Abstract

High-resolution, three-dimensional (3D) imaging of soft tissues requires the solution of two inverse problems: phase retrieval and the reconstruction of the 3D image from a tomographic stack of two-dimensional (2D) projections. The number of projections per stack should be small to accommodate fast tomography of rapid processes and to constrain X-ray radiation dose to optimal levels to either increase the duration of in vivo time-lapse series at a given goal for spatial resolution and/or the conservation of structure under X-ray irradiation. In pursuing the 3D reconstruction problem in the sense of compressive sampling theory, we propose to reduce the number of projections by applying an advanced algebraic technique subject to the minimisation of the total variation (TV) in the reconstructed slice. This problem is formulated in a Lagrangian multiplier fashion with the parameter value determined by appealing to a discrete L-curve in conjunction with a conjugate gradient method. The usefulness of this reconstruction modality is demonstrated for simulated and in vivo data, the latter acquired in parallel-beam imaging experiments using synchrotron radiation.

© 2015 Optical Society of America

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References

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2014 (2)

T. dos Santos Rolo, A. Ershov, T. van de Kamp, and T. Baumbach, “In vivo X-ray cine-tomography for tracking morphological dynamics,” Proc. Natl. Acad. Sci. USA 111, 3921–3926 (2014).
[Crossref] [PubMed]

J. Moosmann, A. Ershov, V. Weinhardt, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, and R. Hofmann, “Time-lapse X-ray phase-contrast microtomography for in vivo imaging and analysis of morphogenesis,” Nat. Protoc. 9, 294–304 (2014).
[Crossref] [PubMed]

2013 (3)

J. Moosmann, A. Ershov, V. Altapova, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, and R. Hofmann, “X-ray phase-contrast in vivo microtomography probes new aspects of Xenopus gastrulation,” Nature 497, 374–377 (2013).
[Crossref] [PubMed]

L. Reichel and G. Rodriguez, “Old and new parameter choice rules for discrete ill-posed problems,” Numer. Algorithms 63, 65–87 (2013).
[Crossref]

A. Kostenko, K. J. Batenburg, H. Suhonen, S. E. Offerman, and L. J. van Vliet, “Phase retrieval in in-line x-ray phase contrast imaging based on total variation minimization,” Opt. Express 21, 710–723 (2013).
[Crossref] [PubMed]

2012 (3)

P. T. Lauzier, J. Tang, and G. Chen, “Prior image constrained compressed sensing: Implementation and performance evaluation,” Med. Phys. 39, 66–80 (2012).
[Crossref] [PubMed]

Y. Hu and M. Jacob, “Higher degree total variation (HDTV) regularization for image recovery,” IEEE Trans. Image Process. 21, 2559–2571 (2012).
[Crossref] [PubMed]

T. L. Jensen, J. H. Jørgensen, P. C. Hansen, and S. H. Jensen, “Implementation of an optimal first-order method for strongly convex total variation regularization,” BIT 52, 329–356 (2012).
[Crossref]

2011 (4)

S. Becker, J. Bobin, and E. Candès, “NESTA: a fast and accurate first-order method for sparse recovery,” SIAM J. Imaging Sci. 4, 1–39 (2011).
[Crossref]

T. van de Kamp, P. Vagovič, T. Baumbach, and A. Riedel, “A biological screw in a beetle’s leg,” Science 333, 52 (2011).
[Crossref]

J. Moosmann, R. Hofmann, and T. Baumbach, “Single-distance phase retrieval at large phase shifts,” Opt. Express 19, 12066–12073 (2011).
[Crossref] [PubMed]

R. Hofmann, J. Moosmann, and T. Baumbach, “Criticality in single-distance phase retrieval,” Opt. Express 19, 25881–25890 (2011).
[Crossref]

2010 (1)

J. Yang, Y. Zhang, and W. Yin, “A fast alternating direction method for TVL1-L2 signal reconstruction from partial Fourier data,” IEEE J. Sel. Top. Signa. 4, 288–297 (2010).
[Crossref]

2009 (4)

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Imaging Sci. 2, 183–202 (2009).
[Crossref]

T. Goldstein and S. Osher, “The split Bregman method for L1-regularized problems,” SIAM J. Imaging Sci. 2, 323–343 (2009).
[Crossref]

E. Esser, “Applications of Lagrangian-based alternating direction methods and connections to split Bregman,” CAM Report 9, 31 (2009).

Z. Wang and A. C. Bovik, “Mean squared error: love it or leave it? A new look at signal fidelity measures,” IEEE Signal Proc. Mag. 26, 98–117 (2009).
[Crossref]

2008 (2)

Y. Wang, J. Yang, W. Yin, and Y. Zhang, “A new alternating minimization algorithm for total variation image reconstruction,” SIAM J. Imaging Sci. 1, 248–272 (2008).
[Crossref]

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

2007 (3)

J. M. Bioucas-Dias and M. A. T. Figueiredo, “A new TwIST: two-step iterative shrinkage / thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16, 2992–3004 (2007).
[Crossref] [PubMed]

J. Song, Q. 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] [PubMed]

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] [PubMed]

2006 (3)

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

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory 52, 489–509 (2006).
[Crossref]

W. W. Hager and H. Zhang, “A survey of nonlinear conjugate gradient methods,” Pac. J. Optim. 2, 35–58 (2006).

2005 (1)

T. Chan, S. Esedoglu, F. Park, and A. Yip, “Recent developments in total variation image restoration,” Mathematical Models of Computer Vision 2005,17 (2005).

