American Institute of Mathematical Sciences

Advanced
November  2010, 4(4): 693-702. doi: 10.3934/ipi.2010.4.693

Remarks on the general Funk transform and thermoacoustic tomography

Victor Palamodov 1,

1. 

School of Mathematical Sciences, Tel Aviv University, Ramat Aviv Tel Aviv 69978, Israel

Received  June 2009 Published  September 2010

We discuss properties of a generalized Minkowski-Funk transform defined for a family of hypersurfaces. We prove two-side estimates and show that the range conditions can be written in terms of the reciprocal Funk transform. Some applications to the spherical mean transform are considered.
Keywords: Kaczmarz method., elliptic PDO, Funk transform of a density, reciprocal Funk transform, partial scan, conjugate points, thermoacoustic tomography, range conditions.
Mathematics Subject Classification: Primary: 53C65; Secondary: 65R32, 78A4.
Citation: Victor Palamodov. Remarks on the general Funk transform and thermoacoustic tomography. Inverse Problems & Imaging, 2010, 4 (4) : 693-702. doi: 10.3934/ipi.2010.4.693
References:
[1]

M. Agranovsky, P. Kuchment and E. T. Quinto, Range descriptions for the spherical mean Radon transform,, J. Funct. Anal., 248 (2007), 344.  doi: doi:10.1016/j.jfa.2007.03.022.  Google Scholar

[2]

J. Boman, On stable inversion of the attenuated Radon transform with half data,, in, (2006), 19.   Google Scholar

[3]

D. Finch and Rakesh, The range of the spherical mean value operator for functions supported in a ball,, Inverse Problems, 22 (2006), 923.  doi: doi:10.1088/0266-5611/22/3/012.  Google Scholar

[4]

P. Funk, Über Flächen mit lauter geschlossenen geodätischen Linien,, Math. Ann., 74 (1913), 278.  doi: doi:10.1007/BF01456044.  Google Scholar

[5]

V. Guillemin, On some results of Gelfand in integral geometry,, in, 43 (1985), 149.   Google Scholar

[6]

L. Hörmander, "The Analysis of Linear Partial Differential Operators IV. Fourier Integral Operators,", Springer, (1985).   Google Scholar

[7]

M. M. Lavrent'ev and A. L. Buhgeim, A certain class of problems of integral geometry,, Dokl. Akad. Nauk SSSR, 211 (1973), 38.   Google Scholar

[8]

R. G. Mukhometov, On a problem of integral geometry on the plane,, in, 180 (1978), 30.   Google Scholar

[9]

F. Natterer, "The Mathematics of Computerized Tomography,", B.G.Teubner, (1986).   Google Scholar

[10]

S. K. Patch, Moment conditions indirectly improve image quality,, in, (2001), 193.   Google Scholar

[11]

S. K. Patch and O. Scherzer, Photo- and thermo-acoustic imaging,, Inverse Problems, 23 (2007).   Google Scholar

[12]

D. A. Popov, The generalized Radon transform on the plane, its inversion, and the Cavalieri conditions,, Funct. Anal. Appl., 35 (2001), 270.  doi: doi:10.1023/A:1013126507543.  Google Scholar

[13]

D. A. Popov and D. V. Sushko, Image restoration in optical-acoustic tomography,, Probl. Inf. Transm., 40 (2004), 254.  doi: doi:10.1023/B:PRIT.0000044261.87490.05.  Google Scholar

[14]

E. T. Quinto, The dependence of the generalized Radon transform on defining measures,, Trans. Amer. Math. Soc., 257 (1980), 331.   Google Scholar

[15]

H. Rullgård, Stability of the inverse problem for the attenuated Radon transform with 180 $^\circ$ data,, Inverse Problems, 20 (2004), 781.  doi: doi:10.1088/0266-5611/20/3/008.  Google Scholar

show all references

References:
[1]

M. Agranovsky, P. Kuchment and E. T. Quinto, Range descriptions for the spherical mean Radon transform,, J. Funct. Anal., 248 (2007), 344.  doi: doi:10.1016/j.jfa.2007.03.022.  Google Scholar

[2]

J. Boman, On stable inversion of the attenuated Radon transform with half data,, in, (2006), 19.   Google Scholar

[3]

D. Finch and Rakesh, The range of the spherical mean value operator for functions supported in a ball,, Inverse Problems, 22 (2006), 923.  doi: doi:10.1088/0266-5611/22/3/012.  Google Scholar

[4]

P. Funk, Über Flächen mit lauter geschlossenen geodätischen Linien,, Math. Ann., 74 (1913), 278.  doi: doi:10.1007/BF01456044.  Google Scholar

[5]

V. Guillemin, On some results of Gelfand in integral geometry,, in, 43 (1985), 149.   Google Scholar

[6]

L. Hörmander, "The Analysis of Linear Partial Differential Operators IV. Fourier Integral Operators,", Springer, (1985).   Google Scholar

[7]

M. M. Lavrent'ev and A. L. Buhgeim, A certain class of problems of integral geometry,, Dokl. Akad. Nauk SSSR, 211 (1973), 38.   Google Scholar

[8]

R. G. Mukhometov, On a problem of integral geometry on the plane,, in, 180 (1978), 30.   Google Scholar

[9]

