Abstract

We consider the application of tomography to the reconstruction of two-dimensional vector fields. The most practical sensor configuration in such problems is the regular positioning along the boundary of the reconstruction domain. However, such a configuration does not result in uniform distribution in the Radon parameter space, which is a necessary requirement to achieve accurate reconstruction results. On the other hand, sampling the projection space uniformly imposes serious constraints on space or time. In this paper, we propose to place the sensors regularly along the boundary of the reconstruction domain and employ probabilistic weights with the purpose of compensating for the lack of uniformity in the distribution of projection space parameters. Simulation results demonstrate that, when the proposed probabilistic weights are employed, an average 27% decrease in the reconstruction error may be achieved, over the case that projection measurements are not weighed (e.g., in one case the error reduces from 3.7% to 2.6%). When compared with the case where actual uniform sampling of the projection space is employed, the proposed method achieves a 90 times reduction in the number of the required sensors or 180 times reduction in the total scanning time, with only 7% increase in the error with which the vector field is estimated.

© 2011 Optical Society of America

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  1. S. P. Juhlin, “Doppler tomography,” in Proceedings of the 15th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC, 1993), pp. 212–213.
    [CrossRef]
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    [CrossRef] [PubMed]
  3. N. P. Efremov, N. P. Poluektov, and V. N. Kharchenko, “Tomography of ion and atom velocities in plasmas,” J. Quant. Spectrosc. Radiat. Transfer 53, 723–728 (1995).
    [CrossRef]
  4. B. M. Howe, P. F. Worcester, and R. C. Spindel, “Ocean acoustic tomography: mesoscale velocity,” J. Geophys. Res. 92, 3785–3806 (1987).
    [CrossRef]
  5. W. Munk and C. Wunsch, “Observing the ocean in the 1990s,” Phil. Trans. R. Soc. A 307, 439–464 (1982).
    [CrossRef]
  6. D. Rouseff, K. B. Winters, and T. E. Ewart, “Reconstruction of oceanic microstructure by tomography: a numerical feasibility study,” J. Geophys. Res. 96, 8823–8833 (1991).
    [CrossRef]
  7. S. A. Johnson, J. F. Greenleaf, M. Tanaka, and G. Flandro, “Reconstructing three-dimensional temperature and fluid velocity vector fields from acoustic transmission measurements,” ISA Trans. 16, 3–15 (1977).
  8. D. M. Kramer and P. C. Lauterbur, “On the problem of reconstructing images of non-scalar parameters from projections. Applications to vector fields,” IEEE Trans. Nucl. Sci. 26, 2674–2677 (1979).
    [CrossRef]
  9. S. J. Norton and M. Linzer, “Correcting for ray refraction in velocity and attenuation tomography: a perturbation approach,” Ultrason. Imag. 4, 201–233 (1982).
    [CrossRef]
  10. S. J. Norton, “Tomographic reconstruction of 2-D vector fields: application to flow imaging,” Geophys. J. Int. 97, 161–168 (1989).
    [CrossRef]
  11. S. J. Norton, “Unique tomographic reconstruction of vector fields using boundary data,” IEEE Trans. Image Process. 1, 406–412 (1992).
    [CrossRef] [PubMed]
  12. H. Braun and A. Hauck, “Tomographic reconstruction of vector fields,” IEEE Trans. Signal Process. 39, 464–471 (1991).
    [CrossRef]
  13. K. B. Winters and D. Rouseff, “A filtered backprojection method for the tomographic reconstruction of fluid vorticity,” Inverse Probl. 6, L33–L38 (1990).
    [CrossRef]
  14. K. B. Winters and D. Rouseff, “Tomographic reconstruction of stratified fluid flow,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 40, 26–33 (1993).
    [CrossRef] [PubMed]
  15. H. K. Aben, “Kerr effect tomography for general axisymmetric field,” Appl. Opt. 26, 2921–2924 (1987).
    [CrossRef] [PubMed]
  16. G. W. Faris and R. L. Byer, “Three-dimensional beam-deflection optical tomography of a supersonic jet,” Appl. Opt. 27, 5202–5212 (1988).
    [CrossRef] [PubMed]
  17. H. M. Hertz, “Kerr effect tomography for nonintrusive spatially resolved measurements of asymmetric electric field distributions,” Appl. Opt. 25, 914–921 (1986).
    [CrossRef] [PubMed]
  18. M. Zahn, “Transform relationship between Kerr-effect optical phase shift and non-uniform electric field distributions,” IEEE Trans. Dielectr. Electr. Insul. 1, 235–246 (1994).
    [CrossRef]
  19. V. A. Sharafutdinov, “Tomographic problem of photoelasticity,” Proc. SPIE 1843, 234–243 (1992).
    [CrossRef]
  20. H. Aben and A. Puro, “Photoelastic tomography for three-dimensional flow birefringence studies,” Inverse Probl. 13, 215–221 (1997).
    [CrossRef]
  21. S. E. Segre, “The measurement of poloidal magnetic field in a tokamak by the change of polarization of an electromagnetic wave,” Plasma Phys. 20, 295–307 (1978).
    [CrossRef]
  22. A. Schwarz, “Three-dimensional reconstruction of temperature and velocity fields in a furnace,” Part. Part. Syst. Charact. 12, 75–80 (1995).
    [CrossRef]
  23. P. Juhlin, “Principles of Doppler tomography,” LUTFD2/(TFMA-92)/7002+17P, Lund Institute of Technology, Sweden, 1992.
  24. L. Desbat and A. Wernsdorfer, “Direct algebraic reconstruction and optimal sampling in vector field tomography,” IEEE Trans. Signal Process. 43, 1798–1808 (1995).
    [CrossRef]
  25. T. Sato, H. Aoki, and O. Ikeda, “Introduction of mass conservation law to improve the tomographic estimation of flow-velocity distribution from differential time-of-flight data,” J. Acoust. Soc. Am. 77, 2104–2106 (1985).
    [CrossRef]
  26. I. Jovanovic, L. Sbaiz, and M. Vetterli, “Acoustic tomography for scalar and vector fields: theory and application to temperature and wind estimation,” J. Atmos. Ocean. Technol. 26, 1475–1492(2009).
    [CrossRef]
  27. D. Rouseff and K. B. Winters, “Two-dimensional vector flow inversion by diffraction tomography,” Inverse Probl. 10, 687–697 (1994).
    [CrossRef]
  28. M. Petrou and A. Giannakidis, “Complete tomographic reconstruction of 2-D vector fields using discrete integral data,” Comp. J. (to be published).
    [CrossRef]
  29. S. R. Deans, The Radon Transform and Some of Its Applications (Wiley, 1983).
  30. A. Papoulis, Probability, Random Variables, and Stochastic Processes, 2nd ed. (McGraw-Hill, 1984).
  31. A. Giannakidis and M. Petrou, “Sampling bounds for 2D vector field tomography,” J. Math. Imaging Vision 37, 151–165 (2010).
    [CrossRef]
  32. J. Hadamard, “Sur les problèmes aux dérivées partielles et leur signification physique,” Princeton University Bulletin 13, 49–52(1902).
  33. C. F. Gauss, Theoria Motus Corporum Coelestium (Perthes, 1809).
  34. I. N. Bronshtein, K. A. Semendyayev, G. Musiol, and H. Muehlig, Handbook of Mathematics (Springer, 2003).
  35. H. R. Schwarz, Numerische Mathematik (B. G. Teubner, 1986).
  36. H. Schwetlick and H. Kretzschmar, Numerische Verfahren fur Naturwissenschaftler und Ingenieure (Fachbuchverlag, 1991).
  37. M. Petrou and A. Kadyrov, “Affine invariant features from the trace transform,” IEEE Trans. Pattern Anal. Machine Intell. 26, 30–44 (2004).
    [CrossRef]
  38. A. Kadyrov and M. Petrou, “Affine parameter estimation from the trace transform,” IEEE Trans. Pattern Anal. Machine Intell. 28, 1631–1645 (2006).
    [CrossRef]

