Abstract

Propagation-based X-ray phase-contrast tomography (PCT) seeks to reconstruct information regarding the complex-valued refractive index distribution of an object. In many applications, a boundary-enhanced image is sought that reveals the locations of discontinuities in the real-valued component of the refractive index distribution. We investigate two iterative algorithms for few-view image reconstruction in boundary-enhanced PCT that exploit the fact that a boundary-enhanced PCT image, or its gradient, is often sparse. In order to exploit object sparseness, the reconstruction algorithms seek to minimize the l 1-norm or TV-norm of the image, subject to data consistency constraints. We demonstrate that the algorithms can reconstruct accurate boundary-enhanced images from highly incomplete few-view projection data.

© 2010 Optical Society of America

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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
  28. F. Santosa and W. Symes, “Linear inversion of band-limited reflection histograms,” SIAM J. Sci. Stat. Comput. 7, 1307–1330 (1986).
    [Crossref]
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    [Crossref] [PubMed]
  33. R. Chartrand, “Exact reconstruction of sparse signals via nonconvex minimization,” IEEE Sig. Process. Lett. 14, 707–710 (2007).
    [Crossref]
  34. T. Blumensath and M. Davies, “Iterative hard thresholding for compressed sensing,” Applied and Computational Harmonic Analysis 27, 265–274 (2009).
    [Crossref]
  35. E. J. Candès and M. B. Wakin, “An Introduction To Compressive Sampling,” IEEE Sig. Proc. Mag. 25, 21–30 (2008).
    [Crossref]
  36. R. Garg and K. Khandekar, “Gradient Descent with Sparsification: An iterative algorithm for sparse recovery with restricted isometry property,” Proceedings of the 26th Annual International Conference on Machine Learning  382, 337–344 (2009).
  37. C. Hamaker and D. C. Solmon, “Angles between null spaces of x-rays,” J. Math. An. App. 62, 1–23 (1978).
    [Crossref]
  38. G. T. Herman and L. B. Meyer, “Algebraic reconstruction techniques can be made computationally efficient,” IEEE Trans. Med. Imag. 12, 600–609 (1993).
    [Crossref]
  39. H. Q. Guan and R. Gordon, “A projection access order for speedy convergence of ART (algebraic reconstruction technique) - a multilevel scheme for computed-tomography,” Phys. Med. Biol. 39, 2005–2022 (1994).
    [Crossref] [PubMed]
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    [Crossref]

2009 (3)

T. Blumensath and M. Davies, “Iterative hard thresholding for compressed sensing,” Applied and Computational Harmonic Analysis 27, 265–274 (2009).
[Crossref]

R. Garg and K. Khandekar, “Gradient Descent with Sparsification: An iterative algorithm for sparse recovery with restricted isometry property,” Proceedings of the 26th Annual International Conference on Machine Learning  382, 337–344 (2009).

X. Pan, E. Y. Sidky, and M. Vannier, “Why do commercial CT scanners still employ traditional, filtered back-projection for image reconstruction?” Inv. Prob. 25, 123009 (2009).
[Crossref]

2008 (5)

E. J. Candès and M. B. Wakin, “An Introduction To Compressive Sampling,” IEEE Sig. Proc. Mag. 25, 21–30 (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] [PubMed]

G.-H. Chen, J. Tang, and S. Leng, “Prior image constrained compressed sensing (PICCS): A method to accurately reconstruct dynamic CT images from highly undersampled projection data sets,” Med. Phys. 35(2), 660–663 (2008).
[Crossref] [PubMed]

G. R. Myers, D. M. Paganin, T. Gureyev, and S. C. Mayo, “Phase-contrast tomography of single-material objects from few projections,” Opt. Express 16, 908–919 (2008).
[Crossref] [PubMed]

G. R. Myers, T. E. Gureyev, D. M. Paganin, and S. C. Mayo, “The binary dissector: phase contrast tomography of two- and three-material objects from few projections,” Opt. Express 16, 10736–10749 (2008).
[Crossref] [PubMed]

2007 (6)

E. F. Donnelly, R. R. Price, K. G. Lewis, and D. R. Pickens, “Polychromatic phase-contrast computed tomography,” Med. Phys. 34(8), 3165–3168 (2007).
[Crossref] [PubMed]

R. Chartrand, “Exact reconstruction of sparse signals via nonconvex minimization,” IEEE Sig. Process. Lett. 14, 707–710 (2007).
[Crossref]

W. Souidene, A. Aissa-El-Bey, K. Abed-Meraim, and A. Beghdadi, “Blind Image Separation using Sparse Representation,” in ICIP 2007. IEEE International Conference on Image Processing,  vol. 3, pp. III -125–III -128 (2007).

E. Y. Sidky, R. Chartrand, and X. Pan, “Image reconstruction from few views by non-convex optimization,” IEEE Nuc. Sci. Conf. Rec. 5, 3526–3530 (2007).

R. Baraniuk, “Compressive Sensing,” IEEE Sig. Process. Mag. 24, 118–121 (2007).
[Crossref]

M. A. Figueiredo, R. D. Nowak, and S. J. Wright, “Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems,” IEEE J. Sel. Top. Sig. Process. 1, 586–598 (2007).
[Crossref]

2006 (4)

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

J. Tropp, “Just relax: Convex programming methods for identifying sparse signals,” IEEE Trans. Info. Theory 51, 1030–1051 (2006).
[Crossref]

T. E. Gureyev, D. M. Paganin, G. R. Myers, Y. I. Nesterest, and S. W. Wilkins, “Phase-and-amplitude computer tomography,” Appl. Phys. Lett. 89, 034,102 (2006).
[Crossref]

E. Candes, J. Romberg, and T. Tao, “Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 3526–3530 (2006).
[Crossref]

2005 (2)

D. Shi, M. Anastasio, and X. Pan, “Reconstruction of refractive index discontinuities from truncated phase-contrast tomography projections,” Appl. Phys. Lett. 86, 034102 (2005).
[Crossref]

M. A. Anastasio, D. Shi, Y. Huang, and G. Gbur, “Image Reconstruction in Spherical Wave Intensity Diffraction Tomography,” J. Opt. Soc. Am. A 22, 2651–2661 (2005).
[Crossref]

2004 (1)

M. A. Anastasio, D. Shi, F. D. Carlo, and X. Pan, “Analytic image reconstruction in local phase-contrast tomography,” Phys. Med. Biol. 49, 121–144 (2004).
[Crossref] [PubMed]

2003 (1)

M. Liebling, T. Blu, and M. Unser, “Fresnelets: New Multiresolution Wavelet Bases for Digital Holography,” IEEE Trans. Image Process. 12(1), 29–43 (2003).
[Crossref]

2002 (3)

T. Weitkamp, C. Rau, A. Snigirev, B. Brenner, T. Gunzler, M. Kuhlmann, and C. Schroer, “In-line phase contrast in synchrotron-radiation microradiography and tomography,” in Developments in X-ray Tomography III, Proceedings of the SPIE,  vol. 4503, pp. 92–102 (2002).

A. V. Bronnikov, “Theory of quantitative phase-contrast computed tomography,” J. Opt. Soc. Am. A 19(3), 472–480 (2002).
[Crossref]

M. Li, H. 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)

Y. Wang, F. D. Carlo, D. Mancini, I. McNulty, B. Tieman, J. Bresnahan, I. Foster, J. Insley, P. Lane, G. von Laszewski, C. Kesselman, M. Su, and M. Thiebaux, “A high-throughput x-ray microtomography system at the Advanced Photon Source,” Rev. Sci. Instrum. 72, 2062–2068 (2001).
[Crossref]

2000 (1)

A. Barty, K. Nugent, A. Roberts, and D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175(4), 329–336 (2000).
[Crossref]

1999 (1)

P. Spanne, C. Raven, I. Snigireva, and A. Snigirev, “In-line holography and phase-contrast microtomography with high energy x-rays,” Phys. Med. Biol. 44(3), 741–749 (1999).
[Crossref] [PubMed]

1998 (2)

T. E. Gureyev and S. W. Wilkins, “On x-ray phase imaging with a point source,” J. Opt. Soc. Am. A 15(3), 579–585 (1998).
[Crossref]

S. Chen, D. Donoho, and M. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. Sci. Comput. 20, 33–61 (1998).
[Crossref]

1997 (2)

A. Pogany, D. Gao, and S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
[Crossref]

P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81, 5878–5886 (1997).
[Crossref]

1996 (1)

P. Cloetens, R. Barrett, J.-P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging,” J. Phys. D: Appl. Phys. 29, 133–146 (1996).
[Crossref]

1994 (1)

H. Q. Guan and R. Gordon, “A projection access order for speedy convergence of ART (algebraic reconstruction technique) - a multilevel scheme for computed-tomography,” Phys. Med. Biol. 39, 2005–2022 (1994).
[Crossref] [PubMed]

1993 (1)

G. T. Herman and L. B. Meyer, “Algebraic reconstruction techniques can be made computationally efficient,” IEEE Trans. Med. Imag. 12, 600–609 (1993).
[Crossref]

1986 (1)

F. Santosa and W. Symes, “Linear inversion of band-limited reflection histograms,” SIAM J. Sci. Stat. Comput. 7, 1307–1330 (1986).
[Crossref]

1978 (1)

C. Hamaker and D. C. Solmon, “Angles between null spaces of x-rays,” J. Math. An. App. 62, 1–23 (1978).
[Crossref]

Abed-Meraim, K.

W. Souidene, A. Aissa-El-Bey, K. Abed-Meraim, and A. Beghdadi, “Blind Image Separation using Sparse Representation,” in ICIP 2007. IEEE International Conference on Image Processing,  vol. 3, pp. III -125–III -128 (2007).

Aissa-El-Bey, A.

W. Souidene, A. Aissa-El-Bey, K. Abed-Meraim, and A. Beghdadi, “Blind Image Separation using Sparse Representation,” in ICIP 2007. IEEE International Conference on Image Processing,  vol. 3, pp. III -125–III -128 (2007).

Anastasio, M.

D. Shi, M. Anastasio, and X. Pan, “Reconstruction of refractive index discontinuities from truncated phase-contrast tomography projections,” Appl. Phys. Lett. 86, 034102 (2005).
[Crossref]

Anastasio, M. A.

M. A. Anastasio, D. Shi, Y. Huang, and G. Gbur, “Image Reconstruction in Spherical Wave Intensity Diffraction Tomography,” J. Opt. Soc. Am. A 22, 2651–2661 (2005).
[Crossref]

M. A. Anastasio, D. Shi, F. D. Carlo, and X. Pan, “Analytic image reconstruction in local phase-contrast tomography,” Phys. Med. Biol. 49, 121–144 (2004).
[Crossref] [PubMed]

Baraniuk, R.

R. Baraniuk, “Compressive Sensing,” IEEE Sig. Process. Mag. 24, 118–121 (2007).
[Crossref]

Barrett, H.

H. Barrett and K. Myers, Foundations of Image Science (Wiley Series in Pure and Applied Optics, 2004).

Barrett, R.

P. Cloetens, R. Barrett, J.-P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging,” J. Phys. D: Appl. Phys. 29, 133–146 (1996).
[Crossref]

Barty, A.

A. Barty, K. Nugent, A. Roberts, and D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175(4), 329–336 (2000).
[Crossref]

Baruchel, J.

P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81, 5878–5886 (1997).
[Crossref]

Beghdadi, A.

W. Souidene, A. Aissa-El-Bey, K. Abed-Meraim, and A. Beghdadi, “Blind Image Separation using Sparse Representation,” in ICIP 2007. IEEE International Conference on Image Processing,  vol. 3, pp. III -125–III -128 (2007).

Blu, T.

M. Liebling, T. Blu, and M. Unser, “Fresnelets: New Multiresolution Wavelet Bases for Digital Holography,” IEEE Trans. Image Process. 12(1), 29–43 (2003).
[Crossref]

Blumensath, T.

T. Blumensath and M. Davies, “Iterative hard thresholding for compressed sensing,” Applied and Computational Harmonic Analysis 27, 265–274 (2009).
[Crossref]

Brenner, B.

T. Weitkamp, C. Rau, A. Snigirev, B. Brenner, T. Gunzler, M. Kuhlmann, and C. Schroer, “In-line phase contrast in synchrotron-radiation microradiography and tomography,” in Developments in X-ray Tomography III, Proceedings of the SPIE,  vol. 4503, pp. 92–102 (2002).

Bresnahan, J.

Y. Wang, F. D. Carlo, D. Mancini, I. McNulty, B. Tieman, J. Bresnahan, I. Foster, J. Insley, P. Lane, G. von Laszewski, C. Kesselman, M. Su, and M. Thiebaux, “A high-throughput x-ray microtomography system at the Advanced Photon Source,” Rev. Sci. Instrum. 72, 2062–2068 (2001).
[Crossref]

Bronnikov, A. V.

Buffiere, J. Y.

P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81, 5878–5886 (1997).
[Crossref]

Candes, E.

E. Candes, J. Romberg, and T. Tao, “Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 3526–3530 (2006).
[Crossref]

Candès, E. J.

E. J. Candès and M. B. Wakin, “An Introduction To Compressive Sampling,” IEEE Sig. Proc. Mag. 25, 21–30 (2008).
[Crossref]

Carlo, F. D.

M. A. Anastasio, D. Shi, F. D. Carlo, and X. Pan, “Analytic image reconstruction in local phase-contrast tomography,” Phys. Med. Biol. 49, 121–144 (2004).
[Crossref] [PubMed]

Y. Wang, F. D. Carlo, D. Mancini, I. McNulty, B. Tieman, J. Bresnahan, I. Foster, J. Insley, P. Lane, G. von Laszewski, C. Kesselman, M. Su, and M. Thiebaux, “A high-throughput x-ray microtomography system at the Advanced Photon Source,” Rev. Sci. Instrum. 72, 2062–2068 (2001).
[Crossref]

Chartrand, R.

R. Chartrand, “Exact reconstruction of sparse signals via nonconvex minimization,” IEEE Sig. Process. Lett. 14, 707–710 (2007).
[Crossref]

E. Y. Sidky, R. Chartrand, and X. Pan, “Image reconstruction from few views by non-convex optimization,” IEEE Nuc. Sci. Conf. Rec. 5, 3526–3530 (2007).

Chen, G.-H.

G.-H. Chen, J. Tang, and S. Leng, “Prior image constrained compressed sensing (PICCS): A method to accurately reconstruct dynamic CT images from highly undersampled projection data sets,” Med. Phys. 35(2), 660–663 (2008).
[Crossref] [PubMed]

Chen, S.

S. Chen, D. Donoho, and M. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. Sci. Comput. 20, 33–61 (1998).
[Crossref]

Cloetens, P.

P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81, 5878–5886 (1997).
[Crossref]

P. Cloetens, R. Barrett, J.-P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging,” J. Phys. D: Appl. Phys. 29, 133–146 (1996).
[Crossref]

Davies, M.

T. Blumensath and M. Davies, “Iterative hard thresholding for compressed sensing,” Applied and Computational Harmonic Analysis 27, 265–274 (2009).
[Crossref]

Donnelly, E. F.

E. F. Donnelly, R. R. Price, K. G. Lewis, and D. R. Pickens, “Polychromatic phase-contrast computed tomography,” Med. Phys. 34(8), 3165–3168 (2007).
[Crossref] [PubMed]

Donoho, D.

S. Chen, D. Donoho, and M. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. Sci. Comput. 20, 33–61 (1998).
[Crossref]

Figueiredo, M. A.

M. A. Figueiredo, R. D. Nowak, and S. J. Wright, “Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems,” IEEE J. Sel. Top. Sig. Process. 1, 586–598 (2007).
[Crossref]

Foster, I.

Y. Wang, F. D. Carlo, D. Mancini, I. McNulty, B. Tieman, J. Bresnahan, I. Foster, J. Insley, P. Lane, G. von Laszewski, C. Kesselman, M. Su, and M. Thiebaux, “A high-throughput x-ray microtomography system at the Advanced Photon Source,” Rev. Sci. Instrum. 72, 2062–2068 (2001).
[Crossref]

Gao, D.

A. Pogany, D. Gao, and S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
[Crossref]

Garg, R.

R. Garg and K. Khandekar, “Gradient Descent with Sparsification: An iterative algorithm for sparse recovery with restricted isometry property,” Proceedings of the 26th Annual International Conference on Machine Learning  382, 337–344 (2009).

Gbur, G.

Gordon, R.

H. Q. Guan and R. Gordon, “A projection access order for speedy convergence of ART (algebraic reconstruction technique) - a multilevel scheme for computed-tomography,” Phys. Med. Biol. 39, 2005–2022 (1994).
[Crossref] [PubMed]

Guan, H. Q.

