar

[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.   Google Scholar

[5]

K. Astala, T. Iwaniec and G. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane,, Princeton University Press, (2009).   Google Scholar

[6]

G. Bal, Ray transforms in hyperbolic geometry,, J. Math. Pures Appl., 84 (2005), 1362.  doi: 10.1016/j.matpur.2005.02.001.  Google Scholar

[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.  Google Scholar

[8]

H. Begehr, Complex Analytic Methods for Partial Differential Equations,, World Scientific Publishing Co., (1994).   Google Scholar

[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.  Google Scholar

[10]

P. Colwell, Blaschke Products: Bounded Analytical Functions,, University of Michigan Press, (1985).   Google Scholar

[11]

L. Ehrenpreis, The Universality of the Radon Transform,, Oxford Mathematical Monographs, (2003).  doi: 10.1093/acprof:oso/9780198509783.001.0001.  Google Scholar

[12]

L. C. Evans, Partial Differential Equations,, 19 of Graduate Studies in Mathematics, (1998).   Google Scholar

[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.  Google Scholar

[14]

M. J. Ablowitz and A. S. Fokas, Complex Variables: Introduction and Applications,, Cambridge Texts in Applied Mathematics, (2003).  doi: 10.1017/CBO9780511791246.  Google Scholar

[15]

J. B. Garnett, Bounded Analytic Functions,, Springer New York, (1981).   Google Scholar

[16]

R. E. Greene and S. G. Krantz, Function Theory of One Complex Variable,, 40 of Graduate Studies in Mathematics, (2006).   Google Scholar

[17]

D. Griffiths, Introduction to Elementary Particles,, Wiley-VCH, (2008).  doi: 10.1002/9783527618460.  Google Scholar

[18]

S. Helgason, The Radon Transform,, 5 of Progress in Mathematics, (1980).   Google Scholar

[19]

______, Groups and Geometric Analysis (Integral Geometry, Invariant Differential Operators and Spherical Functions),, American Mathematical Society, (2000).   Google Scholar

[20]

______, The inversion of the x-ray transform on a compact symmetric space,, Journal of Lie Theory, 17 (2007), 307.   Google Scholar

[21]

L. Hormander, Complex Analysis in Several Variables,, North Holland, (1990).   Google Scholar

[22]

S. S. Romesh Kumar, Inner functions and substitution operators,, Acta Sci. Math. (Szegal), 58 (1993), 509.   Google Scholar

[23]

J. M. Lee, Riemannian Manifolds: An Introduction to Curvature,, 176 in Graduate Texts in Mathematics, (1997).   Google Scholar

[24]

F. Natterer, Inversion of the attenuated radon transform,, Inverse Problems, 17 (2001), 113.  doi: 10.1088/0266-5611/17/1/309.  Google Scholar

[25]

F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction,, (Monographs on Mathematical Modeling and Computation), (2007).  doi: 10.1118/1.1455744.  Google Scholar

[26]

Z. Nehari, Conformal Mappings,, McGraw-Hill Book Company, (1952).   Google Scholar

[27]

R. Novikov, An inversion formula for the attenuated x-ray transformation,, Ark. Math, 40 (2002), 145.  doi: 10.1007/BF02384507.  Google Scholar

[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.  Google Scholar

[29]

H. Renelt, Elliptic Systems and Quasiconformal Mappings,, John Wiley & Sons Inc, (1988).   Google Scholar

[30]

V. Rubakov and S. S. Wilson, Classical Theory of Gauge Fields,, Princeton University Press, (2002).   Google Scholar

[31]

B. Rubin, Notes on radon transforms in integral geometry,, Fract. Calc. Appl. Anal., 6 (2003), 25.   Google Scholar

[32]

M. Salo and G. Uhlmann, The attenuated ray transform on simple surfaces,, J. Diff. Geom., 88 (2011), 161.   Google Scholar

[33]

D. Sarason, Complex Function Theory,, American Mathematical Society, (2007).   Google Scholar

[34]

V. Sharafudtinov, Integral Geometry of Tensor Fields,, VSP, (1994).  doi: 10.1515/9783110900095.  Google Scholar

[35]

G. Uhlmann, Inside Out: Inverse Problems and Applications,, Cambridge University Press, (2003).  doi: 10.1090/conm/333.  Google Scholar

[36]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton University Press, (1970).   Google Scholar

[37]

M. E. Taylor, Partial Differential Equations,, vol. 115-117 of Applied Mathematical Sciences, (1996), 115.   Google Scholar

show all references

References:
[1]

L. Ahlfors, Complex Analysis,, McGraw-Hill, (1978).   Google Scholar

[2]

L. V. Ahlfors, Lectures on Quasiconformal Mappings,, University Lecture Series, (2006).   Google Scholar

[3]