2010 (1)

A. Giannakidis and M. Petrou, “Sampling bounds for 2D vector field tomography,” J. Math. Imaging Vision 37, 151–165 (2010).
[CrossRef]

2009 (1)

I. Jovanovic, L. Sbaiz, and M. Vetterli, “Acoustic tomography for scalar and vector fields: theory and application to temperature and wind estimation,” J. Atmos. Ocean. Technol. 26, 1475–1492(2009).
[CrossRef]

2008 (1)

2006 (1)

A. Kadyrov and M. Petrou, “Affine parameter estimation from the trace transform,” IEEE Trans. Pattern Anal. Machine Intell. 28, 1631–1645 (2006).
[CrossRef]

2004 (1)

M. Petrou and A. Kadyrov, “Affine invariant features from the trace transform,” IEEE Trans. Pattern Anal. Machine Intell. 26, 30–44 (2004).
[CrossRef]

1997 (1)

H. Aben and A. Puro, “Photoelastic tomography for three-dimensional flow birefringence studies,” Inverse Probl. 13, 215–221 (1997).
[CrossRef]

1995 (3)

A. Schwarz, “Three-dimensional reconstruction of temperature and velocity fields in a furnace,” Part. Part. Syst. Charact. 12, 75–80 (1995).
[CrossRef]

L. Desbat and A. Wernsdorfer, “Direct algebraic reconstruction and optimal sampling in vector field tomography,” IEEE Trans. Signal Process. 43, 1798–1808 (1995).
[CrossRef]

N. P. Efremov, N. P. Poluektov, and V. N. Kharchenko, “Tomography of ion and atom velocities in plasmas,” J. Quant. Spectrosc. Radiat. Transfer 53, 723–728 (1995).
[CrossRef]

1994 (2)

M. Zahn, “Transform relationship between Kerr-effect optical phase shift and non-uniform electric field distributions,” IEEE Trans. Dielectr. Electr. Insul. 1, 235–246 (1994).
[CrossRef]

D. Rouseff and K. B. Winters, “Two-dimensional vector flow inversion by diffraction tomography,” Inverse Probl. 10, 687–697 (1994).
[CrossRef]

1993 (1)

K. B. Winters and D. Rouseff, “Tomographic reconstruction of stratified fluid flow,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 40, 26–33 (1993).
[CrossRef] [PubMed]

1992 (2)

V. A. Sharafutdinov, “Tomographic problem of photoelasticity,” Proc. SPIE 1843, 234–243 (1992).
[CrossRef]

S. J. Norton, “Unique tomographic reconstruction of vector fields using boundary data,” IEEE Trans. Image Process. 1, 406–412 (1992).
[CrossRef] [PubMed]

