Abstract

We investigate new sampling strategies for projection tomography, enabling one to employ fewer measurements than expected from classical sampling theory without significant loss of information. Inspired by compressed sensing, our approach is based on the understanding that many real objects are compressible in some known representation, implying that the number of degrees of freedom defining an object is often much smaller than the number of pixels/voxels. We propose a new approach based on quasi-random detector subsampling, whereas previous approaches only addressed subsampling with respect to source location (view angle). The performance of different sampling strategies is considered using object-independent figures of merit, and also based on reconstructions for specific objects, with synthetic and real data. The proposed approach can be implemented using a structured illumination of the interrogated object or the detector array by placing a coded aperture/mask at the source or detector side, respectively. Advantages of the proposed approach include (i) for structured illumination of the detector array, it leads to fewer detector pixels and allows one to integrate detectors for scattered radiation in the unused space; (ii) for structured illumination of the object, it leads to a reduced radiation dose for patients in medical scans; (iii) in the latter case, the blocking of rays reduces scattered radiation while keeping the same energy in the transmitted rays, resulting in a higher signal-to-noise ratio than that achieved by lowering exposure times or the energy of the source; (iv) compared to view-angle subsampling, it allows one to use fewer measurements for the same image quality, or leads to better image quality for the same number of measurements. The proposed approach can also be combined with view-angle subsampling.

© 2014 Optical Society of America

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  1. M. Slaney and A. Kak, Principles of Computerized Tomographic Imaging (SIAM, 1988).
  2. F. Natterer, The Mathematics of Computerized Tomography (Wiley, 1986).
  3. E. J. Candès, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59, 1207–1223 (2006).
    [CrossRef]
  4. D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
    [CrossRef]
  5. R. Rangayyan, A. P. Dhawan, and R. Gordon, “Algorithms for limited-view computed-tomography—an annotated bibliography and a challenge,” Appl. Opt. 24, 4000–4012 (1985).
    [CrossRef]
  6. A. K. Louis, “Incomplete data problems in x-ray computerized tomography,” Numer. Math. 48, 251–262 (1986).
    [CrossRef]
  7. E. Y. Sidky, C. M. Kao, and X. H. Pan, “Accurate image reconstruction from few-views and limited-angle data in divergent-beam CT,” J. X-Ray Sci. Technol. 14, 119–139 (2006).
  8. 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]
  9. J. Song, Q. H. Liu, G. A. Johnson, and C. T. Badea, “Sparseness prior based iterative image reconstruction for retrospectively gated cardiac micro-CT,” Med. Phys. 34, 4476–4483 (2007).
    [CrossRef]
  10. E. Hansis, D. Schafer, O. Dossel, and M. Grass, “Evaluation of iterative sparse object reconstruction from few projections for 3-D rotational coronary angiography,” IEEE Trans. Med. Imaging 27, 1548–1555 (2008).
    [CrossRef]
  11. D. C. Youla and H. Webb, “Image restoration by the method of convex projections: part 1 theory,” IEEE Trans. Med. Imaging 1, 81–94 (1982).
    [CrossRef]
  12. M. I. Sezan and H. Stark, “Image restoration by the method of convex projections: part 2 applications and numerical results,” IEEE Trans. Med. Imaging 1, 95–101 (1982).
    [CrossRef]
  13. E. Candès, 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]
  14. E. Y. Sidky and X. C. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,” Phys. Med. Biol. 53, 4777–4807 (2008).
    [CrossRef]
  15. G. H. Chen, J. Tang, and S. H. Leng, “Prior image constrained compressed sensing: a method to accurately reconstruct dynamic CT images from highly undersampled projection data sets,” Med. Phys. 35, 660–663 (2008).
    [CrossRef]
  16. J. W. Stayman, W. Zbijewski, Y. Otake, A. Uneri, S. Schafer, J. Lee, J. L. Prince, and J. H. Siewerdsen, “Penalized-likelihood reconstruction for sparse data acquisitions with unregistered prior images and compressed sensing penalties,” Proc. SPIE 7961, 79611L (2011).
    [CrossRef]
  17. K. Choi, J. Wang, L. Zhu, T. S. Suh, S. Boyd, and L. Xing, “Compressed sensing based cone-beam computed tomography reconstruction with a first-order method,” Med. Phys. 37, 5113–5125 (2010).
    [CrossRef]
  18. A. A. Wagadarikar, N. P. Pitsianis, X. Sun, and D. J. Brady, “Video rate spectral imaging using a coded aperture snapshot spectral imager,” Opt. Express 17, 6368–6388 (2009).
    [CrossRef]
  19. J. Hahn, S. Lim, K. Choi, R. Horisaki, and D. J. Brady, “Video-rate compressive holographic microscopic tomography,” Opt. Express 19, 7289–7298 (2011).
    [CrossRef]
  20. K. P. MacCabe, A. D. Holmgren, M. P. Tornai, and D. J. Brady, “Snapshot 2D tomography via coded aperture x-ray scatter imaging,” Appl. Opt. 52, 4582–4589 (2013).
    [CrossRef]
  21. J. Greenberg, K. Krishnamurthy, and D. Brady, “Compressive single-pixel snapshot x-ray diffraction imaging,” Opt. Lett. 39, 111–114 (2014).
    [CrossRef]
  22. D. Brady and D. Marks, “Coding for compressive focal tomography,” Appl. Opt. 50, 4436–4449 (2011).
    [CrossRef]
  23. P. Llull, X. Liao, X. Yuan, J. Yang, D. Kittle, L. Carin, G. Sapiro, and D. J. Brady, “Coded aperture compressive temporal imaging,” Opt. Express 21, 10526–10545 (2013).
    [CrossRef]
  24. D. J. Brady, N. Pitsianis, and X. Sun, “Reference structure tomography,” J. Opt. Soc. Am. A 21, 1140–1147 (2004).
    [CrossRef]
  25. K. Choi and D. J. Brady, “Coded aperture computed tomography,” Proc. SPIE 7468, 74680B (2009).
    [CrossRef]
  26. R. Ning, X. Tang, and D. L. Conover, “X-ray scatter suppression algorithm for cone-beam volume CT,” Proc. SPIE 4685, 774–781 (2002).
    [CrossRef]
  27. R. Ning, X. Tang, and D. Conover, “X-ray scatter correction algorithm for cone beam CT imaging,” Med. Phys. 31, 1195–1202 (2004).
    [CrossRef]
  28. G. Harding, J. Kosanetzky, and U. Neitzel, “X-ray diffraction computed tomography,” Med. Phys. 14, 515–525 (1987).
    [CrossRef]
  29. H. Strecker, “Automatic detection of explosives in airline baggage using elastic x-ray scatter,” Medicamundi 42, 30–33 (1998).
  30. C. Cozzini, S. Olesinski, and G. Harding, “Modeling scattering for security applications: a multiple beam x-ray diffraction imaging system,” in IEEE Nuclear Science Symposium and Medical Imaging Conference (NSS/MIC) (IEEE, 2012), pp. 74–77.
  31. F. Smith, Industrial Applications of X-Ray Diffraction (CRC Press, 1999).
  32. S. Pani, E. Cook, J. Horrocks, L. George, S. Hardwick, and R. Speller, “Modelling an energy-dispersive x-ray diffraction system for drug detection,” IEEE Trans. Nucl. Sci. 56, 1238–1241 (2009).
    [CrossRef]
  33. J. Hsieh, Computed Tomography: Principles, Design, Artifacts, and Recent Advances (SPIE, 2009).
  34. E. Candès and J. Romberg, “Sparsity and incoherence in compressive sampling,” Inverse Probl. 23, 969–985 (2007).
    [CrossRef]
  35. F. Krahmer and R. Ward, “Beyond incoherence: stable and robust sampling strategies for compressive imaging,” arXiv:1210.2380 (2012).
  36. B. Adcock, A. C. Hansen, C. Poon, and B. Roman, “Breaking the coherence barrier: asymptotic incoherence and asymptotic sparsity in compressed sensing,” arXiv:1302.0561 (2013).
  37. H. Peng and H. Stark, “Direct fourier reconstruction in fan-beam tomography,” IEEE Trans. Med. Imaging 6, 209–219 (1987).
    [CrossRef]
  38. G.-H. Chen, S. Leng, and C. A. Mistretta, “A novel extension of the parallel-beam projection-slice theorem to divergent fan-beam and cone-beam projections,” Med. Phys. 32, 654–665 (2005).
    [CrossRef]
  39. F. Natterer, “Sampling in fan beam tomography,” SIAM J. Appl. Math. 53, 358–380 (1993).
    [CrossRef]
  40. http://www.mathworks.com/matlabcentral/fileexchange/27375-plot-wavelet-image-2d-decomposition/content/plotwavelet2.m .
  41. D. L. Donoho and M. Elad, “Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ1 minimization,” Proc. Natl. Acad. Sci. USA 100, 2197–2202 (2003).
    [CrossRef]
  42. M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI,” IEEE Signal Process. Mag. 25(2), 72–82 (2008).
    [CrossRef]
  43. M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: the application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58, 1182–1195 (2007).
    [CrossRef]
  44. M. A. Davenport, M. F. Duarte, Y. C. Eldar, and G. Kutyniok, “Introduction to compressed sensing,” in Compressed Sensing: Theory and Applications (Cambridge University, 2012), Chap. 1.
  45. J. A. Tropp, “Just relax: convex programming methods for identifying sparse signals in noise,” IEEE Trans. Inf. Theory 52, 1030–1051 (2006).
    [CrossRef]
  46. M. Sonka and J. M. Fitzpatrick, Handbook of Medical Imaging, Vol. 2 of Medical Image Processing and Analysis (SPIE, 2000).
  47. G. T. Herman, Image Reconstruction from Projections (Academic, 1980).
  48. M. E. Tipping, “Sparse Bayesian learning and the relevance vector machine,” J. Mach. Learn. Res. 1, 211–244 (2001).
  49. S. Ji, Y. Xue, and L. Carin, “Bayesian compressive sensing,” IEEE Trans. Signal Process. 56, 2346–2356 (2008).
    [CrossRef]
  50. J. M. Bernardo and A. F. Smith, Bayesian Theory, Vol. 405 of Wiley Series in Probability and Statistics (Wiley, 2009).
  51. J. A. O’Sullivan and J. Benac, “Alternating minimization algorithms for transmission tomography,” IEEE Trans. Med. Imaging 26, 283–297 (2007).
    [CrossRef]
  52. M. E. Davison, “The ill-conditioned nature of the limited angle tomography problem,” SIAM J. Appl. Math. 43, 428–448 (1983).
    [CrossRef]
  53. D. P. Petersen and D. Middleton, “Sampling and reconstruction of wave-number-limited functions in N-dimensional euclidean spaces,” Inf. Control 5, 279–323 (1962).
    [CrossRef]
  54. A. G. Lindgren and P. A. Rattey, “The inverse discrete Radon transform with applications to tomographic imaging using projection data,” Adv. Electron. Electron Phys. 56, 359–410 (1981).
    [CrossRef]
  55. S. H. Izen, “Sampling in flat detector fan beam tomography,” SIAM J. Appl. Math. 72, 61–84 (2012).
    [CrossRef]
  56. D. J. Crotty, R. L. McKinley, and M. P. Tornai, “Experimental spectral measurements of heavy k-edge filtered beams for x-ray computed mammotomography,” Phys. Med. Biol. 52, 603–616 (2007).
    [CrossRef]
  57. C. W. Dodge, A Rapid Method for the Simulation of Filtered X-Ray Spectra in Diagnostic Imaging Systems (ProQuest, 2008).

