J. Adler and O. Öktem, Solving ill-posed inverse problems using iterative deep neural networks, Inverse Problems, vol.33, issue.12, p.124007, 2017.

A. D. Agaltsov, Imaging and integral geometry. Essay, 2014.

A. D. Agaltsov, A characterization theorem for a generalized Radon transform arising in a model of mathematical economics, Functional Analysis and Its Applications, vol.49, pp.201-204, 2015.

A. D. Agaltsov, E. G. Molchanov, and A. A. Shananin, Inverse Problems in Models of Resource Distribution, The Journal of Geometric Analysis, vol.28, issue.1, pp.726-765, 2018.

S. Alinhac and P. Gérard, Pseudo-differential operators and the Nash-Moser theorem, American Mathematical Soc, vol.82, 2007.

E. V. Arbuzov, A. L. Bukhgeim, and S. G. Kazantsev, Two-dimensional tomography problems and the theory of A-analytic functions, Siberian Advances in Mathematics, vol.8, issue.4, pp.1-20, 1998.

G. Bal, Ray transforms in hyperbolic geometry, Journal de mathématiques pures et appliquées, vol.84, issue.10, pp.1362-1392, 2005.

G. Bal, Inverse transport theory and applications, Inverse Problems, vol.25, p.53001, 2009.

G. Bal and A. Jollivet, Combined source and attenuation reconstructions in SPECT. Tomography and Inverse Transport Theory, vol.559, pp.13-27, 2011.

H. H. Barrett and K. K. Myers, Foundation of Imaging Science, 2004.

G. Beylkin, The inversion problem and applications of the generalized Radon transform, Communications on pure and applied mathematics, vol.37, issue.5, pp.579-599, 1984.

R. N. Bracewell, Strip integration in radio astronomy, Australian Journal of Physics, vol.9, issue.2, pp.198-217, 1956.

H. Bockwinkel, On the propagation of light in a biaxial crystal about a midpoint of oscillation, Verh. Konink Acad. V. Wet. Wissen. Natur, vol.14, p.636, 1906.

J. Boman and E. T. Quinto, Support theorems for real-analytic Radon transforms, 1987.

J. Boman, An example of non-uniqueness for a generalized Radon transform, Journal d'Analyse Mathématique, vol.61, issue.1, pp.395-401, 1993.

J. Boman and J. Strömberg, Novikov's inversion formula for the attenuated Radon transform-a new approach, The Journal of Geometric Analysis, vol.14, issue.2, 2004.

J. Boman, Local non-injectivity for weighted Radon transforms, Contemp. Math, vol.559, pp.39-47, 2011.

A. V. Bronnikov, Reconstruction of attenuation map using discrete consistency conditions, IEEE Transactions on medical imaging, vol.19, issue.5, pp.451-462, 2000.

L. Chang, A method for attenuation correction in radionuclide computed tomography, IEEE Transactions on Nuclear Science, vol.25, issue.1, pp.638-643, 1978.

A. M. Cormack, Representation of a function by its line integrals, with some radiological applications, Journal of applied physics, vol.34, issue.9, pp.2722-2727, 1963.

A. M. Cormack, Representation of a function by its line integrals, with some radiological applications, Journal of applied physics, vol.II, issue.10, pp.2908-2913, 1963.

M. Allan, Cormack -Biographical. NobelPrize.org. NobelMedia AB, 2019.

A. M. Cormack, Recollections of my work with computer assisted tomography, Molecular and cellular biochemistry, vol.32, issue.2, pp.57-61, 1980.

F. Crepaldi, A. , and R. D. Pierro, Activity and attenuation reconstruction for positron emission tomography using emission data only via maximum likelihood and iterative data refinement, IEEE Transactions on Nuclear Science, vol.54, issue.1, pp.100-106, 2007.

S. Dann, On the Minkowski-Funk transform. arXiv preprint: 1003.5565, 2010.

S. R. Deans, The Radon transform and some of its applications, Courier Corporation, 2007.

E. History-of and L. ,

A. Faridani, K. A. Buglione, P. Huabsomboon, O. D. Iancu, and J. Mcgrath, Introduction to local tomography, Contemporary Mathematics, vol.278, pp.29-48, 2001.

D. Finch, Uniqueness for attenuated X-ray transform in the physical range, Inverse Problems, vol.2, issue.2, 1986.

D. V. Finch, The attenuated X-ray transform: recent developments. Inverse Problems and Applications, pp.47-66, 2003.

K. Fritz, Inversion of k-plane Transforms and Applications in Computer Tomographies, SIAM Review, vol.31, issue.2, pp.273-298, 1989.

