November  2010, 4(4): 721-734. doi: 10.3934/ipi.2010.4.721

Local Sobolev estimates of a function by means of its Radon transform

1. 

Department of Mathematics, Stockholm University, 10691 Stockholm, Sweden

2. 

Department of Mathematics, Tufts University, Medford, MA 02155, United States

Received  September 2008 Revised  June 2009 Published  September 2010

In this article, we will define local and microlocal Sobolev seminorms and prove local and microlocal inverse continuity estimates for the Radon hyperplane transform in these seminorms. The relation between the Sobolev wavefront set of a function $f$ and of its Radon transform is well-known [18]. However, Sobolev wavefront is qualitative and therefore the relation in [18] is qualitative. Our results will make the relation between singularities of a function and those of its Radon transform quantitative. This could be important for practical applications, such as tomography, in which the data are smooth but can have large derivatives.
Citation: Hans Rullgård, Eric Todd Quinto. Local Sobolev estimates of a function by means of its Radon transform. Inverse Problems and Imaging, 2010, 4 (4) : 721-734. doi: 10.3934/ipi.2010.4.721
References:
[1]

M. A. Anastasio, Y. Zou, E. Y. Sidky and X. Pan, Local cone-beam tomography image reconstruction on chords, Journal of the Optical Society of America A, 24 (2007), 1569-1579. doi: doi:10.1364/JOSAA.24.001569.

[2]

E. Candès and L. Demanet, Curvelets and Fourier Integral Operators, C. R. Math. Acad. Sci. Paris. Serie I, 336 (2003), 395-398.

[3]

E. J. Candès and D. L. Donoho, Curvelets and Reconstruction of Images from Noisy Radon Data, in "Wavelet Applications in Signal and Image Processing VIII'' (eds. M. A. U. A. Aldroubi, A. F. Laine), Proc. SPIE. 4119 (2000).

[4]

D. V. Finch, I.-R. Lan and G. Uhlmann, Microlocal Analysis of the restricted X-ray transform with sources on a curve, in "Inside Out, Inverse Problems and Applications,'' (ed. G. Uhlmann), MSRI Publications, Cambridge University Press, 47 (2003), 193-218.

[5]

A. Greenleaf and G. Uhlmann, Non-local inversion formulas for the X-ray transform, Duke Math. J., 58 (1989), 205-240. doi: doi:10.1215/S0012-7094-89-05811-0.

[6]

A. Greenleaf and G. Uhlmann, Estimates for singular Radon transforms and pseudodifferential operators with singular symbols, J. Funct. Anal., 89 (1990), 202-232. doi: doi:10.1016/0022-1236(90)90011-9.

[7]

A. Greenleaf and G. Uhlmann, Microlocal techniques in integral geometry, Contemp. Math., 113 (1990), 121-136.

[8]

V. Guillemin and D. Schaeffer, Fourier integral operators from the Radon transform point of view, Proc. Sympos. Pure Math., 27 (1975), 297-300.

[9]

V. Guillemin and S. Sternberg, "Geometric Asymptotics,'' American Mathematical Society, Providence, RI, 1977.

[10]

M. G. Hahn and E. T. Quinto, Distances between measures from 1-dimensional projections as implied by continuity of the inverse Radon transform, Zeit. Wahr., 70 (1985), 361-380. doi: doi:10.1007/BF00534869.

[11]

A. Hertle, Continuity of the Radon transform and its inverse on Euclidean space, Math. Z., 184 (1983), 165-192. doi: doi:10.1007/BF01252856.

[12]

A. Katsevich, Improved cone beam local tomography, Inverse Problems, 22 (2006), 627-643. doi: doi:10.1088/0266-5611/22/2/015.

[13]

A. I. Katsevich, Cone beam local tomography, SIAM J. Appl. Math., 59 (1999), 2224-2246. doi: doi:10.1137/S0036139998336043.

[14]

A. K. Louis, "Analytische Methoden in der Computer Tomographie," Habilitationsschrift, Universität Münster, 1981.

[15]

F. Natterer, The mathematics of computerized tomography, in "Classics in Mathematics," Society for Industrial and Applied Mathematics, New York, 2001.

[16]

F. Natterer and F. Wübbeling, Mathematical methods in image reconstruction, in "Monographs on Mathematical Modeling and Computation," Society for Industrial and Applied Mathematics, New York, 2001.

[17]

B. Petersen, "Introduction to the Fourier Transform and Pseudo-Differential Operators," Pittman, Boston, 1983.

[18]

E. T. Quinto, Singularities of the X-ray transform and limited data tomography in $R^2$ and $R^3$, SIAM J. Math. Anal., 24 (1993), 1215-1225. doi: doi:10.1137/0524069.

[19]

E. T. Quinto, T. Bakhos and S. Chung, A local algorithm for Slant Hole SPECT, in "Mathematical Methods in Biomedical Imaging and Intensity-Modulated Radiation Therapy (IMRT),'' 321-348, CRM Series, 7, Ed. Norm., Pisa, 2008. Centro De Georgi.

[20]

E. T. Quinto and O. Öktem, Local tomography in electron microscopy, SIAM J. Appl. Math., 68 (2008), 1282-1303. doi: doi:10.1137/07068326X.

[21]

A. G. Ramm and A. I. Zaslavsky, Singularities of the Radon transform, Bull. Amer. Math. Soc., 25 (1993), 109-115. doi: doi:10.1090/S0273-0979-1993-00350-1.

[22]

M. Beals and M. Reed, Propagation of singularities for hyperbolic pseudodifferential operators with nonsmooth coefficients, Comm. Pure Appl. Math., 35 (1982), 169-184. doi: doi:10.1002/cpa.3160350203.

[23]

M. Beals and M. Reed, Microlocal regularity theorems for nonsmooth pseudodifferential operators and applications to nonlinear problems, Trans. Amer. Math. Soc. 285 (1984), 159-184.

[24]

H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,'' Second edition. Johann Ambrosius Barth, Heidelberg, 1995.

[25]

Y. Ye, H. Yu and G. Wang, Cone beam pseudo-lambda tomography, Inverse Problems, 23 (2007), 203-215. doi: doi:10.1088/0266-5611/23/1/010.

show all references

References:
[1]

M. A. Anastasio, Y. Zou, E. Y. Sidky and X. Pan, Local cone-beam tomography image reconstruction on chords, Journal of the Optical Society of America A, 24 (2007), 1569-1579. doi: doi:10.1364/JOSAA.24.001569.

[2]

E. Candès and L. Demanet, Curvelets and Fourier Integral Operators, C. R. Math. Acad. Sci. Paris. Serie I, 336 (2003), 395-398.

[3]

E. J. Candès and D. L. Donoho, Curvelets and Reconstruction of Images from Noisy Radon Data, in "Wavelet Applications in Signal and Image Processing VIII'' (eds. M. A. U. A. Aldroubi, A. F. Laine), Proc. SPIE. 4119 (2000).

[4]

D. V. Finch, I.-R. Lan and G. Uhlmann, Microlocal Analysis of the restricted X-ray transform with sources on a curve, in "Inside Out, Inverse Problems and Applications,'' (ed. G. Uhlmann), MSRI Publications, Cambridge University Press, 47 (2003), 193-218.

[5]

A. Greenleaf and G. Uhlmann, Non-local inversion formulas for the X-ray transform, Duke Math. J., 58 (1989), 205-240. doi: doi:10.1215/S0012-7094-89-05811-0.

[6]

A. Greenleaf and G. Uhlmann, Estimates for singular Radon transforms and pseudodifferential operators with singular symbols, J. Funct. Anal., 89 (1990), 202-232. doi: doi:10.1016/0022-1236(90)90011-9.

[7]

A. Greenleaf and G. Uhlmann, Microlocal techniques in integral geometry, Contemp. Math., 113 (1990), 121-136.

[8]

V. Guillemin and D. Schaeffer, Fourier integral operators from the Radon transform point of view, Proc. Sympos. Pure Math., 27 (1975), 297-300.

[9]

V. Guillemin and S. Sternberg, "Geometric Asymptotics,'' American Mathematical Society, Providence, RI, 1977.

[10]

M. G. Hahn and E. T. Quinto, Distances between measures from 1-dimensional projections as implied by continuity of the inverse Radon transform, Zeit. Wahr., 70 (1985), 361-380. doi: doi:10.1007/BF00534869.

[11]

A. Hertle, Continuity of the Radon transform and its inverse on Euclidean space, Math. Z., 184 (1983), 165-192. doi: doi:10.1007/BF01252856.

[12]

A. Katsevich, Improved cone beam local tomography, Inverse Problems, 22 (2006), 627-643. doi: doi:10.1088/0266-5611/22/2/015.

[13]

A. I. Katsevich, Cone beam local tomography, SIAM J. Appl. Math., 59 (1999), 2224-2246. doi: doi:10.1137/S0036139998336043.

[14]

A. K. Louis, "Analytische Methoden in der Computer Tomographie," Habilitationsschrift, Universität Münster, 1981.

[15]

F. Natterer, The mathematics of computerized tomography, in "Classics in Mathematics," Society for Industrial and Applied Mathematics, New York, 2001.

[16]

F. Natterer and F. Wübbeling, Mathematical methods in image reconstruction, in "Monographs on Mathematical Modeling and Computation," Society for Industrial and Applied Mathematics, New York, 2001.

[17]

B. Petersen, "Introduction to the Fourier Transform and Pseudo-Differential Operators," Pittman, Boston, 1983.

[18]

E. T. Quinto, Singularities of the X-ray transform and limited data tomography in $R^2$ and $R^3$, SIAM J. Math. Anal., 24 (1993), 1215-1225. doi: doi:10.1137/0524069.

[19]

E. T. Quinto, T. Bakhos and S. Chung, A local algorithm for Slant Hole SPECT, in "Mathematical Methods in Biomedical Imaging and Intensity-Modulated Radiation Therapy (IMRT),'' 321-348, CRM Series, 7, Ed. Norm., Pisa, 2008. Centro De Georgi.

[20]

E. T. Quinto and O. Öktem, Local tomography in electron microscopy, SIAM J. Appl. Math., 68 (2008), 1282-1303. doi: doi:10.1137/07068326X.

[21]

A. G. Ramm and A. I. Zaslavsky, Singularities of the Radon transform, Bull. Amer. Math. Soc., 25 (1993), 109-115. doi: doi:10.1090/S0273-0979-1993-00350-1.

[22]

M. Beals and M. Reed, Propagation of singularities for hyperbolic pseudodifferential operators with nonsmooth coefficients, Comm. Pure Appl. Math., 35 (1982), 169-184. doi: doi:10.1002/cpa.3160350203.

[23]

M. Beals and M. Reed, Microlocal regularity theorems for nonsmooth pseudodifferential operators and applications to nonlinear problems, Trans. Amer. Math. Soc. 285 (1984), 159-184.

[24]

H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,'' Second edition. Johann Ambrosius Barth, Heidelberg, 1995.

[25]

Y. Ye, H. Yu and G. Wang, Cone beam pseudo-lambda tomography, Inverse Problems, 23 (2007), 203-215. doi: doi:10.1088/0266-5611/23/1/010.

[1]

Alberto Ibort, Alberto López-Yela. Quantum tomography and the quantum Radon transform. Inverse Problems and Imaging, 2021, 15 (5) : 893-928. doi: 10.3934/ipi.2021021

[2]

Michael Krause, Jan Marcel Hausherr, Walter Krenkel. Computing the fibre orientation from Radon data using local Radon transform. Inverse Problems and Imaging, 2011, 5 (4) : 879-891. doi: 10.3934/ipi.2011.5.879

[3]

Jean-François Crouzet. 3D coded aperture imaging, ill-posedness and link with incomplete data radon transform. Inverse Problems and Imaging, 2011, 5 (2) : 341-353. doi: 10.3934/ipi.2011.5.341

[4]

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

[5]

Sunghwan Moon. Inversion of the spherical Radon transform on spheres through the origin using the regular Radon transform. Communications on Pure and Applied Analysis, 2016, 15 (3) : 1029-1039. doi: 10.3934/cpaa.2016.15.1029

[6]

Ali Gholami, Mauricio D. Sacchi. Time-invariant radon transform by generalized Fourier slice theorem. Inverse Problems and Imaging, 2017, 11 (3) : 501-519. doi: 10.3934/ipi.2017023

[7]

Victor Palamodov. Remarks on the general Funk transform and thermoacoustic tomography. Inverse Problems and Imaging, 2010, 4 (4) : 693-702. doi: 10.3934/ipi.2010.4.693

[8]

Masaru Ikehata, Esa Niemi, Samuli Siltanen. Inverse obstacle scattering with limited-aperture data. Inverse Problems and Imaging, 2012, 6 (1) : 77-94. doi: 10.3934/ipi.2012.6.77

[9]

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 and Imaging, 2007, 1 (3) : 457-479. doi: 10.3934/ipi.2007.1.457

[10]

Benjamin Palacios. Photoacoustic tomography in attenuating media with partial data. Inverse Problems and Imaging, , () : -. doi: 10.3934/ipi.2022013

[11]

Lassi Päivärinta, Valery Serov. Recovery of jumps and singularities in the multidimensional Schrodinger operator from limited data. Inverse Problems and Imaging, 2007, 1 (3) : 525-535. doi: 10.3934/ipi.2007.1.525

[12]

Habib Ammari, Josselin Garnier, Vincent Jugnon. Detection, reconstruction, and characterization algorithms from noisy data in multistatic wave imaging. Discrete and Continuous Dynamical Systems - S, 2015, 8 (3) : 389-417. doi: 10.3934/dcdss.2015.8.389

[13]

Fioralba Cakoni, Rainer Kress. Integral equations for inverse problems in corrosion detection from partial Cauchy data. Inverse Problems and Imaging, 2007, 1 (2) : 229-245. doi: 10.3934/ipi.2007.1.229

[14]

Austin Lawson, Tyler Hoffman, Yu-Min Chung, Kaitlin Keegan, Sarah Day. A density-based approach to feature detection in persistence diagrams for firn data. Foundations of Data Science, 2021  doi: 10.3934/fods.2021012

[15]

Kazuhiro Ishige. On the existence of solutions of the Cauchy problem for porous medium equations with radon measure as initial data. Discrete and Continuous Dynamical Systems, 1995, 1 (4) : 521-546. doi: 10.3934/dcds.1995.1.521

[16]

Michael V. Klibanov. Travel time tomography with formally determined incomplete data in 3D. Inverse Problems and Imaging, 2019, 13 (6) : 1367-1393. doi: 10.3934/ipi.2019060

[17]

Chengxiang Wang, Li Zeng, Yumeng Guo, Lingli Zhang. Wavelet tight frame and prior image-based image reconstruction from limited-angle projection data. Inverse Problems and Imaging, 2017, 11 (6) : 917-948. doi: 10.3934/ipi.2017043

[18]

Oleg Yu. Imanuvilov, Masahiro Yamamoto. Stability for determination of Riemannian metrics by spectral data and Dirichlet-to-Neumann map limited on arbitrary subboundary. Inverse Problems and Imaging, 2019, 13 (6) : 1213-1258. doi: 10.3934/ipi.2019054

[19]

Lei Zhang, Luming Jia. Near-field imaging for an obstacle above rough surfaces with limited aperture data. Inverse Problems and Imaging, 2021, 15 (5) : 975-997. doi: 10.3934/ipi.2021024

[20]

Venkateswaran P. Krishnan, Vladimir A. Sharafutdinov. Ray transform on Sobolev spaces of symmetric tensor fields, I: Higher order Reshetnyak formulas. Inverse Problems and Imaging, 2022, 16 (4) : 787-826. doi: 10.3934/ipi.2021076

2020 Impact Factor: 1.639

Metrics

  • PDF downloads (59)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]