2014 (1)

2013 (2)

2012 (1)

S. H. Izen, “Sampling in flat detector fan beam tomography,” SIAM J. Appl. Math. 72, 61–84 (2012).
[CrossRef]

2011 (3)

D. Brady and D. Marks, “Coding for compressive focal tomography,” Appl. Opt. 50, 4436–4449 (2011).
[CrossRef]

J. Hahn, S. Lim, K. Choi, R. Horisaki, and D. J. Brady, “Video-rate compressive holographic microscopic tomography,” Opt. Express 19, 7289–7298 (2011).
[CrossRef]

J. W. Stayman, W. Zbijewski, Y. Otake, A. Uneri, S. Schafer, J. Lee, J. L. Prince, and J. H. Siewerdsen, “Penalized-likelihood reconstruction for sparse data acquisitions with unregistered prior images and compressed sensing penalties,” Proc. SPIE 7961, 79611L (2011).
[CrossRef]

2010 (1)

K. Choi, J. Wang, L. Zhu, T. S. Suh, S. Boyd, and L. Xing, “Compressed sensing based cone-beam computed tomography reconstruction with a first-order method,” Med. Phys. 37, 5113–5125 (2010).
[CrossRef]

2009 (3)

A. A. Wagadarikar, N. P. Pitsianis, X. Sun, and D. J. Brady, “Video rate spectral imaging using a coded aperture snapshot spectral imager,” Opt. Express 17, 6368–6388 (2009).
[CrossRef]

K. Choi and D. J. Brady, “Coded aperture computed tomography,” Proc. SPIE 7468, 74680B (2009).
[CrossRef]

S. Pani, E. Cook, J. Horrocks, L. George, S. Hardwick, and R. Speller, “Modelling an energy-dispersive x-ray diffraction system for drug detection,” IEEE Trans. Nucl. Sci. 56, 1238–1241 (2009).
[CrossRef]

2008 (5)

M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI,” IEEE Signal Process. Mag. 25(2), 72–82 (2008).
[CrossRef]

E. Y. Sidky and X. C. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,” Phys. Med. Biol. 53, 4777–4807 (2008).
[CrossRef]

G. H. Chen, J. Tang, and S. H. Leng, “Prior image constrained compressed sensing: a method to accurately reconstruct dynamic CT images from highly undersampled projection data sets,” Med. Phys. 35, 660–663 (2008).
[CrossRef]

E. Hansis, D. Schafer, O. Dossel, and M. Grass, “Evaluation of iterative sparse object reconstruction from few projections for 3-D rotational coronary angiography,” IEEE Trans. Med. Imaging 27, 1548–1555 (2008).
[CrossRef]

S. Ji, Y. Xue, and L. Carin, “Bayesian compressive sensing,” IEEE Trans. Signal Process. 56, 2346–2356 (2008).
[CrossRef]

2007 (5)

J. A. O’Sullivan and J. Benac, “Alternating minimization algorithms for transmission tomography,” IEEE Trans. Med. Imaging 26, 283–297 (2007).
[CrossRef]

D. J. Crotty, R. L. McKinley, and M. P. Tornai, “Experimental spectral measurements of heavy k-edge filtered beams for x-ray computed mammotomography,” Phys. Med. Biol. 52, 603–616 (2007).
[CrossRef]

J. Song, Q. H. Liu, G. A. Johnson, and C. T. Badea, “Sparseness prior based iterative image reconstruction for retrospectively gated cardiac micro-CT,” Med. Phys. 34, 4476–4483 (2007).
[CrossRef]

M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: the application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58, 1182–1195 (2007).
[CrossRef]

E. Candès and J. Romberg, “Sparsity and incoherence in compressive sampling,” Inverse Probl. 23, 969–985 (2007).
[CrossRef]

2006 (5)

J. A. Tropp, “Just relax: convex programming methods for identifying sparse signals in noise,” IEEE Trans. Inf. Theory 52, 1030–1051 (2006).
[CrossRef]

E. J. Candès, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59, 1207–1223 (2006).
[CrossRef]

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
[CrossRef]

E. Y. Sidky, C. M. Kao, and X. H. Pan, “Accurate image reconstruction from few-views and limited-angle data in divergent-beam CT,” J. X-Ray Sci. Technol. 14, 119–139 (2006).

E. Candès, 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]

2005 (1)

G.-H. Chen, S. Leng, and C. A. Mistretta, “A novel extension of the parallel-beam projection-slice theorem to divergent fan-beam and cone-beam projections,” Med. Phys. 32, 654–665 (2005).
[CrossRef]

2004 (2)

D. J. Brady, N. Pitsianis, and X. Sun, “Reference structure tomography,” J. Opt. Soc. Am. A 21, 1140–1147 (2004).
[CrossRef]

R. Ning, X. Tang, and D. Conover, “X-ray scatter correction algorithm for cone beam CT imaging,” Med. Phys. 31, 1195–1202 (2004).
[CrossRef]

2003 (1)

D. L. Donoho and M. Elad, “Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ1 minimization,” Proc. Natl. Acad. Sci. USA 100, 2197–2202 (2003).
[CrossRef]

2002 (2)

R. Ning, X. Tang, and D. L. Conover, “X-ray scatter suppression algorithm for cone-beam volume CT,” Proc. SPIE 4685, 774–781 (2002).
[CrossRef]

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]

2001 (1)

M. E. Tipping, “Sparse Bayesian learning and the relevance vector machine,” J. Mach. Learn. Res. 1, 211–244 (2001).

1998 (1)

H. Strecker, “Automatic detection of explosives in airline baggage using elastic x-ray scatter,” Medicamundi 42, 30–33 (1998).

1993 (1)

F. Natterer, “Sampling in fan beam tomography,” SIAM J. Appl. Math. 53, 358–380 (1993).
[CrossRef]

1987 (2)

H. Peng and H. Stark, “Direct fourier reconstruction in fan-beam tomography,” IEEE Trans. Med. Imaging 6, 209–219 (1987).
[CrossRef]

G. Harding, J. Kosanetzky, and U. Neitzel, “X-ray diffraction computed tomography,” Med. Phys. 14, 515–525 (1987).
[CrossRef]

1986 (1)

A. K. Louis, “Incomplete data problems in x-ray computerized tomography,” Numer. Math. 48, 251–262 (1986).
[CrossRef]

1985 (1)

1983 (1)

M. E. Davison, “The ill-conditioned nature of the limited angle tomography problem,” SIAM J. Appl. Math. 43, 428–448 (1983).
[CrossRef]

1982 (2)

D. C. Youla and H. Webb, “Image restoration by the method of convex projections: part 1 theory,” IEEE Trans. Med. Imaging 1, 81–94 (1982).
[CrossRef]

M. I. Sezan and H. Stark, “Image restoration by the method of convex projections: part 2 applications and numerical results,” IEEE Trans. Med. Imaging 1, 95–101 (1982).
[CrossRef]

1981 (1)

A. G. Lindgren and P. A. Rattey, “The inverse discrete Radon transform with applications to tomographic imaging using projection data,” Adv. Electron. Electron Phys. 56, 359–410 (1981).
[CrossRef]

1962 (1)

D. P. Petersen and D. Middleton, “Sampling and reconstruction of wave-number-limited functions in N-dimensional euclidean spaces,” Inf. Control 5, 279–323 (1962).
[CrossRef]

Adcock, B.

B. Adcock, A. C. Hansen, C. Poon, and B. Roman, “Breaking the coherence barrier: asymptotic incoherence and asymptotic sparsity in compressed sensing,” arXiv:1302.0561 (2013).

Badea, C. T.

J. Song, Q. H. Liu, G. A. Johnson, and C. T. Badea, “Sparseness prior based iterative image reconstruction for retrospectively gated cardiac micro-CT,” Med. Phys. 34, 4476–4483 (2007).
[CrossRef]

Benac, J.

J. A. O’Sullivan and J. Benac, “Alternating minimization algorithms for transmission tomography,” IEEE Trans. Med. Imaging 26, 283–297 (2007).
[CrossRef]

Bernardo, J. M.

J. M. Bernardo and A. F. Smith, Bayesian Theory, Vol. 405 of Wiley Series in Probability and Statistics (Wiley, 2009).

Boyd, S.

K. Choi, J. Wang, L. Zhu, T. S. Suh, S. Boyd, and L. Xing, “Compressed sensing based cone-beam computed tomography reconstruction with a first-order method,” Med. Phys. 37, 5113–5125 (2010).
[CrossRef]

Brady, D.

Brady, D. J.

Candès, E.

E. Candès and J. Romberg, “Sparsity and incoherence in compressive sampling,” Inverse Probl. 23, 969–985 (2007).
[CrossRef]

E. Candès, 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]

Candès, E. J.

E. J. Candès, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59, 1207–1223 (2006).
[CrossRef]

Carin, L.

Chen, G. H.

G. H. Chen, J. Tang, and S. H. Leng, “Prior image constrained compressed sensing: a method to accurately reconstruct dynamic CT images from highly undersampled projection data sets,” Med. Phys. 35, 660–663 (2008).
[CrossRef]

Chen, G.-H.

G.-H. Chen, S. Leng, and C. A. Mistretta, “A novel extension of the parallel-beam projection-slice theorem to divergent fan-beam and cone-beam projections,” Med. Phys. 32, 654–665 (2005).
[CrossRef]

Choi, K.

J. Hahn, S. Lim, K. Choi, R. Horisaki, and D. J. Brady, “Video-rate compressive holographic microscopic tomography,” Opt. Express 19, 7289–7298 (2011).
[CrossRef]

K. Choi, J. Wang, L. Zhu, T. S. Suh, S. Boyd, and L. Xing, “Compressed sensing based cone-beam computed tomography reconstruction with a first-order method,” Med. Phys. 37, 5113–5125 (2010).
[CrossRef]

K. Choi and D. J. Brady, “Coded aperture computed tomography,” Proc. SPIE 7468, 74680B (2009).
[CrossRef]

Conover, D.

R. Ning, X. Tang, and D. Conover, “X-ray scatter correction algorithm for cone beam CT imaging,” Med. Phys. 31, 1195–1202 (2004).
[CrossRef]

Conover, D. L.

R. Ning, X. Tang, and D. L. Conover, “X-ray scatter suppression algorithm for cone-beam volume CT,” Proc. SPIE 4685, 774–781 (2002).
[CrossRef]

Cook, E.

S. Pani, E. Cook, J. Horrocks, L. George, S. Hardwick, and R. Speller, “Modelling an energy-dispersive x-ray diffraction system for drug detection,” IEEE Trans. Nucl. Sci. 56, 1238–1241 (2009).
[CrossRef]

Cozzini, C.

C. Cozzini, S. Olesinski, and G. Harding, “Modeling scattering for security applications: a multiple beam x-ray diffraction imaging system,” in IEEE Nuclear Science Symposium and Medical Imaging Conference (NSS/MIC) (IEEE, 2012), pp. 74–77.

Crotty, D. J.

D. J. Crotty, R. L. McKinley, and M. P. Tornai, “Experimental spectral measurements of heavy k-edge filtered beams for x-ray computed mammotomography,” Phys. Med. Biol. 52, 603–616 (2007).
[CrossRef]

Davenport, M. A.

M. A. Davenport, M. F. Duarte, Y. C. Eldar, and G. Kutyniok, “Introduction to compressed sensing,” in Compressed Sensing: Theory and Applications (Cambridge University, 2012), Chap. 1.

Davison, M. E.

M. E. Davison, “The ill-conditioned nature of the limited angle tomography problem,” SIAM J. Appl. Math. 43, 428–448 (1983).
[CrossRef]

Dhawan, A. P.

Dodge, C. W.

C. W. Dodge, A Rapid Method for the Simulation of Filtered X-Ray Spectra in Diagnostic Imaging Systems (ProQuest, 2008).

Donoho, D.

M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: the application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58, 1182–1195 (2007).
[CrossRef]

Donoho, D. L.

M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI,” IEEE Signal Process. Mag. 25(2), 72–82 (2008).
[CrossRef]

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
[CrossRef]

D. L. Donoho and M. Elad, “Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ1 minimization,” Proc. Natl. Acad. Sci. USA 100, 2197–2202 (2003).
[CrossRef]

Dossel, O.

E. Hansis, D. Schafer, O. Dossel, and M. Grass, “Evaluation of iterative sparse object reconstruction from few projections for 3-D rotational coronary angiography,” IEEE Trans. Med. Imaging 27, 1548–1555 (2008).
[CrossRef]

Duarte, M. F.

M. A. Davenport, M. F. Duarte, Y. C. Eldar, and G. Kutyniok, “Introduction to compressed sensing,” in Compressed Sensing: Theory and Applications (Cambridge University, 2012), Chap. 1.

Elad, M.

D. L. Donoho and M. Elad, “Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ1 minimization,” Proc. Natl. Acad. Sci. USA 100, 2197–2202 (2003).
[CrossRef]

Eldar, Y. C.

M. A. Davenport, M. F. Duarte, Y. C. Eldar, and G. Kutyniok, “Introduction to compressed sensing,” in Compressed Sensing: Theory and Applications (Cambridge University, 2012), Chap. 1.

Fitzpatrick, J. M.

M. Sonka and J. M. Fitzpatrick, Handbook of Medical Imaging, Vol. 2 of Medical Image Processing and Analysis (SPIE, 2000).