H. Q. Guan and R. Gordon, “A projection access order for speedy convergence of ART (algebraic reconstruction technique) - a multilevel scheme for computed-tomography,” Phys. Med. Biol. 39, 2005–2022 (1994).
[Crossref] [PubMed]

Guigay, J.-P.

P. Cloetens, R. Barrett, J.-P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging,” J. Phys. D: Appl. Phys. 29, 133–146 (1996).
[Crossref]

Gunzler, T.

T. Weitkamp, C. Rau, A. Snigirev, B. Brenner, T. Gunzler, M. Kuhlmann, and C. Schroer, “In-line phase contrast in synchrotron-radiation microradiography and tomography,” in Developments in X-ray Tomography III, Proceedings of the SPIE,  vol. 4503, pp. 92–102 (2002).

Gureyev, T.

Gureyev, T. E.

Hamaker, C.

C. Hamaker and D. C. Solmon, “Angles between null spaces of x-rays,” J. Math. An. App. 62, 1–23 (1978).
[Crossref]

Herman, G. T.

G. T. Herman and L. B. Meyer, “Algebraic reconstruction techniques can be made computationally efficient,” IEEE Trans. Med. Imag. 12, 600–609 (1993).
[Crossref]

Huang, Y.

Insley, J.

Y. Wang, F. D. Carlo, D. Mancini, I. McNulty, B. Tieman, J. Bresnahan, I. Foster, J. Insley, P. Lane, G. von Laszewski, C. Kesselman, M. Su, and M. Thiebaux, “A high-throughput x-ray microtomography system at the Advanced Photon Source,” Rev. Sci. Instrum. 72, 2062–2068 (2001).
[Crossref]

Kak, A.

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

Kao, C.-M.

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

Kesselman, C.

Y. Wang, F. D. Carlo, D. Mancini, I. McNulty, B. Tieman, J. Bresnahan, I. Foster, J. Insley, P. Lane, G. von Laszewski, C. Kesselman, M. Su, and M. Thiebaux, “A high-throughput x-ray microtomography system at the Advanced Photon Source,” Rev. Sci. Instrum. 72, 2062–2068 (2001).
[Crossref]

Khandekar, K.

R. Garg and K. Khandekar, “Gradient Descent with Sparsification: An iterative algorithm for sparse recovery with restricted isometry property,” Proceedings of the 26th Annual International Conference on Machine Learning  382, 337–344 (2009).

Kudo, H.

M. Li, H. 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]

Kuhlmann, M.

T. Weitkamp, C. Rau, A. Snigirev, B. Brenner, T. Gunzler, M. Kuhlmann, and C. Schroer, “In-line phase contrast in synchrotron-radiation microradiography and tomography,” in Developments in X-ray Tomography III, Proceedings of the SPIE,  vol. 4503, pp. 92–102 (2002).

Lane, P.

Y. Wang, F. D. Carlo, D. Mancini, I. McNulty, B. Tieman, J. Bresnahan, I. Foster, J. Insley, P. Lane, G. von Laszewski, C. Kesselman, M. Su, and M. Thiebaux, “A high-throughput x-ray microtomography system at the Advanced Photon Source,” Rev. Sci. Instrum. 72, 2062–2068 (2001).
[Crossref]

Leng, S.

G.-H. Chen, J. Tang, and S. Leng, “Prior image constrained compressed sensing (PICCS): A method to accurately reconstruct dynamic CT images from highly undersampled projection data sets,” Med. Phys. 35(2), 660–663 (2008).
[Crossref] [PubMed]

Lewis, K. G.

E. F. Donnelly, R. R. Price, K. G. Lewis, and D. R. Pickens, “Polychromatic phase-contrast computed tomography,” Med. Phys. 34(8), 3165–3168 (2007).
[Crossref] [PubMed]

Li, M.

M. Li, H. 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]

Liebling, M.

M. Liebling, T. Blu, and M. Unser, “Fresnelets: New Multiresolution Wavelet Bases for Digital Holography,” IEEE Trans. Image Process. 12(1), 29–43 (2003).
[Crossref]

Mancini, D.

Y. Wang, F. D. Carlo, D. Mancini, I. McNulty, B. Tieman, J. Bresnahan, I. Foster, J. Insley, P. Lane, G. von Laszewski, C. Kesselman, M. Su, and M. Thiebaux, “A high-throughput x-ray microtomography system at the Advanced Photon Source,” Rev. Sci. Instrum. 72, 2062–2068 (2001).
[Crossref]

Mayo, S. C.

McNulty, I.

Y. Wang, F. D. Carlo, D. Mancini, I. McNulty, B. Tieman, J. Bresnahan, I. Foster, J. Insley, P. Lane, G. von Laszewski, C. Kesselman, M. Su, and M. Thiebaux, “A high-throughput x-ray microtomography system at the Advanced Photon Source,” Rev. Sci. Instrum. 72, 2062–2068 (2001).
[Crossref]

Meyer, L. B.

G. T. Herman and L. B. Meyer, “Algebraic reconstruction techniques can be made computationally efficient,” IEEE Trans. Med. Imag. 12, 600–609 (1993).
[Crossref]

Myers, G. R.

Myers, K.

H. Barrett and K. Myers, Foundations of Image Science (Wiley Series in Pure and Applied Optics, 2004).

Natterer, F.

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

Nesterest, Y. I.

T. E. Gureyev, D. M. Paganin, G. R. Myers, Y. I. Nesterest, and S. W. Wilkins, “Phase-and-amplitude computer tomography,” Appl. Phys. Lett. 89, 034,102 (2006).
[Crossref]

Nowak, R. D.

M. A. Figueiredo, R. D. Nowak, and S. J. Wright, “Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems,” IEEE J. Sel. Top. Sig. Process. 1, 586–598 (2007).
[Crossref]

Nugent, K.

