2010, 4(4): 713-720. doi: 10.3934/ipi.2010.4.713

X-ray transform on Damek-Ricci spaces

1. 

Laboratoire J.A. Dieudonné, Université de Nice, Parc Valrose, 06108 Nice cedex 2, France

Received  December 2008 Revised  May 2009 Published  September 2010

Damek-Ricci spaces, also called harmonic $NA$ groups, make up a large class of harmonic Riemannian manifolds including all hyperbolic spaces. We prove here an inversion formula and a support theorem for the X-ray transform, i.e. integration along geodesics, on those spaces.
   Using suitably chosen totally geodesic submanifolds we reduce the problems to similar questions on low-dimensional hyperbolic spaces.
Citation: François Rouvière. X-ray transform on Damek-Ricci spaces. Inverse Problems & Imaging, 2010, 4 (4) : 713-720. doi: 10.3934/ipi.2010.4.713
References:
[1]

M. Cowling, A. Dooley, A. Korányi and F. Ricci, $H$-type groups and Iwasawa decompositions,, Adv. Math., 87 (1991), 1. doi: doi:10.1016/0001-8708(91)90060-K.

[2]

M. Cowling, A. Dooley, A. Korányi and F. Ricci, An approach to symmetric spaces of rank one via groups of Heisenberg type,, J. Geom. Anal., 8 (1998), 199.

[3]

E. Damek and F. Ricci, A class of nonsymmetric harmonic Riemannian spaces,, Bull. Amer. Math. Soc., 27 (1992), 139. doi: doi:10.1090/S0273-0979-1992-00293-8.

[4]

E. Damek and F. Ricci, Harmonic analysis on solvable extensions of $H$-type groups,, J. Geom. Anal., 2 (1992), 213.

[5]

S. Helgason, "Differential Geometry, Lie Groups and Symmetric Spaces,", Academic Press, (1978).

[6]

S. Helgason, "The Radon Transform," second edition,, Birkhäuser, (1999).

[7]

S. Helgason, The Abel, Fourier and Radon transforms on symmetric spaces,, Indag. Math., 16 (2005), 531.

[8]

F. Rouvière, Espaces de Damek-Ricci, géométrie et analyse,, Sémin. Congr., 7 (2003), 45.

[9]

, revised version of [8] at, \url{http://math.unice.fr/ frou/recherche/Damek-Ricci.pdf}, ().

[10]

F. Rouvière, Transformation aux rayons X sur un espace symétrique,, C. R. Math. Acad. Sci. Paris, 342 (2006), 1.

show all references

References:
[1]

M. Cowling, A. Dooley, A. Korányi and F. Ricci, $H$-type groups and Iwasawa decompositions,, Adv. Math., 87 (1991), 1. doi: doi:10.1016/0001-8708(91)90060-K.

[2]

M. Cowling, A. Dooley, A. Korányi and F. Ricci, An approach to symmetric spaces of rank one via groups of Heisenberg type,, J. Geom. Anal., 8 (1998), 199.

[3]

E. Damek and F. Ricci, A class of nonsymmetric harmonic Riemannian spaces,, Bull. Amer. Math. Soc., 27 (1992), 139. doi: doi:10.1090/S0273-0979-1992-00293-8.

[4]

E. Damek and F. Ricci, Harmonic analysis on solvable extensions of $H$-type groups,, J. Geom. Anal., 2 (1992), 213.

[5]

S. Helgason, "Differential Geometry, Lie Groups and Symmetric Spaces,", Academic Press, (1978).

[6]

S. Helgason, "The Radon Transform," second edition,, Birkhäuser, (1999).

[7]

S. Helgason, The Abel, Fourier and Radon transforms on symmetric spaces,, Indag. Math., 16 (2005), 531.

[8]

F. Rouvière, Espaces de Damek-Ricci, géométrie et analyse,, Sémin. Congr., 7 (2003), 45.

[9]

, revised version of [8] at, \url{http://math.unice.fr/ frou/recherche/Damek-Ricci.pdf}, ().