1991 (2)

H. Braun and A. Hauck, “Tomographic reconstruction of vector fields,” IEEE Trans. Signal Process. 39, 464–471 (1991).
[CrossRef]

D. Rouseff, K. B. Winters, and T. E. Ewart, “Reconstruction of oceanic microstructure by tomography: a numerical feasibility study,” J. Geophys. Res. 96, 8823–8833 (1991).
[CrossRef]

1990 (1)

K. B. Winters and D. Rouseff, “A filtered backprojection method for the tomographic reconstruction of fluid vorticity,” Inverse Probl. 6, L33–L38 (1990).
[CrossRef]

1989 (1)

S. J. Norton, “Tomographic reconstruction of 2-D vector fields: application to flow imaging,” Geophys. J. Int. 97, 161–168 (1989).
[CrossRef]

1988 (1)

1987 (2)

H. K. Aben, “Kerr effect tomography for general axisymmetric field,” Appl. Opt. 26, 2921–2924 (1987).
[CrossRef] [PubMed]

B. M. Howe, P. F. Worcester, and R. C. Spindel, “Ocean acoustic tomography: mesoscale velocity,” J. Geophys. Res. 92, 3785–3806 (1987).
[CrossRef]

1986 (1)

1985 (1)

T. Sato, H. Aoki, and O. Ikeda, “Introduction of mass conservation law to improve the tomographic estimation of flow-velocity distribution from differential time-of-flight data,” J. Acoust. Soc. Am. 77, 2104–2106 (1985).
[CrossRef]

1982 (2)

W. Munk and C. Wunsch, “Observing the ocean in the 1990s,” Phil. Trans. R. Soc. A 307, 439–464 (1982).
[CrossRef]

S. J. Norton and M. Linzer, “Correcting for ray refraction in velocity and attenuation tomography: a perturbation approach,” Ultrason. Imag. 4, 201–233 (1982).
[CrossRef]

1979 (1)

D. M. Kramer and P. C. Lauterbur, “On the problem of reconstructing images of non-scalar parameters from projections. Applications to vector fields,” IEEE Trans. Nucl. Sci. 26, 2674–2677 (1979).
[CrossRef]

1978 (1)

S. E. Segre, “The measurement of poloidal magnetic field in a tokamak by the change of polarization of an electromagnetic wave,” Plasma Phys. 20, 295–307 (1978).
[CrossRef]

1977 (1)

S. A. Johnson, J. F. Greenleaf, M. Tanaka, and G. Flandro, “Reconstructing three-dimensional temperature and fluid velocity vector fields from acoustic transmission measurements,” ISA Trans. 16, 3–15 (1977).

1902 (1)

J. Hadamard, “Sur les problèmes aux dérivées partielles et leur signification physique,” Princeton University Bulletin 13, 49–52(1902).

Aben, H.

H. Aben and A. Puro, “Photoelastic tomography for three-dimensional flow birefringence studies,” Inverse Probl. 13, 215–221 (1997).
[CrossRef]

Aben, H. K.

Aoki, H.

T. Sato, H. Aoki, and O. Ikeda, “Introduction of mass conservation law to improve the tomographic estimation of flow-velocity distribution from differential time-of-flight data,” J. Acoust. Soc. Am. 77, 2104–2106 (1985).
[CrossRef]

Braun, H.

H. Braun and A. Hauck, “Tomographic reconstruction of vector fields,” IEEE Trans. Signal Process. 39, 464–471 (1991).
[CrossRef]

Bronshtein, I. N.

I. N. Bronshtein, K. A. Semendyayev, G. Musiol, and H. Muehlig, Handbook of Mathematics (Springer, 2003).

Byer, R. L.

Davis, A. M.

Deans, S. R.

S. R. Deans, The Radon Transform and Some of Its Applications (Wiley, 1983).

Desbat, L.

L. Desbat and A. Wernsdorfer, “Direct algebraic reconstruction and optimal sampling in vector field tomography,” IEEE Trans. Signal Process. 43, 1798–1808 (1995).
[CrossRef]

Efremov, N. P.

N. P. Efremov, N. P. Poluektov, and V. N. Kharchenko, “Tomography of ion and atom velocities in plasmas,” J. Quant. Spectrosc. Radiat. Transfer 53, 723–728 (1995).
[CrossRef]

Ewart, T. E.

D. Rouseff, K. B. Winters, and T. E. Ewart, “Reconstruction of oceanic microstructure by tomography: a numerical feasibility study,” J. Geophys. Res. 96, 8823–8833 (1991).
[CrossRef]

Faris, G. W.

Flandro, G.

S. A. Johnson, J. F. Greenleaf, M. Tanaka, and G. Flandro, “Reconstructing three-dimensional temperature and fluid velocity vector fields from acoustic transmission measurements,” ISA Trans. 16, 3–15 (1977).

Gauss, C. F.

C. F. Gauss, Theoria Motus Corporum Coelestium (Perthes, 1809).

Giannakidis, A.

A. Giannakidis and M. Petrou, “Sampling bounds for 2D vector field tomography,” J. Math. Imaging Vision 37, 151–165 (2010).
[CrossRef]

M. Petrou and A. Giannakidis, “Complete tomographic reconstruction of 2-D vector fields using discrete integral data,” Comp. J. (to be published).
[CrossRef]

Greenleaf, J. F.