George, L.

S. Pani, E. Cook, J. Horrocks, L. George, S. Hardwick, and R. Speller, “Modelling an energy-dispersive x-ray diffraction system for drug detection,” IEEE Trans. Nucl. Sci. 56, 1238–1241 (2009).
[CrossRef]

Gordon, R.

Grass, M.

E. Hansis, D. Schafer, O. Dossel, and M. Grass, “Evaluation of iterative sparse object reconstruction from few projections for 3-D rotational coronary angiography,” IEEE Trans. Med. Imaging 27, 1548–1555 (2008).
[CrossRef]

Greenberg, J.

Hahn, J.

Hansen, A. C.

B. Adcock, A. C. Hansen, C. Poon, and B. Roman, “Breaking the coherence barrier: asymptotic incoherence and asymptotic sparsity in compressed sensing,” arXiv:1302.0561 (2013).

Hansis, E.

E. Hansis, D. Schafer, O. Dossel, and M. Grass, “Evaluation of iterative sparse object reconstruction from few projections for 3-D rotational coronary angiography,” IEEE Trans. Med. Imaging 27, 1548–1555 (2008).
[CrossRef]

Harding, G.

G. Harding, J. Kosanetzky, and U. Neitzel, “X-ray diffraction computed tomography,” Med. Phys. 14, 515–525 (1987).
[CrossRef]

C. Cozzini, S. Olesinski, and G. Harding, “Modeling scattering for security applications: a multiple beam x-ray diffraction imaging system,” in IEEE Nuclear Science Symposium and Medical Imaging Conference (NSS/MIC) (IEEE, 2012), pp. 74–77.

Hardwick, S.

S. Pani, E. Cook, J. Horrocks, L. George, S. Hardwick, and R. Speller, “Modelling an energy-dispersive x-ray diffraction system for drug detection,” IEEE Trans. Nucl. Sci. 56, 1238–1241 (2009).
[CrossRef]

Herman, G. T.

G. T. Herman, Image Reconstruction from Projections (Academic, 1980).

Holmgren, A. D.

Horisaki, R.

Horrocks, J.

S. Pani, E. Cook, J. Horrocks, L. George, S. Hardwick, and R. Speller, “Modelling an energy-dispersive x-ray diffraction system for drug detection,” IEEE Trans. Nucl. Sci. 56, 1238–1241 (2009).
[CrossRef]

Hsieh, J.

J. Hsieh, Computed Tomography: Principles, Design, Artifacts, and Recent Advances (SPIE, 2009).

Izen, S. H.

S. H. Izen, “Sampling in flat detector fan beam tomography,” SIAM J. Appl. Math. 72, 61–84 (2012).
[CrossRef]

Ji, S.

S. Ji, Y. Xue, and L. Carin, “Bayesian compressive sensing,” IEEE Trans. Signal Process. 56, 2346–2356 (2008).
[CrossRef]

Johnson, G. A.

J. Song, Q. H. Liu, G. A. Johnson, and C. T. Badea, “Sparseness prior based iterative image reconstruction for retrospectively gated cardiac micro-CT,” Med. Phys. 34, 4476–4483 (2007).
[CrossRef]

Kak, A.

M. Slaney and A. Kak, Principles of Computerized Tomographic Imaging (SIAM, 1988).

Kao, C. M.

E. Y. Sidky, C. M. Kao, and X. H. Pan, “Accurate image reconstruction from few-views and limited-angle data in divergent-beam CT,” J. X-Ray Sci. Technol. 14, 119–139 (2006).

Kittle, D.

Kosanetzky, J.

G. Harding, J. Kosanetzky, and U. Neitzel, “X-ray diffraction computed tomography,” Med. Phys. 14, 515–525 (1987).
[CrossRef]

Krahmer, F.

F. Krahmer and R. Ward, “Beyond incoherence: stable and robust sampling strategies for compressive imaging,” arXiv:1210.2380 (2012).

Krishnamurthy, K.

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]

Kutyniok, G.

M. A. Davenport, M. F. Duarte, Y. C. Eldar, and G. Kutyniok, “Introduction to compressed sensing,” in Compressed Sensing: Theory and Applications (Cambridge University, 2012), Chap. 1.

Lee, J.

J. W. Stayman, W. Zbijewski, Y. Otake, A. Uneri, S. Schafer, J. Lee, J. L. Prince, and J. H. Siewerdsen, “Penalized-likelihood reconstruction for sparse data acquisitions with unregistered prior images and compressed sensing penalties,” Proc. SPIE 7961, 79611L (2011).
[CrossRef]

Leng, S.

G.-H. Chen, S. Leng, and C. A. Mistretta, “A novel extension of the parallel-beam projection-slice theorem to divergent fan-beam and cone-beam projections,” Med. Phys. 32, 654–665 (2005).
[CrossRef]

Leng, S. H.

G. H. Chen, J. Tang, and S. H. Leng, “Prior image constrained compressed sensing: a method to accurately reconstruct dynamic CT images from highly undersampled projection data sets,” Med. Phys. 35, 660–663 (2008).
[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]

Liao, X.

Lim, S.

Lindgren, A. G.

A. G. Lindgren and P. A. Rattey, “The inverse discrete Radon transform with applications to tomographic imaging using projection data,” Adv. Electron. Electron Phys. 56, 359–410 (1981).
[CrossRef]

Liu, Q. H.

J. Song, Q. H. Liu, G. A. Johnson, and C. T. Badea, “Sparseness prior based iterative image reconstruction for retrospectively gated cardiac micro-CT,” Med. Phys. 34, 4476–4483 (2007).
[CrossRef]

Llull, P.

Louis, A. K.

A. K. Louis, “Incomplete data problems in x-ray computerized tomography,” Numer. Math. 48, 251–262 (1986).
[CrossRef]

Lustig, M.

M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI,” IEEE Signal Process. Mag. 25(2), 72–82 (2008).
[CrossRef]

M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: the application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58, 1182–1195 (2007).
[CrossRef]

MacCabe, K. P.

Marks, D.

McKinley, R. L.

D. J. Crotty, R. L. McKinley, and M. P. Tornai, “Experimental spectral measurements of heavy k-edge filtered beams for x-ray computed mammotomography,” Phys. Med. Biol. 52, 603–616 (2007).
[CrossRef]

Middleton, D.

D. P. Petersen and D. Middleton, “Sampling and reconstruction of wave-number-limited functions in N-dimensional euclidean spaces,” Inf. Control 5, 279–323 (1962).
[CrossRef]

Mistretta, C. A.

G.-H. Chen, S. Leng, and C. A. Mistretta, “A novel extension of the parallel-beam projection-slice theorem to divergent fan-beam and cone-beam projections,” Med. Phys. 32, 654–665 (2005).
[CrossRef]

Natterer, F.

F. Natterer, “Sampling in fan beam tomography,” SIAM J. Appl. Math. 53, 358–380 (1993).
[CrossRef]

F. Natterer, The Mathematics of Computerized Tomography (Wiley, 1986).

Neitzel, U.

G. Harding, J. Kosanetzky, and U. Neitzel, “X-ray diffraction computed tomography,” Med. Phys. 14, 515–525 (1987).
[CrossRef]

Ning, R.

R. Ning, X. Tang, and D. Conover, “X-ray scatter correction algorithm for cone beam CT imaging,” Med. Phys. 31, 1195–1202 (2004).
[CrossRef]

R. Ning, X. Tang, and D. L. Conover, “X-ray scatter suppression algorithm for cone-beam volume CT,” Proc. SPIE 4685, 774–781 (2002).
[CrossRef]

O’Sullivan, J. A.

J. A. O’Sullivan and J. Benac, “Alternating minimization algorithms for transmission tomography,” IEEE Trans. Med. Imaging 26, 283–297 (2007).
[CrossRef]

Olesinski, S.

C. Cozzini, S. Olesinski, and G. Harding, “Modeling scattering for security applications: a multiple beam x-ray diffraction imaging system,” in IEEE Nuclear Science Symposium and Medical Imaging Conference (NSS/MIC) (IEEE, 2012), pp. 74–77.

Otake, Y.

J. W. Stayman, W. Zbijewski, Y. Otake, A. Uneri, S. Schafer, J. Lee, J. L. Prince, and J. H. Siewerdsen, “Penalized-likelihood reconstruction for sparse data acquisitions with unregistered prior images and compressed sensing penalties,” Proc. SPIE 7961, 79611L (2011).
[CrossRef]

Pan, X. C.

E. Y. Sidky and X. C. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,” Phys. Med. Biol. 53, 4777–4807 (2008).
[CrossRef]

Pan, X. H.

E. Y. Sidky, C. M. Kao, and X. H. Pan, “Accurate image reconstruction from few-views and limited-angle data in divergent-beam CT,” J. X-Ray Sci. Technol. 14, 119–139 (2006).

Pani, S.

S. Pani, E. Cook, J. Horrocks, L. George, S. Hardwick, and R. Speller, “Modelling an energy-dispersive x-ray diffraction system for drug detection,” IEEE Trans. Nucl. Sci. 56, 1238–1241 (2009).
[CrossRef]

Pauly, J. M.

M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI,” IEEE Signal Process. Mag. 25(2), 72–82 (2008).
[CrossRef]

M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: the application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58, 1182–1195 (2007).
[CrossRef]

Peng, H.

H. Peng and H. Stark, “Direct fourier reconstruction in fan-beam tomography,” IEEE Trans. Med. Imaging 6, 209–219 (1987).
[CrossRef]

Petersen, D. P.

D. P. Petersen and D. Middleton, “Sampling and reconstruction of wave-number-limited functions in N-dimensional euclidean spaces,” Inf. Control 5, 279–323 (1962).
[CrossRef]

Pitsianis, N.

Pitsianis, N. P.

Poon, C.

B. Adcock, A. C. Hansen, C. Poon, and B. Roman, “Breaking the coherence barrier: asymptotic incoherence and asymptotic sparsity in compressed sensing,” arXiv:1302.0561 (2013).

Prince, J. L.

J. W. Stayman, W. Zbijewski, Y. Otake, A. Uneri, S. Schafer, J. Lee, J. L. Prince, and J. H. Siewerdsen, “Penalized-likelihood reconstruction for sparse data acquisitions with unregistered prior images and compressed sensing penalties,” Proc. SPIE 7961, 79611L (2011).
[CrossRef]

Rangayyan, R.

Rattey, P. A.

A. G. Lindgren and P. A. Rattey, “The inverse discrete Radon transform with applications to tomographic imaging using projection data,” Adv. Electron. Electron Phys. 56, 359–410 (1981).
[CrossRef]

Roman, B.

B. Adcock, A. C. Hansen, C. Poon, and B. Roman, “Breaking the coherence barrier: asymptotic incoherence and asymptotic sparsity in compressed sensing,” arXiv:1302.0561 (2013).

Romberg, J.

E. Candès and J. Romberg, “Sparsity and incoherence in compressive sampling,” Inverse Probl. 23, 969–985 (2007).
[CrossRef]

E. Candès, 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]

Romberg, J. K.

E. J. Candès, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59, 1207–1223 (2006).
[CrossRef]

Santos, J. M.

M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI,” IEEE Signal Process. Mag. 25(2), 72–82 (2008).
[CrossRef]

Sapiro, G.

Schafer, D.

E. Hansis, D. Schafer, O. Dossel, and M. Grass, “Evaluation of iterative sparse object reconstruction from few projections for 3-D rotational coronary angiography,” IEEE Trans. Med. Imaging 27, 1548–1555 (2008).
[CrossRef]

Schafer, S.

J. W. Stayman, W. Zbijewski, Y. Otake, A. Uneri, S. Schafer, J. Lee, J. L. Prince, and J. H. Siewerdsen, “Penalized-likelihood reconstruction for sparse data acquisitions with unregistered prior images and compressed sensing penalties,” Proc. SPIE 7961, 79611L (2011).
[CrossRef]

Sezan, M. I.

M. I. Sezan and H. Stark, “Image restoration by the method of convex projections: part 2 applications and numerical results,” IEEE Trans. Med. Imaging 1, 95–101 (1982).
[CrossRef]

Sidky, E. Y.

E. Y. Sidky and X. C. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,” Phys. Med. Biol. 53, 4777–4807 (2008).
[CrossRef]

E. Y. Sidky, C. M. Kao, and X. H. Pan, “Accurate image reconstruction from few-views and limited-angle data in divergent-beam CT,” J. X-Ray Sci. Technol. 14, 119–139 (2006).

Siewerdsen, J. H.

J. W. Stayman, W. Zbijewski, Y. Otake, A. Uneri, S. Schafer, J. Lee, J. L. Prince, and J. H. Siewerdsen, “Penalized-likelihood reconstruction for sparse data acquisitions with unregistered prior images and compressed sensing penalties,” Proc. SPIE 7961, 79611L (2011).
[CrossRef]

Slaney, M.

M. Slaney and A. Kak, Principles of Computerized Tomographic Imaging (SIAM, 1988).

Smith, A. F.

J. M. Bernardo and A. F. Smith, Bayesian Theory, Vol. 405 of Wiley Series in Probability and Statistics (Wiley, 2009).

Smith, F.

F. Smith, Industrial Applications of X-Ray Diffraction (CRC Press, 1999).

Song, J.

J. Song, Q. H. Liu, G. A. Johnson, and C. T. Badea, “Sparseness prior based iterative image reconstruction for retrospectively gated cardiac micro-CT,” Med. Phys. 34, 4476–4483 (2007).
[CrossRef]

Sonka, M.

M. Sonka and J. M. Fitzpatrick, Handbook of Medical Imaging, Vol. 2 of Medical Image Processing and Analysis (SPIE, 2000).

Speller, R.

S. Pani, E. Cook, J. Horrocks, L. George, S. Hardwick, and R. Speller, “Modelling an energy-dispersive x-ray diffraction system for drug detection,” IEEE Trans. Nucl. Sci. 56, 1238–1241 (2009).
[CrossRef]

Stark, H.

H. Peng and H. Stark, “Direct fourier reconstruction in fan-beam tomography,” IEEE Trans. Med. Imaging 6, 209–219 (1987).
[CrossRef]

M. I. Sezan and H. Stark, “Image restoration by the method of convex projections: part 2 applications and numerical results,” IEEE Trans. Med. Imaging 1, 95–101 (1982).
[CrossRef]

Stayman, J. W.

J. W. Stayman, W. Zbijewski, Y. Otake, A. Uneri, S. Schafer, J. Lee, J. L. Prince, and J. H. Siewerdsen, “Penalized-likelihood reconstruction for sparse data acquisitions with unregistered prior images and compressed sensing penalties,” Proc. SPIE 7961, 79611L (2011).
[CrossRef]

Strecker, H.

H. Strecker, “Automatic detection of explosives in airline baggage using elastic x-ray scatter,” Medicamundi 42, 30–33 (1998).

Suh, T. S.

K. Choi, J. Wang, L. Zhu, T. S. Suh, S. Boyd, and L. Xing, “Compressed sensing based cone-beam computed tomography reconstruction with a first-order method,” Med. Phys. 37, 5113–5125 (2010).
[CrossRef]

Sun, X.

Tang, J.

G. H. Chen, J. Tang, and S. H. Leng, “Prior image constrained compressed sensing: a method to accurately reconstruct dynamic CT images from highly undersampled projection data sets,” Med. Phys. 35, 660–663 (2008).
[CrossRef]

Tang, X.

R. Ning, X. Tang, and D. Conover, “X-ray scatter correction algorithm for cone beam CT imaging,” Med. Phys. 31, 1195–1202 (2004).
[CrossRef]

R. Ning, X. Tang, and D. L. Conover, “X-ray scatter suppression algorithm for cone-beam volume CT,” Proc. SPIE 4685, 774–781 (2002).
[CrossRef]

Tao, T.

E. Candès, 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. J. Candès, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59, 1207–1223 (2006).
[CrossRef]

Tipping, M. E.

M. E. Tipping, “Sparse Bayesian learning and the relevance vector machine,” J. Mach. Learn. Res. 1, 211–244 (2001).

Tornai, M. P.

K. P. MacCabe, A. D. Holmgren, M. P. Tornai, and D. J. Brady, “Snapshot 2D tomography via coded aperture x-ray scatter imaging,” Appl. Opt. 52, 4582–4589 (2013).
[CrossRef]

D. J. Crotty, R. L. McKinley, and M. P. Tornai, “Experimental spectral measurements of heavy k-edge filtered beams for x-ray computed mammotomography,” Phys. Med. Biol. 52, 603–616 (2007).
[CrossRef]

Tropp, J. A.

