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May  2013, 7(2): 585-609. doi: 10.3934/ipi.2013.7.585

## Local singularity reconstruction from integrals over curves in $\mathbb{R}^3$

 1 Department of Mathematics, Tufts University, Medford, MA 02155, United States 2 Department of Mathematics, Stockholm University, 10691 Stockholm

Received  December 2011 Revised  September 2012 Published  May 2013

We define a general curvilinear Radon transform in $\mathbb{R}^3$, and we develop its microlocal properties. We prove that singularities can be added (or masked) in any backprojection reconstruction method for this transform. We use the microlocal properties of the transform to develop a local backprojection reconstruction algorithm that decreases the effect of the added singularities and reconstructs the shape of the object. This work was motivated by new models in electron microscope tomography in which the electrons travel over curves such as helices or spirals, and we provide reconstructions for a specific transform motivated by this electron microscope tomography problem.
Citation: Eric Todd Quinto, Hans Rullgård. Local singularity reconstruction from integrals over curves in $\mathbb{R}^3$. Inverse Problems and Imaging, 2013, 7 (2) : 585-609. doi: 10.3934/ipi.2013.7.585
##### References:
 [1] M. Anastasio, Y. Zou, E. Sudley and X. Pan, Local cone-beam tomography image reconstruction on chords, Journal of the Optical Society of America A, 24 (2007), 1569-1579. doi: 10.1364/JOSAA.24.001569. [2] J. Boman and E. T. Quinto, Support theorems for real analytic Radon transforms, Duke Math. J., 55 (1987), 943-948. doi: 10.1215/S0012-7094-87-05547-5. [3] J. Boman and E. T. Quinto, Support theorems for Radon transforms on real analytic on line complexes in three-space, Trans. Amer. Math. Soc., 335 (1993), 877-890. doi: 10.1090/S0002-9947-1993-1080733-8. [4] A. Cormack, The Radon transform on a family of curves in the plane, Proc. Amer. Math. Soc., 83 (1981), 325-330. doi: 10.1090/S0002-9939-1981-0624923-1. [5] A. Cormack, The Radon transform on a family of curves in the plane. II, Proc. Amer. Math. Soc., 86 (1982), 293-298. doi: 10.1090/S0002-9939-1982-0667292-4. [6] N. Dairbekov, G. Paternain, P. Stefanov and G. Uhlmann, The boundary rigidity problem in the presence of a magnetic field, Advances in Mathematics, 216 (2007), 535-609. doi: 10.1016/j.aim.2007.05.014. [7] M. deHoop, H. Smith, G. Uhlmann and R. van der Hilst, Seismic imaging with the generalized Radon transform: A curvelet transform perspective, Inverse Problems, 25 (2009), 21 pp. doi: 10.1088/0266-5611/25/2/025005. [8] D. Fanelli and O. Öktem, Electron tomography: A short overview with an emphasis on the absorption potential model for the forward problem, Inverse Problems, 24 (2008), 013001, 51 pp. doi: 10.1088/0266-5611/24/1/013001. [9] A. Faridani, D. Finch, E. L. Ritman and K. T. Smith, Local tomography II, SIAM Journal of Applied Mathematics, 57 (1997), 1095-1127. doi: 10.1137/S0036139995286357. [10] A. Faridani, E. L. Ritman and K. T. Smith, Local tomography, SIAM Journal of Applied Mathematics, 52 (1992), 459-484. doi: 10.1137/0152026. [11] R. Felea, Composition of Fourier integral operators with fold and blowdown singularities, Comm. Partial Differential Equations, 30 (2005), 1717-1740. doi: 10.1080/03605300500299968. [12] R. Felea and E. T. Quinto, The microlocal properties of the Local 3-D SPECT operator, SIAM J. Math. Anal., 43 (2011), 1145-1157. doi: 10.1137/100807703. [13] D. 