S. A. Johnson, J. F. Greenleaf, M. Tanaka, and G. Flandro, “Reconstructing three-dimensional temperature and fluid velocity vector fields from acoustic transmission measurements,” ISA Trans. 16, 3–15 (1977).

Hadamard, J.

J. Hadamard, “Sur les problèmes aux dérivées partielles et leur signification physique,” Princeton University Bulletin 13, 49–52(1902).

Hauck, A.

H. Braun and A. Hauck, “Tomographic reconstruction of vector fields,” IEEE Trans. Signal Process. 39, 464–471 (1991).
[CrossRef]

Hertz, H. M.

Howe, B. M.

B. M. Howe, P. F. Worcester, and R. C. Spindel, “Ocean acoustic tomography: mesoscale velocity,” J. Geophys. Res. 92, 3785–3806 (1987).
[CrossRef]

Ikeda, O.

T. Sato, H. Aoki, and O. Ikeda, “Introduction of mass conservation law to improve the tomographic estimation of flow-velocity distribution from differential time-of-flight data,” J. Acoust. Soc. Am. 77, 2104–2106 (1985).
[CrossRef]

Izatt, J. A.

Johnson, S. A.

S. A. Johnson, J. F. Greenleaf, M. Tanaka, and G. Flandro, “Reconstructing three-dimensional temperature and fluid velocity vector fields from acoustic transmission measurements,” ISA Trans. 16, 3–15 (1977).

Jovanovic, I.

I. Jovanovic, L. Sbaiz, and M. Vetterli, “Acoustic tomography for scalar and vector fields: theory and application to temperature and wind estimation,” J. Atmos. Ocean. Technol. 26, 1475–1492(2009).
[CrossRef]

Juhlin, P.

P. Juhlin, “Principles of Doppler tomography,” LUTFD2/(TFMA-92)/7002+17P, Lund Institute of Technology, Sweden, 1992.

Juhlin, S. P.

S. P. Juhlin, “Doppler tomography,” in Proceedings of the 15th Annual International Conference of the IEEE Engineering in Medicine and Biology Society (EMBC, 1993), pp. 212–213.
[CrossRef]

Kadyrov, A.

A. Kadyrov and M. Petrou, “Affine parameter estimation from the trace transform,” IEEE Trans. Pattern Anal. Machine Intell. 28, 1631–1645 (2006).
[CrossRef]

M. Petrou and A. Kadyrov, “Affine invariant features from the trace transform,” IEEE Trans. Pattern Anal. Machine Intell. 26, 30–44 (2004).
[CrossRef]

Kharchenko, V. N.

N. P. Efremov, N. P. Poluektov, and V. N. Kharchenko, “Tomography of ion and atom velocities in plasmas,” J. Quant. Spectrosc. Radiat. Transfer 53, 723–728 (1995).
[CrossRef]

Kramer, D. M.

D. M. Kramer and P. C. Lauterbur, “On the problem of reconstructing images of non-scalar parameters from projections. Applications to vector fields,” IEEE Trans. Nucl. Sci. 26, 2674–2677 (1979).
[CrossRef]

Kretzschmar, H.

H. Schwetlick and H. Kretzschmar, Numerische Verfahren fur Naturwissenschaftler und Ingenieure (Fachbuchverlag, 1991).

Lauterbur, P. C.

D. M. Kramer and P. C. Lauterbur, “On the problem of reconstructing images of non-scalar parameters from projections. Applications to vector fields,” IEEE Trans. Nucl. Sci. 26, 2674–2677 (1979).
[CrossRef]

Linzer, M.

S. J. Norton and M. Linzer, “Correcting for ray refraction in velocity and attenuation tomography: a perturbation approach,” Ultrason. Imag. 4, 201–233 (1982).
[CrossRef]

Muehlig, H.

I. N. Bronshtein, K. A. Semendyayev, G. Musiol, and H. Muehlig, Handbook of Mathematics (Springer, 2003).

Munk, W.

W. Munk and C. Wunsch, “Observing the ocean in the 1990s,” Phil. Trans. R. Soc. A 307, 439–464 (1982).
[CrossRef]

Musiol, G.

I. N. Bronshtein, K. A. Semendyayev, G. Musiol, and H. Muehlig, Handbook of Mathematics (Springer, 2003).

Norton, S. J.

S. J. Norton, “Unique tomographic reconstruction of vector fields using boundary data,” IEEE Trans. Image Process. 1, 406–412 (1992).
[CrossRef] [PubMed]

S. J. Norton, “Tomographic reconstruction of 2-D vector fields: application to flow imaging,” Geophys. J. Int. 97, 161–168 (1989).
[CrossRef]

S. J. Norton and M. Linzer, “Correcting for ray refraction in velocity and attenuation tomography: a perturbation approach,” Ultrason. Imag. 4, 201–233 (1982).
[CrossRef]

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 2nd ed. (McGraw-Hill, 1984).

Petrou, M.