B. Frigyik, P. Stefanov, and G. Uhlmann, The X-ray transform for a generic family of curves and weights, Journal of Geometric Analysis, vol.18, issue.1, 2008.

P. Funk, Uber Flächen mit lauter geschlossenen geodätischen Linien. Mathematische Annalen, vol.74, 1913.

P. Funk, Nachruf auf Prof. Johann Radon. Mathematische Nachrichten, vol.62, pp.191-199, 1913.

H. M. Gach, C. Tanase, and F. Boada, 2D & 3D Shepp-Logan phantom standards for MRI. 19th International Conference on Systems Engineering, pp.521-526, 2008.

I. M. Gelfand, S. Gindikin, and M. Graev, Selected topics in integral geometry, American Mathematical Soc, vol.220, 2003.

I. Gelfand and M. I. Graev, Geometry of homogeneous spaces, representations of groups in homogeneous spaces and related questions of integral geometry, Trudy Moskov. Mat. Obshch, vol.8, pp.321-390, 1959.

I. M. Gelfand, M. I. Graev, and Z. Y. Shapiro, Differential forms and integral geometry, Functional Anal. and Appl, vol.3, pp.24-40, 1969.

I. M. Gelfand, M. I. Graev, and N. J. Vilenkin, Obobshchennye funktsii, Vyp. 5. Integralnaya geometriya i svyazannye s nei voprosy teorii predstavlenii, Gosudarstv. Izdat. Fiz.-Mat. Lit, 1962.

S. Gindikin, A remark on the weighted Radon transform on the plane, Inverse Problems & Imaging, vol.4, issue.4, pp.649-653, 2010.

F. O. Goncharov, An iterative inversion of weighted Radon transforms along hyperplanes, Inverse Problems, vol.33, issue.12, p.124005, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01405387

F. O. Goncharov and R. G. Novikov, An analog of Chang inversion formula for weighted Radon transforms in multidimensions, Eurasian Journal of Mathematical and Computer Applications, vol.4, issue.2, pp.23-32, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01304312

F. O. Goncharov and R. G. Novikov, An example of non-uniqueness for Radon transforms with continuous positive rotation invariant weights, The Journal of Geometric Analysis, vol.28, issue.4, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01593781

F. O. Goncharov and R. G. Novikov, A breakdown of injectivity for weighted ray transforms in multidimensions. arXiv preprint, 2019.
URL : https://hal.archives-ouvertes.fr/hal-01635188

F. O. Goncharov and R. G. Novikov, An example of non-uniqueness for weighted Radon transforms along hyperplanes in multidimensions, Inverse Problems, pp.34-054001, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01583755

P. Grangeat, Mathematical framework of cone beam 3D reconstruction via the first derivative of the Radon transform. Mathematical methods in tomography, pp.66-97, 1991.

J. Guillement, F. Jauberteau, L. Kunyansky, R. G. Novikov, and R. Trebossen, On single-photon emission computed tomography imaging based on an exact formula for the nonuniform attenuation correction, Inverse Problems, vol.18, issue.6, p.11, 2002.

J. Guillement and R. G. Novikov, On Wiener type filters in SPECT, Inverse Problems, vol.24, issue.2, 2008.
URL : https://hal.archives-ouvertes.fr/hal-00150503

J. Guillement and R. G. Novikov, On the data dependent filtration techniques in single-photon emission computed tomography. hal archives, p.9611, 2005.
URL : https://hal.archives-ouvertes.fr/hal-00009611

J. Guillement and R. G. Novikov, Optimized analytic reconstruction for SPECT, Journal of Invers and Ill-Posed Problems, vol.20, issue.4, pp.489-500, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00662909

J. Guillement and R. G. Novikov, Inversion of weighted Radon transforms via finite Fourier series weight approximations, Inverse Problems in Science and Engineering, vol.22, issue.5, pp.787-802, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00794083

C. Guillarmou, G. P. Paternain, M. Salo, and G. Gunther-uhlmann, The X-ray transform for connections in negative curvature, Communications in Mathematical Physics, vol.343, issue.1, pp.83-127, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01236244

S. Helgason, The Radon transform, vol.2, 1999.

G. M. Henkin and A. A. Shananin, Application to the theory of production functions, AMS: Translation of mathematical monographs, vol.81, pp.189-223, 1990.

. Harish-chandra, Spherical functions on a semi-simple Lie group I, American Journal of Mathematics, vol.80, pp.241-310, 1958.

