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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 & 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. doi: 10.1364/JOSAA.24.001569. Google Scholar [2] J. Boman and E. T. Quinto, Support theorems for real analytic Radon transforms,, Duke Math. J., 55 (1987), 943. doi: 10.1215/S0012-7094-87-05547-5. Google Scholar [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. doi: 10.1090/S0002-9947-1993-1080733-8. Google Scholar [4] A. Cormack, The Radon transform on a family of curves in the plane,, Proc. Amer. Math. Soc., 83 (1981), 325. doi: 10.1090/S0002-9939-1981-0624923-1. Google Scholar [5] A. Cormack, The Radon transform on a family of curves in the plane. II,, Proc. Amer. Math. Soc., 86 (1982), 293. doi: 10.1090/S0002-9939-1982-0667292-4. Google Scholar [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. doi: 10.1016/j.aim.2007.05.014. Google Scholar [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). doi: 10.1088/0266-5611/25/2/025005. Google Scholar [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). doi: 10.1088/0266-5611/24/1/013001. Google Scholar [9] A. Faridani, D. Finch, E. L. Ritman and K. T. Smith, Local tomography II,, SIAM Journal of Applied Mathematics, 57 (1997), 1095. doi: 10.1137/S0036139995286357. Google Scholar [10] A. Faridani, E. L. Ritman and K. T. Smith, Local tomography,, SIAM Journal of Applied Mathematics, 52 (1992), 459. doi: 10.1137/0152026. Google Scholar [11] R. Felea, Composition of Fourier integral operators with fold and blowdown singularities,, Comm. Partial Differential Equations, 30 (2005), 1717. doi: 10.1080/03605300500299968. Google Scholar [12] R. Felea and E. T. Quinto, The microlocal properties of the Local 3-D SPECT operator,, SIAM J. Math. Anal., 43 (2011), 1145. doi: 10.1137/100807703. Google Scholar [13] D. 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 [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. doi: 10.1007/s12220-007-9007-6. Google Scholar [15] I. Gelfand, M. Graev and N. Vilenkin, "Generalized Functions,", 5 Academic Press, 5 (1966). Google Scholar [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. Google Scholar [17] I. Gelfand and M. I. Graev, Complexes of straight lines in the space $\mathbbC^n$,, Functional Analysis and its Applications, 2 (1968), 219. Google Scholar [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. Google Scholar [19] A. Greenleaf, A. Seeger and S. Waigner, Estimates for generalized Radon transforms in three and four dimensions,, Contemporary Mathematics, 251 (2000), 243. doi: 10.1090/conm/251/03873. Google Scholar [20] A. Greenleaf. and G. Uhlmann, Non-local inversion formulas for the X-ray transform,, Duke Mathematical Journal, 58 (1989), 205. doi: 10.1215/S0012-7094-89-05811-0. Google Scholar [21] A. Greenleaf and G. Uhlmann, Composition of some singular Fourier integral operators and estimates for restricted X-ray transforms,, Ann. Inst. Fourier, 40 (1990), 443. doi: 10.5802/aif.1220. Google Scholar [22] A. Greenleaf and G. Uhlmann, Microlocal Techniques in Integral Geometry,, Contemporary Math., 113 (1990), 121. doi: 10.1090/conm/113/1108649. Google Scholar [23] V. Guillemin, "Some Remarks on Integral Geometry,", Technical report, (1975). Google Scholar [24] V. Guillemin and S. Sternberg, "Geometric Asymptotics,", American Mathematical Society, (1977). Google Scholar [25] P. W. Hawkes and E. Kasper, "Principles of Electron Optics. Volume 3. Wave Optics,", Academic Press, (1994). Google Scholar [26] S. Helgason, The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassman manifolds,, Acta Math., 113 (1965), 153. doi: 10.1007/BF02391776. Google Scholar [27] S. Holman and P. Stefanov, The weighted Doppler transform,, Inverse Probl. Imaging, 4 (2010), 111. doi: 10.3934/ipi.2010.4.111. Google Scholar [28] L. Hörmander, Fourier integral operators, I,, Acta Mathematica, 127 (1971), 79. doi: 10.1007/BF02392052. Google Scholar [29] L. Hörmander, "The Analysis of Linear Partial Differential Operators,", I. Springer Verlag, I (1983). Google Scholar [30] A. Katsevich, Cone beam local tomography,, SIAM J. Appl. Math., 59 (1999), 2224. doi: 10.1137/S0036139998336043. Google Scholar [31] A. Katsevich, Improved Cone Beam Local Tomography,, Inverse Problems, 22 (2006), 627. doi: 10.1088/0266-5611/22/2/015. Google Scholar [32] V. P. Krishnan and E. T. Quinto, Microlocal aspects of bistatic synthetic aperture radar imaging,, Inverse