A. Giannakidis and M. Petrou, “Sampling bounds for 2D vector field tomography,” J. Math. Imaging Vision 37, 151–165 (2010).
[CrossRef]

A. Kadyrov and M. Petrou, “Affine parameter estimation from the trace transform,” IEEE Trans. Pattern Anal. Machine Intell. 28, 1631–1645 (2006).
[CrossRef]

M. Petrou and A. Kadyrov, “Affine invariant features from the trace transform,” IEEE Trans. Pattern Anal. Machine Intell. 26, 30–44 (2004).
[CrossRef]

M. Petrou and A. Giannakidis, “Complete tomographic reconstruction of 2-D vector fields using discrete integral data,” Comp. J. (to be published).
[CrossRef]

Poluektov, N. P.

N. P. Efremov, N. P. Poluektov, and V. N. Kharchenko, “Tomography of ion and atom velocities in plasmas,” J. Quant. Spectrosc. Radiat. Transfer 53, 723–728 (1995).
[CrossRef]

Puro, A.

H. Aben and A. Puro, “Photoelastic tomography for three-dimensional flow birefringence studies,” Inverse Probl. 13, 215–221 (1997).
[CrossRef]

Rouseff, D.

D. Rouseff and K. B. Winters, “Two-dimensional vector flow inversion by diffraction tomography,” Inverse Probl. 10, 687–697 (1994).
[CrossRef]

K. B. Winters and D. Rouseff, “Tomographic reconstruction of stratified fluid flow,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 40, 26–33 (1993).
[CrossRef] [PubMed]

D. Rouseff, K. B. Winters, and T. E. Ewart, “Reconstruction of oceanic microstructure by tomography: a numerical feasibility study,” J. Geophys. Res. 96, 8823–8833 (1991).
[CrossRef]

K. B. Winters and D. Rouseff, “A filtered backprojection method for the tomographic reconstruction of fluid vorticity,” Inverse Probl. 6, L33–L38 (1990).
[CrossRef]

Sato, T.

T. Sato, H. Aoki, and O. Ikeda, “Introduction of mass conservation law to improve the tomographic estimation of flow-velocity distribution from differential time-of-flight data,” J. Acoust. Soc. Am. 77, 2104–2106 (1985).
[CrossRef]

Sbaiz, L.

I. Jovanovic, L. Sbaiz, and M. Vetterli, “Acoustic tomography for scalar and vector fields: theory and application to temperature and wind estimation,” J. Atmos. Ocean. Technol. 26, 1475–1492(2009).
[CrossRef]

Schwarz, A.

A. Schwarz, “Three-dimensional reconstruction of temperature and velocity fields in a furnace,” Part. Part. Syst. Charact. 12, 75–80 (1995).
[CrossRef]

Schwarz, H. R.

H. R. Schwarz, Numerische Mathematik (B. G. Teubner, 1986).

Schwetlick, H.

H. Schwetlick and H. Kretzschmar, Numerische Verfahren fur Naturwissenschaftler und Ingenieure (Fachbuchverlag, 1991).

Segre, S. E.

S. E. Segre, “The measurement of poloidal magnetic field in a tokamak by the change of polarization of an electromagnetic wave,” Plasma Phys. 20, 295–307 (1978).
[CrossRef]

Semendyayev, K. A.

I. N. Bronshtein, K. A. Semendyayev, G. Musiol, and H. Muehlig, Handbook of Mathematics (Springer, 2003).

Sharafutdinov, V. A.

V. A. Sharafutdinov, “Tomographic problem of photoelasticity,” Proc. SPIE 1843, 234–243 (1992).
[CrossRef]

Spindel, R. C.

B. M. Howe, P. F. Worcester, and R. C. Spindel, “Ocean acoustic tomography: mesoscale velocity,” J. Geophys. Res. 92, 3785–3806 (1987).
[CrossRef]

Tanaka, M.

S. A. Johnson, J. F. Greenleaf, M. Tanaka, and G. Flandro, “Reconstructing three-dimensional temperature and fluid velocity vector fields from acoustic transmission measurements,” ISA Trans. 16, 3–15 (1977).

Tao, Y. K.

Vetterli, M.

I. Jovanovic, L. Sbaiz, and M. Vetterli, “Acoustic tomography for scalar and vector fields: theory and application to temperature and wind estimation,” J. Atmos. Ocean. Technol. 26, 1475–1492(2009).
[CrossRef]

Wernsdorfer, A.

L. Desbat and A. Wernsdorfer, “Direct algebraic reconstruction and optimal sampling in vector field tomography,” IEEE Trans. Signal Process. 43, 1798–1808 (1995).
[CrossRef]

Winters, K. B.

D. Rouseff and K. B. Winters, “Two-dimensional vector flow inversion by diffraction tomography,” Inverse Probl. 10, 687–697 (1994).
[CrossRef]

K. B. Winters and D. Rouseff, “Tomographic reconstruction of stratified fluid flow,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 40, 26–33 (1993).
[CrossRef] [PubMed]

D. Rouseff, K. B. Winters, and T. E. Ewart, “Reconstruction of oceanic microstructure by tomography: a numerical feasibility study,” J. Geophys. Res. 96, 8823–8833 (1991).
[CrossRef]

K. B. Winters and D. Rouseff, “A filtered backprojection method for the tomographic reconstruction of fluid vorticity,” Inverse Probl. 6, L33–L38 (1990).
[CrossRef]

Worcester, P. F.

B. M. Howe, P. F. Worcester, and R. C. Spindel, “Ocean acoustic tomography: mesoscale velocity,” J. Geophys. Res. 92, 3785–3806 (1987).
[CrossRef]

Wunsch, C.