A. Barty, K. Nugent, A. Roberts, and D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175(4), 329–336 (2000).
[Crossref]

Paganin, D.

A. Barty, K. Nugent, A. Roberts, and D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175(4), 329–336 (2000).
[Crossref]

Paganin, D. M.

Pan, X.

X. Pan, E. Y. Sidky, and M. Vannier, “Why do commercial CT scanners still employ traditional, filtered back-projection for image reconstruction?” Inv. Prob. 25, 123009 (2009).
[Crossref]

E. Y. Sidky, R. Chartrand, and X. Pan, “Image reconstruction from few views by non-convex optimization,” IEEE Nuc. Sci. Conf. Rec. 5, 3526–3530 (2007).

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

D. Shi, M. Anastasio, and X. Pan, “Reconstruction of refractive index discontinuities from truncated phase-contrast tomography projections,” Appl. Phys. Lett. 86, 034102 (2005).
[Crossref]

M. A. Anastasio, D. Shi, F. D. Carlo, and X. Pan, “Analytic image reconstruction in local phase-contrast tomography,” Phys. Med. Biol. 49, 121–144 (2004).
[Crossref] [PubMed]

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] [PubMed]

Pateyron-Salome, M.

P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81, 5878–5886 (1997).
[Crossref]

Peix, G.

P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81, 5878–5886 (1997).
[Crossref]

Peyrin, F.

P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81, 5878–5886 (1997).
[Crossref]

Pickens, D. R.

E. F. Donnelly, R. R. Price, K. G. Lewis, and D. R. Pickens, “Polychromatic phase-contrast computed tomography,” Med. Phys. 34(8), 3165–3168 (2007).
[Crossref] [PubMed]

Pogany, A.

A. Pogany, D. Gao, and S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
[Crossref]

Price, R. R.

E. F. Donnelly, R. R. Price, K. G. Lewis, and D. R. Pickens, “Polychromatic phase-contrast computed tomography,” Med. Phys. 34(8), 3165–3168 (2007).
[Crossref] [PubMed]

Rau, C.

T. Weitkamp, C. Rau, A. Snigirev, B. Brenner, T. Gunzler, M. Kuhlmann, and C. Schroer, “In-line phase contrast in synchrotron-radiation microradiography and tomography,” in Developments in X-ray Tomography III, Proceedings of the SPIE,  vol. 4503, pp. 92–102 (2002).

Raven, C.

P. Spanne, C. Raven, I. Snigireva, and A. Snigirev, “In-line holography and phase-contrast microtomography with high energy x-rays,” Phys. Med. Biol. 44(3), 741–749 (1999).
[Crossref] [PubMed]

Roberts, A.

A. Barty, K. Nugent, A. Roberts, and D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175(4), 329–336 (2000).
[Crossref]

Romberg, J.

E. Candes, J. Romberg, and T. Tao, “Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 3526–3530 (2006).
[Crossref]

Santosa, F.

F. Santosa and W. Symes, “Linear inversion of band-limited reflection histograms,” SIAM J. Sci. Stat. Comput. 7, 1307–1330 (1986).
[Crossref]

Saunders, M.

S. Chen, D. Donoho, and M. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. Sci. Comput. 20, 33–61 (1998).
[Crossref]

Schlenker, M.

P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81, 5878–5886 (1997).
[Crossref]

P. Cloetens, R. Barrett, J.-P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging,” J. Phys. D: Appl. Phys. 29, 133–146 (1996).
[Crossref]

Schroer, C.

T. Weitkamp, C. Rau, A. Snigirev, B. Brenner, T. Gunzler, M. Kuhlmann, and C. Schroer, “In-line phase contrast in synchrotron-radiation microradiography and tomography,” in Developments in X-ray Tomography III, Proceedings of the SPIE,  vol. 4503, pp. 92–102 (2002).

Shi, D.

D. Shi, M. Anastasio, and X. Pan, “Reconstruction of refractive index discontinuities from truncated phase-contrast tomography projections,” Appl. Phys. Lett. 86, 034102 (2005).
[Crossref]

M. A. Anastasio, D. Shi, Y. Huang, and G. Gbur, “Image Reconstruction in Spherical Wave Intensity Diffraction Tomography,” J. Opt. Soc. Am. A 22, 2651–2661 (2005).
[Crossref]

M. A. Anastasio, D. Shi, F. D. Carlo, and X. Pan, “Analytic image reconstruction in local phase-contrast tomography,” Phys. Med. Biol. 49, 121–144 (2004).
[Crossref] [PubMed]

Sidky, E. Y.

X. Pan, E. Y. Sidky, and M. Vannier, “Why do commercial CT scanners still employ traditional, filtered back-projection for image reconstruction?” Inv. Prob. 25, 123009 (2009).
[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] [PubMed]

E. Y. Sidky, R. Chartrand, and X. Pan, “Image reconstruction from few views by non-convex optimization,” IEEE Nuc. Sci. Conf. Rec. 5, 3526–3530 (2007).

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

Slaney, M.

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

Snigirev, A.

T. Weitkamp, C. Rau, A. Snigirev, B. Brenner, T. Gunzler, M. Kuhlmann, and C. Schroer, “In-line phase contrast in synchrotron-radiation microradiography and tomography,” in Developments in X-ray Tomography III, Proceedings of the SPIE,  vol. 4503, pp. 92–102 (2002).

P. Spanne, C. Raven, I. Snigireva, and A. Snigirev, “In-line holography and phase-contrast microtomography with high energy x-rays,” Phys. Med. Biol. 44(3), 741–749 (1999).
[Crossref] [PubMed]

Snigireva, I.

P. Spanne, C. Raven, I. Snigireva, and A. Snigirev, “In-line holography and phase-contrast microtomography with high energy x-rays,” Phys. Med. Biol. 44(3), 741–749 (1999).
[Crossref] [PubMed]

Solmon, D. C.

C. Hamaker and D. C. Solmon, “Angles between null spaces of x-rays,” J. Math. An. App. 62, 1–23 (1978).
[Crossref]

Souidene, W.

W. Souidene, A. Aissa-El-Bey, K. Abed-Meraim, and A. Beghdadi, “Blind Image Separation using Sparse Representation,” in ICIP 2007. IEEE International Conference on Image Processing,  vol. 3, pp. III -125–III -128 (2007).

Spanne, P.

P. Spanne, C. Raven, I. Snigireva, and A. Snigirev, “In-line holography and phase-contrast microtomography with high energy x-rays,” Phys. Med. Biol. 44(3), 741–749 (1999).
[Crossref] [PubMed]

Su, M.

Y. Wang, F. D. Carlo, D. Mancini, I. McNulty, B. Tieman, J. Bresnahan, I. Foster, J. Insley, P. Lane, G. von Laszewski, C. Kesselman, M. Su, and M. Thiebaux, “A high-throughput x-ray microtomography system at the Advanced Photon Source,” Rev. Sci. Instrum. 72, 2062–2068 (2001).
[Crossref]

Symes, W.

F. Santosa and W. Symes, “Linear inversion of band-limited reflection histograms,” SIAM J. Sci. Stat. Comput. 7, 1307–1330 (1986).
[Crossref]

Tang, J.

G.-H. Chen, J. Tang, and S. Leng, “Prior image constrained compressed sensing (PICCS): A method to accurately reconstruct dynamic CT images from highly undersampled projection data sets,” Med. Phys. 35(2), 660–663 (2008).
[Crossref] [PubMed]

Tao, T.

E. Candes, J. Romberg, and T. Tao, “Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 3526–3530 (2006).
[Crossref]

Thiebaux, M.

Y. Wang, F. D. Carlo, D. Mancini, I. McNulty, B. Tieman, J. Bresnahan, I. Foster, J. Insley, P. Lane, G. von Laszewski, C. Kesselman, M. Su, and M. Thiebaux, “A high-throughput x-ray microtomography system at the Advanced Photon Source,” Rev. Sci. Instrum. 72, 2062–2068 (2001).
[Crossref]

Tieman, B.

Y. Wang, F. D. Carlo, D. Mancini, I. McNulty, B. Tieman, J. Bresnahan, I. Foster, J. Insley, P. Lane, G. von Laszewski, C. Kesselman, M. Su, and M. Thiebaux, “A high-throughput x-ray microtomography system at the Advanced Photon Source,” Rev. Sci. Instrum. 72, 2062–2068 (2001).
[Crossref]

Tropp, J.

J. Tropp, “Just relax: Convex programming methods for identifying sparse signals,” IEEE Trans. Info. Theory 51, 1030–1051 (2006).
[Crossref]

Unser, M.

M. Liebling, T. Blu, and M. Unser, “Fresnelets: New Multiresolution Wavelet Bases for Digital Holography,” IEEE Trans. Image Process. 12(1), 29–43 (2003).
[Crossref]

Vannier, M.

X. Pan, E. Y. Sidky, and M. Vannier, “Why do commercial CT scanners still employ traditional, filtered back-projection for image reconstruction?” Inv. Prob. 25, 123009 (2009).
[Crossref]

von Laszewski, G.

Y. Wang, F. D. Carlo, D. Mancini, I. McNulty, B. Tieman, J. Bresnahan, I. Foster, J. Insley, P. Lane, G. von Laszewski, C. Kesselman, M. Su, and M. Thiebaux, “A high-throughput x-ray microtomography system at the Advanced Photon Source,” Rev. Sci. Instrum. 72, 2062–2068 (2001).
[Crossref]

Wakin, M. B.

E. J. Candès and M. B. Wakin, “An Introduction To Compressive Sampling,” IEEE Sig. Proc. Mag. 25, 21–30 (2008).
[Crossref]

Wang, Y.

Y. Wang, F. D. Carlo, D. Mancini, I. McNulty, B. Tieman, J. Bresnahan, I. Foster, J. Insley, P. Lane, G. von Laszewski, C. Kesselman, M. Su, and M. Thiebaux, “A high-throughput x-ray microtomography system at the Advanced Photon Source,” Rev. Sci. Instrum. 72, 2062–2068 (2001).
[Crossref]

Weitkamp, T.

T. Weitkamp, C. Rau, A. Snigirev, B. Brenner, T. Gunzler, M. Kuhlmann, and C. Schroer, “In-line phase contrast in synchrotron-radiation microradiography and tomography,” in Developments in X-ray Tomography III, Proceedings of the SPIE,  vol. 4503, pp. 92–102 (2002).

Wilkins, S. W.

T. E. Gureyev, D. M. Paganin, G. R. Myers, Y. I. Nesterest, and S. W. Wilkins, “Phase-and-amplitude computer tomography,” Appl. Phys. Lett. 89, 034,102 (2006).
[Crossref]

T. E. Gureyev and S. W. Wilkins, “On x-ray phase imaging with a point source,” J. Opt. Soc. Am. A 15(3), 579–585 (1998).
[Crossref]

A. Pogany, D. Gao, and S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
[Crossref]

Wright, S. J.

M. A. Figueiredo, R. D. Nowak, and S. J. Wright, “Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems,” IEEE J. Sel. Top. Sig. Process. 1, 586–598 (2007).
[Crossref]

Yang, H.

