Abstract

Grating-based X-ray phase-contrast (gbPC) is an X-ray phase-contrast imaging method involving optical gratings that typically employs the Talbot self-imaging effect. X-ray phase contrast is known to provide significant benefits for biomedical imaging. To investigate these benefits for gbPC, a high-sensitivity gbPC micro-CT setup for small biological samples has been constructed. A gbPC projection measurement simultaneously retrieves the transmittance, differential-phase and dark-field modalities of a sample. Phase stepping, the most common gbPC acquisition technique, involves several acquisitions at different lateral positions of one of the gratings. The three modalities can then be retrieved by least-squares- or FFT-based methods. Unfortunately, increasing differential-phase sensitivity also leads to an increased magnitude of artifacts introduced during retrieval of the modalities from the phase-stepping data, which limits image quality. Most importantly, processing of phase-stepping data with incorrect stepping positions (i.e., spatial sampling jitter) can introduce artifacts to the modalities. Using data from the high-sensitivity gbPC setup, as well as simulations, we show that an artifact is introduced by the jitter which is correlated with the phase of the stepping curve. We present a theoretical explanation for this correlation by introducing small deviations to an equidistant sampling of a stepping curve and approximating the effect on the calculation of the three gbPC modalities with a first-order Taylor approximation. Finally, we present an algorithm for the detection and removal of these artifacts that exploits these correlations. We show that this algorithm is able to eliminate these artifacts without degrading true image information.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2017 (2)

Y. Zhang, X. Tian, and R. Liang, “Fringe-print-through error analysis and correction in snapshot phase-shifting interference microscope,” Opt. Express 25, 26554–26566 (2017).
[Crossref] [PubMed]

S. Kaeppler, J. Rieger, G. Pelzer, F. Horn, T. Michel, A. Maier, G. Anton, and C. Riess, “Improved reconstruction of phase-stepping data for Talbot–Lau x-ray imaging,” J. Med. Imaging 4, 034005 (2017).
[Crossref]

2016 (3)

M. Seifert, S. Kaeppler, C. Hauke, F. Horn, G. Pelzer, J. Rieger, T. Michel, C. Riess, and G. Anton, “Optimisation of image reconstruction for phase-contrast x-ray Talbot-Lau imaging with regard to mechanical robustness,” Phys. Med. Biol. 61, 6441–6464 (2016).
[Crossref] [PubMed]

M. Marschner, M. Willner, G. Potdevin, A. Fehringer, P. B. Noël, F. Pfeiffer, and J. Herzen, “Helical X-ray phase-contrast computed tomography without phase stepping,” Sci. Rep. 6, 23953 (2016).
[Crossref] [PubMed]

L. Birnbacher, M. Willner, A. Velroyen, M. Marschner, A. Hipp, J. Meiser, F. Koch, T. Schröter, D. Kunka, J. Mohr, F. Pfeiffer, and J. Herzen, “Experimental Realisation of High-sensitivity Laboratory X-ray Grating-based Phase-contrast Computed Tomography,” Sci. Rep. 6, 24022 (2016).
[Crossref] [PubMed]

2015 (1)

G. Pelzer, J. Rieger, C. Hauke, F. Horn, T. Michel, M. Seifert, and G. Anton, “Reconstruction method for grating-based x-ray phase-contrast images without knowledge of the grating positions,” J. Instrum. 10, P12017 (2015).
[Crossref]

2014 (5)

H. Hetterich, M. Willner, S. Fill, J. Herzen, F. Bamberg, A. Hipp, U. Schüller, S. Adam-Neumair, S. Wirth, M. Reiser, F. Pfeiffer, and T. Saam, “Phase-contrast CT: qualitative and quantitative evaluation of atherosclerotic carotid artery plaque,” Radiology 271, 870–878 (2014).
[Crossref] [PubMed]

S. Grandl, M. Willner, J. Herzen, A. Sztrókay-Gaul, D. Mayr, S. D. Auweter, A. Hipp, L. Birnbacher, M. Marschner, M. Chabior, M. Reiser, F. Pfeiffer, F. Bamberg, and K. Hellerhoff, “Visualizing typical features of breast fibroadenomas using phase-contrast CT: An ex-vivo study,” PLOS ONE 9, e97101 (2014).
[Crossref] [PubMed]

A. A. Fingerle, M. Willner, J. Herzen, D. Münzel, D. Hahn, E. J. Rummeny, P. B. Noël, and F. Pfeiffer, “Simulated Cystic Renal Lesions: Quantitative X-ray Phase-Contrast CT – An in Vitro Phantom Study,” Radiology 272, 739–748 (2014).
[Crossref] [PubMed]

A. Malecki, G. Potdevin, T. Biernath, E. Eggl, K. Willer, T. Lasser, J. Maisenbacher, J. Gibmeier, A. Wanner, and F. Pfeiffer, “X-ray tensor tomography,” EPL 105, 38002 (2014).
[Crossref]

M. Strobl, “General solution for quantitative dark-field contrast imaging with grating interferometers,” Sci. Rep. 4, 7243 (2014).
[Crossref] [PubMed]

2013 (4)

A. Bravin, P. Coan, and P. Suortti, “X-ray phase-contrast imaging: from pre-clinical applications towards clinics,” Phys. Med. Biol. 58, R1–R35 (2013).
[Crossref]

A. Malecki, G. Potdevin, T. Biernath, E. Eggl, E. Grande Garcia, T. Baum, P. B. Noël, J. S. Bauer, and F. Pfeiffer, “Coherent Superposition in Grating-Based Directional Dark-Field Imaging,” PLOS ONE 8, e61218 (2013).
[Crossref]

J. Vargas, C. O. S. Sorzano, J. C. Estrada, and J. M. Carazo, “Generalization of the principal component analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
[Crossref]

H. Wen, H. Miao, E. E. Bennett, N. M. Adamo, and L. Chen, “Flexible Retrospective Phase Stepping in X-ray Scatter Correction and Phase Contrast Imaging Using Structured Illumination,” PLOS ONE 8, e78276 (2013).
[Crossref] [PubMed]

2011 (2)

2010 (3)

W. Yashiro, Y. Terui, K. Kawabata, and A. Momose, “On the origin of visibility contrast in x-ray Talbot interferometry,” Opt. Express 18, 16890–16901 (2010).
[Crossref] [PubMed]

M. Bech, O. Bunk, T. Donath, R. Feidenhans’l, C. David, and F. Pfeiffer, “Quantitative x-ray dark-field computed tomography,” Phys. Med. Biol. 55, 5529–5539 (2010).
[Crossref] [PubMed]

V. Revol, C. Kottler, R. Kaufmann, U. Straumann, and C. Urban, “Noise analysis of grating-based x-ray differential phase contrast imaging,” Rev. Sci. Instrum. 81, 073709 (2010).
[Crossref] [PubMed]

2009 (1)

2008 (1)

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
[Crossref] [PubMed]

2007 (1)

F. Pfeiffer, O. Bunk, C. Kottler, and C. David, “Tomographic reconstruction of three-dimensional objects from hard X-ray differential phase contrast projection images,” Nucl. Instrum. Methods Phys. Res. A 580, 925–928 (2007).
[Crossref]

2006 (2)

F. Eng and F. Gustafsson, “Bias compensated least squares estimation of continuous time output error models in the case of stochastic sampling time jitter,” IFAC Proc. Vol.  39, 612–617 (2006).
[Crossref]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2, 258–261 (2006).
[Crossref]

2005 (2)

2004 (1)

2003 (1)

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot Interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003).
[Crossref]

1988 (1)

Adam-Neumair, S.

H. Hetterich, M. Willner, S. Fill, J. Herzen, F. Bamberg, A. Hipp, U. Schüller, S. Adam-Neumair, S. Wirth, M. Reiser, F. Pfeiffer, and T. Saam, “Phase-contrast CT: qualitative and quantitative evaluation of atherosclerotic carotid artery plaque,” Radiology 271, 870–878 (2014).
[Crossref] [PubMed]

Adamo, N. M.

H. Wen, H. Miao, E. E. Bennett, N. M. Adamo, and L. Chen, “Flexible Retrospective Phase Stepping in X-ray Scatter Correction and Phase Contrast Imaging Using Structured Illumination,” PLOS ONE 8, e78276 (2013).
[Crossref] [PubMed]

Anton, G.

S. Kaeppler, J. Rieger, G. Pelzer, F. Horn, T. Michel, A. Maier, G. Anton, and C. Riess, “Improved reconstruction of phase-stepping data for Talbot–Lau x-ray imaging,” J. Med. Imaging 4, 034005 (2017).
[Crossref]

M. Seifert, S. Kaeppler, C. Hauke, F. Horn, G. Pelzer, J. Rieger, T. Michel, C. Riess, and G. Anton, “Optimisation of image reconstruction for phase-contrast x-ray Talbot-Lau imaging with regard to mechanical robustness,” Phys. Med. Biol. 61, 6441–6464 (2016).
[Crossref] [PubMed]

G. Pelzer, J. Rieger, C. Hauke, F. Horn, T. Michel, M. Seifert, and G. Anton, “Reconstruction method for grating-based x-ray phase-contrast images without knowledge of the grating positions,” J. Instrum. 10, P12017 (2015).
[Crossref]

Auweter, S. D.

S. Grandl, M. Willner, J. Herzen, A. Sztrókay-Gaul, D. Mayr, S. D. Auweter, A. Hipp, L. Birnbacher, M. Marschner, M. Chabior, M. Reiser, F. Pfeiffer, F. Bamberg, and K. Hellerhoff, “Visualizing typical features of breast fibroadenomas using phase-contrast CT: An ex-vivo study,” PLOS ONE 9, e97101 (2014).
[Crossref] [PubMed]

Auxier, J.

Bamberg, F.

S. Grandl, M. Willner, J. Herzen, A. Sztrókay-Gaul, D. Mayr, S. D. Auweter, A. Hipp, L. Birnbacher, M. Marschner, M. Chabior, M. Reiser, F. Pfeiffer, F. Bamberg, and K. Hellerhoff, “Visualizing typical features of breast fibroadenomas using phase-contrast CT: An ex-vivo study,” PLOS ONE 9, e97101 (2014).
[Crossref] [PubMed]

H. Hetterich, M. Willner, S. Fill, J. Herzen, F. Bamberg, A. Hipp, U. Schüller, S. Adam-Neumair, S. Wirth, M. Reiser, F. Pfeiffer, and T. Saam, “Phase-contrast CT: qualitative and quantitative evaluation of atherosclerotic carotid artery plaque,” Radiology 271, 870–878 (2014).
[Crossref] [PubMed]

Bauer, J. S.

A. Malecki, G. Potdevin, T. Biernath, E. Eggl, E. Grande Garcia, T. Baum, P. B. Noël, J. S. Bauer, and F. Pfeiffer, “Coherent Superposition in Grating-Based Directional Dark-Field Imaging,” PLOS ONE 8, e61218 (2013).
[Crossref]

Baum, T.

A. Malecki, G. Potdevin, T. Biernath, E. Eggl, E. Grande Garcia, T. Baum, P. B. Noël, J. S. Bauer, and F. Pfeiffer, “Coherent Superposition in Grating-Based Directional Dark-Field Imaging,” PLOS ONE 8, e61218 (2013).
[Crossref]

Bech, M.

M. Bech, O. Bunk, T. Donath, R. Feidenhans’l, C. David, and F. Pfeiffer, “Quantitative x-ray dark-field computed tomography,” Phys. Med. Biol. 55, 5529–5539 (2010).
[Crossref] [PubMed]

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
[Crossref] [PubMed]

Belenguer, T.

Bennett, E. E.

H. Wen, H. Miao, E. E. Bennett, N. M. Adamo, and L. Chen, “Flexible Retrospective Phase Stepping in X-ray Scatter Correction and Phase Contrast Imaging Using Structured Illumination,” PLOS ONE 8, e78276 (2013).
[Crossref] [PubMed]

S. K. Lynch, V. Pai, J. Auxier, A. F. Stein, E. E. Bennett, C. K. Kemble, X. Xiao, W.-K. Lee, N. Y. Morgan, and H. H. Wen, “Interpretation of dark-field contrast and particle-size selectivity in grating interferometers,” Appl. Opt. 50, 4310–4319 (2011).
[Crossref] [PubMed]

Biernath, T.

A. Malecki, G. Potdevin, T. Biernath, E. Eggl, K. Willer, T. Lasser, J. Maisenbacher, J. Gibmeier, A. Wanner, and F. Pfeiffer, “X-ray tensor tomography,” EPL 105, 38002 (2014).
[Crossref]

A. Malecki, G. Potdevin, T. Biernath, E. Eggl, E. Grande Garcia, T. Baum, P. B. Noël, J. S. Bauer, and F. Pfeiffer, “Coherent Superposition in Grating-Based Directional Dark-Field Imaging,” PLOS ONE 8, e61218 (2013).
[Crossref]

Birnbacher, L.

L. Birnbacher, M. Willner, A. Velroyen, M. Marschner, A. Hipp, J. Meiser, F. Koch, T. Schröter, D. Kunka, J. Mohr, F. Pfeiffer, and J. Herzen, “Experimental Realisation of High-sensitivity Laboratory X-ray Grating-based Phase-contrast Computed Tomography,” Sci. Rep. 6, 24022 (2016).
[Crossref] [PubMed]

S. Grandl, M. Willner, J. Herzen, A. Sztrókay-Gaul, D. Mayr, S. D. Auweter, A. Hipp, L. Birnbacher, M. Marschner, M. Chabior, M. Reiser, F. Pfeiffer, F. Bamberg, and K. Hellerhoff, “Visualizing typical features of breast fibroadenomas using phase-contrast CT: An ex-vivo study,” PLOS ONE 9, e97101 (2014).
[Crossref] [PubMed]

Bravin, A.

A. Bravin, P. Coan, and P. Suortti, “X-ray phase-contrast imaging: from pre-clinical applications towards clinics,” Phys. Med. Biol. 58, R1–R35 (2013).
[Crossref]

Brönnimann, C.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
[Crossref] [PubMed]

Bunk, O.

M. Bech, O. Bunk, T. Donath, R. Feidenhans’l, C. David, and F. Pfeiffer, “Quantitative x-ray dark-field computed tomography,” Phys. Med. Biol. 55, 5529–5539 (2010).
[Crossref] [PubMed]

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
[Crossref] [PubMed]

F. Pfeiffer, O. Bunk, C. Kottler, and C. David, “Tomographic reconstruction of three-dimensional objects from hard X-ray differential phase contrast projection images,” Nucl. Instrum. Methods Phys. Res. A 580, 925–928 (2007).
[Crossref]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2, 258–261 (2006).
[Crossref]

Byer, R. L.

Carazo, J. M.

J. Vargas, C. O. S. Sorzano, J. C. Estrada, and J. M. Carazo, “Generalization of the principal component analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
[Crossref]

Chabior, M.

S. Grandl, M. Willner, J. Herzen, A. Sztrókay-Gaul, D. Mayr, S. D. Auweter, A. Hipp, L. Birnbacher, M. Marschner, M. Chabior, M. Reiser, F. Pfeiffer, F. Bamberg, and K. Hellerhoff, “Visualizing typical features of breast fibroadenomas using phase-contrast CT: An ex-vivo study,” PLOS ONE 9, e97101 (2014).
[Crossref] [PubMed]

Chen, L.

H. Wen, H. Miao, E. E. Bennett, N. M. Adamo, and L. Chen, “Flexible Retrospective Phase Stepping in X-ray Scatter Correction and Phase Contrast Imaging Using Structured Illumination,” PLOS ONE 8, e78276 (2013).
[Crossref] [PubMed]

Cloetens, P.

Coan, P.

A. Bravin, P. Coan, and P. Suortti, “X-ray phase-contrast imaging: from pre-clinical applications towards clinics,” Phys. Med. Biol. 58, R1–R35 (2013).
[Crossref]

David, C.

M. Bech, O. Bunk, T. Donath, R. Feidenhans’l, C. David, and F. Pfeiffer, “Quantitative x-ray dark-field computed tomography,” Phys. Med. Biol. 55, 5529–5539 (2010).
[Crossref] [PubMed]

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
[Crossref] [PubMed]

F. Pfeiffer, O. Bunk, C. Kottler, and C. David, “Tomographic reconstruction of three-dimensional objects from hard X-ray differential phase contrast projection images,” Nucl. Instrum. Methods Phys. Res. A 580, 925–928 (2007).
[Crossref]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2, 258–261 (2006).
[Crossref]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13, 6296–6304 (2005).
[Crossref] [PubMed]

Diaz, A.

Donath, T.

M. Bech, O. Bunk, T. Donath, R. Feidenhans’l, C. David, and F. Pfeiffer, “Quantitative x-ray dark-field computed tomography,” Phys. Med. Biol. 55, 5529–5539 (2010).
[Crossref] [PubMed]

Eggl, E.

A. Malecki, G. Potdevin, T. Biernath, E. Eggl, K. Willer, T. Lasser, J. Maisenbacher, J. Gibmeier, A. Wanner, and F. Pfeiffer, “X-ray tensor tomography,” EPL 105, 38002 (2014).
[Crossref]

A. Malecki, G. Potdevin, T. Biernath, E. Eggl, E. Grande Garcia, T. Baum, P. B. Noël, J. S. Bauer, and F. Pfeiffer, “Coherent Superposition in Grating-Based Directional Dark-Field Imaging,” PLOS ONE 8, e61218 (2013).
[Crossref]

Eikenberry, E. F.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
[Crossref] [PubMed]

Eng, F.

F. Eng and F. Gustafsson, “Bias compensated least squares estimation of continuous time output error models in the case of stochastic sampling time jitter,” IFAC Proc. Vol.  39, 612–617 (2006).
[Crossref]

Estrada, J. C.

J. Vargas, C. O. S. Sorzano, J. C. Estrada, and J. M. Carazo, “Generalization of the principal component analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
[Crossref]

Faris, G. W.

Fehringer, A.

M. Marschner, M. Willner, G. Potdevin, A. Fehringer, P. B. Noël, F. Pfeiffer, and J. Herzen, “Helical X-ray phase-contrast computed tomography without phase stepping,” Sci. Rep. 6, 23953 (2016).
[Crossref] [PubMed]

Feidenhans’l, R.

M. Bech, O. Bunk, T. Donath, R. Feidenhans’l, C. David, and F. Pfeiffer, “Quantitative x-ray dark-field computed tomography,” Phys. Med. Biol. 55, 5529–5539 (2010).
[Crossref] [PubMed]

Fill, S.

H. Hetterich, M. Willner, S. Fill, J. Herzen, F. Bamberg, A. Hipp, U. Schüller, S. Adam-Neumair, S. Wirth, M. Reiser, F. Pfeiffer, and T. Saam, “Phase-contrast CT: qualitative and quantitative evaluation of atherosclerotic carotid artery plaque,” Radiology 271, 870–878 (2014).
[Crossref] [PubMed]

Fingerle, A. A.

A. A. Fingerle, M. Willner, J. Herzen, D. Münzel, D. Hahn, E. J. Rummeny, P. B. Noël, and F. Pfeiffer, “Simulated Cystic Renal Lesions: Quantitative X-ray Phase-Contrast CT – An in Vitro Phantom Study,” Radiology 272, 739–748 (2014).
[Crossref] [PubMed]

Flach, D.

T. Souders, D. Flach, C. Hagwood, and G. Yang, “The effects of timing jitter in sampling systems,” in “6th IEEE Conference Record., Instrumentation and Measurement Technology Conference,” (IEEE, 1989), pp. 199–203.

Gibmeier, J.

A. Malecki, G. Potdevin, T. Biernath, E. Eggl, K. Willer, T. Lasser, J. Maisenbacher, J. Gibmeier, A. Wanner, and F. Pfeiffer, “X-ray tensor tomography,” EPL 105, 38002 (2014).
[Crossref]

Grande Garcia, E.

A. Malecki, G. Potdevin, T. Biernath, E. Eggl, E. Grande Garcia, T. Baum, P. B. Noël, J. S. Bauer, and F. Pfeiffer, “Coherent Superposition in Grating-Based Directional Dark-Field Imaging,” PLOS ONE 8, e61218 (2013).
[Crossref]

Grandl, S.

S. Grandl, M. Willner, J. Herzen, A. Sztrókay-Gaul, D. Mayr, S. D. Auweter, A. Hipp, L. Birnbacher, M. Marschner, M. Chabior, M. Reiser, F. Pfeiffer, F. Bamberg, and K. Hellerhoff, “Visualizing typical features of breast fibroadenomas using phase-contrast CT: An ex-vivo study,” PLOS ONE 9, e97101 (2014).
[Crossref] [PubMed]

Grünzweig, C.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
[Crossref] [PubMed]

Gustafsson, F.

F. Eng and F. Gustafsson, “Bias compensated least squares estimation of continuous time output error models in the case of stochastic sampling time jitter,” IFAC Proc. Vol.  39, 612–617 (2006).
[Crossref]

Hagwood, C.

T. Souders, D. Flach, C. Hagwood, and G. Yang, “The effects of timing jitter in sampling systems,” in “6th IEEE Conference Record., Instrumentation and Measurement Technology Conference,” (IEEE, 1989), pp. 199–203.

Hahn, D.

A. A. Fingerle, M. Willner, J. Herzen, D. Münzel, D. Hahn, E. J. Rummeny, P. B. Noël, and F. Pfeiffer, “Simulated Cystic Renal Lesions: Quantitative X-ray Phase-Contrast CT – An in Vitro Phantom Study,” Radiology 272, 739–748 (2014).
[Crossref] [PubMed]

Hamaishi, Y.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot Interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003).
[Crossref]

Han, B.

Hauke, C.