W. Munk and C. Wunsch, “Observing the ocean in the 1990s,” Phil. Trans. R. Soc. A 307, 439–464 (1982).
[CrossRef]

Zahn, M.

M. Zahn, “Transform relationship between Kerr-effect optical phase shift and non-uniform electric field distributions,” IEEE Trans. Dielectr. Electr. Insul. 1, 235–246 (1994).
[CrossRef]

Appl. Opt. (3)

Comp. J. (1)

M. Petrou and A. Giannakidis, “Complete tomographic reconstruction of 2-D vector fields using discrete integral data,” Comp. J. (to be published).
[CrossRef]

Geophys. J. Int. (1)

S. J. Norton, “Tomographic reconstruction of 2-D vector fields: application to flow imaging,” Geophys. J. Int. 97, 161–168 (1989).
[CrossRef]

IEEE Trans. Dielectr. Electr. Insul. (1)

M. Zahn, “Transform relationship between Kerr-effect optical phase shift and non-uniform electric field distributions,” IEEE Trans. Dielectr. Electr. Insul. 1, 235–246 (1994).
[CrossRef]

IEEE Trans. Image Process. (1)

S. J. Norton, “Unique tomographic reconstruction of vector fields using boundary data,” IEEE Trans. Image Process. 1, 406–412 (1992).
[CrossRef] [PubMed]

IEEE Trans. Nucl. Sci. (1)

D. M. Kramer and P. C. Lauterbur, “On the problem of reconstructing images of non-scalar parameters from projections. Applications to vector fields,” IEEE Trans. Nucl. Sci. 26, 2674–2677 (1979).
[CrossRef]

IEEE Trans. Pattern Anal. Machine Intell. (2)

M. Petrou and A. Kadyrov, “Affine invariant features from the trace transform,” IEEE Trans. Pattern Anal. Machine Intell. 26, 30–44 (2004).
[CrossRef]

A. Kadyrov and M. Petrou, “Affine parameter estimation from the trace transform,” IEEE Trans. Pattern Anal. Machine Intell. 28, 1631–1645 (2006).
[CrossRef]

IEEE Trans. Signal Process. (2)

L. Desbat and A. Wernsdorfer, “Direct algebraic reconstruction and optimal sampling in vector field tomography,” IEEE Trans. Signal Process. 43, 1798–1808 (1995).
[CrossRef]

H. Braun and A. Hauck, “Tomographic reconstruction of vector fields,” IEEE Trans. Signal Process. 39, 464–471 (1991).
[CrossRef]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (1)

K. B. Winters and D. Rouseff, “Tomographic reconstruction of stratified fluid flow,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 40, 26–33 (1993).
[CrossRef] [PubMed]

Inverse Probl. (3)

K. B. Winters and D. Rouseff, “A filtered backprojection method for the tomographic reconstruction of fluid vorticity,” Inverse Probl. 6, L33–L38 (1990).
[CrossRef]

D. Rouseff and K. B. Winters, “Two-dimensional vector flow inversion by diffraction tomography,” Inverse Probl. 10, 687–697 (1994).
[CrossRef]

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Figures (17)

Fig. 1
Fig. 1

Digitized reconstruction region is a square of size 2 U . The size of the pixels, with which we sample the 2D space, is P × P . Open circles represent the known and predetermined sensor positions from which we obtain the line-integral data. These positions are the middle points of the boundary edges of all boundary pixels. A scanning line segment A B is sampled with sampling step Δ s . The angle between the line segment and the positive direction of the x axis is w. Also shown are the two parameters ρ and θ used to define the scanning line (projection space coordinates) and the unit vectors s ^ and ρ ^ , which are parallel and perpendicular, respectively, to line segment A B .

Fig. 2
Fig. 2

First case of scanning lines.

Fig. 3
Fig. 3

Second case of scanning lines.

Fig. 4
Fig. 4

Third case of scanning lines.

Fig. 5
Fig. 5

Fourth case of scanning lines.

Fig. 6
Fig. 6

Fifth case of scanning lines.

Fig. 7
Fig. 7

Sixth case of scanning lines.

Fig. 8
Fig. 8

Areas that the six individual and the overall probability densities cover in the projection space.

Fig. 9
Fig. 9

Left, map of probability mass for the binning of the numerical example; right, same map expressed in x y coordinates.

Fig. 10
Fig. 10

Simulation results when the location of the source of the electric field was (from top to bottom) at ( 19 , 19 ) , ( 16 , 21 ) , ( 24 , 11.5 ) , and ( 21 , 12 ) ): (a) estimated vector field when estimation was based on weighted linear equations that approximate uniform sampling of the Radon space; (b) theoretical electric field as computed from Coulomb’s law; (c) estimated vector field when the estimation was based on linear equations that correspond to actual uniform sampling of the Radon space.

Fig. 11
Fig. 11

Left three columns, the relative errors in magnitude for case (i) uniform sensor placement along the boundary of the domain, as proposed in [28]; case (ii) same sensor arrangement as in (i), but also using weights to approximate uniform sampling in the ( ρ , θ ) space, as proposed in this paper; (iii) actual uniform sampling in the ( ρ , θ ) space. Right three columns, error in vector field orientation for the same cases. The location of the source of the electric field was (from top to bottom) at ( 19 , 19 ) , ( 16 , 21 ) , ( 24 , 11.5 ) , and ( 21 , 12 ) , respectively. We note that the histograms of the first and the fourth columns have heavier tails toward higher values.

