American Institute of Mathematical Sciences

Advanced
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

Hans Rullgård 1, and  Eric Todd Quinto 2,

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.
Keywords: Sobolev seminorms, limited data tomography, Radon transform, wavefront set, singularity detection..
Mathematics Subject Classification: Primary: 44A12, 92C55; Secondary: 65R30, 35S3.
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]

Anna Anop, Robert Denk, Aleksandr Murach. Elliptic problems with rough boundary data in generalized Sobolev spaces. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020286
[2]

Haruki Umakoshi. A semilinear heat equation with initial data in negative Sobolev spaces. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 745-767. doi: 10.3934/dcdss.2020365
[3]

Philippe G. Lefloch, Cristinel Mardare, Sorin Mardare. Isometric immersions into the Minkowski spacetime for Lorentzian manifolds with limited regularity. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 341-365. doi: 10.3934/dcds.2009.23.341
[4]

Michiel Bertsch, Flavia Smarrazzo, Andrea Terracina, Alberto Tesei. Signed Radon measure-valued solutions of flux saturated scalar conservation laws. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3143-3169. doi: 10.3934/dcds.2020041
[5]

Alessandro Carbotti, Giovanni E. Comi. A note on Riemann-Liouville fractional Sobolev spaces. Communications on Pure & Applied Analysis, 2021, 20 (1) : 17-54. doi: 10.3934/cpaa.2020255
[6]

Teresa D'Aprile. Bubbling solutions for the Liouville equation around a quantized singularity in symmetric domains. Communications on Pure & Applied Analysis, 2021, 20 (1) : 159-191. doi: 10.3934/cpaa.2020262
[7]

Jintai Ding, Zheng Zhang, Joshua Deaton. The singularity attack to the multivariate signature scheme HIMQ-3. Advances in Mathematics of Communications, 2021, 15 (1) : 65-72. doi: 10.3934/amc.2020043
[8]

Xin Zhong. Singularity formation to the nonhomogeneous magneto-micropolar fluid equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021021
[9]

Noah Stevenson, Ian Tice. A truncated real interpolation method and characterizations of screened Sobolev spaces. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5509-5566. doi: 10.3934/cpaa.2020250
[10]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049
[11]

Evelyn Sander, Thomas Wanner. Equilibrium validation in models for pattern formation based on Sobolev embeddings. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 603-632. doi: 10.3934/dcdsb.2020260
[12]

Huu-Quang Nguyen, Ya-Chi Chu, Ruey-Lin Sheu. On the convexity for the range set of two quadratic functions. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020169
[13]

Sumit Kumar Debnath, Pantelimon Stǎnicǎ, Nibedita Kundu, Tanmay Choudhury. Secure and efficient multiparty private set intersection cardinality. Advances in Mathematics of Communications, 2021, 15 (2) : 365-386. doi: 10.3934/amc.2020071
[14]

Paul E. Anderson, Timothy P. Chartier, Amy N. Langville, Kathryn E. Pedings-Behling. The rankability of weighted data from pairwise comparisons. Foundations of Data Science, 2021  doi: 10.3934/fods.2021002
[15]

Yasmine Cherfaoui, Mustapha Moulaï. Biobjective optimization over the efficient set of multiobjective integer programming problem. Journal of Industrial & Management Optimization, 2021, 17 (1) : 117-131. doi: 10.3934/jimo.2019102
[16]

Xin Guo, Lei Shi. Preface of the special issue on analysis in data science: Methods and applications. Mathematical Foundations of Computing, 2020, 3 (4) : i-ii. doi: 10.3934/mfc.2020026
[17]

Gongbao Li, Tao Yang. Improved Sobolev inequalities involving weighted Morrey norms and the existence of nontrivial solutions to doubly critical elliptic systems involving fractional Laplacian and Hardy terms. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020469
[18]

Shasha Hu, Yihong Xu, Yuhan Zhang. Second-Order characterizations for set-valued equilibrium problems with variable ordering structures. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020164
[19]

Wenbin Li, Jianliang Qian. Simultaneously recovering both domain and varying density in inverse gravimetry by efficient level-set methods. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020073
[20]

Ali Mahmoodirad, Harish Garg, Sadegh Niroomand. Solving fuzzy linear fractional set covering problem by a goal programming based solution approach. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020162

2019 Impact Factor: 1.373

Tools

Metrics

  • PDF downloads (30)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences