Abstract

We present a method for obtaining accurate image reconstruction from highly sparse data in diffraction tomography (DT). A practical need exists for reconstruction from few-view and limited-angle data, as this can greatly reduce required scan times in DT. Our method does this by minimizing the total variation (TV) of the estimated image, subject to the constraint that the Fourier transform of the estimated image matches the measured Fourier data samples. Using simulation studies, we show that the TV-minimization algorithm allows accurate reconstruction in a variety of few-view and limited-angle situations in DT. Accurate image reconstruction is obtained from far fewer data samples than are required by common algorithms such as the filtered-backpropagation algorithm. Overall our results indicate that the TV-minimization algorithm can be successfully applied to DT image reconstruction under a variety of scan configurations and data conditions of practical significance.

© 2008 Optical Society of America

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  1. M. Slaney and A. C. Kak, “Diffraction tomography,” in Inverse Optics, Vol. 14, A.J.Devaney, ed. (SPIE, 1983), pp. 2-19.
  2. M. P. Andre, P. J. Martin, G. P. Otto, L. K. Olson, T. K. Barrett, B. A. Spivey, and D. A. Palmer, “A new consideration of diffraction computed tomography for breast imaging: Studies in phantoms and patients,” Acoust. Imaging 21, 379-390 (1995).
  3. A. Devaney, “Geophysical diffraction tomography,” IEEE Trans. Geosci. Remote Sens. 22, 3-13 (1984).
    [CrossRef]
  4. V. E. Kunitsyn, E. S. Andreeva, E. D. Tereschenko, B. Z. Khudukon, and T. Nygren, “Investigations of the ionosphere by satellite radiotomography,” Int. J. Imaging Syst. Technol. 5, 112-127 (1994).
    [CrossRef]
  5. R. Mueller, M. Kaveh, and G. Wade, “Reconstructive tomography and applications to ultrasonics,” in Proc. IEEE 67, 567-587 (1979).
    [CrossRef]
  6. S. X. Pan and A. C. Kak, “A computational study of reconstruction algorithms for diffraction tomography: Interpolation versus filtered backpropagation,” IEEE Trans. Acoust., Speech, Signal Process. 31, 1262-1275 (1983).
    [CrossRef]
  7. A. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336-350 (1982).
    [CrossRef] [PubMed]
  8. P. Guo and A. J. Devaney, “Comparison of reconstruction algorithms for optical diffraction tomography,” J. Opt. Soc. Am. A 22, 2338-2347 (2005).
    [CrossRef]
  9. E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489-509 (2006).
    [CrossRef]
  10. E. Y. Sidky, C. Kao, and X. Pan, “Accurate image reconstruction from few-views and limited-angle data in divergent-beam CT,” J. X-Ray Sci. Technol. 14, 1-21 (2006).
  11. E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153-156 (1969).
    [CrossRef]
  12. X. Pan, “Unified reconstruction theory for diffraction tomography, with consideration of noise control,” J. Opt. Soc. Am. A 15, 2312-2326 (1998).
    [CrossRef]
  13. A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (SIAM, 2001).
    [CrossRef]
  14. X. Pan and M. A. Anastasio, “Minimal-scan filtered backpropagation algorithms for diffraction tomography,” J. Opt. Soc. Am. A 16, 2896-2903 (1999).
    [CrossRef]
  15. M. H. Li, H. Q. Yang, and H. Kudo, “An accurate iterative reconstruction algorithm for sparse objects: Application to 3D blood vessel reconstruction from a limited number of projections,” Phys. Med. Biol. 47, 2599-2609 (2002).
    [CrossRef] [PubMed]
  16. E. Candes and T. Tao, “Near optimal signal recovery from random projections: Universal encoding strategies,” IEEE Trans. Inf. Theory 52, 5406-5425 (2004).
    [CrossRef]
  17. E. Y. Sidky, University of Chicago, Department of Radiology, 5841 S. Maryland Ave. MC2026, Chicago, Illinois 60637, USA, and X. Pan are preparing a manuscript to be called “Image reconstruction in circular cone-beam computed tomography by total variation minimization.”

2006 (2)

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489-509 (2006).
[CrossRef]

E. Y. Sidky, C. Kao, and X. Pan, “Accurate image reconstruction from few-views and limited-angle data in divergent-beam CT,” J. X-Ray Sci. Technol. 14, 1-21 (2006).

2005 (1)

2004 (1)

E. Candes and T. Tao, “Near optimal signal recovery from random projections: Universal encoding strategies,” IEEE Trans. Inf. Theory 52, 5406-5425 (2004).
[CrossRef]

2002 (1)

M. H. Li, H. Q. Yang, and H. Kudo, “An accurate iterative reconstruction algorithm for sparse objects: Application to 3D blood vessel reconstruction from a limited number of projections,” Phys. Med. Biol. 47, 2599-2609 (2002).
[CrossRef] [PubMed]

2001 (1)

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (SIAM, 2001).
[CrossRef]

1999 (1)

1998 (1)

1995 (1)

M. P. Andre, P. J. Martin, G. P. Otto, L. K. Olson, T. K. Barrett, B. A. Spivey, and D. A. Palmer, “A new consideration of diffraction computed tomography for breast imaging: Studies in phantoms and patients,” Acoust. Imaging 21, 379-390 (1995).

1994 (1)

V. E. Kunitsyn, E. S. Andreeva, E. D. Tereschenko, B. Z. Khudukon, and T. Nygren, “Investigations of the ionosphere by satellite radiotomography,” Int. J. Imaging Syst. Technol. 5, 112-127 (1994).
[CrossRef]

1984 (1)

A. Devaney, “Geophysical diffraction tomography,” IEEE Trans. Geosci. Remote Sens. 22, 3-13 (1984).
[CrossRef]

1983 (2)

M. Slaney and A. C. Kak, “Diffraction tomography,” in Inverse Optics, Vol. 14, A.J.Devaney, ed. (SPIE, 1983), pp. 2-19.

S. X. Pan and A. C. Kak, “A computational study of reconstruction algorithms for diffraction tomography: Interpolation versus filtered backpropagation,” IEEE Trans. Acoust., Speech, Signal Process. 31, 1262-1275 (1983).
[CrossRef]

1982 (1)

A. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336-350 (1982).
[CrossRef] [PubMed]

1979 (1)

R. Mueller, M. Kaveh, and G. Wade, “Reconstructive tomography and applications to ultrasonics,” in Proc. IEEE 67, 567-587 (1979).
[CrossRef]

1969 (1)

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153-156 (1969).
[CrossRef]

Anastasio, M. A.

Andre, M. P.

M. P. Andre, P. J. Martin, G. P. Otto, L. K. Olson, T. K. Barrett, B. A. Spivey, and D. A. Palmer, “A new consideration of diffraction computed tomography for breast imaging: Studies in phantoms and patients,” Acoust. Imaging 21, 379-390 (1995).

Andreeva, E. S.

V. E. Kunitsyn, E. S. Andreeva, E. D. Tereschenko, B. Z. Khudukon, and T. Nygren, “Investigations of the ionosphere by satellite radiotomography,” Int. J. Imaging Syst. Technol. 5, 112-127 (1994).
[CrossRef]

Barrett, T. K.

M. P. Andre, P. J. Martin, G. P. Otto, L. K. Olson, T. K. Barrett, B. A. Spivey, and D. A. Palmer, “A new consideration of diffraction computed tomography for breast imaging: Studies in phantoms and patients,” Acoust. Imaging 21, 379-390 (1995).

Candes, E.

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489-509 (2006).
[CrossRef]

E. Candes and T. Tao, “Near optimal signal recovery from random projections: Universal encoding strategies,” IEEE Trans. Inf. Theory 52, 5406-5425 (2004).
[CrossRef]

Devaney, A.

A. Devaney, “Geophysical diffraction tomography,” IEEE Trans. Geosci. Remote Sens. 22, 3-13 (1984).
[CrossRef]

A. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336-350 (1982).
[CrossRef] [PubMed]

Devaney, A. J.

Guo, P.

Kak, A. C.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (SIAM, 2001).
[CrossRef]

S. X. Pan and A. C. Kak, “A computational study of reconstruction algorithms for diffraction tomography: Interpolation versus filtered backpropagation,” IEEE Trans. Acoust., Speech, Signal Process. 31, 1262-1275 (1983).
[CrossRef]

M. Slaney and A. C. Kak, “Diffraction tomography,” in Inverse Optics, Vol. 14, A.J.Devaney, ed. (SPIE, 1983), pp. 2-19.

Kao, C.

E. Y. Sidky, C. Kao, and X. Pan, “Accurate image reconstruction from few-views and limited-angle data in divergent-beam CT,” J. X-Ray Sci. Technol. 14, 1-21 (2006).

Kaveh, M.