2004 (1)

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[Crossref] [PubMed]

2000 (1)

D. Calvetti, S. Morigi, L. Reichel, and F. Sgallari, “Tikhonov regularization and the L-curve for large discrete ill-posed problems,” J. Comput. Appl. Math. 123, 423–446 (2000).
[Crossref]

1999 (1)

Y. Dai and Y. Yuan, “A nonlinear conjugate gradient method with a strong global convergence property,” SIAM J. Optimiz. 10, 177–182 (1999).
[Crossref]

1996 (1)

C. R. Vogel, “Non-convergence of the L-curve regularization parameter selection method,” Inverse Probl. 12, 535 (1996).
[Crossref]

1993 (1)

P. C. Hansen and D. P. O’Leary, “The use of the L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14, 1487–1503 (1993).
[Crossref]

1992 (1)

M. Fukushima, “Application of the alternating direction method of multipliers to separable convex programming problems,” Comput. Optim. Appl. 1, 93–111 (1992).
[Crossref]

1985 (1)

R. L. Siddon, “Fast calculation of the exact radiological path for a three-dimensional CT array,” Med. Phys. 12, 252–255 (1985).
[Crossref] [PubMed]

1979 (1)

G. H. Golub, M. Heath, and G. Wahba, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
[Crossref]

1970 (1)

R. Gordon, R. Bender, and G. T. Herman, “Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and X-ray photography,” J. Theor. Biol. 29, 471–481 (1970).
[Crossref] [PubMed]

1966 (1)

V. A. Morozov, “On the solution of functional equations by the method of regularization,” “Soviet Math. Dokl,”,  7, pp. 414–417 (1966.

1937 (1)

S. Kaczmarz, “Angenäherte Auflösung von Systemen Linearer Gleichungen,” Bulletin International de l’Academie Polonaise des Sciences et des Lettres 35, 355–357 (1937).

Altapova, V.

J. Moosmann, A. Ershov, V. Altapova, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, and R. Hofmann, “X-ray phase-contrast in vivo microtomography probes new aspects of Xenopus gastrulation,” Nature 497, 374–377 (2013).
[Crossref] [PubMed]

Badea, C. T.

J. Song, Q. 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] [PubMed]

Batenburg, K. J.

Baumbach, T.

T. dos Santos Rolo, A. Ershov, T. van de Kamp, and T. Baumbach, “In vivo X-ray cine-tomography for tracking morphological dynamics,” Proc. Natl. Acad. Sci. USA 111, 3921–3926 (2014).
[Crossref] [PubMed]

J. Moosmann, A. Ershov, V. Weinhardt, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, and R. Hofmann, “Time-lapse X-ray phase-contrast microtomography for in vivo imaging and analysis of morphogenesis,” Nat. Protoc. 9, 294–304 (2014).
[Crossref] [PubMed]

J. Moosmann, A. Ershov, V. Altapova, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, and R. Hofmann, “X-ray phase-contrast in vivo microtomography probes new aspects of Xenopus gastrulation,” Nature 497, 374–377 (2013).
[Crossref] [PubMed]

T. van de Kamp, P. Vagovič, T. Baumbach, and A. Riedel, “A biological screw in a beetle’s leg,” Science 333, 52 (2011).
[Crossref]

R. Hofmann, J. Moosmann, and T. Baumbach, “Criticality in single-distance phase retrieval,” Opt. Express 19, 25881–25890 (2011).
[Crossref]

J. Moosmann, R. Hofmann, and T. Baumbach, “Single-distance phase retrieval at large phase shifts,” Opt. Express 19, 12066–12073 (2011).
[Crossref] [PubMed]

R. Hofmann, A. Schober, J. Moosmann, M. Hertel, S. Hahn, V. Weinhardt, D. Hänschke, L. Helfen, X. Xiao, and T. Baumbach, “Single-distance livecell imaging with propagation-based X-ray phase contrast,” to be submitted.

Beck, A.

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Imaging Sci. 2, 183–202 (2009).
[Crossref]

Becker, S.

S. Becker, J. Bobin, and E. Candès, “NESTA: a fast and accurate first-order method for sparse recovery,” SIAM J. Imaging Sci. 4, 1–39 (2011).
[Crossref]

Bender, R.

R. Gordon, R. Bender, and G. T. Herman, “Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and X-ray photography,” J. Theor. Biol. 29, 471–481 (1970).
[Crossref] [PubMed]

Bioucas-Dias, J. M.

J. M. Bioucas-Dias and M. A. T. Figueiredo, “A new TwIST: two-step iterative shrinkage / thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16, 2992–3004 (2007).
[Crossref] [PubMed]

Bobin, J.

S. Becker, J. Bobin, and E. Candès, “NESTA: a fast and accurate first-order method for sparse recovery,” SIAM J. Imaging Sci. 4, 1–39 (2011).
[Crossref]

Bovik, A. C.

Z. Wang and A. C. Bovik, “Mean squared error: love it or leave it? A new look at signal fidelity measures,” IEEE Signal Proc. Mag. 26, 98–117 (2009).
[Crossref]

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[Crossref] [PubMed]

Calvetti, D.

D. Calvetti, S. Morigi, L. Reichel, and F. Sgallari, “Tikhonov regularization and the L-curve for large discrete ill-posed problems,” J. Comput. Appl. Math. 123, 423–446 (2000).
[Crossref]

Candès, E.

S. Becker, J. Bobin, and E. Candès, “NESTA: a fast and accurate first-order method for sparse recovery,” SIAM J. Imaging Sci. 4, 1–39 (2011).
[Crossref]

Candès, E. J.