M. Seifert, S. Kaeppler, C. Hauke, F. Horn, G. Pelzer, J. Rieger, T. Michel, C. Riess, and G. Anton, “Optimisation of image reconstruction for phase-contrast x-ray Talbot-Lau imaging with regard to mechanical robustness,” Phys. Med. Biol. 61, 6441–6464 (2016).
[Crossref] [PubMed]

G. Pelzer, J. Rieger, C. Hauke, F. Horn, T. Michel, M. Seifert, and G. Anton, “Reconstruction method for grating-based x-ray phase-contrast images without knowledge of the grating positions,” J. Instrum. 10, P12017 (2015).
[Crossref]

Hellerhoff, K.

S. Grandl, M. Willner, J. Herzen, A. Sztrókay-Gaul, D. Mayr, S. D. Auweter, A. Hipp, L. Birnbacher, M. Marschner, M. Chabior, M. Reiser, F. Pfeiffer, F. Bamberg, and K. Hellerhoff, “Visualizing typical features of breast fibroadenomas using phase-contrast CT: An ex-vivo study,” PLOS ONE 9, e97101 (2014).
[Crossref] [PubMed]

Herzen, J.

L. Birnbacher, M. Willner, A. Velroyen, M. Marschner, A. Hipp, J. Meiser, F. Koch, T. Schröter, D. Kunka, J. Mohr, F. Pfeiffer, and J. Herzen, “Experimental Realisation of High-sensitivity Laboratory X-ray Grating-based Phase-contrast Computed Tomography,” Sci. Rep. 6, 24022 (2016).
[Crossref] [PubMed]

M. Marschner, M. Willner, G. Potdevin, A. Fehringer, P. B. Noël, F. Pfeiffer, and J. Herzen, “Helical X-ray phase-contrast computed tomography without phase stepping,” Sci. Rep. 6, 23953 (2016).
[Crossref] [PubMed]

A. A. Fingerle, M. Willner, J. Herzen, D. Münzel, D. Hahn, E. J. Rummeny, P. B. Noël, and F. Pfeiffer, “Simulated Cystic Renal Lesions: Quantitative X-ray Phase-Contrast CT – An in Vitro Phantom Study,” Radiology 272, 739–748 (2014).
[Crossref] [PubMed]

H. Hetterich, M. Willner, S. Fill, J. Herzen, F. Bamberg, A. Hipp, U. Schüller, S. Adam-Neumair, S. Wirth, M. Reiser, F. Pfeiffer, and T. Saam, “Phase-contrast CT: qualitative and quantitative evaluation of atherosclerotic carotid artery plaque,” Radiology 271, 870–878 (2014).
[Crossref] [PubMed]

S. Grandl, M. Willner, J. Herzen, A. Sztrókay-Gaul, D. Mayr, S. D. Auweter, A. Hipp, L. Birnbacher, M. Marschner, M. Chabior, M. Reiser, F. Pfeiffer, F. Bamberg, and K. Hellerhoff, “Visualizing typical features of breast fibroadenomas using phase-contrast CT: An ex-vivo study,” PLOS ONE 9, e97101 (2014).
[Crossref] [PubMed]

Hetterich, H.

H. Hetterich, M. Willner, S. Fill, J. Herzen, F. Bamberg, A. Hipp, U. Schüller, S. Adam-Neumair, S. Wirth, M. Reiser, F. Pfeiffer, and T. Saam, “Phase-contrast CT: qualitative and quantitative evaluation of atherosclerotic carotid artery plaque,” Radiology 271, 870–878 (2014).
[Crossref] [PubMed]

Hipp, A.

L. Birnbacher, M. Willner, A. Velroyen, M. Marschner, A. Hipp, J. Meiser, F. Koch, T. Schröter, D. Kunka, J. Mohr, F. Pfeiffer, and J. Herzen, “Experimental Realisation of High-sensitivity Laboratory X-ray Grating-based Phase-contrast Computed Tomography,” Sci. Rep. 6, 24022 (2016).
[Crossref] [PubMed]

H. Hetterich, M. Willner, S. Fill, J. Herzen, F. Bamberg, A. Hipp, U. Schüller, S. Adam-Neumair, S. Wirth, M. Reiser, F. Pfeiffer, and T. Saam, “Phase-contrast CT: qualitative and quantitative evaluation of atherosclerotic carotid artery plaque,” Radiology 271, 870–878 (2014).
[Crossref] [PubMed]

S. Grandl, M. Willner, J. Herzen, A. Sztrókay-Gaul, D. Mayr, S. D. Auweter, A. Hipp, L. Birnbacher, M. Marschner, M. Chabior, M. Reiser, F. Pfeiffer, F. Bamberg, and K. Hellerhoff, “Visualizing typical features of breast fibroadenomas using phase-contrast CT: An ex-vivo study,” PLOS ONE 9, e97101 (2014).
[Crossref] [PubMed]

Horn, F.

S. Kaeppler, J. Rieger, G. Pelzer, F. Horn, T. Michel, A. Maier, G. Anton, and C. Riess, “Improved reconstruction of phase-stepping data for Talbot–Lau x-ray imaging,” J. Med. Imaging 4, 034005 (2017).
[Crossref]

M. Seifert, S. Kaeppler, C. Hauke, F. Horn, G. Pelzer, J. Rieger, T. Michel, C. Riess, and G. Anton, “Optimisation of image reconstruction for phase-contrast x-ray Talbot-Lau imaging with regard to mechanical robustness,” Phys. Med. Biol. 61, 6441–6464 (2016).
[Crossref] [PubMed]

G. Pelzer, J. Rieger, C. Hauke, F. Horn, T. Michel, M. Seifert, and G. Anton, “Reconstruction method for grating-based x-ray phase-contrast images without knowledge of the grating positions,” J. Instrum. 10, P12017 (2015).
[Crossref]

Kaeppler, S.

S. Kaeppler, J. Rieger, G. Pelzer, F. Horn, T. Michel, A. Maier, G. Anton, and C. Riess, “Improved reconstruction of phase-stepping data for Talbot–Lau x-ray imaging,” J. Med. Imaging 4, 034005 (2017).
[Crossref]

M. Seifert, S. Kaeppler, C. Hauke, F. Horn, G. Pelzer, J. Rieger, T. Michel, C. Riess, and G. Anton, “Optimisation of image reconstruction for phase-contrast x-ray Talbot-Lau imaging with regard to mechanical robustness,” Phys. Med. Biol. 61, 6441–6464 (2016).
[Crossref] [PubMed]

Kaufmann, R.

V. Revol, C. Kottler, R. Kaufmann, U. Straumann, and C. Urban, “Noise analysis of grating-based x-ray differential phase contrast imaging,” Rev. Sci. Instrum. 81, 073709 (2010).
[Crossref] [PubMed]

Kawabata, K.

Kawamoto, S.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot Interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003).
[Crossref]

Kemble, C. K.

Koch, F.

L. Birnbacher, M. Willner, A. Velroyen, M. Marschner, A. Hipp, J. Meiser, F. Koch, T. Schröter, D. Kunka, J. Mohr, F. Pfeiffer, and J. Herzen, “Experimental Realisation of High-sensitivity Laboratory X-ray Grating-based Phase-contrast Computed Tomography,” Sci. Rep. 6, 24022 (2016).
[Crossref] [PubMed]

Kottler, C.

V. Revol, C. Kottler, R. Kaufmann, U. Straumann, and C. Urban, “Noise analysis of grating-based x-ray differential phase contrast imaging,” Rev. Sci. Instrum. 81, 073709 (2010).
[Crossref] [PubMed]

F. Pfeiffer, O. Bunk, C. Kottler, and C. David, “Tomographic reconstruction of three-dimensional objects from hard X-ray differential phase contrast projection images,” Nucl. Instrum. Methods Phys. Res. A 580, 925–928 (2007).
[Crossref]

Koyama, I.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot Interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003).
[Crossref]

Kraft, P.

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
[Crossref] [PubMed]

Kunka, D.

L. Birnbacher, M. Willner, A. Velroyen, M. Marschner, A. Hipp, J. Meiser, F. Koch, T. Schröter, D. Kunka, J. Mohr, F. Pfeiffer, and J. Herzen, “Experimental Realisation of High-sensitivity Laboratory X-ray Grating-based Phase-contrast Computed Tomography,” Sci. Rep. 6, 24022 (2016).
[Crossref] [PubMed]

Lasser, T.

A. Malecki, G. Potdevin, T. Biernath, E. Eggl, K. Willer, T. Lasser, J. Maisenbacher, J. Gibmeier, A. Wanner, and F. Pfeiffer, “X-ray tensor tomography,” EPL 105, 38002 (2014).
[Crossref]

Lee, W.-K.

Liang, R.

Lynch, S. K.

Maier, A.

S. Kaeppler, J. Rieger, G. Pelzer, F. Horn, T. Michel, A. Maier, G. Anton, and C. Riess, “Improved reconstruction of phase-stepping data for Talbot–Lau x-ray imaging,” J. Med. Imaging 4, 034005 (2017).
[Crossref]

Maisenbacher, J.

A. Malecki, G. Potdevin, T. Biernath, E. Eggl, K. Willer, T. Lasser, J. Maisenbacher, J. Gibmeier, A. Wanner, and F. Pfeiffer, “X-ray tensor tomography,” EPL 105, 38002 (2014).
[Crossref]

Malecki, A.

A. Malecki, G. Potdevin, T. Biernath, E. Eggl, K. Willer, T. Lasser, J. Maisenbacher, J. Gibmeier, A. Wanner, and F. Pfeiffer, “X-ray tensor tomography,” EPL 105, 38002 (2014).
[Crossref]

A. Malecki, G. Potdevin, T. Biernath, E. Eggl, E. Grande Garcia, T. Baum, P. B. Noël, J. S. Bauer, and F. Pfeiffer, “Coherent Superposition in Grating-Based Directional Dark-Field Imaging,” PLOS ONE 8, e61218 (2013).
[Crossref]

Marone, F.

Marschner, M.

L. Birnbacher, M. Willner, A. Velroyen, M. Marschner, A. Hipp, J. Meiser, F. Koch, T. Schröter, D. Kunka, J. Mohr, F. Pfeiffer, and J. Herzen, “Experimental Realisation of High-sensitivity Laboratory X-ray Grating-based Phase-contrast Computed Tomography,” Sci. Rep. 6, 24022 (2016).
[Crossref] [PubMed]

M. Marschner, M. Willner, G. Potdevin, A. Fehringer, P. B. Noël, F. Pfeiffer, and J. Herzen, “Helical X-ray phase-contrast computed tomography without phase stepping,” Sci. Rep. 6, 23953 (2016).
[Crossref] [PubMed]

S. Grandl, M. Willner, J. Herzen, A. Sztrókay-Gaul, D. Mayr, S. D. Auweter, A. Hipp, L. Birnbacher, M. Marschner, M. Chabior, M. Reiser, F. Pfeiffer, F. Bamberg, and K. Hellerhoff, “Visualizing typical features of breast fibroadenomas using phase-contrast CT: An ex-vivo study,” PLOS ONE 9, e97101 (2014).
[Crossref] [PubMed]

Mayr, D.

S. Grandl, M. Willner, J. Herzen, A. Sztrókay-Gaul, D. Mayr, S. D. Auweter, A. Hipp, L. Birnbacher, M. Marschner, M. Chabior, M. Reiser, F. Pfeiffer, F. Bamberg, and K. Hellerhoff, “Visualizing typical features of breast fibroadenomas using phase-contrast CT: An ex-vivo study,” PLOS ONE 9, e97101 (2014).
[Crossref] [PubMed]

Meiser, J.

L. Birnbacher, M. Willner, A. Velroyen, M. Marschner, A. Hipp, J. Meiser, F. Koch, T. Schröter, D. Kunka, J. Mohr, F. Pfeiffer, and J. Herzen, “Experimental Realisation of High-sensitivity Laboratory X-ray Grating-based Phase-contrast Computed Tomography,” Sci. Rep. 6, 24022 (2016).
[Crossref] [PubMed]

Miao, H.

H. Wen, H. Miao, E. E. Bennett, N. M. Adamo, and L. Chen, “Flexible Retrospective Phase Stepping in X-ray Scatter Correction and Phase Contrast Imaging Using Structured Illumination,” PLOS ONE 8, e78276 (2013).
[Crossref] [PubMed]

Michel, T.

S. Kaeppler, J. Rieger, G. Pelzer, F. Horn, T. Michel, A. Maier, G. Anton, and C. Riess, “Improved reconstruction of phase-stepping data for Talbot–Lau x-ray imaging,” J. Med. Imaging 4, 034005 (2017).
[Crossref]

M. Seifert, S. Kaeppler, C. Hauke, F. Horn, G. Pelzer, J. Rieger, T. Michel, C. Riess, and G. Anton, “Optimisation of image reconstruction for phase-contrast x-ray Talbot-Lau imaging with regard to mechanical robustness,” Phys. Med. Biol. 61, 6441–6464 (2016).
[Crossref] [PubMed]

G. Pelzer, J. Rieger, C. Hauke, F. Horn, T. Michel, M. Seifert, and G. Anton, “Reconstruction method for grating-based x-ray phase-contrast images without knowledge of the grating positions,” J. Instrum. 10, P12017 (2015).
[Crossref]

Mohr, J.

L. Birnbacher, M. Willner, A. Velroyen, M. Marschner, A. Hipp, J. Meiser, F. Koch, T. Schröter, D. Kunka, J. Mohr, F. Pfeiffer, and J. Herzen, “Experimental Realisation of High-sensitivity Laboratory X-ray Grating-based Phase-contrast Computed Tomography,” Sci. Rep. 6, 24022 (2016).
[Crossref] [PubMed]

Momose, A.

W. Yashiro, Y. Terui, K. Kawabata, and A. Momose, “On the origin of visibility contrast in x-ray Talbot interferometry,” Opt. Express 18, 16890–16901 (2010).
[Crossref] [PubMed]

A. Momose, “Recent Advances in X-ray Phase Imaging,” Jpn. J. Appl. Phys. 44, 6355–6367 (2005).
[Crossref]

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot Interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003).
[Crossref]

Morgan, N. Y.

Münch, B.

Münzel, D.

A. A. Fingerle, M. Willner, J. Herzen, D. Münzel, D. Hahn, E. J. Rummeny, P. B. Noël, and F. Pfeiffer, “Simulated Cystic Renal Lesions: Quantitative X-ray Phase-Contrast CT – An in Vitro Phantom Study,” Radiology 272, 739–748 (2014).
[Crossref] [PubMed]

Noël, P. B.

M. Marschner, M. Willner, G. Potdevin, A. Fehringer, P. B. Noël, F. Pfeiffer, and J. Herzen, “Helical X-ray phase-contrast computed tomography without phase stepping,” Sci. Rep. 6, 23953 (2016).
[Crossref] [PubMed]

A. A. Fingerle, M. Willner, J. Herzen, D. Münzel, D. Hahn, E. J. Rummeny, P. B. Noël, and F. Pfeiffer, “Simulated Cystic Renal Lesions: Quantitative X-ray Phase-Contrast CT – An in Vitro Phantom Study,” Radiology 272, 739–748 (2014).
[Crossref] [PubMed]

A. Malecki, G. Potdevin, T. Biernath, E. Eggl, E. Grande Garcia, T. Baum, P. B. Noël, J. S. Bauer, and F. Pfeiffer, “Coherent Superposition in Grating-Based Directional Dark-Field Imaging,” PLOS ONE 8, e61218 (2013).
[Crossref]

Pai, V.

Pelzer, G.

S. Kaeppler, J. Rieger, G. Pelzer, F. Horn, T. Michel, A. Maier, G. Anton, and C. Riess, “Improved reconstruction of phase-stepping data for Talbot–Lau x-ray imaging,” J. Med. Imaging 4, 034005 (2017).
[Crossref]

M. Seifert, S. Kaeppler, C. Hauke, F. Horn, G. Pelzer, J. Rieger, T. Michel, C. Riess, and G. Anton, “Optimisation of image reconstruction for phase-contrast x-ray Talbot-Lau imaging with regard to mechanical robustness,” Phys. Med. Biol. 61, 6441–6464 (2016).
[Crossref] [PubMed]

G. Pelzer, J. Rieger, C. Hauke, F. Horn, T. Michel, M. Seifert, and G. Anton, “Reconstruction method for grating-based x-ray phase-contrast images without knowledge of the grating positions,” J. Instrum. 10, P12017 (2015).
[Crossref]

Pfeiffer, F.

M. Marschner, M. Willner, G. Potdevin, A. Fehringer, P. B. Noël, F. Pfeiffer, and J. Herzen, “Helical X-ray phase-contrast computed tomography without phase stepping,” Sci. Rep. 6, 23953 (2016).
[Crossref] [PubMed]

L. Birnbacher, M. Willner, A. Velroyen, M. Marschner, A. Hipp, J. Meiser, F. Koch, T. Schröter, D. Kunka, J. Mohr, F. Pfeiffer, and J. Herzen, “Experimental Realisation of High-sensitivity Laboratory X-ray Grating-based Phase-contrast Computed Tomography,” Sci. Rep. 6, 24022 (2016).
[Crossref] [PubMed]

H. Hetterich, M. Willner, S. Fill, J. Herzen, F. Bamberg, A. Hipp, U. Schüller, S. Adam-Neumair, S. Wirth, M. Reiser, F. Pfeiffer, and T. Saam, “Phase-contrast CT: qualitative and quantitative evaluation of atherosclerotic carotid artery plaque,” Radiology 271, 870–878 (2014).
[Crossref] [PubMed]

A. A. Fingerle, M. Willner, J. Herzen, D. Münzel, D. Hahn, E. J. Rummeny, P. B. Noël, and F. Pfeiffer, “Simulated Cystic Renal Lesions: Quantitative X-ray Phase-Contrast CT – An in Vitro Phantom Study,” Radiology 272, 739–748 (2014).
[Crossref] [PubMed]

S. Grandl, M. Willner, J. Herzen, A. Sztrókay-Gaul, D. Mayr, S. D. Auweter, A. Hipp, L. Birnbacher, M. Marschner, M. Chabior, M. Reiser, F. Pfeiffer, F. Bamberg, and K. Hellerhoff, “Visualizing typical features of breast fibroadenomas using phase-contrast CT: An ex-vivo study,” PLOS ONE 9, e97101 (2014).
[Crossref] [PubMed]

A. Malecki, G. Potdevin, T. Biernath, E. Eggl, K. Willer, T. Lasser, J. Maisenbacher, J. Gibmeier, A. Wanner, and F. Pfeiffer, “X-ray tensor tomography,” EPL 105, 38002 (2014).
[Crossref]

A. Malecki, G. Potdevin, T. Biernath, E. Eggl, E. Grande Garcia, T. Baum, P. B. Noël, J. S. Bauer, and F. Pfeiffer, “Coherent Superposition in Grating-Based Directional Dark-Field Imaging,” PLOS ONE 8, e61218 (2013).
[Crossref]

M. Bech, O. Bunk, T. Donath, R. Feidenhans’l, C. David, and F. Pfeiffer, “Quantitative x-ray dark-field computed tomography,” Phys. Med. Biol. 55, 5529–5539 (2010).
[Crossref] [PubMed]

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
[Crossref] [PubMed]

F. Pfeiffer, O. Bunk, C. Kottler, and C. David, “Tomographic reconstruction of three-dimensional objects from hard X-ray differential phase contrast projection images,” Nucl. Instrum. Methods Phys. Res. A 580, 925–928 (2007).
[Crossref]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2, 258–261 (2006).
[Crossref]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13, 6296–6304 (2005).
[Crossref] [PubMed]

Potdevin, G.

M. Marschner, M. Willner, G. Potdevin, A. Fehringer, P. B. Noël, F. Pfeiffer, and J. Herzen, “Helical X-ray phase-contrast computed tomography without phase stepping,” Sci. Rep. 6, 23953 (2016).
[Crossref] [PubMed]

A. Malecki, G. Potdevin, T. Biernath, E. Eggl, K. Willer, T. Lasser, J. Maisenbacher, J. Gibmeier, A. Wanner, and F. Pfeiffer, “X-ray tensor tomography,” EPL 105, 38002 (2014).
[Crossref]

A. Malecki, G. Potdevin, T. Biernath, E. Eggl, E. Grande Garcia, T. Baum, P. B. Noël, J. S. Bauer, and F. Pfeiffer, “Coherent Superposition in Grating-Based Directional Dark-Field Imaging,” PLOS ONE 8, e61218 (2013).
[Crossref]

Quiroga, J. A.

Reiser, M.

H. Hetterich, M. Willner, S. Fill, J. Herzen, F. Bamberg, A. Hipp, U. Schüller, S. Adam-Neumair, S. Wirth, M. Reiser, F. Pfeiffer, and T. Saam, “Phase-contrast CT: qualitative and quantitative evaluation of atherosclerotic carotid artery plaque,” Radiology 271, 870–878 (2014).
[Crossref] [PubMed]

S. Grandl, M. Willner, J. Herzen, A. Sztrókay-Gaul, D. Mayr, S. D. Auweter, A. Hipp, L. Birnbacher, M. Marschner, M. Chabior, M. Reiser, F. Pfeiffer, F. Bamberg, and K. Hellerhoff, “Visualizing typical features of breast fibroadenomas using phase-contrast CT: An ex-vivo study,” PLOS ONE 9, e97101 (2014).
[Crossref] [PubMed]

Revol, V.

V. Revol, C. Kottler, R. Kaufmann, U. Straumann, and C. Urban, “Noise analysis of grating-based x-ray differential phase contrast imaging,” Rev. Sci. Instrum. 81, 073709 (2010).
[Crossref] [PubMed]

Rieger, J.