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" (editor, G. Uhlmann), 47 of MSRI Publications, 193-218. Cambridge University Press, (2003). [14] B. Frigyik, P. Stefanov and G. Uhlmann, The X-Ray transform for a generic family of curves and weights, J. Geom. Anal., 18 (2008), 81-97. doi: 10.1007/s12220-007-9007-6. [15] I. Gelfand, M. Graev and N. Vilenkin, "Generalized Functions," 5 Academic Press, New York, 1966. [16] I. Gelfand and M. I. Graev, Integral transformations connected with straight line complexes in a complex affine space, Soviet Math. Doklady, 2 (1961), 809-812. [17] I. Gelfand and M. I. Graev, Complexes of straight lines in the space $\mathbbC^n$, Functional Analysis and its Applications, 2 (1968), 219-229. [18] I. Gelfand, M. I. Graev and Z. Shapiro, A local problem of integral geometry in a space of curves, Functional Anal. and Appl., 13 (1979), 87-102. [19] A. Greenleaf, A. Seeger and S. Waigner, Estimates for generalized Radon transforms in three and four dimensions, Contemporary Mathematics, 251 (2000), 243-254. doi: 10.1090/conm/251/03873. [20] A. Greenleaf. and G. Uhlmann, Non-local inversion formulas for the X-ray transform, Duke Mathematical Journal, 58 (1989), 205-240. doi: 10.1215/S0012-7094-89-05811-0. [21] A. Greenleaf and G. Uhlmann, Composition of some singular Fourier integral operators and estimates for restricted X-ray transforms, Ann. Inst. Fourier, Grenoble, 40 (1990), 443-466. doi: 10.5802/aif.1220. [22] A. Greenleaf and G. Uhlmann, Microlocal Techniques in Integral Geometry, Contemporary Math., 113 (1990), 121-136. doi: 10.1090/conm/113/1108649. [23] V. Guillemin, "Some Remarks on Integral Geometry," Technical report, MIT, 1975. [24] V. Guillemin and S. Sternberg, "Geometric Asymptotics," American Mathematical Society, Providence, RI, 1977. [25] P. W. Hawkes and E. Kasper, "Principles of Electron Optics. Volume 3. Wave Optics," Academic Press, 1994. [26] S. Helgason, The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassman manifolds, Acta Math., 113 (1965), 153-180. doi: 10.1007/BF02391776. [27] S. Holman and P. Stefanov, The weighted Doppler transform, Inverse Probl. Imaging, 4 (2010), 111-130. doi: 10.3934/ipi.2010.4.111. [28] L. Hörmander, Fourier integral operators, I, Acta Mathematica, 127 (1971), 79-183. doi: 10.1007/BF02392052. [29] L. Hörmander, "The Analysis of Linear Partial Differential Operators," I. Springer Verlag, New York, 1983. [30] A. Katsevich, Cone beam local tomography, SIAM J. Appl. Math., 59 (1999), 2224-2246. doi: 10.1137/S0036139998336043. [31] A. Katsevich, Improved Cone Beam Local Tomography, Inverse Problems, 22 (2006), 627-643. doi: 10.1088/0266-5611/22/2/015. [32] V. P. Krishnan and E. T. Quinto, Microlocal aspects of bistatic synthetic aperture radar imaging, Inverse Problems and Imaging, 5 (2011), 659-674. doi: 10.3934/ipi.2011.5.659. [33] Á. Kurusa, Support curves of invertible Radon transforms, Arch. Math. (Basel), 61 (1993), 448-458. doi: 10.1007/BF01207544. [34] A. Lawrence, J. Bouwer, G. Perkins and M. Ellisman, Transform-based backprojection for volume reconstruction of large format electron microscope tilt series, J. Structural Bio., 154 (2006), 144-167. doi: 10.1016/j.jsb.2005.12.012. [35] A. Louis and P. Maaş, Contour reconstruction in 3-D X-Ray