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory 52, 489–509 (2006).
[Crossref]

Chan, T.

T. Chan, S. Esedoglu, F. Park, and A. Yip, “Recent developments in total variation image restoration,” Mathematical Models of Computer Vision 2005,17 (2005).

Chen, G.

P. T. Lauzier, J. Tang, and G. Chen, “Prior image constrained compressed sensing: Implementation and performance evaluation,” Med. Phys. 39, 66–80 (2012).
[Crossref] [PubMed]

Dai, Y.

Y. Dai and Y. Yuan, “A nonlinear conjugate gradient method with a strong global convergence property,” SIAM J. Optimiz. 10, 177–182 (1999).
[Crossref]

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] [PubMed]

dos Santos Rolo, T.

T. dos Santos Rolo, A. Ershov, T. van de Kamp, and T. Baumbach, “In vivo X-ray cine-tomography for tracking morphological dynamics,” Proc. Natl. Acad. Sci. USA 111, 3921–3926 (2014).
[Crossref] [PubMed]

X. Yang, T. Jejkal, H. Pasic, R. Stotzka, A. Streit, J. van Wezel, and T. dos Santos Rolo, “Data intensive computing of X-ray computed tomography reconstruction at the LSDF,” in “21st Euromicro International Conference on Parallel, Distributed and Network-Based Processing (PDP),” (IEEE, 2013), pp. 86–93.

Ershov, A.

T. dos Santos Rolo, A. Ershov, T. van de Kamp, and T. Baumbach, “In vivo X-ray cine-tomography for tracking morphological dynamics,” Proc. Natl. Acad. Sci. USA 111, 3921–3926 (2014).
[Crossref] [PubMed]

J. Moosmann, A. Ershov, V. Weinhardt, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, and R. Hofmann, “Time-lapse X-ray phase-contrast microtomography for in vivo imaging and analysis of morphogenesis,” Nat. Protoc. 9, 294–304 (2014).
[Crossref] [PubMed]

J. Moosmann, A. Ershov, V. Altapova, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, and R. Hofmann, “X-ray phase-contrast in vivo microtomography probes new aspects of Xenopus gastrulation,” Nature 497, 374–377 (2013).
[Crossref] [PubMed]

Esedoglu, S.

T. Chan, S. Esedoglu, F. Park, and A. Yip, “Recent developments in total variation image restoration,” Mathematical Models of Computer Vision 2005,17 (2005).

Esser, E.

E. Esser, “Applications of Lagrangian-based alternating direction methods and connections to split Bregman,” CAM Report 9, 31 (2009).

Figueiredo, M. A. T.

J. M. Bioucas-Dias and M. A. T. Figueiredo, “A new TwIST: two-step iterative shrinkage / thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16, 2992–3004 (2007).
[Crossref] [PubMed]

Fukushima, M.

M. Fukushima, “Application of the alternating direction method of multipliers to separable convex programming problems,” Comput. Optim. Appl. 1, 93–111 (1992).
[Crossref]

Goldstein, T.

T. Goldstein and S. Osher, “The split Bregman method for L1-regularized problems,” SIAM J. Imaging Sci. 2, 323–343 (2009).
[Crossref]

T. Goldstein, B. O’Donoghue, and S. Setzer, “Fast alternating direction optimization methods,” CAM report pp. 12–35 (2012).

Golub, G. H.

G. H. Golub, M. Heath, and G. Wahba, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
[Crossref]

Gordon, R.

R. Gordon, R. Bender, and G. T. Herman, “Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and X-ray photography,” J. Theor. Biol. 29, 471–481 (1970).
[Crossref] [PubMed]

Hager, W. W.

W. W. Hager and H. Zhang, “A survey of nonlinear conjugate gradient methods,” Pac. J. Optim. 2, 35–58 (2006).

Hahn, S.

R. Hofmann, A. Schober, J. Moosmann, M. Hertel, S. Hahn, V. Weinhardt, D. Hänschke, L. Helfen, X. Xiao, and T. Baumbach, “Single-distance livecell imaging with propagation-based X-ray phase contrast,” to be submitted.

Hänschke, D.

R. Hofmann, A. Schober, J. Moosmann, M. Hertel, S. Hahn, V. Weinhardt, D. Hänschke, L. Helfen, X. Xiao, and T. Baumbach, “Single-distance livecell imaging with propagation-based X-ray phase contrast,” to be submitted.

Hansen, P. C.

T. L. Jensen, J. H. Jørgensen, P. C. Hansen, and S. H. Jensen, “Implementation of an optimal first-order method for strongly convex total variation regularization,” BIT 52, 329–356 (2012).
[Crossref]

P. C. Hansen and D. P. O’Leary, “The use of the L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14, 1487–1503 (1993).
[Crossref]

Heath, M.

G. H. Golub, M. Heath, and G. Wahba, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
[Crossref]

Helfen, L.

R. Hofmann, A. Schober, J. Moosmann, M. Hertel, S. Hahn, V. Weinhardt, D. Hänschke, L. Helfen, X. Xiao, and T. Baumbach, “Single-distance livecell imaging with propagation-based X-ray phase contrast,” to be submitted.

Herman, G. T.

R. Gordon, R. Bender, and G. T. Herman, “Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and X-ray photography,” J. Theor. Biol. 29, 471–481 (1970).
[Crossref] [PubMed]

Hertel, M.

R. Hofmann, A. Schober, J. Moosmann, M. Hertel, S. Hahn, V. Weinhardt, D. Hänschke, L. Helfen, X. Xiao, and T. Baumbach, “Single-distance livecell imaging with propagation-based X-ray phase contrast,” to be submitted.

Hofmann, R.

J. Moosmann, A. Ershov, V. Weinhardt, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, and R. Hofmann, “Time-lapse X-ray phase-contrast microtomography for in vivo imaging and analysis of morphogenesis,” Nat. Protoc. 9, 294–304 (2014).
[Crossref] [PubMed]

J. Moosmann, A. Ershov, V. Altapova, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, and R. Hofmann, “X-ray phase-contrast in vivo microtomography probes new aspects of Xenopus gastrulation,” Nature 497, 374–377 (2013).
[Crossref] [PubMed]

R. Hofmann, J. Moosmann, and T. Baumbach, “Criticality in single-distance phase retrieval,” Opt. Express 19, 25881–25890 (2011).
[Crossref]

J. Moosmann, R. Hofmann, and T. Baumbach, “Single-distance phase retrieval at large phase shifts,” Opt. Express 19, 12066–12073 (2011).
[Crossref] [PubMed]

R. Hofmann, A. Schober, J. Moosmann, M. Hertel, S. Hahn, V. Weinhardt, D. Hänschke, L. Helfen, X. Xiao, and T. Baumbach, “Single-distance livecell imaging with propagation-based X-ray phase contrast,” to be submitted.

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G. N. Hounsfield, “A method and apparatus for examination of a body by radiation such as X-ray or gamma radiation,” (1972). Patent Specification 1283915.

Hu, Y.

Y. Hu and M. Jacob, “Higher degree total variation (HDTV) regularization for image recovery,” IEEE Trans. Image Process. 21, 2559–2571 (2012).
[Crossref] [PubMed]

Jacob, M.

Y. Hu and M. Jacob, “Higher degree total variation (HDTV) regularization for image recovery,” IEEE Trans. Image Process. 21, 2559–2571 (2012).
[Crossref] [PubMed]

Jejkal, T.

X. Yang, T. Jejkal, H. Pasic, R. Stotzka, A. Streit, J. van Wezel, and T. dos Santos Rolo, “Data intensive computing of X-ray computed tomography reconstruction at the LSDF,” in “21st Euromicro International Conference on Parallel, Distributed and Network-Based Processing (PDP),” (IEEE, 2013), pp. 86–93.

Jensen, S. H.

T. L. Jensen, J. H. Jørgensen, P. C. Hansen, and S. H. Jensen, “Implementation of an optimal first-order method for strongly convex total variation regularization,” BIT 52, 329–356 (2012).
[Crossref]

Jensen, T. L.

T. L. Jensen, J. H. Jørgensen, P. C. Hansen, and S. H. Jensen, “Implementation of an optimal first-order method for strongly convex total variation regularization,” BIT 52, 329–356 (2012).
[Crossref]

Johnson, G. A.

J. Song, Q. 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] [PubMed]

Jørgensen, J. H.

T. L. Jensen, J. H. Jørgensen, P. C. Hansen, and S. H. Jensen, “Implementation of an optimal first-order method for strongly convex total variation regularization,” BIT 52, 329–356 (2012).
[Crossref]

Kaczmarz, S.

S. Kaczmarz, “Angenäherte Auflösung von Systemen Linearer Gleichungen,” Bulletin International de l’Academie Polonaise des Sciences et des Lettres 35, 355–357 (1937).

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A. C. Kak and M. Slaney, Principles of computerized tomographic imaging (Society for Industrial and Applied Mathematics, 2001).
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Kao, C.

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

Kashef, J.

J. Moosmann, A. Ershov, V. Weinhardt, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, and R. Hofmann, “Time-lapse X-ray phase-contrast microtomography for in vivo imaging and analysis of morphogenesis,” Nat. Protoc. 9, 294–304 (2014).
[Crossref] [PubMed]

J. Moosmann, A. Ershov, V. Altapova, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, and R. Hofmann, “X-ray phase-contrast in vivo microtomography probes new aspects of Xenopus gastrulation,” Nature 497, 374–377 (2013).
[Crossref] [PubMed]

Kostenko, A.

LaBonne, C.

J. Moosmann, A. Ershov, V. Weinhardt, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, and R. Hofmann, “Time-lapse X-ray phase-contrast microtomography for in vivo imaging and analysis of morphogenesis,” Nat. Protoc. 9, 294–304 (2014).
[Crossref] [PubMed]

J. Moosmann, A. Ershov, V. Altapova, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, and R. Hofmann, “X-ray phase-contrast in vivo microtomography probes new aspects of Xenopus gastrulation,” Nature 497, 374–377 (2013).
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P. T. Lauzier, J. Tang, and G. Chen, “Prior image constrained compressed sensing: Implementation and performance evaluation,” Med. Phys. 39, 66–80 (2012).
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C. Li, Compressive Sensing for 3D Data Processing Tasks: Applications, Models and Algorithms, (PhD Thesis in Rice University, 2011).

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Liu, Q.

J. Song, Q. 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).
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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).
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Moosmann, J.

J. Moosmann, A. Ershov, V. Weinhardt, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, and R. Hofmann, “Time-lapse X-ray phase-contrast microtomography for in vivo imaging and analysis of morphogenesis,” Nat. Protoc. 9, 294–304 (2014).
[Crossref] [PubMed]

J. Moosmann, A. Ershov, V. Altapova, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, and R. Hofmann, “X-ray phase-contrast in vivo microtomography probes new aspects of Xenopus gastrulation,” Nature 497, 374–377 (2013).
[Crossref] [PubMed]

J. Moosmann, R. Hofmann, and T. Baumbach, “Single-distance phase retrieval at large phase shifts,” Opt. Express 19, 12066–12073 (2011).
[Crossref] [PubMed]

R. Hofmann, J. Moosmann, and T. Baumbach, “Criticality in single-distance phase retrieval,” Opt. Express 19, 25881–25890 (2011).
[Crossref]

R. Hofmann, A. Schober, J. Moosmann, M. Hertel, S. Hahn, V. Weinhardt, D. Hänschke, L. Helfen, X. Xiao, and T. Baumbach, “Single-distance livecell imaging with propagation-based X-ray phase contrast,” to be submitted.

Morigi, S.

D. Calvetti, S. Morigi, L. Reichel, and F. Sgallari, “Tikhonov regularization and the L-curve for large discrete ill-posed problems,” J. Comput. Appl. Math. 123, 423–446 (2000).
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P. C. Hansen and D. P. O’Leary, “The use of the L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14, 1487–1503 (1993).
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T. Goldstein and S. Osher, “The split Bregman method for L1-regularized problems,” SIAM J. Imaging Sci. 2, 323–343 (2009).
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E. Y. Sidky and X. Pan, “Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization,” Phys. Med. Biol. 53, 4777–4807 (2008).
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E. Y. Sidky, C. Kao, and X. Pan, “Accurate image reconstruction from few-views and limited-angle data in divergent-beam CT,” J. X-ray Sci. Technol. 14, 119–139 (2006).

Park, F.

T. Chan, S. Esedoglu, F. Park, and A. Yip, “Recent developments in total variation image restoration,” Mathematical Models of Computer Vision 2005,17 (2005).

Pasic, H.

X. Yang, T. Jejkal, H. Pasic, R. Stotzka, A. Streit, J. van Wezel, and T. dos Santos Rolo, “Data intensive computing of X-ray computed tomography reconstruction at the LSDF,” in “21st Euromicro International Conference on Parallel, Distributed and Network-Based Processing (PDP),” (IEEE, 2013), pp. 86–93.

Pauly, J. M.

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] [PubMed]

Prasad, M. S.

J. Moosmann, A. Ershov, V. Weinhardt, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, and R. Hofmann, “Time-lapse X-ray phase-contrast microtomography for in vivo imaging and analysis of morphogenesis,” Nat. Protoc. 9, 294–304 (2014).
[Crossref] [PubMed]

J. Moosmann, A. Ershov, V. Altapova, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, and R. Hofmann, “X-ray phase-contrast in vivo microtomography probes new aspects of Xenopus gastrulation,” Nature 497, 374–377 (2013).
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L. Reichel and G. Rodriguez, “Old and new parameter choice rules for discrete ill-posed problems,” Numer. Algorithms 63, 65–87 (2013).
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D. Calvetti, S. Morigi, L. Reichel, and F. Sgallari, “Tikhonov regularization and the L-curve for large discrete ill-posed problems,” J. Comput. Appl. Math. 123, 423–446 (2000).
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Riedel, A.

T. van de Kamp, P. Vagovič, T. Baumbach, and A. Riedel, “A biological screw in a beetle’s leg,” Science 333, 52 (2011).
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L. Reichel and G. Rodriguez, “Old and new parameter choice rules for discrete ill-posed problems,” Numer. Algorithms 63, 65–87 (2013).
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E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory 52, 489–509 (2006).
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Schober, A.

R. Hofmann, A. Schober, J. Moosmann, M. Hertel, S. Hahn, V. Weinhardt, D. Hänschke, L. Helfen, X. Xiao, and T. Baumbach, “Single-distance livecell imaging with propagation-based X-ray phase contrast,” to be submitted.

Setzer, S.

T. Goldstein, B. O’Donoghue, and S. Setzer, “Fast alternating direction optimization methods,” CAM report pp. 12–35 (2012).

Sgallari, F.

D. Calvetti, S. Morigi, L. Reichel, and F. Sgallari, “Tikhonov regularization and the L-curve for large discrete ill-posed problems,” J. Comput. Appl. Math. 123, 423–446 (2000).
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Sheikh, H. R.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
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R. L. Siddon, “Fast calculation of the exact radiological path for a three-dimensional CT array,” Med. Phys. 12, 252–255 (1985).
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Sidky, E. Y.

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

Simoncelli, E. P.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[Crossref] [PubMed]

Slaney, M.

A. C. Kak and M. Slaney, Principles of computerized tomographic imaging (Society for Industrial and Applied Mathematics, 2001).
[Crossref]

Song, J.

J. Song, Q. 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] [PubMed]

Stotzka, R.

X. Yang, T. Jejkal, H. Pasic, R. Stotzka, A. Streit, J. van Wezel, and T. dos Santos Rolo, “Data intensive computing of X-ray computed tomography reconstruction at the LSDF,” in “21st Euromicro International Conference on Parallel, Distributed and Network-Based Processing (PDP),” (IEEE, 2013), pp. 86–93.

Streit, A.

X. Yang, T. Jejkal, H. Pasic, R. Stotzka, A. Streit, J. van Wezel, and T. dos Santos Rolo, “Data intensive computing of X-ray computed tomography reconstruction at the LSDF,” in “21st Euromicro International Conference on Parallel, Distributed and Network-Based Processing (PDP),” (IEEE, 2013), pp. 86–93.

Suhonen, H.

Tai, X.

X. Tai and C. Wu, “Augmented Lagrangian method, dual methods and split Bregman iteration for ROF model,” in “Scale Space and Variational Methods in Computer Vision,” (Springer, 2009), pp. 502–513.
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Tang, J.

P. T. Lauzier, J. Tang, and G. Chen, “Prior image constrained compressed sensing: Implementation and performance evaluation,” Med. Phys. 39, 66–80 (2012).
[Crossref] [PubMed]

Tao, T.

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory 52, 489–509 (2006).
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A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Imaging Sci. 2, 183–202 (2009).
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T. van de Kamp, P. Vagovič, T. Baumbach, and A. Riedel, “A biological screw in a beetle’s leg,” Science 333, 52 (2011).
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van de Kamp, T.

T. dos Santos Rolo, A. Ershov, T. van de Kamp, and T. Baumbach, “In vivo X-ray cine-tomography for tracking morphological dynamics,” Proc. Natl. Acad. Sci. USA 111, 3921–3926 (2014).
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T. van de Kamp, P. Vagovič, T. Baumbach, and A. Riedel, “A biological screw in a beetle’s leg,” Science 333, 52 (2011).
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van Wezel, J.

X. Yang, T. Jejkal, H. Pasic, R. Stotzka, A. Streit, J. van Wezel, and T. dos Santos Rolo, “Data intensive computing of X-ray computed tomography reconstruction at the LSDF,” in “21st Euromicro International Conference on Parallel, Distributed and Network-Based Processing (PDP),” (IEEE, 2013), pp. 86–93.

Vogel, C. R.

C. R. Vogel, “Non-convergence of the L-curve regularization parameter selection method,” Inverse Probl. 12, 535 (1996).
[Crossref]

Wahba, G.

G. H. Golub, M. Heath, and G. Wahba, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
[Crossref]

Wang, Y.

Y. Wang, J. Yang, W. Yin, and Y. Zhang, “A new alternating minimization algorithm for total variation image reconstruction,” SIAM J. Imaging Sci. 1, 248–272 (2008).
[Crossref]

Wang, Z.

Z. Wang and A. C. Bovik, “Mean squared error: love it or leave it? A new look at signal fidelity measures,” IEEE Signal Proc. Mag. 26, 98–117 (2009).
[Crossref]

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[Crossref] [PubMed]

Weinhardt, V.

J. Moosmann, A. Ershov, V. Weinhardt, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, and R. Hofmann, “Time-lapse X-ray phase-contrast microtomography for in vivo imaging and analysis of morphogenesis,” Nat. Protoc. 9, 294–304 (2014).
[Crossref] [PubMed]

R. Hofmann, A. Schober, J. Moosmann, M. Hertel, S. Hahn, V. Weinhardt, D. Hänschke, L. Helfen, X. Xiao, and T. Baumbach, “Single-distance livecell imaging with propagation-based X-ray phase contrast,” to be submitted.

Wu, C.

X. Tai and C. Wu, “Augmented Lagrangian method, dual methods and split Bregman iteration for ROF model,” in “Scale Space and Variational Methods in Computer Vision,” (Springer, 2009), pp. 502–513.
[Crossref]

Xiao, X.

J. Moosmann, A. Ershov, V. Weinhardt, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, and R. Hofmann, “Time-lapse X-ray phase-contrast microtomography for in vivo imaging and analysis of morphogenesis,” Nat. Protoc. 9, 294–304 (2014).
[Crossref] [PubMed]

J. Moosmann, A. Ershov, V. Altapova, T. Baumbach, M. S. Prasad, C. LaBonne, X. Xiao, J. Kashef, and R. Hofmann, “X-ray phase-contrast in vivo microtomography probes new aspects of Xenopus gastrulation,” Nature 497, 374–377 (2013).
[Crossref] [PubMed]

R. Hofmann, A. Schober, J. Moosmann, M. Hertel, S. Hahn, V. Weinhardt, D. Hänschke, L. Helfen, X. Xiao, and T. Baumbach, “Single-distance livecell imaging with propagation-based X-ray phase contrast,” to be submitted.

Yang, J.

J. Yang, Y. Zhang, and W. Yin, “A fast alternating direction method for TVL1-L2 signal reconstruction from partial Fourier data,” IEEE J. Sel. Top. Signa. 4, 288–297 (2010).
[Crossref]

Y. Wang, J. Yang, W. Yin, and Y. Zhang, “A new alternating minimization algorithm for total variation image reconstruction,” SIAM J. Imaging Sci. 1, 248–272 (2008).
[Crossref]

Yang, X.

X. Yang, T. Jejkal, H. Pasic, R. Stotzka, A. Streit, J. van Wezel, and T. dos Santos Rolo, “Data intensive computing of X-ray computed tomography reconstruction at the LSDF,” in “21st Euromicro International Conference on Parallel, Distributed and Network-Based Processing (PDP),” (IEEE, 2013), pp. 86–93.

Yin, W.

J. Yang, Y. Zhang, and W. Yin, “A fast alternating direction method for TVL1-L2 signal reconstruction from partial Fourier data,” IEEE J. Sel. Top. Signa. 4, 288–297 (2010).
[Crossref]

Y. Wang, J. Yang, W. Yin, and Y. Zhang, “A new alternating minimization algorithm for total variation image reconstruction,” SIAM J. Imaging Sci. 1, 248–272 (2008).
[Crossref]

C. Li, W. Yin, and Y. Zhang, “User’s guide for TVAL3: TV minimization by augmented lagrangian and alternating direction algorithms,” CAAM Report (2009).

Yip, A.

T. Chan, S. Esedoglu, F. Park, and A. Yip, “Recent developments in total variation image restoration,” Mathematical Models of Computer Vision 2005,17 (2005).