. Harish-chandra, Spherical functions on a semi-simple Lie group II, American Journal of Mathematics, vol.80, pp.553-613, 1958.

S. Helgason, Radon-Fourier transforms on symmetric spaces and related group representations, Bull. Amer. Math. Soc, vol.71, pp.757-763, 1965.

S. Helgason, The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassman manifolds, Acta Math, vol.113, pp.153-180, 1965.

S. Helgason, A duality in integral geometry on symmetric spaces, Proc. U.S.-Japan Seminar in Differential Geometry, 1965.

S. Helgason, A duality for symmetric spaces, with applications to group representations, Advances in Math, vol.5, pp.1-154, 1970.

T. Hohage and F. Werner, Inverse problems with Poisson data: statistical regularization theory. Applications and algorithms. Inverse Problems, vol.32, p.93001, 2016.

G. Hounsfield, Method and Apparatus for measuring X-or gamma-radiation absorption or transmission at plural angles and analyzing the data. United States Patent US3778614A, 1971.

G. Hounsfield and N. Godfrey, Hounsfield -Biographical. NobelPrize.org. NobelMedia AB 2019. Mon, 2019.

J. Ilmavirta, On Radon transforms on compact Lie groups, Proceedings of the American Mathematical Society, vol.144, pp.681-691, 2016.

J. Ilmavirta and J. Rassalo, Geodesic ray transform with matrix weights for piecewise constant functions arXiv preprint, 2019.

F. John, Plane Waves and Spherical Means Applied to Partial Differential Equations, 1955.

J. Ilmavirta, Coherent quantum tomography, SIAM Journal on Mathematical Analysis, vol.48, issue.5, pp.3039-3064, 2016.

A. Katsevich, Theoretically exact filtered backprojection-type inversion algorithm for spiral CT, SIAM Journal on Applied Mathematics, vol.62, issue.6, pp.2012-2026, 2002.

L. Kaufman, Maximum likelihood, least squares, and penalized least squares for PET, IEEE Transactions on Medical Imaging, vol.12, issue.2, pp.200-214, 1993.

S. Kazantsev, Funk-Minkowski transform and spherical convolution of Hilbert type and reconstructing functions on the sphere, 2018.

J. Keiner, S. Kunis, and D. Potts, Using NFFT 3 -a software library for various nonequispaced fast Fourier transforms, ACM Trans. Math. Software, vol.36, 2009.

P. Kuchement, Generalized transforms of Radon-type and their applications, Proceedings of Symposia in Applied Mathematics, 2006.

P. Kuchement, The Radon transform and medical imaging, SIAM, vol.85, 2014.

P. Kuchment, K. Lancaster, and L. Mogilevskaya, On local tomography, Inverse Problems, vol.11, issue.3, pp.571-589, 1995.

L. Kunyansky, Generalized and attenuated Radon transforms: restorative approach to the numerical inversion, Inverse Problems, vol.8, issue.5, pp.809-819, 1992.

L. Kunyansky, A new SPECT reconstruction algorithm based on Novikov explicit inversion formula, Inverse Problems, vol.17, issue.2, 2001.

M. M. Lavrent'ev and A. L. Bukhgeim, A class of operator equations of the first kind, Functional Analysis and Its Applications, vol.7, issue.4, pp.290-298, 1973.

M. Lavrent'ev, V. G. Romanov, and S. P. Shishatski, Ill-posed problems of mathematical physics, vol.64, 1986.

. Ph and . Mader, Über die Darstellung von Punktfunktionen im n-dimensionalen euklidischen Raum durch Ebenenintegrale, Math. Zeit, vol.26, pp.646-652, 1927.

A. Markoe, Fourier inversion of the attenuated X-ray transform, SIAM Journal on Mathematical Analysis, vol.15, issue.4, pp.718-722, 1984.

A. Markoe and E. T. Quinto, An elementary proof of local invertibility for generalized and attenuated Radon transforms, SIAM Journal on Mathematical Analysis, vol.16, issue.5, pp.1114-1119, 1985.

D. Miller, M. Oristaglio, and G. Beylkin, A new slant on seismic imaging: Migration and integral geometry, Geophysics, vol.52, issue.7, pp.943-964, 1987.

H. Minkowski, About bodies of constant width, Mathematics Sbornik, vol.25, pp.505-508, 1904.

E. Miqueles and A. R. De-pierro, Iterative reconstruction in X-ray fluorescence tomography based on Radon transform, IEEE Transactions on medical imaging, vol.30, issue.2, pp.438-450, 2011.