Fig. 12
Fig. 12

Maps of relative magnitude error for the corresponding cases in Fig. 10.

Fig. 13
Fig. 13

Maps of absolute error in orientation in degrees for the corresponding cases in Fig. 10.

Fig. 14
Fig. 14

Simulation results for more complicated vector fields. Two top rows, fields created by two static sources, placed at ( 19 , 19 ) and ( 16 , 21 ) (top) and ( 11 , 24.5 ) and ( 19 , 19 ) (bottom). Bottom two rows, six sources placed at ( 19 , 19 ) , ( 16 , 21 ) , ( 24 , 11.5 ) , ( 21 , 12 ) , ( 19 , 19 ) , and ( 11 , 24.5 ) . Third row, strengths of the sources are 4.1027 8.9365 0.5789 3.5287 8.1317 0.0986 (first case). Fourth row, strengths of the sources are 0.1530 7.4680 4.4510 9.3180 4.6600 4.1860 (second case). (a) Results with the proposed correction. (b) Theoretically computed result. (c) Results with uniform sampling of the Radon space.

Fig. 15
Fig. 15

Histograms of errors for the estimations of Fig. 14, arranged as in Fig. 11.

Fig. 16
Fig. 16

Maps of relative magnitude error for the estimations of Fig. 14, arranged as in Fig. 12.

Fig. 17
Fig. 17

Maps of absolute error in orientation, measured in degrees, for the estimations of Fig. 14, arranged as in Fig. 13.

Tables (2)

Tables Icon

Table 1 Average Relative Magnitude Estimation Error (%) per Pixel (ME) and the Average Absolute Angular Estimation Error (in Degrees) per Pixel (AE) for the Three Methods a

Tables Icon

Table 2 Average Errors for the Estimations of Fig. 14, Presented as the Results in Table 1

Equations (55)