J. A. Tropp, “Just relax: convex programming methods for identifying sparse signals in noise,” IEEE Trans. Inf. Theory 52, 1030–1051 (2006).
[CrossRef]

Uneri, A.

J. W. Stayman, W. Zbijewski, Y. Otake, A. Uneri, S. Schafer, J. Lee, J. L. Prince, and J. H. Siewerdsen, “Penalized-likelihood reconstruction for sparse data acquisitions with unregistered prior images and compressed sensing penalties,” Proc. SPIE 7961, 79611L (2011).
[CrossRef]

Wagadarikar, A. A.

Wang, J.

K. Choi, J. Wang, L. Zhu, T. S. Suh, S. Boyd, and L. Xing, “Compressed sensing based cone-beam computed tomography reconstruction with a first-order method,” Med. Phys. 37, 5113–5125 (2010).
[CrossRef]

Ward, R.

F. Krahmer and R. Ward, “Beyond incoherence: stable and robust sampling strategies for compressive imaging,” arXiv:1210.2380 (2012).

Webb, H.

D. C. Youla and H. Webb, “Image restoration by the method of convex projections: part 1 theory,” IEEE Trans. Med. Imaging 1, 81–94 (1982).
[CrossRef]

Xing, L.

K. Choi, J. Wang, L. Zhu, T. S. Suh, S. Boyd, and L. Xing, “Compressed sensing based cone-beam computed tomography reconstruction with a first-order method,” Med. Phys. 37, 5113–5125 (2010).
[CrossRef]

Xue, Y.

S. Ji, Y. Xue, and L. Carin, “Bayesian compressive sensing,” IEEE Trans. Signal Process. 56, 2346–2356 (2008).
[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]

Yang, J.

Youla, D. C.

D. C. Youla and H. Webb, “Image restoration by the method of convex projections: part 1 theory,” IEEE Trans. Med. Imaging 1, 81–94 (1982).
[CrossRef]

Yuan, X.

Zbijewski, W.

J. W. Stayman, W. Zbijewski, Y. Otake, A. Uneri, S. Schafer, J. Lee, J. L. Prince, and J. H. Siewerdsen, “Penalized-likelihood reconstruction for sparse data acquisitions with unregistered prior images and compressed sensing penalties,” Proc. SPIE 7961, 79611L (2011).
[CrossRef]

Zhu, L.

K. Choi, J. Wang, L. Zhu, T. S. Suh, S. Boyd, and L. Xing, “Compressed sensing based cone-beam computed tomography reconstruction with a first-order method,” Med. Phys. 37, 5113–5125 (2010).
[CrossRef]

Adv. Electron. Electron Phys. (1)

A. G. Lindgren and P. A. Rattey, “The inverse discrete Radon transform with applications to tomographic imaging using projection data,” Adv. Electron. Electron Phys. 56, 359–410 (1981).
[CrossRef]

Appl. Opt. (3)

Commun. Pure Appl. Math. (1)

E. J. Candès, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59, 1207–1223 (2006).
[CrossRef]

IEEE Signal Process. Mag. (1)

M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI,” IEEE Signal Process. Mag. 25(2), 72–82 (2008).
[CrossRef]

IEEE Trans. Inf. Theory (3)

J. A. Tropp, “Just relax: convex programming methods for identifying sparse signals in noise,” IEEE Trans. Inf. Theory 52, 1030–1051 (2006).
[CrossRef]

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
[CrossRef]

E. Candès, 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]

IEEE Trans. Med. Imaging (5)

E. Hansis, D. Schafer, O. Dossel, and M. Grass, “Evaluation of iterative sparse object reconstruction from few projections for 3-D rotational coronary angiography,” IEEE Trans. Med. Imaging 27, 1548–1555 (2008).
[CrossRef]

D. C. Youla and H. Webb, “Image restoration by the method of convex projections: part 1 theory,” IEEE Trans. Med. Imaging 1, 81–94 (1982).
[CrossRef]

M. I. Sezan and H. Stark, “Image restoration by the method of convex projections: part 2 applications and numerical results,” IEEE Trans. Med. Imaging 1, 95–101 (1982).
[CrossRef]

H. Peng and H. Stark, “Direct fourier reconstruction in fan-beam tomography,” IEEE Trans. Med. Imaging 6, 209–219 (1987).
[CrossRef]

J. A. O’Sullivan and J. Benac, “Alternating minimization algorithms for transmission tomography,” IEEE Trans. Med. Imaging 26, 283–297 (2007).
[CrossRef]

IEEE Trans. Nucl. Sci. (1)

S. Pani, E. Cook, J. Horrocks, L. George, S. Hardwick, and R. Speller, “Modelling an energy-dispersive x-ray diffraction system for drug detection,” IEEE Trans. Nucl. Sci. 56, 1238–1241 (2009).
[CrossRef]

IEEE Trans. Signal Process. (1)

S. Ji, Y. Xue, and L. Carin, “Bayesian compressive sensing,” IEEE Trans. Signal Process. 56, 2346–2356 (2008).
[CrossRef]

Inf. Control (1)

D. P. Petersen and D. Middleton, “Sampling and reconstruction of wave-number-limited functions in N-dimensional euclidean spaces,” Inf. Control 5, 279–323 (1962).
[CrossRef]

Inverse Probl. (1)

E. Candès and J. Romberg, “Sparsity and incoherence in compressive sampling,” Inverse Probl. 23, 969–985 (2007).
[CrossRef]

J. Mach. Learn. Res. (1)

M. E. Tipping, “Sparse Bayesian learning and the relevance vector machine,” J. Mach. Learn. Res. 1, 211–244 (2001).

J. Opt. Soc. Am. A (1)

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

E. Y. Sidky, C. M. Kao, and X. H. Pan, “Accurate image reconstruction from few-views and limited-angle data in divergent-beam CT,” J. X-Ray Sci. Technol. 14, 119–139 (2006).

Magn. Reson. Med. (1)

M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: the application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58, 1182–1195 (2007).
[CrossRef]

Med. Phys. (6)

J. Song, Q. H. Liu, G. A. Johnson, and C. T. Badea, “Sparseness prior based iterative image reconstruction for retrospectively gated cardiac micro-CT,” Med. Phys. 34, 4476–4483 (2007).
[CrossRef]

G. H. Chen, J. Tang, and S. H. Leng, “Prior image constrained compressed sensing: a method to accurately reconstruct dynamic CT images from highly undersampled projection data sets,” Med. Phys. 35, 660–663 (2008).
[CrossRef]

K. Choi, J. Wang, L. Zhu, T. S. Suh, S. Boyd, and L. Xing, “Compressed sensing based cone-beam computed tomography reconstruction with a first-order method,” Med. Phys. 37, 5113–5125 (2010).
[CrossRef]

G.-H. Chen, S. Leng, and C. A. Mistretta, “A novel extension of the parallel-beam projection-slice theorem to divergent fan-beam and cone-beam projections,” Med. Phys. 32, 654–665 (2005).
[CrossRef]

R. Ning, X. Tang, and D. Conover, “X-ray scatter correction algorithm for cone beam CT imaging,” Med. Phys. 31, 1195–1202 (2004).
[CrossRef]

G. Harding, J. Kosanetzky, and U. Neitzel, “X-ray diffraction computed tomography,” Med. Phys. 14, 515–525 (1987).
[CrossRef]

Medicamundi (1)

H. Strecker, “Automatic detection of explosives in airline baggage using elastic x-ray scatter,” Medicamundi 42, 30–33 (1998).

Numer. Math. (1)

A. K. Louis, “Incomplete data problems in x-ray computerized tomography,” Numer. Math. 48, 251–262 (1986).
[CrossRef]

Opt. Express (3)

Opt. Lett. (1)

Phys. Med. Biol. (3)

E. Y. Sidky and X. C. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,” Phys. Med. Biol. 53, 4777–4807 (2008).
[CrossRef]

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]

D. J. Crotty, R. L. McKinley, and M. P. Tornai, “Experimental spectral measurements of heavy k-edge filtered beams for x-ray computed mammotomography,” Phys. Med. Biol. 52, 603–616 (2007).
[CrossRef]

Proc. Natl. Acad. Sci. USA (1)

D. L. Donoho and M. Elad, “Optimally sparse representation in general (nonorthogonal) dictionaries via ℓ1 minimization,” Proc. Natl. Acad. Sci. USA 100, 2197–2202 (2003).
[CrossRef]

Proc. SPIE (3)

J. W. Stayman, W. Zbijewski, Y. Otake, A. Uneri, S. Schafer, J. Lee, J. L. Prince, and J. H. Siewerdsen, “Penalized-likelihood reconstruction for sparse data acquisitions with unregistered prior images and compressed sensing penalties,” Proc. SPIE 7961, 79611L (2011).
[CrossRef]

K. Choi and D. J. Brady, “Coded aperture computed tomography,” Proc. SPIE 7468, 74680B (2009).
[CrossRef]

R. Ning, X. Tang, and D. L. Conover, “X-ray scatter suppression algorithm for cone-beam volume CT,” Proc. SPIE 4685, 774–781 (2002).
[CrossRef]

SIAM J. Appl. Math. (3)

F. Natterer, “Sampling in fan beam tomography,” SIAM J. Appl. Math. 53, 358–380 (1993).
[CrossRef]

S. H. Izen, “Sampling in flat detector fan beam tomography,” SIAM J. Appl. Math. 72, 61–84 (2012).
[CrossRef]

M. E. Davison, “The ill-conditioned nature of the limited angle tomography problem,” SIAM J. Appl. Math. 43, 428–448 (1983).
[CrossRef]

Other (13)

J. M. Bernardo and A. F. Smith, Bayesian Theory, Vol. 405 of Wiley Series in Probability and Statistics (Wiley, 2009).

F. Krahmer and R. Ward, “Beyond incoherence: stable and robust sampling strategies for compressive imaging,” arXiv:1210.2380 (2012).

B. Adcock, A. C. Hansen, C. Poon, and B. Roman, “Breaking the coherence barrier: asymptotic incoherence and asymptotic sparsity in compressed sensing,” arXiv:1302.0561 (2013).

M. A. Davenport, M. F. Duarte, Y. C. Eldar, and G. Kutyniok, “Introduction to compressed sensing,” in Compressed Sensing: Theory and Applications (Cambridge University, 2012), Chap. 1.

M. Sonka and J. M. Fitzpatrick, Handbook of Medical Imaging, Vol. 2 of Medical Image Processing and Analysis (SPIE, 2000).

G. T. Herman, Image Reconstruction from Projections (Academic, 1980).

C. W. Dodge, A Rapid Method for the Simulation of Filtered X-Ray Spectra in Diagnostic Imaging Systems (ProQuest, 2008).

http://www.mathworks.com/matlabcentral/fileexchange/27375-plot-wavelet-image-2d-decomposition/content/plotwavelet2.m .

J. Hsieh, Computed Tomography: Principles, Design, Artifacts, and Recent Advances (SPIE, 2009).

C. Cozzini, S. Olesinski, and G. Harding, “Modeling scattering for security applications: a multiple beam x-ray diffraction imaging system,” in IEEE Nuclear Science Symposium and Medical Imaging Conference (NSS/MIC) (IEEE, 2012), pp. 74–77.

F. Smith, Industrial Applications of X-Ray Diffraction (CRC Press, 1999).

M. Slaney and A. Kak, Principles of Computerized Tomographic Imaging (SIAM, 1988).

F. Natterer, The Mathematics of Computerized Tomography (Wiley, 1986).

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Figures (28)

Fig. 1.
Fig. 1.

Different scanning geometries in 2D and the corresponding parameters for the line integrals. The disk of radius ρ defines the image domain where the function f is to be reconstructed from measurements of the line integrals. (a) Parallel beam (translate–rotate) geometry, (b) curved detector fan-beam geometry, and (c) flat detector fan-beam geometry.

Fig. 2.
Fig. 2.

Different sampling strategies under consideration demonstrated on a fan-beam geometry. The blue circles denote source locations (corresponding to view angles), and lines denote source–detector pairs (for clarity, only the lines for one source location are shown for each case). The solid and dashed lines correspond to measured and nonmeasured line integrals, respectively. The large circle represents the source trajectory, and the small circle represents the imaged domain. (a) Uniform view (UV) subsampling, (b) random view (RV)-angle subsampling, (c) uniform detector (UD) sampling, and (d) random detector (RD) subsampling.

Fig. 3.
Fig. 3.

Fan-beam geometry with flat detector panel used throughout the analysis and simulations, which corresponds to the experimental system described in Section 7. ρ=58mm, Δ=1.2mm, r=484.6mm, Rx=290.2mm.

Fig. 4.
Fig. 4.

Singular values for undersampling by a factor of 16 using the different random subsampling strategies described in Section 2.B. RV, random view; SRD, static random detector; DRD, dynamic random detector. The mean and standard deviation (std) for 20 different random selections are indicated by solid lines and shaded strips, respectively. For clarity the boundaries of the shaded strips correspond to 1 std above and below the mean (gray strip for RV and yellow strip for SRD). Note that the means for DRD and SDRD are indistinguishable from each other in this scale and therefore the latter is omitted here. Also, the std for DRD and SDRD are indistinguishable from the mean of DRD and are therefore omitted here. Also included are the singular values for the complete dataset (“Full”). The forward model is described in Fig. 3, and a 128×128 image resolution is used. A circular domain was used to be consistent with classical sampling theory. (a) All singular components and (b) only first 5000 components.

Fig. 5.
Fig. 5.

Singular values for ×16 undersampling using the different sampling strategies described in Section 2.B. Here we compare uniform sampling to random sampling in the cases of (a) view subsampling and (b) detector subsampling. UV, uniform view; RV, random view; UD, uniform detector; DRD, dynamic random detector; SRD, static random detector. For strategies involving random sampling, the mean and standard deviation (std) for 20 different random selections are indicated by solid lines and shaded strips, respectively. For clarity the boundaries of the shaded strips correspond to 1 std above and below the mean (gray strip for RV and yellow strip for SRD). Note that the means for DRD and SDRD are indistinguishable from each other in this scale and therefore the latter is omitted here. Also, the std for DRD and SDRD are indistinguishable from the mean of DRD and are therefore omitted here. The forward model is described in Fig. 3, and a 128×128 image resolution is used.

Fig. 6.
Fig. 6.

Right singular vectors with lowest singular values in the case of uniform detector (UD) subsampling by a factor of 16. Note that a circular domain was used.

Fig. 7.
Fig. 7.

Right singular vectors with lowest singular values in the case of dynamic random detector (DRD) subsampling by a factor of 16. Note that a circular domain was used.

Fig. 8.
Fig. 8.

Right singular vectors with lowest singular values in the case of uniform view-angle (UV) subsampling by a factor of 16. Note that a circular domain was used.

Fig. 9.
Fig. 9.

Multiresolution image decomposition using Haar wavelets for the UD right singular vectors shown in Fig. 6. Each block contains the magnitudes of wavelet coefficients corresponding to a different spatial scale, with the horizontal scale decreasing from left to right and the vertical scale decreasing from top to bottom, e.g., the block near the bottom-right corner contains the wavelet coefficients for the finest scale in both directions (the support of these wavelets is 2×2 pixels). Within a block, each entry corresponds to a different spatial translation of the wavelet. The magnitude of the wavelet coefficients is presented in logarithmic scale. These plots were generated using the MATLAB command “plotwavelet2” and with a rescaling value of 100 [40].

Fig. 10.
Fig. 10.

Multiresolution image decomposition using Haar wavelets for the DRD right singular vectors shown in Fig. 7. See further details in the caption of Fig. 9.

Fig. 11.
Fig. 11.

Multiresolution image decomposition using Haar wavelets for the UV right singular vectors shown in Fig. 8. See further details in the caption of Fig. 9.

Fig. 12.
Fig. 12.

(a) Truth, (b)–(d) reconstructed images using RVM with four iterations. Values represent the attenuation per unit length divided by the attenuation per unit length of water. (b) Using all measurements; (c)–(f) measurements downsampled by a factor of 32 using the following sampling strategies: (c) dynamic random detector (DRD), (d) random view (RV), (e) semidynamic random detector (SDRD), (f) static random detector (SRD). Data have been generated according to Eqs. (14) and (15) with I0=105. Colored hatch marks in (b) and (c) indicate the cross sections shown in Figs. 14(a) and 14(b), respectively (with matching colors).

Fig. 13.
Fig. 13.