S. Kaeppler, J. Rieger, G. Pelzer, F. Horn, T. Michel, A. Maier, G. Anton, and C. Riess, “Improved reconstruction of phase-stepping data for Talbot–Lau x-ray imaging,” J. Med. Imaging 4, 034005 (2017).
[Crossref]

M. Seifert, S. Kaeppler, C. Hauke, F. Horn, G. Pelzer, J. Rieger, T. Michel, C. Riess, and G. Anton, “Optimisation of image reconstruction for phase-contrast x-ray Talbot-Lau imaging with regard to mechanical robustness,” Phys. Med. Biol. 61, 6441–6464 (2016).
[Crossref] [PubMed]

G. Pelzer, J. Rieger, C. Hauke, F. Horn, T. Michel, M. Seifert, and G. Anton, “Reconstruction method for grating-based x-ray phase-contrast images without knowledge of the grating positions,” J. Instrum. 10, P12017 (2015).
[Crossref]

Riess, C.

S. Kaeppler, J. Rieger, G. Pelzer, F. Horn, T. Michel, A. Maier, G. Anton, and C. Riess, “Improved reconstruction of phase-stepping data for Talbot–Lau x-ray imaging,” J. Med. Imaging 4, 034005 (2017).
[Crossref]

M. Seifert, S. Kaeppler, C. Hauke, F. Horn, G. Pelzer, J. Rieger, T. Michel, C. Riess, and G. Anton, “Optimisation of image reconstruction for phase-contrast x-ray Talbot-Lau imaging with regard to mechanical robustness,” Phys. Med. Biol. 61, 6441–6464 (2016).
[Crossref] [PubMed]

Rummeny, E. J.

A. A. Fingerle, M. Willner, J. Herzen, D. Münzel, D. Hahn, E. J. Rummeny, P. B. Noël, and F. Pfeiffer, “Simulated Cystic Renal Lesions: Quantitative X-ray Phase-Contrast CT – An in Vitro Phantom Study,” Radiology 272, 739–748 (2014).
[Crossref] [PubMed]

Saam, T.

H. Hetterich, M. Willner, S. Fill, J. Herzen, F. Bamberg, A. Hipp, U. Schüller, S. Adam-Neumair, S. Wirth, M. Reiser, F. Pfeiffer, and T. Saam, “Phase-contrast CT: qualitative and quantitative evaluation of atherosclerotic carotid artery plaque,” Radiology 271, 870–878 (2014).
[Crossref] [PubMed]

Schröter, T.

L. Birnbacher, M. Willner, A. Velroyen, M. Marschner, A. Hipp, J. Meiser, F. Koch, T. Schröter, D. Kunka, J. Mohr, F. Pfeiffer, and J. Herzen, “Experimental Realisation of High-sensitivity Laboratory X-ray Grating-based Phase-contrast Computed Tomography,” Sci. Rep. 6, 24022 (2016).
[Crossref] [PubMed]

Schüller, U.

H. Hetterich, M. Willner, S. Fill, J. Herzen, F. Bamberg, A. Hipp, U. Schüller, S. Adam-Neumair, S. Wirth, M. Reiser, F. Pfeiffer, and T. Saam, “Phase-contrast CT: qualitative and quantitative evaluation of atherosclerotic carotid artery plaque,” Radiology 271, 870–878 (2014).
[Crossref] [PubMed]

Seifert, M.

M. Seifert, S. Kaeppler, C. Hauke, F. Horn, G. Pelzer, J. Rieger, T. Michel, C. Riess, and G. Anton, “Optimisation of image reconstruction for phase-contrast x-ray Talbot-Lau imaging with regard to mechanical robustness,” Phys. Med. Biol. 61, 6441–6464 (2016).
[Crossref] [PubMed]

G. Pelzer, J. Rieger, C. Hauke, F. Horn, T. Michel, M. Seifert, and G. Anton, “Reconstruction method for grating-based x-ray phase-contrast images without knowledge of the grating positions,” J. Instrum. 10, P12017 (2015).
[Crossref]

Sorzano, C. O. S.

J. Vargas, C. O. S. Sorzano, J. C. Estrada, and J. M. Carazo, “Generalization of the principal component analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
[Crossref]

Souders, T.

T. Souders, D. Flach, C. Hagwood, and G. Yang, “The effects of timing jitter in sampling systems,” in “6th IEEE Conference Record., Instrumentation and Measurement Technology Conference,” (IEEE, 1989), pp. 199–203.

Stampanoni, M.

Stein, A. F.

Straumann, U.

V. Revol, C. Kottler, R. Kaufmann, U. Straumann, and C. Urban, “Noise analysis of grating-based x-ray differential phase contrast imaging,” Rev. Sci. Instrum. 81, 073709 (2010).
[Crossref] [PubMed]

Strobl, M.

M. Strobl, “General solution for quantitative dark-field contrast imaging with grating interferometers,” Sci. Rep. 4, 7243 (2014).
[Crossref] [PubMed]

Suortti, P.

A. Bravin, P. Coan, and P. Suortti, “X-ray phase-contrast imaging: from pre-clinical applications towards clinics,” Phys. Med. Biol. 58, R1–R35 (2013).
[Crossref]

Suzuki, Y.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot Interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003).
[Crossref]

Sztrókay-Gaul, A.

S. Grandl, M. Willner, J. Herzen, A. Sztrókay-Gaul, D. Mayr, S. D. Auweter, A. Hipp, L. Birnbacher, M. Marschner, M. Chabior, M. Reiser, F. Pfeiffer, F. Bamberg, and K. Hellerhoff, “Visualizing typical features of breast fibroadenomas using phase-contrast CT: An ex-vivo study,” PLOS ONE 9, e97101 (2014).
[Crossref] [PubMed]

Takai, K.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot Interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003).
[Crossref]

Terui, Y.

Tian, X.

Trtik, P.

Urban, C.

V. Revol, C. Kottler, R. Kaufmann, U. Straumann, and C. Urban, “Noise analysis of grating-based x-ray differential phase contrast imaging,” Rev. Sci. Instrum. 81, 073709 (2010).
[Crossref] [PubMed]

Vargas, J.

J. Vargas, C. O. S. Sorzano, J. C. Estrada, and J. M. Carazo, “Generalization of the principal component analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
[Crossref]

J. Vargas, J. A. Quiroga, and T. Belenguer, “Phase-shifting interferometry based on principal component analysis,” Opt. Lett. 36, 1326–1328 (2011).
[Crossref] [PubMed]

Velroyen, A.

L. Birnbacher, M. Willner, A. Velroyen, M. Marschner, A. Hipp, J. Meiser, F. Koch, T. Schröter, D. Kunka, J. Mohr, F. Pfeiffer, and J. Herzen, “Experimental Realisation of High-sensitivity Laboratory X-ray Grating-based Phase-contrast Computed Tomography,” Sci. Rep. 6, 24022 (2016).
[Crossref] [PubMed]

Wang, Z.

Wanner, A.

A. Malecki, G. Potdevin, T. Biernath, E. Eggl, K. Willer, T. Lasser, J. Maisenbacher, J. Gibmeier, A. Wanner, and F. Pfeiffer, “X-ray tensor tomography,” EPL 105, 38002 (2014).
[Crossref]

Weitkamp, T.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2, 258–261 (2006).
[Crossref]

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13, 6296–6304 (2005).
[Crossref] [PubMed]

Wen, H.