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J 1 = L f ¯ ( x , y ) · s ^ d s = L f d s ,
J 2 = L f ¯ ( x , y ) · ρ ^ d s = L f d s .
J i = A B f ¯ ( x , y ) · s i ^ d s .
J i = l f ¯ l · Δ s ¯ i ,
ρ = x cos θ + y sin θ ,
ρ = x 1 cos θ U sin θ ,
ρ = U cos θ + y 2 sin θ .
f x 1 ( x 1 ) = 1 2 U [ H ( x 1 + U ) H ( x 1 U ) ] ,
f y 2 ( y 2 ) = 1 2 U [ H ( y 2 + U ) H ( y 2 U ) ] .
f x 1 y 2 ( x 1 , y 2 ) = f x 1 ( x 1 ) f y 2 ( y 2 ) = 1 4 U 2 [ H ( x 1 + U ) H ( x 1 U ) ] [ H ( y 2 + U ) H ( y 2 U ) ] .
θ = arctan x 1 U y 2 + U ,
ρ = x 1 cos ( arctan x 1 U y 2 + U ) U sin ( arctan x 1 U y 2 + U ) .
ρ = h ( x 1 , y 2 ) ,
θ = g ( x 1 , y 2 ) ,
f ρ θ ( ρ , θ ) = f x 1 y 2 ( x 1 a , y 2 a ) | J ( x 1 a , y 2 a ) | + + f x 1 y 2 ( x 1 k , y 2 k ) | J ( x 1 k , y 2 k ) | ,
J ( x 1 , y 2 ) = | ρ x 1 ρ y 2 θ x 1 θ y 2 | = | x 1 ρ x 1 θ y 2 ρ y 2 θ | 1
( x 1 a , y 2 a ) = ( ρ cos θ + U tan θ , ρ sin θ U cot θ ) .
J ( x 1 a , y 2 a ) = | 1 cos θ ρ sin θ cos 2 θ + U cos 2 θ 1 sin θ ρ cos θ sin 2 θ + U sin 2 θ | 1 = [ U ρ cos θ cos θ sin 2 θ U + ρ sin θ sin θ cos 2 θ ] 1 = [ U ( cos θ sin θ ) ρ cos 2 θ sin 2 θ ] 1 .
J ( x 1 a , y 2 a ) = cos 2 θ sin 2 θ U ( cos θ sin θ ) ρ .
f ρ θ 1 ( ρ , θ ) = | U ( cos θ sin θ ) ρ cos 2 θ sin 2 θ | f x 1 y 2 ( ρ cos θ + U tan θ , ρ sin θ U cot θ ) .
f ρ θ 1 ( ρ , θ ) = | U ( cos θ sin θ ) ρ cos 2 θ sin 2 θ | 4 U 2 [ H ( ρ cos θ + U tan θ + U ) H ( ρ cos θ + U tan θ U ) ] × [ H ( ρ sin θ U cot θ + U ) H ( ρ sin θ U cot θ U ) ] .
f ρ θ 2 ( ρ , θ ) = | U ( cos θ + sin θ ) + ρ cos 2 θ sin 2 θ | 4 U 2 [ H ( ρ cos θ + U tan θ + U ) H ( ρ cos θ + U tan θ U ) ] × [ H ( ρ sin θ + U cot θ + U ) H ( ρ sin θ + U cot θ U ) ] .
f ρ θ 3 ( ρ , θ ) = | U ( cos θ + sin θ ) ρ cos 2 θ sin 2 θ | 4 U 2 [ H ( ρ cos θ U tan θ + U ) H ( ρ cos θ U tan θ U ) ] × [ H ( ρ sin θ U cot θ + U ) H ( ρ sin θ U cot θ U ) ] .
f ρ θ 4 ( ρ , θ ) = | U ( sin θ cos θ ) ρ cos 2 θ sin 2 θ | 4 U 2 [ H ( ρ cos θ U tan θ + U ) H ( ρ cos θ U tan θ U ) ] × [ H ( ρ sin θ + U cot θ + U ) H ( ρ sin θ + U cot θ U ) ] .
ρ = x 1 cos θ U sin θ ,
ρ = x 2 cos θ + U sin θ .
f x 2 ( x 2 ) = 1 2 U [ H ( x 2 + U ) H ( x 2 U ) ] .
f x 1 x 2 ( x 1 , x 2 ) = f x 1 ( x 1 ) f x 2 ( x 2 ) = 1 4 U 2 [ H ( x 1 + U ) H ( x 1 U ) ] [ H ( x 2 + U ) H ( x 2 U ) ] .
θ = arctan x 1 x 2 2 U π ,
ρ = x 1 cos ( arctan x 1 x 2 2 U π ) U sin ( arctan x 1 x 2 2 U π ) .
θ = arctan x 1 x 2 2 U ,
ρ = x 1 cos ( arctan x 1 x 2 2 U ) U sin ( arctan x 1 x 2 2 U ) .
θ = arctan x 1 x 2 2 U + π ,
ρ = x 1 cos ( arctan x 1 x 2 2 U + π ) U sin ( arctan x 1 x 2 2 U + π ) .
( x 1 a , x 2 a ) = ( ρ cos θ + U tan θ , ρ cos θ U tan θ )
J ( x 1 a , x 2 a ) = | ρ x 1 a ρ x 2 a θ x 1 a θ x 2 a | = | x 1 a ρ x 1 a θ x 2 a ρ x 2 a θ | 1 = | 1 cos θ ρ sin θ cos 2 θ + U cos 2 θ 1 cos θ ρ sin θ cos 2 θ U cos 2 θ | 1 = cos 3 θ 2 U .
f ρ θ 5 ( ρ , θ ) = | 2 U cos 3 θ | f x 1 x 2 ( ρ cos θ + U tan θ , ρ cos θ U tan θ ) .
f ρ θ 5 ( ρ , θ ) = | 2 U cos 3 θ | 4 U 2 [ H ( ρ cos θ + U tan θ + U ) H ( ρ cos θ + U tan θ U ) ] × [ H ( ρ cos θ U tan θ + U ) H ( ρ cos θ U tan θ U ) ] .
ρ = U cos θ + y 1 sin θ ,
ρ = U cos θ + y 2 sin θ .
f y 1 ( y 1 ) = 1 2 U [ H ( y 1 + U ) H ( y 1 U ) ] .
f y 1 y 2 ( y 1 , y 2 ) = f y 1 ( y 1 ) f y 2 ( y 2 ) = 1 4 U 2 [ H ( y 1 + U ) H ( y 1 U ) ] [ H ( y 2 + U ) H ( y 2 U ) ] .
θ = arccot y 2 y 1 2 U π ,
ρ = U cos ( arccot y 2 y 1 2 U π ) + y 1 sin ( arccot y 2 y 1 2 U π ) .
θ = arccot y 2 y 1 2 U ,
ρ = U cos ( arccot y 2 y 1 2 U ) + y 1 sin ( arccot y 2 y 1 2 U ) .
( y 1 a , y 2 a ) = ( ρ sin θ U cot θ , ρ sin θ + U cot θ ) ,
J ( y 1 a , y 2 a ) = | ρ y 1 a ρ y 2 a θ y 1 a θ y 2 a | = | y 1 a ρ y 1 a θ y 2 a ρ y 2 a θ | 1 = sin 3 θ 2 U .
f ρ θ 6 ( ρ , θ ) = | 2 U sin 3 θ | f y 1 y 2 ( ρ sin θ U cot θ , ρ sin θ + U cot θ ) .
f ρ θ 6 ( ρ , θ ) = | 2 U sin 3 θ | 4 U 2 [ H ( ρ sin θ U cot θ + U ) H ( ρ sin θ U cot θ U ) ] × [ H ( ρ sin θ + U cot θ + U ) H ( ρ sin θ + U cot θ U ) ] .
f ρ θ ( ρ , θ ) = i = 1 6 f ρ θ i ( ρ , θ | i th case ) Prob ( i th case )
f ρ θ ( ρ , θ ) = 1 6 ( f ρ θ 1 ( ρ , θ ) + f ρ θ 2 ( ρ , θ ) + f ρ θ 3 ( ρ , θ ) + f ρ θ 4 ( ρ , θ ) + f ρ θ 5 ( ρ , θ ) + f ρ θ 6 ( ρ , θ ) ) .
p b = θ b l θ b u ρ b l ρ b u f ρ θ ( ρ , θ ) d ρ d θ , b = 1 , 2 , , R T ,
+ + f ρ θ ( ρ , θ ) d ρ d θ = 1.
w i = 1 p b 1 R T , i = 1 , 2 , , L ,

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