Reconstructed images for uniform undersampling by a factor of 32. (a), (b) RVM after four iterations; (c), (d) filtered backprojection (FBP). (a), (c) uniform detector (UD) sampling; (b), (d) uniform view (UV) sampling. The data have been generated according to the model in Eqs. (14) and (15) with I0=105. Cross sections of the image in (b) along the horizontal and vertical lines are shown in Figs. 14(a) and 14(b), respectively (with matching colors).

Fig. 14.
Fig. 14.

1D Cuts of the images in Fig. 12(b) (blue) for complete data, Fig. 12(c) (green) for dynamic random detector sampling (DRD), and Fig. 13(b) (red) for uniform view (UV) sampling. These cuts are indicated in the images by matching colored hatch marks. (a) Horizontal cut at the middle of the image and (b) vertical cut at the middle of the image.

Fig. 15.
Fig. 15.

Reconstructed images using RVM for experimental data. Values represent the attenuation per unit length divided by the attenuation per unit length of water and are limited to the window [0,1.8]. Undersampling ratios are relative to the minimal sampling rate dictated by classical theory. (a) The object is ×2 uniformly undersampled with respect to both detectors and views; (b)–(d) the available data are downsampled by an additional factor of 4 with respect to detectors or views (×8 undersampled for either detectors or views and ×2 undersampled for either views or detectors, respectively). (b) Dynamic random detector sampling (DRD), (c) uniform view (UV) sampling; (d) semidynamic random detector (SDRD) sampling. Only eight iterations of RVM have been used to generate all images. All strategies have the same number of measurements (line integrals). Colored hatch marks in (a) indicate the cross sections shown in Fig. 17.

Fig. 16.
Fig. 16.

Reconstructed images using RVM for experimental data. Values represent the attenuation per unit length divided by the attenuation per unit length of water. Undersampling ratios are relative to the minimal sampling rates dictated by classical theory. (a) View angles are undersampled by a factor of 16, and all available detectors are used, which are oversampled by 1.5; (b) detectors are ×16 undersampled using the DRD strategy, and views are ×2 undersampled due to limited data; (c) in the combined approach detectors are ×4 undersampled using the DRD strategy, and view angles are ×4 undersampled uniformly (object is undersampled by a total factor of 16). Unlike in Fig. 15, detector selection is done from all available detectors, which are ×1.5 oversampled, and to obtain the same undersampling relative to classical theory, the detectors were downsampled ×3 more than views, so in (b) and (c) there are ×3 fewer measurements (line integrals) than in (a). All other specifications are the same as in Fig. 15.

Fig. 17.
Fig. 17.

Linear attenuation coefficients (relative to water) along the columns indicated by matching colored hatch marks in the images in Figs. 15 and 16. Blue represents Fig. 15(a) for full data, red represents Fig. 16(a) for UV sampling, and green represents Fig. 16(c) for combined DRD-UV. Arrows point to regions where significant differences between methods exist.

Fig. 18.
Fig. 18.

Reconstructed images for experimental data using the filtered backprojection (FBP) algorithm. (a) View angles are undersampled uniformly by a factor of 16, and detectors are undersampled uniformly by a factor of 2 (relative to classical sampling theory) and (b) detectors are undersampled uniformly by a factor of 16, and view angles are undersampled uniformly by a factor of 2 (relative to classical sampling theory).

Fig. 19.
Fig. 19.

Cross sections through the reconstruction of pediatric patient data using all the data, after 50 iterations of the AM algorithm, using 145 ordered subsets (to speed convergence) and a neighborhood penalty with β=3200. The axial (red frame), coronal (green frame), and sagittal (blue frame) views are shown from left to right. The color-coded hatch marks within the enclosing frames indicate the planes from which the orthogonal views are taken. Linear attenuation coefficients from 0 (air) to 0.45cm1 (approximately bone) are shown in the range from black to white, respectively. The image volume is 512×512×135 voxels, with voxel dimensions of 1mm×1mm×2mm. A cut along the column indicated by yellow hatch marks in the axial view is shown in Fig. 22.

Fig. 20.
Fig. 20.

Cross sections through the reconstructions of patient data using 1/29 of the views, uniformly spaced in angle of rotation around the patient, but all the detectors, and using 10 ordered subsets. All other specifications are the same as in Fig. 19.

Fig. 21.
Fig. 21.

Cross sections through the reconstructions of patient data using all of the views, but only 1/29 of the detector measurements, and using 10 ordered subsets. All other specifications are the same as in Fig. 19.

Fig. 22.
Fig. 22.

Linear attenuation coefficients along the columns indicated by yellow hatch marks in Fig. 19 for full data (blue), Fig. 20 for UV sampling (red), and Fig. 21 for dynamic random detector (DRD) sampling (green), plotted on the same axis. Arrows indicate portions of the graph in which UV sampling led to substantial reconstruction errors. Some of these errors were in regions of sudden transition, indicating that some high-frequency content was lost.

Fig. 23.
Fig. 23.

Sum of squared differences between reconstructions using various sampling strategies and our gold standard, which used full data and regularization (shown in Fig. 19). The sum extends only over voxels within the 512×512×135 image volume, which were inside the patient. The units are mm2. DRD, dynamic random detector; UV, uniform view; UD, uniform detector.

Fig. 24.
Fig. 24.

Values represent the attenuation per unit length divided by the attenuation per unit length of water. (a) Truth; (b)–(f) reconstructed images via RVM: (b) using all measurements, (c)–(f) after measurements have been downsampled by ×32. (c) Dynamic random detector sampling (DRD), (d) random view sampling (RV), (e) SDRD sampling, (f) static random detector sampling (SRD). Only four iterations of RVM have been used. The data have been generated according to the model in Eqs. (14) and (15) with I0=105. Showing values in [0.5,2];.

Fig. 25.
Fig. 25.

Reconstructed images for uniform undersampling ×32. (a), (b) Using RVM after four iterations; (c), (d) using filtered backprojection (FBP). (a),(c) UD sampling; (b), (d) UV sampling. Only four iterations in RVM have been used. The data have been generated according to the model in Eqs. (14) and (15) with I0=105. The attenuation values are limited to the range [0.5,2], where the differences are more apparent.

Fig. 26.
Fig. 26.

Left: the scanned object, 3D letters “DUKE.” Right: positioning of object. We used tape, and other means, to suspend objects in the air as well as to minimize scatter compared to other potential holders.

Fig. 27.
Fig. 27.

Simulation of the flux-energy curve for the x-ray source with the 0.55 mm Cerium filter used in the experiment, obtained by XSPECT software.

Fig. 28.
Fig. 28.

We collimated the source into a fan beam using two lead blocks that were both made flat and parallel with a fly cutter. The length of collimator blocks helped ensure the beam had a very limited divergence.

Tables (4)

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Table 1. Sufficient Sampling Conditions for Reconstructing Details down to Size δ for the Scanning Geometries in Fig. 1 According to Classical Sampling Theory

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Table 2. Mutual Coherence of Different Random Sampling Strategies with Respect to the Haar Wavelet Basis and for Different Undersampling Ratiosa

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Table 3. Mutual Coherence of Different Uniform Sampling Strategies with Respect to the Haar Wavelet Basis and for Different Undersampling Ratiosa

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Table 4. Root Mean-Squared Error (RMSE) for RVM Reconstructions Averaged over 50 Different Random Selections from the Complete Dataset and for Different Sampling Strategiesa

Equations (20)

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Rf(φ,s)=x·θ=sf(x)dx,θ=(cosφ,sinφ).
Df(β,α)=L(β,α)f(x)dx,
Df(β,α)=Rf(β+απ/2,rsinα),
y=Hf,
y=Φx,
Φx1Φx2Φ(x1x2)0
f=Ψx,
y=Hf+ϵ=HΨx+ϵ,
(1δs)x22Φx22(1+δs)x22,
TPSF(i,j)=(ΨTHTHΨ)i,j,
μmaxij|TPSF(i,j)|/TPSF(i,i)TPSF(j,j).
Ii=Ii0exp(Lif(x)dz),
yi=lndi0di=lndi0lndi.
yHf+ϵ,ϵN(0,Σ),
Σ=diag[I01exp(Hf)].
Ψ(f)=ijNiwijψ(fifj),
ψ(t)=1δ2(|δt|ln(1+|δt|)).
|ξ|>b|f^(ξ)|2dξϵ0,
S{γ=Wl,lZ2},
RfLC1eλbfL1+C2ϵ0,

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