CT, IEEE Trans. Medical Imaging, 12 (1993), 764-769. doi: 10.1109/42.251129. [36] C. Nolan and M. Cheney, Synthetic Aperture inversion, Inverse Problems, 18 (2002), 221-235. doi: 10.1088/0266-5611/18/1/315. [37] S. Phan, J. Bouwer, J. Lanman, M. Terada and A. Lawrence, Non-linear bundle adjustment for electron tomography, in "Computer Science and Information Engineering, 2009 WRI World Congress on Issue Date: March 31 2009-April 2 2009," 1 (2009), 604-612. doi: 10.1109/CSIE.2009.864. [38] S. Phan and A. Lawrence, Tomography of large format electron microscope tilt series: Image alignment and volume reconstruction, in "Proceedings of the 2008 Congress on Image and Signal Processing," 2 (2008), 176-182. doi: 10.1109/CISP.2008.535. [39] D. Popov, The generalized radon transform on the plane, the inverse transform, and the Cavalieri conditions, Functional Analysis and Its Applications, 35 (2001), 270-283. doi: 10.1023/A:1013126507543. [40] E. T. Quinto, The dependence of the generalized Radon transform on defining measures, Trans. Amer. Math. Soc., 257 (1980), 331-346. doi: 10.1090/S0002-9947-1980-0552261-8. [41] 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, Pisa, Italy, (2008). Centro De Georgi. CRM Series, 7. [42] E. T. Quinto and O. Öktem, Local tomography in electron microscopy, SIAM Journal of Applied Mathematics, 68 (2008), 1282-1303. doi: 10.1137/07068326X. [43] E. T. Quinto, A. Rieder and T. Schuster, Local Algorithms in Sonar, Inverse Problems, 27 (2011), 035006, 18p. doi: 10.1088/0266-5611/27/3/035006. [44] L. Reimer, "Transmission Electron Microscopy," 36 of Springer series in optical sciences. Springer Verlag, New York, 4 edition, 1997. doi: 10.1007/978-3-662-14824-2. [45] V. Romanov, "Integral Geometry and Inverse Problems for Hyperbolic Equations," 26 of Springer Tracts in Natural Philosophy. Springer Verlag, Berlin, New York, 1969. doi: 10.1007/978-3-642-80781-7. [46] P. Stefanov and G. Uhlmann, Integral geometry of tensor fields on a class of non-simple Riemannian manifolds, Amer. J. Math., 130 (2008), 239-268. doi: 10.1353/ajm.2008.0003. [47] E. Vainberg, I. Kazak and V. Kurozaev, Reconstruction of the internal three-dimensional structure of objects based on real-time integral projections, Soviet Journal of Nondestructive Testing, 17 (1981), 415-423.

show all references

##### References:
 [1] M. Anastasio, Y. Zou, E. Sudley and X. Pan, Local cone-beam tomography image reconstruction on chords, Journal of the Optical Society of America A, 24 (2007), 1569-1579. doi: 10.1364/JOSAA.24.001569. [2] J. Boman and E. T. Quinto, Support theorems for real analytic Radon transforms, Duke Math. J., 55 (1987), 943-948. doi: 10.1215/S0012-7094-87-05547-5. [3] J. Boman and E. T. Quinto, Support theorems for Radon transforms on real analytic on line complexes in three-space, Trans. Amer. Math. Soc., 335 (1993), 877-890. doi: 10.1090/S0002-9947-1993-1080733-8. [4] A. Cormack, The Radon transform on a family of curves in the plane, Proc. Amer. Math. Soc., 83 (1981), 325-330. doi: 10.1090/S0002-9939-1981-0624923-1. [5] A. Cormack, The Radon transform on a family of curves in the plane. II, Proc. Amer. Math. Soc., 86 (1982), 293-298. doi: 10.1090/S0002-9939-1982-0667292-4. [6] N. Dairbekov, G. Paternain, P. Stefanov and G. Uhlmann, The boundary rigidity problem in the presence of a magnetic field, Advances in Mathematics, 216 (2007), 