Problems and Imaging, 5 (2011), 659. doi: 10.3934/ipi.2011.5.659. Google Scholar [33] Á. Kurusa, Support curves of invertible Radon transforms,, Arch. Math. (Basel), 61 (1993), 448. doi: 10.1007/BF01207544. Google Scholar [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. doi: 10.1016/j.jsb.2005.12.012. Google Scholar [35] A. Louis and P. Maaş, Contour reconstruction in 3-D X-Ray CT,, IEEE Trans. Medical Imaging, 12 (1993), 764. doi: 10.1109/42.251129. Google Scholar [36] C. Nolan and M. Cheney, Synthetic Aperture inversion,, Inverse Problems, 18 (2002), 221. doi: 10.1088/0266-5611/18/1/315. Google Scholar [37] S. Phan, J. Bouwer, J. Lanman, M. Terada and A. Lawrence, Non-linear bundle adjustment for electron tomography,, in, 1 (2009), 604. doi: 10.1109/CSIE.2009.864. Google Scholar [38] S. Phan and A. Lawrence, Tomography of large format electron microscope tilt series: Image alignment and volume reconstruction,, in, 2 (2008), 176. doi: 10.1109/CISP.2008.535. Google Scholar [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. doi: 10.1023/A:1013126507543. Google Scholar [40] E. T. Quinto, The dependence of the generalized Radon transform on defining measures,, Trans. Amer. Math. Soc., 257 (1980), 331. doi: 10.1090/S0002-9947-1980-0552261-8. Google Scholar [41] E. T. Quinto, T. Bakhos and S. Chung, A local algorithm for Slant Hole SPECT,, in, 7 (2008), 321. Google Scholar [42] E. T. Quinto and O. Öktem, Local tomography in electron microscopy,, SIAM Journal of Applied Mathematics, 68 (2008), 1282. doi: 10.1137/07068326X. Google Scholar [43] E. T. Quinto, A. Rieder and T. Schuster, Local Algorithms in Sonar,, Inverse Problems, 27 (2011). doi: 10.1088/0266-5611/27/3/035006. Google Scholar [44] L. Reimer, "Transmission Electron Microscopy,", 36 of Springer series in optical sciences. Springer Verlag, 36 (1997). doi: 10.1007/978-3-662-14824-2. Google Scholar [45] V. Romanov, "Integral Geometry and Inverse Problems for Hyperbolic Equations,", 26 of Springer Tracts in Natural Philosophy. Springer Verlag, 26 (1969). doi: 10.1007/978-3-642-80781-7. Google Scholar [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. doi: 10.1353/ajm.2008.0003. Google Scholar [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. Google Scholar

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. doi: 10.1364/JOSAA.24.001569. Google Scholar [2] J. Boman and E. T. Quinto, Support theorems for real analytic Radon transforms,, Duke Math. J., 55 (1987), 943. doi: 10.1215/S0012-7094-87-05547-5. Google Scholar [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. doi: 10.1090/S0002-9947-1993-1080733-8. Google Scholar [4] A. Cormack, The Radon transform on a family of curves in the plane,, Proc. Amer. Math. Soc., 83 (1981), 325. doi: 10.1090/S0002-9939-1981-0624923-1. Google Scholar [5] A. Cormack, The Radon transform on a family of curves in the plane. II,, Proc. Amer. Math. Soc., 86 (1982), 293. doi: 10.1090/S0002-9939-1982-0667292-4. Google Scholar [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. doi: 10.1016/j.aim.2007.05.014. Google Scholar [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). doi: 10.1088/0266-5611/25/2/025005. Google Scholar [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). doi: 10.1088/0266-5611/24/1/013001. Google Scholar [9] A. Faridani, D. Finch, E. L. Ritman and K. T. Smith, Local tomography II,, SIAM Journal of Applied Mathematics, 57 (1997), 1095. doi: 10.1137/S0036139995286357. Google Scholar [10] A. Faridani, E. L. Ritman and K. T. Smith, Local tomography,, SIAM Journal of Applied Mathematics, 52 (1992), 459. doi: 10.1137/0152026. Google Scholar [11] R. Felea, Composition of Fourier integral operators with fold and blowdown singularities,, Comm. Partial Differential Equations, 30 (2005), 1717. doi: 10.1080/03605300500299968. Google Scholar [12] R. Felea and E. T. Quinto, The microlocal properties of the Local 3-D SPECT operator,, SIAM J. Math. Anal., 43 (2011), 1145. doi: 10.1137/100807703. Google Scholar [13] D. 