L. Ahlfors and L. Bers, Riemann's mapping theorem for variable metrics,, Ann. of Math, 72 (1960), 385.   Google Scholar

[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.   Google Scholar

[5]

K. Astala, T. Iwaniec and G. Martin, Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane,, Princeton University Press, (2009).   Google Scholar

[6]

G. Bal, Ray transforms in hyperbolic geometry,, J. Math. Pures Appl., 84 (2005), 1362.  doi: 10.1016/j.matpur.2005.02.001.  Google Scholar

[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.  Google Scholar

[8]

H. Begehr, Complex Analytic Methods for Partial Differential Equations,, World Scientific Publishing Co., (1994).   Google Scholar

[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.  Google Scholar

[10]

P. Colwell, Blaschke Products: Bounded Analytical Functions,, University of Michigan Press, (1985).   Google Scholar

[11]

L. Ehrenpreis, The Universality of the Radon Transform,, Oxford Mathematical Monographs, (2003).  doi: 10.1093/acprof:oso/9780198509783.001.0001.  Google Scholar

[12]

L. C. Evans, Partial Differential Equations,, 19 of Graduate Studies in Mathematics, (1998).   Google Scholar

[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.  Google Scholar

[14]

M. J. Ablowitz and A. S. Fokas, Complex Variables: Introduction and Applications,, Cambridge Texts in Applied Mathematics, (2003).  doi: 10.1017/CBO9780511791246.  Google Scholar

[15]

J. B. Garnett, Bounded Analytic Functions,, Springer New York, (1981).   Google Scholar

[16]

R. E. Greene and S. G. Krantz, Function Theory of One Complex Variable,, 40 of Graduate Studies in Mathematics, (2006).   Google Scholar

[17]

D. Griffiths, Introduction to Elementary Particles,, Wiley-VCH, (2008).  doi: 10.1002/9783527618460.  Google Scholar

[18]

S. Helgason, The Radon Transform,, 5 of Progress in Mathematics, (1980).   Google Scholar

[19]

______, Groups and Geometric Analysis (Integral Geometry, Invariant Differential Operators and Spherical Functions),, American Mathematical Society, (2000).   Google Scholar

[20]

______, The inversion of the x-ray transform on a compact symmetric space,, Journal of Lie Theory, 17 (2007), 307.   Google Scholar

[21]

L. Hormander, Complex Analysis in Several Variables,, North Holland, (1990).   Google Scholar

[22]

S. S. Romesh Kumar, Inner functions and substitution operators,, Acta Sci. Math. (Szegal), 58 (1993), 509.   Google Scholar

[23]

J. M. Lee, Riemannian Manifolds: An Introduction to Curvature,, 176 in Graduate Texts in Mathematics, (1997).   Google Scholar

[24]

F. Natterer, Inversion of the attenuated radon transform,, Inverse Problems, 17 (2001), 113.  doi: 10.1088/0266-5611/17/1/309.  Google Scholar

[25]

F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction,, (Monographs on Mathematical Modeling and Computation), (2007).  doi: 10.1118/1.1455744.  Google Scholar

[26]

Z. Nehari, Conformal Mappings,, McGraw-Hill Book Company, (1952).   Google Scholar

[27]

R. Novikov, An inversion formula for the attenuated x-ray transformation,, Ark. Math, 40 (2002), 145.  doi: 10.1007/BF02384507.  Google Scholar

[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.  Google Scholar

[29]

H. Renelt, Elliptic Systems and Quasiconformal Mappings,, John Wiley & Sons Inc, (1988).   Google Scholar

[30]

V. Rubakov and S. S. Wilson, Classical Theory of Gauge Fields,, Princeton University Press, (2002).   Google Scholar

[31]

B. Rubin, Notes on radon transforms in integral geometry,, Fract. Calc. Appl. Anal., 6 (2003), 25.   Google Scholar

[32]

M. Salo and G. Uhlmann, The attenuated ray transform on simple surfaces,, J. Diff. Geom., 88 (2011), 161.   Google Scholar

[33]

D. Sarason, Complex Function Theory,, American Mathematical Society, (2007).   Google Scholar

[34]

V. Sharafudtinov, Integral Geometry of Tensor Fields,, VSP, (1994).  doi: 10.1515/9783110900095.  Google Scholar

[35]

G. Uhlmann, Inside Out: Inverse Problems and Applications,, Cambridge University Press, (2003).  doi: 10.1090/conm/333.  Google Scholar

[36]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions,, Princeton University Press, (1970).   Google Scholar

[37]

M. E. Taylor, Partial Differential Equations,, vol. 115-117 of Applied Mathematical Sciences, (1996), 115.   Google Scholar

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