F. Natterer, "The Mathematics of Computerized Tomography,", B.G.Teubner, (1986).   Google Scholar

[10]

S. K. Patch, Moment conditions indirectly improve image quality,, in, (2001), 193.   Google Scholar

[11]

S. K. Patch and O. Scherzer, Photo- and thermo-acoustic imaging,, Inverse Problems, 23 (2007).   Google Scholar

[12]

D. A. Popov, The generalized Radon transform on the plane, its inversion, and the Cavalieri conditions,, Funct. Anal. Appl., 35 (2001), 270.  doi: doi:10.1023/A:1013126507543.  Google Scholar

[13]

D. A. Popov and D. V. Sushko, Image restoration in optical-acoustic tomography,, Probl. Inf. Transm., 40 (2004), 254.  doi: doi:10.1023/B:PRIT.0000044261.87490.05.  Google Scholar

[14]

E. T. Quinto, The dependence of the generalized Radon transform on defining measures,, Trans. Amer. Math. Soc., 257 (1980), 331.   Google Scholar

[15]

H. Rullgård, Stability of the inverse problem for the attenuated Radon transform with 180 $^\circ$ data,, Inverse Problems, 20 (2004), 781.  doi: doi:10.1088/0266-5611/20/3/008.  Google Scholar

[1]

Mark Agranovsky, David Finch, Peter Kuchment. Range conditions for a spherical mean transform. Inverse Problems & Imaging, 2009, 3 (3) : 373-382. doi: 10.3934/ipi.2009.3.373
[2]

Figen Özpinar, Fethi Bin Muhammad Belgacem. The discrete homotopy perturbation Sumudu transform method for solving partial difference equations. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 615-624. doi: 10.3934/dcdss.2019039
[3]

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
[4]

Linh V. Nguyen. A family of inversion formulas in thermoacoustic tomography. Inverse Problems & Imaging, 2009, 3 (4) : 649-675. doi: 10.3934/ipi.2009.3.649
[5]

Aleksander Denisiuk. On range condition of the tensor x-ray transform in $ \mathbb R^n $. Inverse Problems & Imaging, 2020, 14 (3) : 423-435. doi: 10.3934/ipi.2020020
[6]

Daniel Fusca. The Madelung transform as a momentum map. Journal of Geometric Mechanics, 2017, 9 (2) : 157-165. doi: 10.3934/jgm.2017006
[7]

Doǧan Çömez. The modulated ergodic Hilbert transform. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 325-336. doi: 10.3934/dcdss.2009.2.325
[8]

Sean Holman, Plamen Stefanov. The weighted Doppler transform. Inverse Problems & Imaging, 2010, 4 (1) : 111-130. doi: 10.3934/ipi.2010.4.111
[9]

James W. Webber, Sean Holman. Microlocal analysis of a spindle transform. Inverse Problems & Imaging, 2019, 13 (2) : 231-261. doi: 10.3934/ipi.2019013
[10]

Leonid Kunyansky. Fast reconstruction algorithms for the thermoacoustic tomography in certain domains with cylindrical or spherical symmetries. Inverse Problems & Imaging, 2012, 6 (1) : 111-131. doi: 10.3934/ipi.2012.6.111
[11]

C E Yarman, B Yazıcı. A new exact inversion method for exponential Radon transform using the harmonic analysis of the Euclidean motion group. Inverse Problems & Imaging, 2007, 1 (3) : 457-479. doi: 10.3934/ipi.2007.1.457
[12]

Simon Gindikin. A remark on the weighted Radon transform on the plane. Inverse Problems & Imaging, 2010, 4 (4) : 649-653. doi: 10.3934/ipi.2010.4.649
[13]

Sebastian Reich, Seoleun Shin. On the consistency of ensemble transform filter formulations. Journal of Computational Dynamics, 2014, 1 (1) : 177-189. doi: 10.3934/jcd.2014.1.177
[14]

Gareth Ainsworth. The attenuated magnetic ray transform on surfaces. Inverse Problems & Imaging, 2013, 7 (1) : 27-46. doi: 10.3934/ipi.2013.7.27
[15]

Linh V. Nguyen. Spherical mean transform: A PDE approach. Inverse Problems & Imaging, 2013, 7 (1) : 243-252. doi: 10.3934/ipi.2013.7.243
[16]

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
[17]

Fahd Jarad, Thabet Abdeljawad. Generalized fractional derivatives and Laplace transform. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 709-722. doi: 10.3934/dcdss.2020039
[18]

Melody Alsaker, Sarah Jane Hamilton, Andreas Hauptmann. A direct D-bar method for partial boundary data electrical impedance tomography with a priori information. Inverse Problems & Imaging, 2017, 11 (3) : 427-454. doi: 10.3934/ipi.2017020
[19]

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
[20]

Ingrid Beltiţă, Anders Melin. The quadratic contribution to the backscattering transform in the rotation invariant case. Inverse Problems & Imaging, 2010, 4 (4) : 599-618. doi: 10.3934/ipi.2010.4.599

2018 Impact Factor: 1.469

Tools

Metrics

  • PDF downloads (43)
  • HTML views (0)
  • Cited by (16)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences