# American Institute of Mathematical Sciences

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), 1569-1579. 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), 395-398.  Google Scholar [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). 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 "Inside Out, Inverse Problems and Applications,'' (ed. G. Uhlmann), MSRI Publications, Cambridge University Press, 47 (2003), 193-218. Google Scholar [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.  Google Scholar [6] A. Greenleaf and G. Uhlmann, Estimates for singular Radon transforms and pseudodifferential operators with singular symbols, J. Funct. Anal., 89 (1990), 202-232. 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), 121-136.  Google Scholar [8] V. Guillemin and D. Schaeffer, Fourier integral operators from the Radon transform point of view, Proc. Sympos. Pure Math., 27 (1975), 297-300.  Google Scholar [9] V. Guillemin and S. Sternberg, "Geometric Asymptotics,'' American Mathematical Society, Providence, RI, 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), 361-380. 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), 165-192. doi: doi:10.1007/BF01252856.  Google Scholar [12] A. Katsevich, Improved cone beam local tomography, Inverse Problems, 22 (2006), 627-643. 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), 2224-2246. doi: doi:10.1137/S0036139998336043.  Google Scholar [14] A. K. Louis, "Analytische Methoden in der Computer Tomographie," Habilitationsschrift, Universität Münster, 1981. Google Scholar [15] F. Natterer, The mathematics of computerized tomography, in "Classics in Mathematics," Society for Industrial and Applied Mathematics, New York, 2001.  Google Scholar [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.  Google Scholar [17] B. Petersen, "Introduction to the Fourier Transform and Pseudo-Differential Operators," Pittman, Boston, 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), 1215-1225. doi: doi:10.1137/0524069.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [22] M. Beals and M. Reed, Propagation of singularities for hyperbolic pseudodifferential operators with nonsmooth coefficients, Comm. Pure Appl. Math., 35 (1982), 169-184. 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), 159-184.  Google Scholar [24] H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,'' Second edition. Johann Ambrosius Barth, Heidelberg, 1995.  Google Scholar [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.  Google Scholar

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##### 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.  Google Scholar [2] E. Candès and L. Demanet, Curvelets and Fourier Integral Operators, C. R. Math. Acad. Sci. Paris. Serie I, 336 (2003), 395-398.  Google Scholar [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). 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 "Inside Out, Inverse Problems and Applications,'' (ed. G. Uhlmann), MSRI Publications, Cambridge University Press, 47 (2003), 193-218. Google Scholar [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.  Google Scholar [6] A. Greenleaf and G. Uhlmann, Estimates for singular Radon transforms and pseudodifferential operators with singular symbols, J. Funct. Anal., 89 (1990), 202-232. 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), 121-136.  Google Scholar [8] V. Guillemin and D. Schaeffer, Fourier integral operators from the Radon transform point of view, Proc. Sympos. Pure Math., 27 (1975), 297-300.  Google Scholar [9] V. Guillemin and S. Sternberg, "Geometric Asymptotics,'' American Mathematical Society, Providence, RI, 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), 361-380. 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), 165-192. doi: doi:10.1007/BF01252856.  Google Scholar [12] A. Katsevich, Improved cone beam local tomography, Inverse Problems, 22 (2006), 627-643. 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), 2224-2246. doi: doi:10.1137/S0036139998336043.  Google Scholar [14] A. K. Louis, "Analytische Methoden in der Computer Tomographie," Habilitationsschrift, Universität Münster, 1981. Google Scholar [15] F. Natterer, The mathematics of computerized tomography, in "Classics in Mathematics," Society for Industrial and Applied Mathematics, New York, 2001.  Google Scholar [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.  Google Scholar [17] B. Petersen, "Introduction to the Fourier Transform and Pseudo-Differential Operators," Pittman, Boston, 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), 1215-1225. doi: doi:10.1137/0524069.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [22] M. Beals and M. Reed, Propagation of singularities for hyperbolic pseudodifferential operators with nonsmooth coefficients, Comm. Pure Appl. Math., 35 (1982), 169-184. 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), 159-184.  Google Scholar [24] H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,'' Second edition. Johann Ambrosius Barth, Heidelberg, 1995.  Google Scholar [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.  Google Scholar

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