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Y. Dai and Y. Yuan, “A nonlinear conjugate gradient method with a strong global convergence property,” SIAM J. Optimiz. 10, 177–182 (1999).
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W. W. Hager and H. Zhang, “A survey of nonlinear conjugate gradient methods,” Pac. J. Optim. 2, 35–58 (2006).

Zhang, Y.

J. Yang, Y. Zhang, and W. Yin, “A fast alternating direction method for TVL1-L2 signal reconstruction from partial Fourier data,” IEEE J. Sel. Top. Signa. 4, 288–297 (2010).
[Crossref]

Y. Wang, J. Yang, W. Yin, and Y. Zhang, “A new alternating minimization algorithm for total variation image reconstruction,” SIAM J. Imaging Sci. 1, 248–272 (2008).
[Crossref]

C. Li, W. Yin, and Y. Zhang, “User’s guide for TVAL3: TV minimization by augmented lagrangian and alternating direction algorithms,” CAAM Report (2009).

BIT (1)

T. L. Jensen, J. H. Jørgensen, P. C. Hansen, and S. H. Jensen, “Implementation of an optimal first-order method for strongly convex total variation regularization,” BIT 52, 329–356 (2012).
[Crossref]

Bulletin International de l’Academie Polonaise des Sciences et des Lettres (1)

S. Kaczmarz, “Angenäherte Auflösung von Systemen Linearer Gleichungen,” Bulletin International de l’Academie Polonaise des Sciences et des Lettres 35, 355–357 (1937).

CAM Report (1)

E. Esser, “Applications of Lagrangian-based alternating direction methods and connections to split Bregman,” CAM Report 9, 31 (2009).

Comput. Optim. Appl. (1)

M. Fukushima, “Application of the alternating direction method of multipliers to separable convex programming problems,” Comput. Optim. Appl. 1, 93–111 (1992).
[Crossref]

IEEE J. Sel. Top. Signa. (1)

J. Yang, Y. Zhang, and W. Yin, “A fast alternating direction method for TVL1-L2 signal reconstruction from partial Fourier data,” IEEE J. Sel. Top. Signa. 4, 288–297 (2010).
[Crossref]

IEEE Signal Proc. Mag. (1)

Z. Wang and A. C. Bovik, “Mean squared error: love it or leave it? A new look at signal fidelity measures,” IEEE Signal Proc. Mag. 26, 98–117 (2009).
[Crossref]

IEEE Trans. Image Process. (3)

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[Crossref] [PubMed]

Y. Hu and M. Jacob, “Higher degree total variation (HDTV) regularization for image recovery,” IEEE Trans. Image Process. 21, 2559–2571 (2012).
[Crossref] [PubMed]

J. M. Bioucas-Dias and M. A. T. Figueiredo, “A new TwIST: two-step iterative shrinkage / thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16, 2992–3004 (2007).
[Crossref] [PubMed]

IEEE Trans. Inform. Theory (1)

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inform. Theory 52, 489–509 (2006).
[Crossref]

Inverse Probl. (1)

C. R. Vogel, “Non-convergence of the L-curve regularization parameter selection method,” Inverse Probl. 12, 535 (1996).
[Crossref]

J. Comput. Appl. Math. (1)

D. Calvetti, S. Morigi, L. Reichel, and F. Sgallari, “Tikhonov regularization and the L-curve for large discrete ill-posed problems,” J. Comput. Appl. Math. 123, 423–446 (2000).
[Crossref]

J. Theor. Biol. (1)

R. Gordon, R. Bender, and G. T. Herman, “Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and X-ray photography,” J. Theor. Biol. 29, 471–481 (1970).
[Crossref] [PubMed]

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

E. Y. Sidky, C. Kao, and X. Pan, “Accurate image reconstruction from few-views and limited-angle data in divergent-beam CT,” J. X-ray Sci. Technol. 14, 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] [PubMed]

Mathematical Models of Computer Vision (1)

T. Chan, S. Esedoglu, F. Park, and A. Yip, “Recent developments in total variation image restoration,” Mathematical Models of Computer Vision 2005,17 (2005).

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Figures (17)