K. Murase, H. Itoh, H. Mogami, M. Ishine, M. Kawamura et al., A comparative study of attenuation correction algorithms in single photon emission computed tomography (SPECT), European Journal on Nuclear Medicine, vol.13, pp.55-62, 1987.

F. Natterer, The mathematics of computerized tomography, SIAM, vol.32, 1986.

F. Natterer, Mathematical methods in tomography, Acta Numerica, vol.8, pp.107-141, 1999.

F. Natterer, Inversion of the attenuated Radon transform. Inverse problems, vol.17, 2001.

M. K. Nguyen, T. T. Truong, D. Driol, and H. Zaidi, On a novel approach to Compton scattered emission imaging, IEEE Transactions on Nuclear Science, vol.56, issue.3, pp.1430-1437, 2009.
URL : https://hal.archives-ouvertes.fr/hal-00483666

R. G. Novikov, Weighted Radon transforms for which Chang's approximate inversion formula is exact, Russian Mathematical Surveys, vol.66, issue.2, 2011.

R. G. Novikov, Weighted Radon transforms and first order differential systems on the plane, Moscow Mathematical Journal, vol.14, issue.4, pp.807-823, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00714524

R. G. Novikov, On determination of a gauge field on R d from its non-abelian Radon transform along oriented straight lines, Journal of the Institute of Mathematics of Jussieu, vol.1, issue.4, pp.559-629, 2002.

G. P. Paternain, M. Salo, and G. Uhlmann, The attenuated ray transform for connections and Higgs fields. Geometric and functional analysis, vol.22, pp.1460-1489, 2012.

R. G. Novikov, An inversion formula for the attenuated X-ray transform, Arkiv für Mathematik, vol.40, issue.1, pp.145-167, 2002.

J. Pietzsch, With a Little Help from My Friends. NobelPrize.org. Nobel Media AB 2019. Fri, 2019.

A. Pli?, On non-uniqueness in Cauchy problem for an elliptic second order differential equation, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys, vol.11, 1963.

J. Radon, Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten, Ber. Saechs Akad. Wiss. Leipzig, Math-Phys, vol.69, pp.262-267, 1917.

R. Ramlau and O. Scherzer, The first 100 years of Radon transform, Inverse Problems, pp.34-090201, 2018.

V. A. Sharafutdinov, Uniqueness theorems for the exponential X-ray transform, Journal of Inverse and Ill-Posed Problems, vol.1, issue.4, pp.355-372, 1993.

V. A. Sharafutdinov, Integral geometry of tensor fields, vol.1, 2012.

L. Shepp and B. Logan, The Fourier reconstruction of a head section, IEEE Transactions on nuclear science, vol.21, issue.3, pp.21-43, 1974.

E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces (PMS-32), 2016.

R. S. Strichartz, Radon inversion-variations on a theme, The American Mathematical Monthly, vol.89, issue.6, pp.377-423, 1982.

R. S. Strichartz, L p harmonic analysis and Radon transforms on the Heisenberg group, Journal of functional Analysis, vol.96, issue.2, pp.350-406, 1991.

A. N. Tikhonov, V. Ya, and . Arsenin, Methods of solving ill-posed problems, 1974.

P. Toft, The Radon Transform: Theory and Implementation, 1996.

O. Tretiak and C. Metz, The exponential Radon transform, SIAM Journal on Applied Mathematics, vol.39, issue.2, pp.341-354, 1980.

E. T. Quinto, The dependence of the generalized Radon transforms on defining measures, Transactions on the American Mathematical Society, vol.257, issue.2, pp.331-346, 1980.

E. T. Quinto, The invertibility of rotation invariant Radon transforms, Journal of Mathematical Analysis and Applications, vol.91, issue.2, pp.510-522, 1983.

E. T. Quinto, Singularities of the X-ray transform and limited data tomography in R 2 and R 3, SIAM Journal on Mathematical Analysis, vol.24, issue.5, pp.1215-1225, 1993.

E. T. Quinto, An introduction to X-ray tomography and Radon transforms, Proceedings of symposia in Applied Mathematics, vol.63, 2006.

E. T. Quinto, G. Ambartsoumian, R. Felea, V. Krishnan, and C. Nolan, Microlocal Analysis and Imaging, a short note in "The mathematics of the planet Earth, pp.8-11, 2014.

E. T. Quinto, C. Grathwohl, P. Kunstmann, and A. Rieder, Microlocal Analysis of imaging operators for effective common offset seismic reconstruction, Inverse Problems, vol.34, issue.11, 2018.