535-609. doi: 10.1016/j.aim.2007.05.014. [7] M. deHoop, H. Smith, G. Uhlmann and R. van der Hilst, Seismic imaging with the generalized Radon transform: A curvelet transform perspective, Inverse Problems, 25 (2009), 21 pp. doi: 10.1088/0266-5611/25/2/025005. [8] D. Fanelli and O. Öktem, Electron tomography: A short overview with an emphasis on the absorption potential model for the forward problem, Inverse Problems, 24 (2008), 013001, 51 pp. doi: 10.1088/0266-5611/24/1/013001. [9] A. Faridani, D. Finch, E. L. Ritman and K. T. Smith, Local tomography II, SIAM Journal of Applied Mathematics, 57 (1997), 1095-1127. doi: 10.1137/S0036139995286357. [10] A. Faridani, E. L. Ritman and K. T. Smith, Local tomography, SIAM Journal of Applied Mathematics, 52 (1992), 459-484. doi: 10.1137/0152026. [11] R. Felea, Composition of Fourier integral operators with fold and blowdown singularities, Comm. Partial Differential Equations, 30 (2005), 1717-1740. doi: 10.1080/03605300500299968. [12] R. Felea and E. T. Quinto, The microlocal properties of the Local 3-D SPECT operator, SIAM J. Math. Anal., 43 (2011), 1145-1157. doi: 10.1137/100807703. [13] D. 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" (editor, G. Uhlmann), 47 of MSRI Publications, 193-218. Cambridge University Press, (2003). [14] B. Frigyik, P. Stefanov and G. Uhlmann, The X-Ray transform for a generic family of curves and weights, J. Geom. Anal., 18 (2008), 81-97. doi: 10.1007/s12220-007-9007-6. [15] I. Gelfand, M. Graev and N. Vilenkin, "Generalized Functions," 5 Academic Press, New York, 1966. [16] I. Gelfand and M. I. Graev, Integral transformations connected with straight line complexes in a complex affine space, Soviet Math. Doklady, 2 (1961), 809-812. [17] I. Gelfand and M. I. Graev, Complexes of straight lines in the space $\mathbbC^n$, Functional Analysis and its Applications, 2 (1968), 219-229. [18] I. Gelfand, M. I. Graev and Z. Shapiro, A local problem of integral geometry in a space of curves, Functional Anal. and Appl., 13 (1979), 87-102. [19] A. Greenleaf, A. Seeger and S. Waigner, Estimates for generalized Radon transforms in three and four dimensions, Contemporary Mathematics, 251 (2000), 243-254. doi: 10.1090/conm/251/03873. [20] A. Greenleaf. and G. Uhlmann, Non-local inversion formulas for the X-ray transform, Duke Mathematical Journal, 58 (1989), 205-240. doi: 10.1215/S0012-7094-89-05811-0. [21] A. Greenleaf and G. Uhlmann, Composition of some singular Fourier integral operators and estimates for restricted X-ray transforms, Ann. Inst. Fourier, Grenoble, 40 (1990), 443-466. doi: 10.5802/aif.1220. [22] A. Greenleaf and G. Uhlmann, Microlocal Techniques in Integral Geometry, Contemporary Math., 113 (1990), 121-136. doi: 10.1090/conm/113/1108649. [23] V. Guillemin, "Some Remarks on Integral Geometry," Technical report, MIT, 1975. [24] V. Guillemin and S. Sternberg, "Geometric Asymptotics," American Mathematical Society, Providence, RI, 1977. [25] P. W. Hawkes and E. Kasper, "Principles of Electron Optics. Volume 3. Wave Optics," Academic Press, 1994. [26] S. Helgason, The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassman manifolds, Acta Math., 113 (1965), 153-180. doi: 10.1007/BF02391776. [27] S. Holman and P. Stefanov, The weighted Doppler transform, Inverse Probl. Imaging, 4 (2010), 111-130. doi: 10.3934/ipi.2010.4.111. [28] L. Hörmander, Fourier integral operators, I, Acta