M. Li, H. 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]

Appl. Phys. Lett. (2)

T. E. Gureyev, D. M. Paganin, G. R. Myers, Y. I. Nesterest, and S. W. Wilkins, “Phase-and-amplitude computer tomography,” Appl. Phys. Lett. 89, 034,102 (2006).
[Crossref]

D. Shi, M. Anastasio, and X. Pan, “Reconstruction of refractive index discontinuities from truncated phase-contrast tomography projections,” Appl. Phys. Lett. 86, 034102 (2005).
[Crossref]

Applied and Computational Harmonic Analysis (1)

T. Blumensath and M. Davies, “Iterative hard thresholding for compressed sensing,” Applied and Computational Harmonic Analysis 27, 265–274 (2009).
[Crossref]

Developments in X-ray Tomography III, Proceedings of the SPIE (1)

T. Weitkamp, C. Rau, A. Snigirev, B. Brenner, T. Gunzler, M. Kuhlmann, and C. Schroer, “In-line phase contrast in synchrotron-radiation microradiography and tomography,” in Developments in X-ray Tomography III, Proceedings of the SPIE,  vol. 4503, pp. 92–102 (2002).

ICIP 2007. IEEE International Conference on Image (1)

W. Souidene, A. Aissa-El-Bey, K. Abed-Meraim, and A. Beghdadi, “Blind Image Separation using Sparse Representation,” in ICIP 2007. IEEE International Conference on Image Processing,  vol. 3, pp. III -125–III -128 (2007).

IEEE J. Sel. Top. Sig. Process. (1)

M. A. Figueiredo, R. D. Nowak, and S. J. Wright, “Gradient projection for sparse reconstruction: Application to compressed sensing and other inverse problems,” IEEE J. Sel. Top. Sig. Process. 1, 586–598 (2007).
[Crossref]

IEEE Nuc. Sci. Conf. Rec. (1)

E. Y. Sidky, R. Chartrand, and X. Pan, “Image reconstruction from few views by non-convex optimization,” IEEE Nuc. Sci. Conf. Rec. 5, 3526–3530 (2007).

IEEE Sig. Proc. Mag. (1)

E. J. Candès and M. B. Wakin, “An Introduction To Compressive Sampling,” IEEE Sig. Proc. Mag. 25, 21–30 (2008).
[Crossref]

IEEE Sig. Process. Lett. (1)

R. Chartrand, “Exact reconstruction of sparse signals via nonconvex minimization,” IEEE Sig. Process. Lett. 14, 707–710 (2007).
[Crossref]

IEEE Sig. Process. Mag. (1)

R. Baraniuk, “Compressive Sensing,” IEEE Sig. Process. Mag. 24, 118–121 (2007).
[Crossref]

IEEE Trans. Image Process. (1)

M. Liebling, T. Blu, and M. Unser, “Fresnelets: New Multiresolution Wavelet Bases for Digital Holography,” IEEE Trans. Image Process. 12(1), 29–43 (2003).
[Crossref]

IEEE Trans. Inf. Theory (1)

E. Candes, J. Romberg, and T. Tao, “Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 3526–3530 (2006).
[Crossref]

IEEE Trans. Info. Theory (1)

J. Tropp, “Just relax: Convex programming methods for identifying sparse signals,” IEEE Trans. Info. Theory 51, 1030–1051 (2006).
[Crossref]

IEEE Trans. Med. Imag. (1)

G. T. Herman and L. B. Meyer, “Algebraic reconstruction techniques can be made computationally efficient,” IEEE Trans. Med. Imag. 12, 600–609 (1993).
[Crossref]

Inv. Prob. (1)

X. Pan, E. Y. Sidky, and M. Vannier, “Why do commercial CT scanners still employ traditional, filtered back-projection for image reconstruction?” Inv. Prob. 25, 123009 (2009).
[Crossref]

J. Appl. Phys. (1)

P. Cloetens, M. Pateyron-Salome, J. Y. Buffiere, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, “Observation of microstructure and damage in materials by phase sensitive radiography and tomography,” J. Appl. Phys. 81, 5878–5886 (1997).
[Crossref]

J. Math. An. App. (1)

C. Hamaker and D. C. Solmon, “Angles between null spaces of x-rays,” J. Math. An. App. 62, 1–23 (1978).
[Crossref]

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

J. Phys. D: Appl. Phys. (1)

P. Cloetens, R. Barrett, J.-P. Guigay, and M. Schlenker, “Phase objects in synchrotron radiation hard x-ray imaging,” J. Phys. D: Appl. Phys. 29, 133–146 (1996).
[Crossref]

J. X-ray Sci. Tech. (1)

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

Med. Phys. (2)

G.-H. Chen, J. Tang, and S. Leng, “Prior image constrained compressed sensing (PICCS): A method to accurately reconstruct dynamic CT images from highly undersampled projection data sets,” Med. Phys. 35(2), 660–663 (2008).
[Crossref] [PubMed]

E. F. Donnelly, R. R. Price, K. G. Lewis, and D. R. Pickens, “Polychromatic phase-contrast computed tomography,” Med. Phys. 34(8), 3165–3168 (2007).
[Crossref] [PubMed]

Opt. Commun. (1)

A. Barty, K. Nugent, A. Roberts, and D. Paganin, “Quantitative phase tomography,” Opt. Commun. 175(4), 329–336 (2000).
[Crossref]

Opt. Express (2)

Phys. Med. Biol. (5)

M. A. Anastasio, D. Shi, F. D. Carlo, and X. Pan, “Analytic image reconstruction in local phase-contrast tomography,” Phys. Med. Biol. 49, 121–144 (2004).
[Crossref] [PubMed]

P. Spanne, C. Raven, I. Snigireva, and A. Snigirev, “In-line holography and phase-contrast microtomography with high energy x-rays,” Phys. Med. Biol. 44(3), 741–749 (1999).
[Crossref] [PubMed]

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] [PubMed]

