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November  2016, 10(4): 899-914. doi: 10.3934/ipi.2016026

The localized basis functions for scalar and vector 3D tomography and their ray transforms

1. 

V.M. Matrosov Institute of System Dynamics and Control Theory, Siberian Branch of the Russian Academy of Sciences, Lermontov str. 134, 664033, Irkutsk-33, Russian Federation

Received  December 2015 Revised  July 2016 Published  October 2016

Localized basis set functions related to spherical wave functions are constructed for scalar and vector 3D tomography. The functions are truncated in a spherical domain of radius $r_0$, so that these vanish outside the domain. The analytical form of the respective scalar and vector ray transforms are obtained. So, the inversion algorithm can be reduced to solving the linear system of equations because of orthogonality and completeness of the basis functions.
Citation: Alexander Balandin. The localized basis functions for scalar and vector 3D tomography and their ray transforms. Inverse Problems and Imaging, 2016, 10 (4) : 899-914. doi: 10.3934/ipi.2016026
References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Dover Publications, New York, 1970.

[2]

S. You, H. Tanabe, Y. Ono and A. L. Balandin, Vector and scalar tomography of compact, Toroid Plasmas, J. Fusion Energy, 29 (2010), 592-595. doi: 10.1007/s10894-010-9317-8.

[3]

A. L. Balandin, Tomography of force-free fields, Numerical Analysis and Applications, 8 (2015), 195-207. doi: 10.1134/S1995423915030015.

[4]

B. Knyazev, A. L. Balandin, V. S. Cherkassky, Y. Y. Choporova, V. V. Gerasimov, A. A. Nikitin, V. V. Pickalov, M. G. Vlasenko, D. G. Rodionov, D. G. Esaev, M. A. Dem'yanenko and O. A. Shevchenko, Classic holography, tomography and speckle-metrology using a high-power terahertz FEL and real-time image detectors, 35th International Conference on Infrared, Millimeter, and Terahertz Waves (IRMMW-THz 2010, Rome, Italy, September 5-10, 2010). IEEE, 2010. doi: 10.1109/ICIMW.2010.5612533.

[5]

H. Bateman and A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, New York, 1953.

[6]

L. C. Biedenharn and J. D. Louck, Angular Momentum in Quantum Physics. Theory and Applications, Encyclopedia of Mathematics and its Applications, Addison-Wesley, Reading, MA, 1981.

[7]

A. M. Cormack, Representation of a function by its line integrals, with some radiological applications, J. Appl. Phys., 34 (1963), p2722. doi: 10.1063/1.1729798.

[8]

J. Cantarella, D. DeTurck and H. Gluck, Vector calculus and the topology of domains in 3-space, Amer. Math. Month., 109 (2002), 409-442. doi: 10.2307/2695643.

[9]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer-Verlag, Berlin, 1992.

[10]

M. E. Davison, A singular value decomposition for the Radon transform in $n$-dimensional, Euclidean space, Numer. Func. Anal. Optim., 3 (1981), 321-340. doi: 10.1080/01630568108816093.

[11]

E. Y. Derevtsov, A. Efimov, A. K. Lois and T. Schuster, Singular value decomposition and its application to numerical inversion for ray transforms in 2D vector tomography, J. Inv. Ill-Posed Prob., 19 (2011), 689-715. doi: 10.1515/jiip.2011.047.

[12]

E. Derevtsov, S. Kazantsev and T. Schuster, Polynomial bases for subspaces of potential and solenoidal vector fields in the unit ball of $R^3$, J. Inv. Ill-Posed Prob., 15 (2007), 19-55. doi: 10.1515/JIIP.2007.002.

[13]

A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, 1957.

[14]

W. Freeden, T. Gervens and M. Schreiner, Constructive Approximation on the Sphere with Applications to Geomathematics, Clarendon Press, Oxford, 1998.

[15]

W. Freeden and M. Schreiner, Spherical Functions of Mathematical Geosciences. A Scalar, Vectorial, and Tensorial Setup, Springer-Verlag Berlin Heidelberg, 2009.

[16]

W. W. Hansen, A new type of expansion in radiation problems, Phys. Rev., 47 (1935), 139-143. doi: 10.1103/PhysRev.47.139.

[17]

S. H. Izen, A series inversion for the X-ray transform in $n$ dimensions, Inverse Problems, 4 (1988), 725-748. doi: 10.1088/0266-5611/4/3/012.

[18]

S. H. Izen, Inversion of the $k$-plane transform by orthogonal function series expansions, Inverse Problems, 5 (1989), 181-202. doi: 10.1088/0266-5611/5/2/006.