Fig. 1
Fig. 1 In vivo X-ray tomography experiment with propagation-based contrast. Left panel: experimental setup showing (1) the X-ray source (here: bending magnet), (2) tomographic stage, (3) sample, and (4) detector system. Right panel: parallel-beam projection of attenuation index through 2D slice of object (forward model), for explanations of labels see text.
Fig. 2
Fig. 2 Upper row: reconstruction results for Shepp-Logan phantom (256 × 256) from 60 projections (402 projections required in FBP reconstruction to meet pixel resolution) using the CGTV solver subject to various values of λ. Lower row: Cuts along horizontal through respective images of upper row (marked by red lines). Corresponding cuts through the ground truth are blue lines.
Fig. 3
Fig. 3 Discrete L-curve for the reconstruction of the Shepp-Logan phantom (256 × 256) from 60 projections with respect to λ = 0, 0.001, 0.005, 0.01, 0.05, 0.1, 0.5, 1, 2, 4, 8, 16, 32, 64 (for information preserving FBP reconstruction 402 projections would be required). Green arrow indicates the optimal point on the L-curve, corresponding to λ = 2.
Fig. 4
Fig. 4 Strategy of volume reconstruction using the optimised CGTV in the case of parallel-beam imaging.
Fig. 5
Fig. 5 MSE in (a) and SSIM in (b) upon evaluating the CGTV reconstruction of the Shepp-Logan phantom with fourteen different values of λ. Green arrows indicate the optimal value λ = 2 as determined from the discrete L-curve in Fig. 3.
Fig. 6
Fig. 6 (a) original Barbara (256 × 256) and reconstruction results from 120 projections using the CGTV solver with (b) λ = 0.001, (c) λ = 4, and (d) λ = 64.
Fig. 7
Fig. 7 Discrete L-curve for the reconstruction of the image Fig. 6(a) (256 × 256) from 120 projections (for information preserving FBP reconstruction 402 projections would be required). Values of λ, which are employed, read λ = 0, 0.001, 0.005, 0.01, 0.05, 0.1, 0.5, 1, 2, 4, 8, 16, 32, 64. According to the criterion of shortest distance to the origin, the optimal value is λ = 4, indicated by the green arrow.
Fig. 8
Fig. 8 True image and image reconstructions using FBP and CGTV for the Shepp-Logan phantom (first row, 60 projections) and Barbara (second row, 120 projections).
Fig. 9
Fig. 9 Reference image and λ dependent reconstruction results from intensity data of weevil subject to different levels of Poisson noise and 80 projections: (a) FBP reconstruction using full set of low-noise projections; (b) and (c) CGTV reconstructions for a small value of λ = 0.001 for 0.5 % and 3 % noise data, respectively; (d) and (e) respective CGTV reconstructions for λ = 0.5 which is optimal for both, 0.5 % and 3 % noise data; (f) CGTV reconstruction using λ = 4 for 0.5 % noise data.
Fig. 10
Fig. 10 Discrete L-curves for CGTV reconstruction from 80 projections of the weevil data with 0.5 % (blue curve) and 3 % (lime green curve) Poisson noise using λ = 0, 0.001, 0.005, 0.01, 0.05, 0.1, 0.5, 1, 2, 4, 8, 16, 32, 64. Both curves exhibit minimum distance to the origin at the same value λ = 0.5.
Fig. 11
Fig. 11 Reference image and λ dependent reconstruction results from phase maps retrieved from intensity projections of stage-17 frog embryo: (a) FBP reconstruction using full set of projections; (b)–(d) CGTV reconstructions using 167 projections for different values of λ.
Fig. 12
Fig. 12 Discrete L-curve for CGTV reconstruction from 167 phase maps of stage-17 frog embryo using λ = 0, 0.001, 0.005, 0.01, 0.05, 0.1, 0.5, 1, 2, 4, 8, 16, 32, 64.
Fig. 13
Fig. 13 Comparison of FBP and small-λ CGTV reconstruction of weevil and frog-embryo slices (λ = 0.001 in both cases) using a small number of projections, P = 80 and P = 167 (as opposed to maximum numbers P = 400 and P = 499), respectively. Note that streakline artifacts appear for both FBP and small-λ CGTV reconstruction.
Fig. 14
Fig. 14 (a) Definition of line through reconstructed slice of stage-17 frog embryo, (b) reconstruction results (normalised to [0, 1]) along this line for FBP subject to the full number of projections (P = 499, reference, red curve) and optimised CGTV using P = 167 (blue curve) and P = 499 (green curve).
Fig. 15
Fig. 15 Error of optimised CGTV with respect to full-view FBP reconstruction (P = 400) of weevil slice using error metrics MSE and SSIM, defined in Sec. 3.1, versus the number of projections P. Green arrows mark approximate onset of saturation.
Fig. 16
Fig. 16 Error of optimised CGTV with respect to full-view FBP reconstruction (P = 499) of stage-17 frog-embryo slice using error metrics MSE and SSIM, defined in Sec. 3.1, versus the number of projections P. Green arrows mark approximate onset of saturation.
Fig. 17
Fig. 17 Convergence analysis of CGTV reconstruction of weevil data for five different values of λ. The optimal value is λ* = 0.5 denoted in red, and in its vicinity minimisation saturates for a number of iterations k > 30.

Tables (2)

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Algorithm 1 TV-based conjugate gradient method (CGTV)

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Table 1 IQA measures for reconstructions using FBP and optimised CGTV subject to low numbers of projections through the Shepp-Logan and Barbara phantoms.

Equations (14)

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P FBP = π 2 K .
x * = arg min x x TV , subject to Ax = p ,
x * = arg min x Ax p 2 2 + λ x TV .
DGT 11 , i j ( X ) | Δ i , j h X | + | Δ i , j v X |
DGT 12 , i j ( X ) [ ( Δ i , j h X ) 2 + ( Δ i , j v X ) 2 ] 1 / 2 ,
Δ i , j h X X i , j X i 1 , j
Δ i , j v X X i , j X i , j 1 ,
x 11 , TV i , j DGT 11 , i j ( X )
x 12 , TV i , j DGT 12 , i j ( X ) .
F ( x ) = ( Ax p 2 2 ) = 2 A T ( Ax p ) .
| z | ( z 2 + ε ) 1 / 2 ,
x TV X i , j = 2 X i , j X i 1 , j X i , j 1 [ ( X i , j X i 1 , j ) 2 + ( X i , j X i , j 1 ) 2 + ε ] 1 / 2 + X i , j X i + 1 , j [ ( X i + 1 , j X i , j ) 2 + ( X i + 1 , j X i + 1 , j 1 ) 2 + ε ] 1 / 2 + X i , j X i , j + 1 [ ( X i , j + 1 X i 1 , j + 1 ) 2 + ( X i , j + 1 X i , j ) 2 + ε ] 1 / 2 .
MSE ( X , Y ) = 1 n 2 i , j = 1 n ( X i , j Y i , j ) 2 .
SSIM ( X , Y ) = l ( X , Y ) α × c ( X , Y ) β × s ( X , Y ) γ ,

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