Mathematica, 127 (1971), 79-183. doi: 10.1007/BF02392052. [29] L. Hörmander, "The Analysis of Linear Partial Differential Operators," I. Springer Verlag, New York, 1983. [30] A. Katsevich, Cone beam local tomography, SIAM J. Appl. Math., 59 (1999), 2224-2246. doi: 10.1137/S0036139998336043. [31] A. Katsevich, Improved Cone Beam Local Tomography, Inverse Problems, 22 (2006), 627-643. doi: 10.1088/0266-5611/22/2/015. [32] V. P. Krishnan and E. T. Quinto, Microlocal aspects of bistatic synthetic aperture radar imaging, Inverse Problems and Imaging, 5 (2011), 659-674. doi: 10.3934/ipi.2011.5.659. [33] Á. Kurusa, Support curves of invertible Radon transforms, Arch. Math. (Basel), 61 (1993), 448-458. doi: 10.1007/BF01207544. [34] A. Lawrence, J. Bouwer, G. Perkins and M. Ellisman, Transform-based backprojection for volume reconstruction of large format electron microscope tilt series, J. Structural Bio., 154 (2006), 144-167. doi: 10.1016/j.jsb.2005.12.012. [35] A. Louis and P. Maaş, Contour reconstruction in 3-D X-Ray CT, IEEE Trans. Medical Imaging, 12 (1993), 764-769. doi: 10.1109/42.251129. [36] C. Nolan and M. Cheney, Synthetic Aperture inversion, Inverse Problems, 18 (2002), 221-235. doi: 10.1088/0266-5611/18/1/315. [37] S. Phan, J. Bouwer, J. Lanman, M. Terada and A. Lawrence, Non-linear bundle adjustment for electron tomography, in "Computer Science and Information Engineering, 2009 WRI World Congress on Issue Date: March 31 2009-April 2 2009," 1 (2009), 604-612. doi: 10.1109/CSIE.2009.864. [38] S. Phan and A. Lawrence, Tomography of large format electron microscope tilt series: Image alignment and volume reconstruction, in "Proceedings of the 2008 Congress on Image and Signal Processing," 2 (2008), 176-182. doi: 10.1109/CISP.2008.535. [39] D. Popov, The generalized radon transform on the plane, the inverse transform, and the Cavalieri conditions, Functional Analysis and Its Applications, 35 (2001), 270-283. doi: 10.1023/A:1013126507543. [40] E. T. Quinto, The dependence of the generalized Radon transform on defining measures, Trans. Amer. Math. Soc., 257 (1980), 331-346. doi: 10.1090/S0002-9947-1980-0552261-8. [41] 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, Pisa, Italy, (2008). Centro De Georgi. CRM Series, 7. [42] E. T. Quinto and O. Öktem, Local tomography in electron microscopy, SIAM Journal of Applied Mathematics, 68 (2008), 1282-1303. doi: 10.1137/07068326X. [43] E. T. Quinto, A. Rieder and T. Schuster, Local Algorithms in Sonar, Inverse Problems, 27 (2011), 035006, 18p. doi: 10.1088/0266-5611/27/3/035006. [44] L. Reimer, "Transmission Electron Microscopy," 36 of Springer series in optical sciences. Springer Verlag, New York, 4 edition, 1997. doi: 10.1007/978-3-662-14824-2. [45] V. Romanov, "Integral Geometry and Inverse Problems for Hyperbolic Equations," 26 of Springer Tracts in Natural Philosophy. Springer Verlag, Berlin, New York, 1969. doi: 10.1007/978-3-642-80781-7. [46] P. Stefanov and G. Uhlmann, Integral geometry of tensor fields on a class of non-simple Riemannian manifolds, Amer. J. Math., 130 (2008), 239-268. doi: 10.1353/ajm.2008.0003. [47] E. Vainberg, I. Kazak and V. Kurozaev, Reconstruction of the internal three-dimensional structure of objects based on real-time integral projections, Soviet Journal of Nondestructive Testing, 17 (1981), 415-423.
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