[19]

S. Kazantsev and T. Schuster, Asymptotic inversion formulas in 3D vector field tomography for different geometries, J. Inv. Ill-Posed Prob., 19 (2011), 769-799. doi: 10.1515/jiip.2011.049.

[20]

S. G. Kazantsev and A. A. Bukgheim, Singular value decomposition for 2D fan-beam Radon transform of tensor fields, J. Inv. Ill-Posed Prob., 12 (2004), 245-278. doi: 10.1515/1569394042215865.

[21]

A. K. Louis, Orthogonal function series expansions and the null space of the Radon transform, SIAM J. Math. Anal., 15 (1984), 621-633. doi: 10.1137/0515047.

[22]

A. K. Louis, Incomplete data problems in X-ray computerized tomography I. Singular value decomposition of the limited angle transform, Numer. Math., 48 (1986), 251-262. doi: 10.1007/BF01389474.

[23]

R. B. Marr, On the reconstruction of a function on a circular domain from a sampling of its line integrals, J. Math. Anal. Appl., 45 (1974), 357-374. doi: 10.1016/0022-247X(74)90078-X.

[24]

P. Maass, The X-ray transform: Singular value decomposition and resolution, Inverse Problems, 3 (1987), 729-741. doi: 10.1088/0266-5611/3/4/016.

[25]

P. M. Morse and H. Feshbach, Methods of Theoretical Physics, McGraw-Hill, New York, 1953.

[26]

F. Natterer, The Mathematics of Computerized Tomography, John Wiley & Sons, New York, 1986.

[27]

F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction, SIAM, Philadelphia, 2001. doi: 10.1137/1.9780898718324.

[28]

F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark, NIST Handbook of Mathematical Functions, NIST and Cambridge Univ. Press, 2010.

[29]

M. Rosier, Biorthogonal series expansions of the X-ray and $k$-plane transforms, Inverse Problems, 11 (1995), 231-249. doi: 10.1088/0266-5611/11/1/014.

[30]

S. Stein, Addition theorem for spherical wave functions, Quart. Appl. Math., 19 (1961), 15-24.

[31]

J. A. Stratton, Electromagnetic Theory, McGrow-Hill, New York, 1941. doi: 10.1002/9781119134640.

[32]

D. A. Varshalovich, A. N. Moskalev and V. K. Khersonskii, Quantum Theory of Angular Momentum, World Scientific Publishing, Singapore, 1988. doi: 10.1142/0270.

[33]

L. Wang and R. S. Granetz, Expansion method in three-dimensional tomography, J. Opt. Soc. Am. A, 10 (1993), 2292-2295. doi: 10.1364/JOSAA.10.002292.

[34]

G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Univ. Press, Cambridge, 1995.

show all references

References:
[1]

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables, Dover Publications, New York, 1970.

[2]

S. You, H. Tanabe, Y. Ono and A. L. Balandin, Vector and scalar tomography of compact, Toroid Plasmas, J. Fusion Energy, 29 (2010), 592-595. doi: 10.1007/s10894-010-9317-8.

[3]

A. L. Balandin, Tomography of force-free fields, Numerical Analysis and Applications, 8 (2015), 195-207. doi: 10.1134/S1995423915030015.

[4]

B. Knyazev, A. L. Balandin, V. S. Cherkassky, Y. Y. Choporova, V. V. Gerasimov, A. A. Nikitin, V. V. Pickalov, M. G. Vlasenko, D. G. Rodionov, D. G. Esaev, M. A. Dem'yanenko and O. A. Shevchenko, Classic holography, tomography and speckle-metrology using a high-power terahertz FEL and real-time image detectors, 35th International Conference on Infrared, Millimeter, and Terahertz Waves (IRMMW-THz 2010, Rome, Italy, September 5-10, 2010). IEEE, 2010. doi: 10.1109/ICIMW.2010.5612533.

[5]

H. Bateman and A. Erdelyi, Higher Transcendental Functions, McGraw-Hill, New York, 1953.

[6]

L. C. Biedenharn and J. D. Louck, Angular Momentum in Quantum Physics. Theory and Applications, Encyclopedia of Mathematics and its Applications, Addison-Wesley, Reading, MA, 1981.

[7]

A. M. Cormack, Representation of a function by its line integrals, with some radiological applications, J. Appl. Phys., 34 (1963), p2722. doi: 10.1063/1.1729798.

[8]

J. Cantarella, D. DeTurck and H. Gluck, Vector calculus and the topology of domains in 3-space, Amer. Math. Month., 109 (2002), 409-442. doi: 10.2307/2695643.

[9]

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Springer-Verlag, Berlin, 1992.