M. Li, H. 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]

H. Q. Guan and R. Gordon, “A projection access order for speedy convergence of ART (algebraic reconstruction technique) - a multilevel scheme for computed-tomography,” Phys. Med. Biol. 39, 2005–2022 (1994).
[Crossref] [PubMed]

Rev. Sci. Instrum. (2)

A. Pogany, D. Gao, and S. W. Wilkins, “Contrast and resolution in imaging with a microfocus x-ray source,” Rev. Sci. Instrum. 68, 2774–2782 (1997).
[Crossref]

Y. Wang, F. D. Carlo, D. Mancini, I. McNulty, B. Tieman, J. Bresnahan, I. Foster, J. Insley, P. Lane, G. von Laszewski, C. Kesselman, M. Su, and M. Thiebaux, “A high-throughput x-ray microtomography system at the Advanced Photon Source,” Rev. Sci. Instrum. 72, 2062–2068 (2001).
[Crossref]

SIAM J. Sci. Comput. (1)

S. Chen, D. Donoho, and M. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. Sci. Comput. 20, 33–61 (1998).
[Crossref]

SIAM J. Sci. Stat. Comput. (1)

F. Santosa and W. Symes, “Linear inversion of band-limited reflection histograms,” SIAM J. Sci. Stat. Comput. 7, 1307–1330 (1986).
[Crossref]

Other (4)

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

H. Barrett and K. Myers, Foundations of Image Science (Wiley Series in Pure and Applied Optics, 2004).

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

R. Garg and K. Khandekar, “Gradient Descent with Sparsification: An iterative algorithm for sparse recovery with restricted isometry property,” Proceedings of the 26th Annual International Conference on Machine Learning  382, 337–344 (2009).

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Figures (14)

Fig. 1.
Fig. 1.

The imaging geometry of in-line phase-contrast tomography.

Fig. 2.
Fig. 2.

Phantom composed of ellipses of high eccentricity that roughly resembles, in terms of sparsity, the object in the experimental results.

Fig. 3.
Fig. 3.

Estimated values of the isometry constants δs for the Radon transform with 90, equally-spaced views and 2048 detector bins per projection using 512×512 pixels to represent the scanned object. As only a restricted search is performed these values are interpreted as a lower bound on the true isometry constants.

Fig. 4.
Fig. 4.

Ellipse phantom made more challenging by adding f (min) 100 (bottom, left) and f (max) 100 (top, left). These images are added with a similar total energy as that of the original ellipse phantom.

Fig. 5.
Fig. 5.

Reconstructions of the ellipse phantom by IHT-POCS (left) and IHT (middle) algorithms and semi-log plots of the data and image error of both algorithms.

Fig. 6.
Fig. 6.

Reconstructions of the ellipse phantom, with f (min) 100 and f (max) 100 added, by IHT-POCS (left) and IHT (right) algorithms.

Fig. 7.
Fig. 7.

Image reconstructions of the ellipse phantom by IHT-POCS (left) and IHT (right) algorithms. For these results the data are generated from the continuous Radon transform of the ellipse phantom.

Fig. 8.
Fig. 8.

A plot of data residual vs. image sparsity for reconstruction of the ellipse phantom on a 2048×2048 grid using the IHT-POCS algorithm.

Fig. 9.
Fig. 9.

A plot of data residual vs. threshold parameter employed in the IHT-POCS and IHT-POCS-TV algorithms.

Fig. 10.
Fig. 10.

Boundary enhanced images corresponding to a slice of constant z. The image reconstructed from 1440 tomographic views by use of the FBP algorithm is contained in subfigure (a). Images reconstructed from 90 tomographic views by use of the POCS, IHT-POCS, and IHT-POCS-TV algorithms are displayed in subfigures (b)–(d), respectively.

Fig. 11.
Fig. 11.

550×900 pixel region-of-interest positioned near the center of the images in Fig. 1–(a)–(d) are displayed in subfigures (a)–(d).

Fig. 12.
Fig. 12.

500×900 pixel region-of-interest in reconstructed images corresponding to a slice of constant z. The image reconstructed from 1440 tomographic views by use of the FBP algorithm is contained in subfigure (a). Images reconstructed from 90 tomographic views by use of the IHT-POCS, and IHT-POCS-TV algorithms are displayed in subfigures (b) and (c).

Fig. 13.
Fig. 13.

600×350 pixel region-of-interest in reconstructed images corresponding to a slice of constant z. The image reconstructed from 1440 tomographic views by use of the FBP algorithm is contained in subfigure (a). Images reconstructed from 90 tomographic views by use of the IHT-POCS, and IHT-POCS-TV algorithms are displayed in subfigures (b) and (c).

Equations (16)

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g(xr,z,ϕ)1d[I(xr,z,ϕ)I0(xr,z,ϕ)1],
g(xr,z,ϕ)=xr,z2(r2;z) = xr,z2 d yr δ (r2;z),
g(xr,z,ϕ)=R2 δ (r2;z) ,
g[r,s,t]g(xr,z,ϕ)x=rΔd,z=sΔd,ϕ=tΔθ,
2δ(r2;z=snΔd)l=1M2 m=1M2 bz [l,m] Ψl.m (r2) ,
g=R̂ b ,
b0=argminb0 such that R̂bgε,
bs=argminR̂bg2 such that b0s,
bTV=argmin bTV such that R̂bgεandb0s*,
fs22(1δs)R̂fs22fs22 (1+δs),
f1(min)=argminf1 R̂f1,
f1(max)=argmaxf1 R̂f1.
σ1(min)=R̂f1(min)2 ,
σ1(max)=R̂f1(max)2 .
δ1=σ1(max)σ1(min)σ1(max)+σ1(min) .
fs(min)=argminα,pR̂cosαfs1(min)+R̂sinαp,

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