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

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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

Abstract
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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.
[2]	E. Candès and L. Demanet, Curvelets and Fourier Integral Operators,, C. R. Math. Acad. Sci. Paris. Serie I, 336 (2003), 395.
[3]	E. J. Candès and D. L. Donoho, Curvelets and Reconstruction of Images from Noisy Radon Data,, in, 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, 47 (2003), 193.
[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.
[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.
[7]	A. Greenleaf and G. Uhlmann, Microlocal techniques in integral geometry,, Contemp. Math., 113 (1990), 121.
[8]	V. Guillemin and D. Schaeffer, Fourier integral operators from the Radon transform point of view,, Proc. Sympos. Pure Math., 27 (1975), 297.
[9]	V. Guillemin and S. Sternberg, "Geometric Asymptotics,'', American Mathematical Society, (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. doi: doi:10.1007/BF00534869.
[11]	A. Hertle, Continuity of the Radon transform and its inverse on Euclidean space,, Math. Z., 184 (1983), 165. doi: doi:10.1007/BF01252856.
[12]	A. Katsevich, Improved cone beam local tomography,, Inverse Problems, 22 (2006), 627. doi: doi:10.1088/0266-5611/22/2/015.
[13]	A. I. Katsevich, Cone beam local tomography,, SIAM J. Appl. Math., 59 (1999), 2224. doi: doi:10.1137/S0036139998336043.
[14]	A. K. Louis, "Analytische Methoden in der Computer Tomographie,", Habilitationsschrift, (1981).
[15]	F. Natterer, The mathematics of computerized tomography,, in, (2001).
[16]	F. Natterer and F. Wübbeling, Mathematical methods in image reconstruction,, in, (2001).
[17]	B. Petersen, "Introduction to the Fourier Transform and Pseudo-Differential Operators,", Pittman, (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. doi: doi:10.1137/0524069.
[19]	E. T. Quinto, T. Bakhos and S. Chung, A local algorithm for Slant Hole SPECT,, in, (2008), 321.
[20]	E. T. Quinto and O. Öktem, Local tomography in electron microscopy,, SIAM J. Appl. Math., 68 (2008), 1282. 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. 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. 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), 285* (1984), 159.
[24]	H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,'', Second edition. Johann Ambrosius Barth, (1995).
[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.

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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. 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.
[3]	E. J. Candès and D. L. Donoho, Curvelets and Reconstruction of Images from Noisy Radon Data,, in, 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, 47 (2003), 193.
[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.
[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.
[7]	A. Greenleaf and G. Uhlmann, Microlocal techniques in integral geometry,, Contemp. Math., 113 (1990), 121.
[8]	V. Guillemin and D. Schaeffer, Fourier integral operators from the Radon transform point of view,, Proc. Sympos. Pure Math., 27 (1975), 297.
[9]	V. Guillemin and S. Sternberg, "Geometric Asymptotics,'', American Mathematical Society, (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. doi: doi:10.1007/BF00534869.
[11]	A. Hertle, Continuity of the Radon transform and its inverse on Euclidean space,, Math. Z., 184 (1983), 165. doi: doi:10.1007/BF01252856.
[12]	A. Katsevich, Improved cone beam local tomography,, Inverse Problems, 22 (2006), 627. doi: doi:10.1088/0266-5611/22/2/015.
[13]	A. I. Katsevich, Cone beam local tomography,, SIAM J. Appl. Math., 59 (1999), 2224. doi: doi:10.1137/S0036139998336043.
[14]	A. K. Louis, "Analytische Methoden in der Computer Tomographie,", Habilitationsschrift, (1981).
[15]	F. Natterer, The mathematics of computerized tomography,, in, (2001).
[16]	F. Natterer and F. Wübbeling, Mathematical methods in image reconstruction,, in, (2001).
[17]	B. Petersen, "Introduction to the Fourier Transform and Pseudo-Differential Operators,", Pittman, (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. doi: doi:10.1137/0524069.
[19]	E. T. Quinto, T. Bakhos and S. Chung, A local algorithm for Slant Hole SPECT,, in, (2008), 321.
[20]	E. T. Quinto and O. Öktem, Local tomography in electron microscopy,, SIAM J. Appl. Math., 68 (2008), 1282. 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. 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. 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), 285* (1984), 159.
[24]	H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,'', Second edition. Johann Ambrosius Barth, (1995).
[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.

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