[10]

M. E. Davison, A singular value decomposition for the Radon transform in $n$-dimensional, Euclidean space, Numer. Func. Anal. Optim., 3 (1981), 321-340. doi: 10.1080/01630568108816093.

[11]

E. Y. Derevtsov, A. Efimov, A. K. Lois and T. Schuster, Singular value decomposition and its application to numerical inversion for ray transforms in 2D vector tomography, J. Inv. Ill-Posed Prob., 19 (2011), 689-715. doi: 10.1515/jiip.2011.047.

[12]

E. Derevtsov, S. Kazantsev and T. Schuster, Polynomial bases for subspaces of potential and solenoidal vector fields in the unit ball of $R^3$, J. Inv. Ill-Posed Prob., 15 (2007), 19-55. doi: 10.1515/JIIP.2007.002.

[13]

A. R. Edmonds, Angular Momentum in Quantum Mechanics, Princeton University Press, 1957.

[14]

W. Freeden, T. Gervens and M. Schreiner, Constructive Approximation on the Sphere with Applications to Geomathematics, Clarendon Press, Oxford, 1998.

[15]

W. Freeden and M. Schreiner, Spherical Functions of Mathematical Geosciences. A Scalar, Vectorial, and Tensorial Setup, Springer-Verlag Berlin Heidelberg, 2009.

[16]

W. W. Hansen, A new type of expansion in radiation problems, Phys. Rev., 47 (1935), 139-143. doi: 10.1103/PhysRev.47.139.

[17]

S. H. Izen, A series inversion for the X-ray transform in $n$ dimensions, Inverse Problems, 4 (1988), 725-748. doi: 10.1088/0266-5611/4/3/012.

[18]

S. H. Izen, Inversion of the $k$-plane transform by orthogonal function series expansions, Inverse Problems, 5 (1989), 181-202. doi: 10.1088/0266-5611/5/2/006.

[19]

S. Kazantsev and T. Schuster, Asymptotic inversion formulas in 3D vector field tomography for different geometries, J. Inv. Ill-Posed Prob., 19 (2011), 769-799. doi: 10.1515/jiip.2011.049.

[20]

S. G. Kazantsev and A. A. Bukgheim, Singular value decomposition for 2D fan-beam Radon transform of tensor fields, J. Inv. Ill-Posed Prob., 12 (2004), 245-278. doi: 10.1515/1569394042215865.

[21]

A. K. Louis, Orthogonal function series expansions and the null space of the Radon transform, SIAM J. Math. Anal., 15 (1984), 621-633. doi: 10.1137/0515047.

[22]

A. K. Louis, Incomplete data problems in X-ray computerized tomography I. Singular value decomposition of the limited angle transform, Numer. Math., 48 (1986), 251-262. doi: 10.1007/BF01389474.

[23]

R. B. Marr, On the reconstruction of a function on a circular domain from a sampling of its line integrals, J. Math. Anal. Appl., 45 (1974), 357-374. doi: 10.1016/0022-247X(74)90078-X.

[24]

P. Maass, The X-ray transform: Singular value decomposition and resolution, Inverse Problems, 3 (1987), 729-741. doi: 10.1088/0266-5611/3/4/016.

[25]

P. M. Morse and H. Feshbach, Methods of Theoretical Physics, McGraw-Hill, New York, 1953.

[26]

F. Natterer, The Mathematics of Computerized Tomography, John Wiley & Sons, New York, 1986.

[27]

F. Natterer and F. Wübbeling, Mathematical Methods in Image Reconstruction, SIAM, Philadelphia, 2001. doi: 10.1137/1.9780898718324.

[28]

F. W. J. Olver, D. W. Lozier, R. F. Boisvert and C. W. Clark, NIST Handbook of Mathematical Functions, NIST and Cambridge Univ. Press, 2010.

[29]

M. Rosier, Biorthogonal series expansions of the X-ray and $k$-plane transforms, Inverse Problems, 11 (1995), 231-249. doi: 10.1088/0266-5611/11/1/014.

[30]

S. Stein, Addition theorem for spherical wave functions, Quart. Appl. Math., 19 (1961), 15-24.

[31]

J. A. Stratton, Electromagnetic Theory, McGrow-Hill, New York, 1941. doi: 10.1002/9781119134640.

[32]

D. A. Varshalovich, A. N. Moskalev and V. K. Khersonskii, Quantum Theory of Angular Momentum, World Scientific Publishing, Singapore, 1988. doi: 10.1142/0270.

[33]

L. Wang and R. S. Granetz, Expansion method in three-dimensional tomography, J. Opt. Soc. Am. A, 10 (1993), 2292-2295. doi: 10.1364/JOSAA.10.002292.

[34]

G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Univ. Press, Cambridge, 1995.

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