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 & 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. doi: doi:10.1364/JOSAA.24.001569. Google Scholar

[2]

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

[3]

E. J. Candès and D. L. Donoho, Curvelets and Reconstruction of Images from Noisy Radon Data,, in, 4119 (2000). Google Scholar

[4]

D. V. Finch, I.-R. Lan and G. Uhlmann, Microlocal Analysis of the restricted X-ray transform with sources on a curve,, in, 47 (2003), 193. Google Scholar

[5]

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

[6]

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

[7]

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

[8]

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

[9]

V. Guillemin and S. Sternberg, "Geometric Asymptotics,'', American Mathematical Society, (1977). Google Scholar

[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. doi: doi:10.1007/BF00534869. Google Scholar

[11]

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

[12]

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

[13]

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

[14]

A. K. Louis, "Analytische Methoden in der Computer Tomographie,", Habilitationsschrift, (1981). Google Scholar

[15]

F. Natterer, The mathematics of computerized tomography,, in, (2001). Google Scholar

[16]

F. Natterer and F. Wübbeling, Mathematical methods in image reconstruction,, in, (2001). Google Scholar

[17]

B. Petersen, "Introduction to the Fourier Transform and Pseudo-Differential Operators,", Pittman, (1983). Google Scholar

[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. doi: doi:10.1137/0524069. Google Scholar

[19]

E. T. Quinto, T. Bakhos and S. Chung, A local algorithm for Slant Hole SPECT,, in, (2008), 321. Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

[24]

H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,'', Second edition. Johann Ambrosius Barth, (1995). Google Scholar

[25]

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

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. doi: doi:10.1364/JOSAA.24.001569. Google Scholar

[2]

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

[3]

E. J. Candès and D. L. Donoho, Curvelets and Reconstruction of Images from Noisy Radon Data,, in, 4119 (2000). Google Scholar

[4]

D. V. Finch, I.-R. Lan and G. Uhlmann, Microlocal Analysis of the restricted X-ray transform with sources on a curve,, in, 47 (2003), 193. Google Scholar

[5]

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

[6]

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

[7]

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

[8]

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

[9]

V. Guillemin and S. Sternberg, "Geometric Asymptotics,'', American Mathematical Society, (1977). Google Scholar

[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. doi: doi:10.1007/BF00534869. Google Scholar

[11]

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

[12]

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

[13]

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

[14]

A. K. Louis, "Analytische Methoden in der Computer Tomographie,", Habilitationsschrift, (1981). Google Scholar

[15]

F. Natterer, The mathematics of computerized tomography,, in, (2001). Google Scholar

[16]

F. Natterer and F. Wübbeling, Mathematical methods in image reconstruction,, in, (2001). Google Scholar

[17]

B. Petersen, "Introduction to the Fourier Transform and Pseudo-Differential Operators,", Pittman, (1983). Google Scholar

[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. doi: doi:10.1137/0524069. Google Scholar

[19]

E. T. Quinto, T. Bakhos and S. Chung, A local algorithm for Slant Hole SPECT,, in, (2008), 321. Google Scholar

[20]

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

[21]

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

[22]

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

[23]

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

[24]

H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,'', Second edition. Johann Ambrosius Barth, (1995). Google Scholar

[25]

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

[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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 & Imaging, 2019, 13 (6) : 1213-1258. doi: 10.3934/ipi.2019054

[16]

Alberto Fiorenza, Anna Mercaldo, Jean Michel Rakotoson. Regularity and uniqueness results in grand Sobolev spaces for parabolic equations with measure data. Discrete & Continuous Dynamical Systems - A, 2002, 8 (4) : 893-906. doi: 10.3934/dcds.2002.8.893

[17]

Yohei Fujishima. Blow-up set for a superlinear heat equation and pointedness of the initial data. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4617-4645. doi: 10.3934/dcds.2014.34.4617

[18]

Piernicola Bettiol, Richard Vinter. Estimates on trajectories in a closed set with corners for $(t,x)$ dependent data. Mathematical Control & Related Fields, 2013, 3 (3) : 245-267. doi: 10.3934/mcrf.2013.3.245

[19]

Zhi-An Wang. Wavefront of an angiogenesis model. Discrete & Continuous Dynamical Systems - B, 2012, 17 (8) : 2849-2860. doi: 10.3934/dcdsb.2012.17.2849

[20]

Melody Dodd, Jennifer L. Mueller. A real-time D-bar algorithm for 2-D electrical impedance tomography data. Inverse Problems & Imaging, 2014, 8 (4) : 1013-1031. doi: 10.3934/ipi.2014.8.1013

2018 Impact Factor: 1.469

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]