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 [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. doi: 10.1007/s12220-007-9007-6. Google Scholar [15] I. Gelfand, M. Graev and N. Vilenkin, "Generalized Functions,", 5 Academic Press, 5 (1966). Google Scholar [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. Google Scholar [17] I. Gelfand and M. I. Graev, Complexes of straight lines in the space $\mathbbC^n$,, Functional Analysis and its Applications, 2 (1968), 219. Google Scholar [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. Google Scholar [19] A. Greenleaf, A. Seeger and S. Waigner, Estimates for generalized Radon transforms in three and four dimensions,, Contemporary Mathematics, 251 (2000), 243. doi: 10.1090/conm/251/03873. Google Scholar [20] A. Greenleaf. and G. Uhlmann, Non-local inversion formulas for the X-ray transform,, Duke Mathematical Journal, 58 (1989), 205. doi: 10.1215/S0012-7094-89-05811-0. Google Scholar [21] A. Greenleaf and G. Uhlmann, Composition of some singular Fourier integral operators and estimates for restricted X-ray transforms,, Ann. Inst. Fourier, 40 (1990), 443. doi: 10.5802/aif.1220. Google Scholar [22] A. Greenleaf and G. Uhlmann, Microlocal Techniques in Integral Geometry,, Contemporary Math., 113 (1990), 121. doi: 10.1090/conm/113/1108649. Google Scholar [23] V. Guillemin, "Some Remarks on Integral Geometry,", Technical report, (1975). Google Scholar [24] V. Guillemin and S. Sternberg, "Geometric Asymptotics,", American Mathematical Society, (1977). Google Scholar [25] P. W. Hawkes and E. Kasper, "Principles of Electron Optics. Volume 3. Wave Optics,", Academic Press, (1994). Google Scholar [26] S. Helgason, The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassman manifolds,, Acta Math., 113 (1965), 153. doi: 10.1007/BF02391776. Google Scholar [27] S. Holman and P. Stefanov, The weighted Doppler transform,, Inverse Probl. Imaging, 4 (2010), 111. doi: 10.3934/ipi.2010.4.111. Google Scholar [28] L. Hörmander, Fourier integral operators, I,, Acta Mathematica, 127 (1971), 79. doi: 10.1007/BF02392052. Google Scholar [29] L. Hörmander, "The Analysis of Linear Partial Differential Operators,", I. Springer Verlag, I (1983). Google Scholar [30] A. Katsevich, Cone beam local tomography,, SIAM J. Appl. Math., 59 (1999), 2224. doi: 10.1137/S0036139998336043. Google Scholar [31] A. Katsevich, Improved Cone Beam Local Tomography,, Inverse Problems, 22 (2006), 627. doi: 10.1088/0266-5611/22/2/015. Google Scholar [32] V. P. Krishnan and E. T. Quinto, Microlocal aspects of bistatic synthetic aperture radar imaging,, Inverse Problems and Imaging, 5 (2011), 659. doi: 10.3934/ipi.2011.5.659. Google Scholar [33] Á. Kurusa, Support curves of invertible Radon transforms,, Arch. Math. (Basel), 61 (1993), 448. doi: 10.1007/BF01207544. Google Scholar [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. doi: 10.1016/j.jsb.2005.12.012. Google Scholar [35] A. Louis and P. Maaş, Contour reconstruction in 3-D X-Ray CT,, IEEE Trans. Medical Imaging, 12 (1993), 764. doi: 10.1109/42.251129. Google Scholar [36] C. Nolan and M. Cheney, Synthetic Aperture inversion,, Inverse Problems, 18 (2002), 221. doi: 10.1088/0266-5611/18/1/315. Google Scholar [37] S. Phan, J. Bouwer, J. Lanman, M. Terada and A. Lawrence, Non-linear bundle adjustment for electron tomography,, in, 1 (2009), 604. doi: 10.1109/CSIE.2009.864. Google Scholar [38] S. Phan and A. Lawrence, Tomography of large format electron microscope tilt series: Image alignment and volume reconstruction,, in, 2 (2008), 176. doi: 10.1109/CISP.2008.535. Google Scholar [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. doi: 10.1023/A:1013126507543. Google Scholar [40] E. T. Quinto, The dependence of the generalized Radon transform on defining measures,, Trans. Amer. Math. Soc., 257 (1980), 331. doi: 10.1090/S0002-9947-1980-0552261-8. Google Scholar [41] E. T. Quinto, T. Bakhos and S. Chung, A local algorithm for Slant Hole SPECT,, in, 7 (2008), 321. Google Scholar [42] E. T. Quinto and O. Öktem, Local tomography in electron microscopy,, SIAM Journal of Applied Mathematics, 68 (2008), 1282. doi: 10.1137/07068326X. Google Scholar [43] E. T. Quinto, A. Rieder and T. Schuster, Local Algorithms in Sonar,, Inverse Problems, 27 (2011). doi: 10.1088/0266-5611/27/3/035006. Google Scholar [44] L. Reimer, "Transmission Electron Microscopy,", 36 of Springer series in optical sciences. Springer Verlag, 36 (1997). doi: 10.1007/978-3-662-14824-2. Google Scholar [45] V. Romanov, "Integral Geometry and Inverse Problems for Hyperbolic Equations,", 26 of Springer Tracts in Natural Philosophy. Springer Verlag, 26 (1969). doi: 10.1007/978-3-642-80781-7. Google Scholar [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. doi: 10.1353/ajm.2008.0003. Google Scholar [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. Google Scholar
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