K. Van-slambrouck, S. Stute, C. Comtat, M. Sibomana, F. H. Van-velden et al., Bias reduction for low-statistics PET: maximum likelihood reconstruction with a modified Poisson distribution, IEEE Transactions on medical imaging, vol.34, issue.1, pp.126-136, 2015.

M. Vassholz, B. Koberstein-schwarz, A. Ruhlandt, M. Krenkel, and T. Salditt, New Xray tomography method based on the 3D Radon transform compatible with anisotropic sources, Physical review letters, vol.116, issue.8, p.88101, 2016.

J. Wang, H. Lu, Z. Liang, D. Eremina, G. Zhang et al., An experimental study on the noise properties of X-ray CT sinogram data in Radon space, Physics in Medicine & Biology, vol.53, issue.12, p.3327, 2008.

J. Boman and E. T. Quinto, Support theorems for real-analytic Radon transforms, Duke Mathematical J, vol.55, issue.4, pp.943-948, 1987.

J. Boman and J. Strömberg, Novikov's inversion formula for the attenuated Radon transform -a new approach, The Journal of Geometric Analysis, vol.14, issue.2, pp.185-198, 2004.

L. Chang, A method for attenuation correction in radionuclide computed tomography, IEEE Transactions on Nuclear Science, vol.25, issue.1, pp.638-643, 1978.

S. R. Deans, The Radon Transform and some of Its Applications, Courier Corporation, 2007.

T. Durrani and D. Bisset, The Radon transform and its properties, Geophysics, issue.8, pp.1180-1187, 1984.

I. M. Gel'fand, M. I. Graev, N. Ya, and . Vilenkin, Integral Geometry and Representation Theory, vol.5, 2014.

S. Gindikin, A remark on the weighted Radon transform on the plane, Inverse Problems and Imaging, vol.4, pp.649-653, 2010.

P. Grangeat, Mathematical framework of cone beam 3D reconstruction via the first derivative of the Radon transform, Mathematical methods in tomography, pp.66-97, 1991.

S. Helgason, The Radon Transform on R n, 2011.

L. A. Kunyansky, Generalized and attenuated Radon transforms: restorative approach to the numerical inversion, Inverse Problems, vol.8, issue.5, pp.809-819, 1992.

D. Ludwig, The Radon transform on Euclidean space, Communications on Pure and Applied Mathematics, vol.19, issue.1, pp.49-81, 1996.

F. Natterer, The Mathematics of Computerized Tomography, vol.32, 1986.

R. G. Novikov, An inversion formula for the attenuated X-ray transformation, Arkiv för matematik, vol.40, issue.1, pp.145-167, 2002.

R. G. Novikov, Weighted Radon transforms for which Chang's approximate inversion formula is exact, Russian Mathematical Surveys, vol.66, issue.2, pp.442-443, 2011.

J. Radon, Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten, Ber. Saechs Akad. Wiss. Leipzig, Math-Phys, vol.69, pp.262-267, 1917.

O. Tretiak and C. Metz, The exponential Radon transform, SIAM Journal on Applied Mathematics, vol.39, issue.2, pp.341-354, 1980.

J. Boman and E. T. Quinto, Support theorems for real-analytic Radon transforms, Duke Mathematical J, vol.55, issue.4, pp.943-948, 1987.

G. Beylkin, The inversion problem and applications of the generalized Radon transform, Communications on pure and applied mathematics, vol.37, issue.5, pp.579-599, 1984.

L. Chang, A method for attenuation correction in radionuclide computed tomography, IEEE Transactions on Nuclear Science, vol.25, issue.1, pp.638-643, 1978.

A. Denisiuk, Inversion of the x-ray transform for complexes of lines in R n, Inverse Problems, vol.32, issue.2, 2016.

F. O. Goncharov and R. G. Novikov, An analog of Chang inversion formula for weighted Radon transforms in multidimensions, Eurasian Journal of Mathematical and Computer Applications, vol.4, issue.2, pp.23-32, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01304312

F. O. Goncharov, Integrals of spherical harmonics with Fourier exponents in multidimensions, Eurasian Journal of Mathematical and Computer Applications, vol.4, issue.4, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01415990

P. Grangeat, Mathematical framework of cone beam 3D reconstruction via the first derivative of the Radon transform, Mathematical methods in tomography, pp.66-97, 1991.

J. Guillement and R. G. Novikov, Inversion of weighted Radon transforms via finite Fourier series weight approximations, Inverse Problems in Science and Engineering, vol.22, issue.5, pp.787-802, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00794083

A. Higuchi, Symmetric tensor spherical harmonics on the n-sphere and their application to the de Sitter group SO(N, 1), Journal of mathematical physics, vol.28, issue.7, pp.1553-1566, 1987.