[10]

F. Rouvière, Transformation aux rayons X sur un espace symétrique,, C. R. Math. Acad. Sci. Paris, 342 (2006), 1.

[1]

Dan Jane, Gabriel P. Paternain. On the injectivity of the X-ray transform for Anosov thermostats. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 471-487. doi: 10.3934/dcds.2009.24.471

[2]

Silvia Allavena, Michele Piana, Federico Benvenuto, Anna Maria Massone. An interpolation/extrapolation approach to X-ray imaging of solar flares. Inverse Problems & Imaging, 2012, 6 (2) : 147-162. doi: 10.3934/ipi.2012.6.147

[3]

Venkateswaran P. Krishnan, Plamen Stefanov. A support theorem for the geodesic ray transform of symmetric tensor fields. Inverse Problems & Imaging, 2009, 3 (3) : 453-464. doi: 10.3934/ipi.2009.3.453

[4]

Nuutti Hyvönen, Martti Kalke, Matti Lassas, Henri Setälä, Samuli Siltanen. Three-dimensional dental X-ray imaging by combination of panoramic and projection data. Inverse Problems & Imaging, 2010, 4 (2) : 257-271. doi: 10.3934/ipi.2010.4.257

[5]

Arun K. Kulshreshth, Andreas Alpers, Gabor T. Herman, Erik Knudsen, Lajos Rodek, Henning F. Poulsen. A greedy method for reconstructing polycrystals from three-dimensional X-ray diffraction data. Inverse Problems & Imaging, 2009, 3 (1) : 69-85. doi: 10.3934/ipi.2009.3.69

[6]

Jakob S. Jørgensen, Emil Y. Sidky, Per Christian Hansen, Xiaochuan Pan. Empirical average-case relation between undersampling and sparsity in X-ray CT. Inverse Problems & Imaging, 2015, 9 (2) : 431-446. doi: 10.3934/ipi.2015.9.431

[7]

Zhenhua Zhao, Yining Zhu, Jiansheng Yang, Ming Jiang. Mumford-Shah-TV functional with application in X-ray interior tomography. Inverse Problems & Imaging, 2018, 12 (2) : 331-348. doi: 10.3934/ipi.2018015

[8]

Gareth Ainsworth. The attenuated magnetic ray transform on surfaces. Inverse Problems & Imaging, 2013, 7 (1) : 27-46. doi: 10.3934/ipi.2013.7.27

[9]

Gareth Ainsworth. The magnetic ray transform on Anosov surfaces. Discrete & Continuous Dynamical Systems - A, 2015, 35 (5) : 1801-1816. doi: 10.3934/dcds.2015.35.1801

[10]

Valeria Banica, Rémi Carles, Thomas Duyckaerts. On scattering for NLS: From Euclidean to hyperbolic space. Discrete & Continuous Dynamical Systems - A, 2009, 24 (4) : 1113-1127. doi: 10.3934/dcds.2009.24.1113

[11]

Yiran Wang. Parametrices for the light ray transform on Minkowski spacetime. Inverse Problems & Imaging, 2018, 12 (1) : 229-237. doi: 10.3934/ipi.2018009

[12]

Fabio Punzo. Support properties of solutions to nonlinear parabolic equations with variable density in the hyperbolic space. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 657-670. doi: 10.3934/dcdss.2012.5.657

[13]

Yoshitsugu Kabeya. A unified approach to Matukuma type equations on the hyperbolic space or on a sphere. Conference Publications, 2013, 2013 (special) : 385-391. doi: 10.3934/proc.2013.2013.385

[14]

Yen-Lin Wu, Zhi-You Chen, Jann-Long Chern, Y. Kabeya. Existence and uniqueness of singular solutions for elliptic equation on the hyperbolic space. Communications on Pure & Applied Analysis, 2014, 13 (2) : 949-960. doi: 10.3934/cpaa.2014.13.949

[15]

Elias M. Guio, Ricardo Sa Earp. Existence and non-existence for a mean curvature equation in hyperbolic space. Communications on Pure & Applied Analysis, 2005, 4 (3) : 549-568. doi: 10.3934/cpaa.2005.4.549

[16]

Qinian Jin, YanYan Li. Starshaped compact hypersurfaces with prescribed $k$-th mean curvature in hyperbolic space. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 367-377. doi: 10.3934/dcds.2006.15.367

[17]

Gareth Ainsworth, Yernat M. Assylbekov. On the range of the attenuated magnetic ray transform for connections and Higgs fields. Inverse Problems & Imaging, 2015, 9 (2) : 317-335. doi: 10.3934/ipi.2015.9.317

[18]

Jan Boman. Unique continuation of microlocally analytic distributions and injectivity theorems for the ray transform. Inverse Problems & Imaging, 2010, 4 (4) : 619-630. doi: 10.3934/ipi.2010.4.619

[19]

Mark Hubenthal. The broken ray transform in $n$ dimensions with flat reflecting boundary. Inverse Problems & Imaging, 2015, 9 (1) : 143-161. doi: 10.3934/ipi.2015.9.143

[20]

Siamak RabieniaHaratbar. Support theorem for the Light-Ray transform of vector fields on Minkowski spaces. Inverse Problems & Imaging, 2018, 12 (2) : 293-314. doi: 10.3934/ipi.2018013

2016 Impact Factor: 1.094

Metrics

  • PDF downloads (0)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]