R. Mueller, M. Kaveh, and G. Wade, “Reconstructive tomography and applications to ultrasonics,” in Proc. IEEE 67, 567-587 (1979).
[CrossRef]

Khudukon, B. Z.

V. E. Kunitsyn, E. S. Andreeva, E. D. Tereschenko, B. Z. Khudukon, and T. Nygren, “Investigations of the ionosphere by satellite radiotomography,” Int. J. Imaging Syst. Technol. 5, 112-127 (1994).
[CrossRef]

Kudo, H.

M. H. Li, H. Q. Yang, and H. Kudo, “An accurate iterative reconstruction algorithm for sparse objects: Application to 3D blood vessel reconstruction from a limited number of projections,” Phys. Med. Biol. 47, 2599-2609 (2002).
[CrossRef] [PubMed]

Kunitsyn, V. E.

V. E. Kunitsyn, E. S. Andreeva, E. D. Tereschenko, B. Z. Khudukon, and T. Nygren, “Investigations of the ionosphere by satellite radiotomography,” Int. J. Imaging Syst. Technol. 5, 112-127 (1994).
[CrossRef]

Li, M. H.

M. H. Li, H. Q. Yang, and H. Kudo, “An accurate iterative reconstruction algorithm for sparse objects: Application to 3D blood vessel reconstruction from a limited number of projections,” Phys. Med. Biol. 47, 2599-2609 (2002).
[CrossRef] [PubMed]

Martin, P. J.

M. P. Andre, P. J. Martin, G. P. Otto, L. K. Olson, T. K. Barrett, B. A. Spivey, and D. A. Palmer, “A new consideration of diffraction computed tomography for breast imaging: Studies in phantoms and patients,” Acoust. Imaging 21, 379-390 (1995).

Mueller, R.

R. Mueller, M. Kaveh, and G. Wade, “Reconstructive tomography and applications to ultrasonics,” in Proc. IEEE 67, 567-587 (1979).
[CrossRef]

Nygren, T.

V. E. Kunitsyn, E. S. Andreeva, E. D. Tereschenko, B. Z. Khudukon, and T. Nygren, “Investigations of the ionosphere by satellite radiotomography,” Int. J. Imaging Syst. Technol. 5, 112-127 (1994).
[CrossRef]

Olson, L. K.

M. P. Andre, P. J. Martin, G. P. Otto, L. K. Olson, T. K. Barrett, B. A. Spivey, and D. A. Palmer, “A new consideration of diffraction computed tomography for breast imaging: Studies in phantoms and patients,” Acoust. Imaging 21, 379-390 (1995).

Otto, G. P.

M. P. Andre, P. J. Martin, G. P. Otto, L. K. Olson, T. K. Barrett, B. A. Spivey, and D. A. Palmer, “A new consideration of diffraction computed tomography for breast imaging: Studies in phantoms and patients,” Acoust. Imaging 21, 379-390 (1995).

Palmer, D. A.

M. P. Andre, P. J. Martin, G. P. Otto, L. K. Olson, T. K. Barrett, B. A. Spivey, and D. A. Palmer, “A new consideration of diffraction computed tomography for breast imaging: Studies in phantoms and patients,” Acoust. Imaging 21, 379-390 (1995).

Pan, S. X.

S. X. Pan and A. C. Kak, “A computational study of reconstruction algorithms for diffraction tomography: Interpolation versus filtered backpropagation,” IEEE Trans. Acoust., Speech, Signal Process. 31, 1262-1275 (1983).
[CrossRef]

Pan, X.

Romberg, J.

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489-509 (2006).
[CrossRef]

Sidky, E. Y.

E. Y. Sidky, C. Kao, and X. Pan, “Accurate image reconstruction from few-views and limited-angle data in divergent-beam CT,” J. X-Ray Sci. Technol. 14, 1-21 (2006).

E. Y. Sidky, University of Chicago, Department of Radiology, 5841 S. Maryland Ave. MC2026, Chicago, Illinois 60637, USA, and X. Pan are preparing a manuscript to be called “Image reconstruction in circular cone-beam computed tomography by total variation minimization.”

Slaney, M.

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (SIAM, 2001).
[CrossRef]

M. Slaney and A. C. Kak, “Diffraction tomography,” in Inverse Optics, Vol. 14, A.J.Devaney, ed. (SPIE, 1983), pp. 2-19.

Spivey, B. A.

M. P. Andre, P. J. Martin, G. P. Otto, L. K. Olson, T. K. Barrett, B. A. Spivey, and D. A. Palmer, “A new consideration of diffraction computed tomography for breast imaging: Studies in phantoms and patients,” Acoust. Imaging 21, 379-390 (1995).

Tao, T.

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489-509 (2006).
[CrossRef]

E. Candes and T. Tao, “Near optimal signal recovery from random projections: Universal encoding strategies,” IEEE Trans. Inf. Theory 52, 5406-5425 (2004).
[CrossRef]

Tereschenko, E. D.

V. E. Kunitsyn, E. S. Andreeva, E. D. Tereschenko, B. Z. Khudukon, and T. Nygren, “Investigations of the ionosphere by satellite radiotomography,” Int. J. Imaging Syst. Technol. 5, 112-127 (1994).
[CrossRef]

Wade, G.

R. Mueller, M. Kaveh, and G. Wade, “Reconstructive tomography and applications to ultrasonics,” in Proc. IEEE 67, 567-587 (1979).
[CrossRef]

Wolf, E.

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153-156 (1969).
[CrossRef]

Yang, H. Q.

M. H. Li, H. Q. Yang, and H. Kudo, “An accurate iterative reconstruction algorithm for sparse objects: Application to 3D blood vessel reconstruction from a limited number of projections,” Phys. Med. Biol. 47, 2599-2609 (2002).
[CrossRef] [PubMed]

Acoust. Imaging (1)

M. P. Andre, P. J. Martin, G. P. Otto, L. K. Olson, T. K. Barrett, B. A. Spivey, and D. A. Palmer, “A new consideration of diffraction computed tomography for breast imaging: Studies in phantoms and patients,” Acoust. Imaging 21, 379-390 (1995).

IEEE Trans. Acoust., Speech, Signal Process. (1)

S. X. Pan and A. C. Kak, “A computational study of reconstruction algorithms for diffraction tomography: Interpolation versus filtered backpropagation,” IEEE Trans. Acoust., Speech, Signal Process. 31, 1262-1275 (1983).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

A. Devaney, “Geophysical diffraction tomography,” IEEE Trans. Geosci. Remote Sens. 22, 3-13 (1984).
[CrossRef]

IEEE Trans. Inf. Theory (2)

E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489-509 (2006).
[CrossRef]

E. Candes and T. Tao, “Near optimal signal recovery from random projections: Universal encoding strategies,” IEEE Trans. Inf. Theory 52, 5406-5425 (2004).
[CrossRef]

Int. J. Imaging Syst. Technol. (1)

V. E. Kunitsyn, E. S. Andreeva, E. D. Tereschenko, B. Z. Khudukon, and T. Nygren, “Investigations of the ionosphere by satellite radiotomography,” Int. J. Imaging Syst. Technol. 5, 112-127 (1994).
[CrossRef]

J. Opt. Soc. Am. A (3)

J. X-Ray Sci. Technol. (1)

E. Y. Sidky, C. Kao, and X. Pan, “Accurate image reconstruction from few-views and limited-angle data in divergent-beam CT,” J. X-Ray Sci. Technol. 14, 1-21 (2006).

Opt. Commun. (1)

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153-156 (1969).
[CrossRef]

Phys. Med. Biol. (1)

M. H. Li, H. Q. Yang, and H. Kudo, “An accurate iterative reconstruction algorithm for sparse objects: Application to 3D blood vessel reconstruction from a limited number of projections,” Phys. Med. Biol. 47, 2599-2609 (2002).
[CrossRef] [PubMed]

Proc. IEEE (1)

R. Mueller, M. Kaveh, and G. Wade, “Reconstructive tomography and applications to ultrasonics,” in Proc. IEEE 67, 567-587 (1979).
[CrossRef]

Ultrason. Imaging (1)

A. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imaging 4, 336-350 (1982).
[CrossRef] [PubMed]

Other (3)

M. Slaney and A. C. Kak, “Diffraction tomography,” in Inverse Optics, Vol. 14, A.J.Devaney, ed. (SPIE, 1983), pp. 2-19.

E. Y. Sidky, University of Chicago, Department of Radiology, 5841 S. Maryland Ave. MC2026, Chicago, Illinois 60637, USA, and X. Pan are preparing a manuscript to be called “Image reconstruction in circular cone-beam computed tomography by total variation minimization.”

A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (SIAM, 2001).
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

(a) Classical DT scan configuration, with incident radiation along η axis and scattered field measured along η = . (b) From the FDPT, the ID FT of the scattered data along η = equals the 2D FT of the object function along semicircle AOB with radius ν 0 .

Fig. 2
Fig. 2

(a) Original 128 × 128 high-contrast Shepp–Logan phantom. Grayscale spans range from 0.15 to 0.7 in arbitrary intensity units. The dotted white vertical and dashed white horizontal lines indicate the slicing directions for profiles. (b) The dotted vertical profile corresponds to the dotted white line in panel (a). (c) The dashed horizontal profile corresponds to the dashed white line in panel (a).

Fig. 3
Fig. 3

(a) Original 128 × 128 high-contrast Shepp–Logan phantom. Grayscale spans range from 0.15 to 0.7 in arbitrary intensity units. (b) GMI of image in (a); the GMI is clearly sparse.

Fig. 4
Fig. 4

(a) Data sampling function used for full 360 ° scan, showing 17 views. Scale is arbitrary for illustrative purposes. (b) Data sampling function used for 180 ° scan, 17 views.

Fig. 5
Fig. 5

Reconstruction results from noiseless few-view data using 17 views and 256 samples per view, (a) Image reconstructed using the TV algorithm. (b) Image reconstructed using the FBPP algorithm. (c) Vertical profile, where solid and dashed curves show reconstruction from TV and FBPP algorithms, respectively. For comparison, the true profile is displayed with a dotted curve. It is indistinguishable from the solid curve due to the near-exactness of the reconstruction. (d) Horizontal profile.

Fig. 6
Fig. 6

(a) Reconstructed image (left), vertical profile (center, solid curve), and horizontal profile (right, solid curve) reconstructed with the TV algorithm from noisy data acquired at 129 views evenly spaced over 360 ° . The corresponding profile obtained from inverse FT of the full noisy Fourier dataset (dotted curve) is also shown. (b) Same for 33 views. (c) Same for 17 views.

Fig. 7
Fig. 7

FBPP reconstructions with the same noise level as in Fig. 6 for (a) 129, (b) 33, and (c) 17 views.

Fig. 8
Fig. 8

(a) Reconstructed image (left) and horizontal profile (center, solid curve) reconstructed with the TV algorithm from noisy data acquired for 25 views over 270 ° . The corresponding profile obtained from inverse FT of the full noisy Fourier dataset (dotted curve) is also shown. At right is the FBPP reconstruction for the same noise level. (b) Same for 17 views over 180 ° . (c) Same for 11 views over 120 ° .

Fig. 9
Fig. 9

(a) Reconstructed image (left) and horizontal profile (center, solid curve) reconstructed with the TV algorithm from noisy data acquired at 30 views with 125 samples per view, evenly spaced over 360 ° . The corresponding profile obtained from inverse FT of the full noisy Fourier dataset (dotted curve) is also shown. At right is the FBPP reconstruction for the same noise level. (b) Same for 20 views with 175 samples per view. (c) Same for 15 views and 175 samples per view over 180 ° .

Fig. 10
Fig. 10

(a) Additional phantom used in testing which is a random superposition of ellipses. Grayscale ranges from 0 to 1.2. (b) Reconstruction of the phantom in (a) from noiseless data.

Fig. 11
Fig. 11

(a) Reconstruction with 30 views, 125 samples using a different noise realization. (b) Reconstruction using data generated from the analytic FT of an ellipse rather than the discrete FT.

Equations (9)

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U s ( ν m , ϕ ) = d ξ u s ( ξ , ϕ ) exp { 2 π j ν m ξ } .
U s ( ν m , ϕ ) = { 2 π 2 ν 0 2 U 0 j ν exp ( 2 π j ν l ) d r f ( r ) exp { 2 π j [ ν m ξ + ( ν ν 0 ) η ] } ν ν 0 0 ν > ν 0 ,
M ( ν m , ϕ ) = d r f ( r ) exp { 2 π j [ ν m ξ + ( ν ν 0 ) η ] } .
g k , l = M ( ν k , ϕ l ) ,
Δ f i , j = [ ( f i , j f i 1 , j ) 2 + ( f i , j f i , j 1 ) 2 ] 1 2 .
f TV = i , j { [ ( f i , j f i 1 , j ) 2 + ( f i , j f i , j 1 ) 2 ] 1 2 } ,
( i ) { k , l [ DFT ( f ) k , l g k , l ] 2 } 1 2 ϵ ;
( i i ) f i , j 0 i , j ;
( i i i ) Im ( f i , j ) = 0 i , j ;

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