A. W. Knapp, Advanced real analysis, 2005.

L. A. Kunyansky, Generalized and attenuated Radon transforms: restorative approach to the numerical inversion, Inverse Problems, vol.8, issue.5, p.809, 1992.

G. Lohöfer, Inequalities for the Associated Legendre functions, Journal of Approximation Theory, vol.95, pp.178-193, 1998.

M. Morimoto, Analytic functionals on the sphere, 1998.

F. Natterer, The mathematics of computerized tomography, vol.32, 1986.

A. A. Neves, L. A. Padilha, A. F. , E. Rodriguez, C. Cruz et al., Analytical results for a Bessel function times Legendre polynomials class integrals, Journal of Physics A: Mathematical and General, vol.39, issue.18, p.293, 2006.

R. G. Novikov, Weighted Radon transforms for which Chang's approximate inversion formula is exact, Russian Mathematical Surveys, vol.66, issue.2, pp.442-443, 2011.

R. G. Novikov, Weighted Radon transforms and first order differential systems on the plane, Moscow mathematical journal, vol.14, issue.4, pp.807-823, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00714524

E. Quinto, The invertibility of rotation invariant Radon transforms, Journal of Mathematical Analysis and Applications, vol.91, issue.2, pp.510-522, 1983.

J. Radon, Uber die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten, Ber. Saechs Akad. Wiss. Leipzig, Math-Phys, vol.69, pp.262-267, 1917.

E. M. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces (PMS-32), vol.32, 2016.

N. M. Temme, Special functions: An introduction to the classical functions of mathematical physics, 2011.

S. B. Zhizhiashvili and L. V. Topuriya, Fourier-Laplace series on a sphere, Journal of Soviet Mathematics, vol.12, issue.6, pp.682-714, 1979.

G. Beylkin, The inversion problem and applications of the generalized Radon transform, Communications on pure and applied mathematics, vol.37, issue.5, pp.579-599, 1984.

J. Boman and E. Quinto, Support theorems for real-analytic Radon transforms, Duke Mathematical Journal, vol.55, issue.4, pp.943-948, 1987.

J. Boman, An example of non-uniqueness for a generalized Radon transform, Journal d'Analyse Mathematique, vol.61, issue.1, pp.395-401, 1993.

F. O. Goncharov and R. G. Novikov, An analog of Chang inversion formula for weighted Radon transforms in multidimensions, Eurasian Journal of Mathematical and Computer Applications, vol.4, issue.2, pp.23-32, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01304312

F. O. Goncharov, An iterative inversion of weighted Radon transforms along hyperplanes, Inverse Problems, vol.33, p.5, 2017.

A. Markoe and E. Quinto, An elementary proof of local invertibility for generalized and attenuated Radon transforms, SIAM Journal on Mathematical Analysis, vol.16, issue.5, pp.1114-1119, 1985.

J. Radon, Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten, Ber. Saechs Akad. Wiss. Leipzig, Math-Phys, vol.69, pp.262-267, 1917.

G. Beylkin, The inversion problem and applications of the generalized Radon transform, Communications on pure and applied mathematics, vol.37, issue.5, pp.579-599, 1984.

G. Beylkin, Imaging of discontinuities in the inverse scattering problem by inversion of a causal generalized Radon transform, Journal of Mathematical Physics, vol.26, issue.1, pp.99-108, 1985.

J. Boman and E. T. Quinto, Support theorems for real-analytic Radon transforms, Duke Mathematical J, vol.55, issue.4, pp.943-948, 1987.

J. Boman, An example of non-uniqueness for a generalized Radon transform, Journal d'Analyse Mathematique, vol.61, issue.1, pp.395-401, 1993.

M. P. Do-carmo and R. Geometry, , 1992.

D. Finch, Uniqueness for the attenuated X-ray transform in the physical range, Inverse problems, vol.2, issue.2, 1986.

F. O. Goncharov and R. G. Novikov, An analog of Chang inversion formula for weighted Radon transforms in multidimensions, Eurasian Journal of Mathematical and Computer Applications, vol.4, issue.2, pp.23-32, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01304312

F. O. Goncharov, An iterative inversion of weighted Radon transforms along hyperplanes, Inverse Problems, vol.33, p.124005, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01405387

F. O. Goncharov and R. G. Novikov, An example of non-uniqueness for the weighted Radon transforms along hyperplanes in multidimensions, Inverse Problems, pp.34-054001, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01583755

F. O. Goncharov and R. G. Novikov, A breakdown of injectivity for weighted ray transforms in multidimensions. hal-01635188, version 1, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01635188

J. Guillement and R. G. Novikov, Inversion of weighted Radon transforms via finite Fourier series weight approximations, Inverse Problems in Science and Engineering, vol.22, issue.5, pp.787-802, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00794083

L. Kunyansky, Generalized and attenuated Radon transforms: restorative approach to the numerical inversion, Inverse Problems, vol.8, issue.5, p.809, 1992.

M. M. Lavrent'ev and A. L. Bukhgeim, A class of operator equations of the first kind, Functional Analysis and Its Applications, vol.7, issue.4, pp.290-298, 1973.

A. Markoe and E. T. Quinto, An elementary proof of local invertibility for generalized and attenuated Radon transforms, SIAM Journal on Mathematical Analysis, vol.16, issue.5, pp.1114-1119, 1985.

F. Natterer, The Mathematics of Computerized Tomography. SIAM, 2001.

R. G. Novikov, Weighted Radon transforms for which Chang's approximate inversion formula is exact, Russian Mathematical Surveys, vol.66, issue.2, pp.442-443, 2011.

R. G. Novikov, Weighted Radon transforms and first order differential systems on the plane, Moscow Mathematical Journal, vol.14, issue.4, pp.807-823, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00714524

E. T. Quinto, The invertibility of rotation invariant Radon transforms, Journal of Mathematical Analysis and Applications, vol.91, issue.2, pp.510-522, 1983.

E. T. Quinto, The invertibility of rotation invariant Radon transforms, Erratum. Journal of Mathematical Analysis and Applications, vol.94, issue.2, pp.602-603, 1983.

, Using Lemma 1, one can see that f | ?(x,?) ? C ? 0 (R), f | ?(x,?) (u) = f | ?(x,?) (|u|)

, 139) it follows that ?? 0 > 0 (? 0 < ? 1 ) : W (x 0 ,? 0 ) (x, ?) ? 1/2 for (x, ?) ? ?, Lemma 1, properties of ? (x 0 ,? 0 ) of (7.131) and from formulas (7.135), (7.137), (7.138)

, Let W (x 0 ,? 0 ),? 0 := W (x 0 ,? 0 ) for (x, ?) ? ?(J r(x 0 ,? 0 ),? 0 ), p.141

, Properties (7.136), (7.138), (7.140) imply (7.82) for W (x 0 ,? 0 )

, 141) one can see that where s, s , r, r satisfy assumption (7.178), C is a constant of, vol.135

, 184) we obtain |U 0 (r, s) ? U 0 (r , s )| 16 ? 3 15 C 16 (|s ? s | + |r ? r |), Multiplying the left and the right hand-sides of, vol.181

, 182) we obtain |U 0 (r, s) ? U 0 (r , s )| ? 3C|r ? r |

, /3 , if 1 ? r ? |r ? r |, p.186

, 181) depending only on ?. Using (7.185) and (7.183) for m = 16, a = |s ? s |, b = |r ? r |, we have that |U 0 (r, s) ? U 0 (r , s )| ? 3C(|s ? s | 1/16 + |r ? r | 1/16 )

, 187) imply (7.127) for this case

, 178) for cases 1, 2, 3, respectively, cover all possible choices of s, s , r, r in (7.127), This completes the proof of (7.127). in T (r) (see notations of (7.19), d = 2). Formulas (7.47), (7.48), (7.122), (7.197) imply that ? r ? supp f k = ? if r ? 1 ? 2 ?m , k < m

E. V. Arbuzov, A. L. Bukhgeim, and S. G. Kazantsev, Two-dimensional tomography problems and the theory of A-analytic functions, Siberian Adv. Math, vol.8, pp.1-20, 1998.

G. Bal, Inverse transport theory and applications, Inverse Problems, vol.25, issue.5, p.48, 2009.

G. Bal and A. Jollivet, Combined source and attenuation reconstructions in SPECT. Tomography and Inverse Transport Theory, Contemp. Math, vol.559, pp.13-27, 2011.

G. Beylkin, The inversion problem and applications of the generalized Radon transform, Communications on pure and applied mathematics, vol.37, issue.5, pp.579-599, 1984.

J. Boman and E. T. Quinto, Support theorems for real-analytic Radon transforms, Duke Mathematical J, vol.55, issue.4, pp.943-948, 1987.

J. Boman and E. T. Quinto, Support theorems for Radon transforms on real analytic line complexes in three-space, Trans. Amer. Math. Soc, vol.335, issue.2, pp.877-890, 1993.

J. Boman, An example of non-uniqueness for a generalized Radon transform, Journal d'Analyse Mathematique, vol.61, issue.1, pp.395-401, 1993.

J. Boman and J. Strömberg, Novikov's inversion formula for the attenuated Radon transform-a new approach, The Journal of Geometric Analysis, vol.14, issue.2, pp.185-198, 2004.

A. M. Cormack, Representation of a function by its line integrals, with some radiological applications I, II, J. Appl. Phys, vol.34, pp.2908-2912, 1963.

M. P. Do-carmo and R. Geometry, , 1992.

G. M. Fichtenholz, A course of differential and integral calculus, vol.II, 1959.

D. Finch, Uniqueness for the attenuated X-ray transform in the physical range, Inverse Problems, vol.2, issue.2, 1986.

I. M. Gel&apos;fand, S. G. Gindikin, and M. I. Graev, Integral geometry in affine and projective spaces, Journal of Soviet Mathematics, vol.18, issue.2, pp.39-167, 1982.

S. Gindikin, A remark on the weighted Radon transform on the plane, Inverse Problems and Imaging, vol.4, issue.4, pp.649-653, 2010.

F. O. Goncharov and R. G. Novikov, An example of non-uniqueness for Radon transforms with continuous positive rotation invariant weights, The Journal of Geometric Analysis, vol.28, issue.4, pp.3807-3828, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01593781

S. Helgason, Differential geometry and symmetric spaces, vol.341, 2001.

J. Ilmavirta, Coherent quantum tomography, SIAM Journal on Mathematical Analysis, vol.48, issue.5, pp.3039-3064, 2016.

F. John, The ultrahyperbolic differential equation with 4 independent variables, Duke Math. J, vol.4, pp.300-322, 1938.

P. Kuchment, K. Lancaster, and L. Mogilevskaya, On local tomography, Inverse Problems, vol.11, issue.3, pp.571-589, 1995.

L. Kunyansky, Generalized and attenuated Radon transforms: restorative approach to the numerical inversion, Inverse Problems, vol.8, issue.5, pp.809-819, 1992.

M. M. Lavrent&apos;ev and A. L. Bukhgeim, A class of operator equations of the first kind, Functional Analysis and Its Applications, vol.7, issue.4, pp.290-298, 1973.

A. Markoe and E. T. Quinto, An elementary proof of local invertibility for generalized and attenuated Radon transforms, SIAM Journal on Mathematical Analysis, vol.16, issue.5, pp.1114-1119, 1985.

F. Natterer, The Mathematics of Computerized Tomography, 2001.

L. V. Nguyen, On the strength of streak artifacts in filtered back-projection reconstructions for limited angle weighted X-ray transform, J. Fourier Anal. Appl, vol.23, issue.3, pp.712-728, 2017.

R. G. Novikov, On determination of a gauge field on R d from its non-abelian Radon transform along oriented straight lines, Journal of the Institute of Mathematics of Jussieu, vol.1, issue.4, pp.559-629, 2002.

R. G. Novikov, An inversion formula for the attenuated X-ray transformation, Arkiv för matematik, vol.40, issue.1, pp.145-167, 2002.

R. G. Novikov, Weighted Radon transforms and first order differential systems on the plane, Moscow Mathematical Journal, vol.14, issue.4, pp.807-823, 2014.
URL : https://hal.archives-ouvertes.fr/hal-00714524

V. P. Palamodov, An inversion method for an attenuated x-ray transform, Inverse Problems, vol.12, issue.5, pp.717-729, 1996.

A. Puro and A. Garin, Cormack-type inversion of attenuated Radon transform, Inverse Problems, vol.29, p.65004, 2013.

J. Radon, Uber die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten, Ber. Saechs Akad. Wiss. Leipzig, Math-Phys, vol.69, pp.262-267, 1917.

V. A. Sharafutdinov, On the problem of emission tomography for nonhomogeneous media, Soviet Phys. Dokl, vol.326, pp.469-470, 1992.

V. A. Sharafutdinov, Uniqueness theorems for the exponential X-ray transform, Journal of Inverse and Ill-Posed Problems, vol.1, issue.4, pp.355-372, 1993.

O. J. Tretiak and C. Metz, The exponential Radon transform, SIAM J. Appl. Math, vol.39, pp.341-354, 1980.

E. T. Quinto, The invertibility of rotation invariant Radon transforms, Journal of Mathematical Analysis and Applications, vol.91, issue.2, pp.510-522, 1983.