H. Wen, H. Miao, E. E. Bennett, N. M. Adamo, and L. Chen, “Flexible Retrospective Phase Stepping in X-ray Scatter Correction and Phase Contrast Imaging Using Structured Illumination,” PLOS ONE 8, e78276 (2013).
[Crossref] [PubMed]

Wen, H. H.

Willer, K.

A. Malecki, G. Potdevin, T. Biernath, E. Eggl, K. Willer, T. Lasser, J. Maisenbacher, J. Gibmeier, A. Wanner, and F. Pfeiffer, “X-ray tensor tomography,” EPL 105, 38002 (2014).
[Crossref]

Willner, M.

L. Birnbacher, M. Willner, A. Velroyen, M. Marschner, A. Hipp, J. Meiser, F. Koch, T. Schröter, D. Kunka, J. Mohr, F. Pfeiffer, and J. Herzen, “Experimental Realisation of High-sensitivity Laboratory X-ray Grating-based Phase-contrast Computed Tomography,” Sci. Rep. 6, 24022 (2016).
[Crossref] [PubMed]

M. Marschner, M. Willner, G. Potdevin, A. Fehringer, P. B. Noël, F. Pfeiffer, and J. Herzen, “Helical X-ray phase-contrast computed tomography without phase stepping,” Sci. Rep. 6, 23953 (2016).
[Crossref] [PubMed]

S. Grandl, M. Willner, J. Herzen, A. Sztrókay-Gaul, D. Mayr, S. D. Auweter, A. Hipp, L. Birnbacher, M. Marschner, M. Chabior, M. Reiser, F. Pfeiffer, F. Bamberg, and K. Hellerhoff, “Visualizing typical features of breast fibroadenomas using phase-contrast CT: An ex-vivo study,” PLOS ONE 9, e97101 (2014).
[Crossref] [PubMed]

A. A. Fingerle, M. Willner, J. Herzen, D. Münzel, D. Hahn, E. J. Rummeny, P. B. Noël, and F. Pfeiffer, “Simulated Cystic Renal Lesions: Quantitative X-ray Phase-Contrast CT – An in Vitro Phantom Study,” Radiology 272, 739–748 (2014).
[Crossref] [PubMed]

H. Hetterich, M. Willner, S. Fill, J. Herzen, F. Bamberg, A. Hipp, U. Schüller, S. Adam-Neumair, S. Wirth, M. Reiser, F. Pfeiffer, and T. Saam, “Phase-contrast CT: qualitative and quantitative evaluation of atherosclerotic carotid artery plaque,” Radiology 271, 870–878 (2014).
[Crossref] [PubMed]

Wirth, S.

H. Hetterich, M. Willner, S. Fill, J. Herzen, F. Bamberg, A. Hipp, U. Schüller, S. Adam-Neumair, S. Wirth, M. Reiser, F. Pfeiffer, and T. Saam, “Phase-contrast CT: qualitative and quantitative evaluation of atherosclerotic carotid artery plaque,” Radiology 271, 870–878 (2014).
[Crossref] [PubMed]

Xiao, X.

Yang, G.

T. Souders, D. Flach, C. Hagwood, and G. Yang, “The effects of timing jitter in sampling systems,” in “6th IEEE Conference Record., Instrumentation and Measurement Technology Conference,” (IEEE, 1989), pp. 199–203.

Yashiro, W.

Zhang, Y.

Ziegler, E.

Appl. Opt. (2)

EPL (1)

A. Malecki, G. Potdevin, T. Biernath, E. Eggl, K. Willer, T. Lasser, J. Maisenbacher, J. Gibmeier, A. Wanner, and F. Pfeiffer, “X-ray tensor tomography,” EPL 105, 38002 (2014).
[Crossref]

IFAC Proc. (1)

F. Eng and F. Gustafsson, “Bias compensated least squares estimation of continuous time output error models in the case of stochastic sampling time jitter,” IFAC Proc. Vol.  39, 612–617 (2006).
[Crossref]

J. Instrum. (1)

G. Pelzer, J. Rieger, C. Hauke, F. Horn, T. Michel, M. Seifert, and G. Anton, “Reconstruction method for grating-based x-ray phase-contrast images without knowledge of the grating positions,” J. Instrum. 10, P12017 (2015).
[Crossref]

J. Med. Imaging (1)

S. Kaeppler, J. Rieger, G. Pelzer, F. Horn, T. Michel, A. Maier, G. Anton, and C. Riess, “Improved reconstruction of phase-stepping data for Talbot–Lau x-ray imaging,” J. Med. Imaging 4, 034005 (2017).
[Crossref]

Jpn. J. Appl. Phys. (2)

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-Ray Talbot Interferometry,” Jpn. J. Appl. Phys. 42, L866–L868 (2003).
[Crossref]

A. Momose, “Recent Advances in X-ray Phase Imaging,” Jpn. J. Appl. Phys. 44, 6355–6367 (2005).
[Crossref]

Nat. Mater. (1)

F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Brönnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008).
[Crossref] [PubMed]

Nat. Phys. (1)

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2, 258–261 (2006).
[Crossref]

Nucl. Instrum. Methods Phys. Res. A (1)

F. Pfeiffer, O. Bunk, C. Kottler, and C. David, “Tomographic reconstruction of three-dimensional objects from hard X-ray differential phase contrast projection images,” Nucl. Instrum. Methods Phys. Res. A 580, 925–928 (2007).
[Crossref]

Opt. Commun. (1)

J. Vargas, C. O. S. Sorzano, J. C. Estrada, and J. M. Carazo, “Generalization of the principal component analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
[Crossref]

Opt. Express (4)

Opt. Lett. (2)

Phys. Med. Biol. (3)

M. Seifert, S. Kaeppler, C. Hauke, F. Horn, G. Pelzer, J. Rieger, T. Michel, C. Riess, and G. Anton, “Optimisation of image reconstruction for phase-contrast x-ray Talbot-Lau imaging with regard to mechanical robustness,” Phys. Med. Biol. 61, 6441–6464 (2016).
[Crossref] [PubMed]

A. Bravin, P. Coan, and P. Suortti, “X-ray phase-contrast imaging: from pre-clinical applications towards clinics,” Phys. Med. Biol. 58, R1–R35 (2013).
[Crossref]

M. Bech, O. Bunk, T. Donath, R. Feidenhans’l, C. David, and F. Pfeiffer, “Quantitative x-ray dark-field computed tomography,” Phys. Med. Biol. 55, 5529–5539 (2010).
[Crossref] [PubMed]

PLOS ONE (3)

A. Malecki, G. Potdevin, T. Biernath, E. Eggl, E. Grande Garcia, T. Baum, P. B. Noël, J. S. Bauer, and F. Pfeiffer, “Coherent Superposition in Grating-Based Directional Dark-Field Imaging,” PLOS ONE 8, e61218 (2013).
[Crossref]

S. Grandl, M. Willner, J. Herzen, A. Sztrókay-Gaul, D. Mayr, S. D. Auweter, A. Hipp, L. Birnbacher, M. Marschner, M. Chabior, M. Reiser, F. Pfeiffer, F. Bamberg, and K. Hellerhoff, “Visualizing typical features of breast fibroadenomas using phase-contrast CT: An ex-vivo study,” PLOS ONE 9, e97101 (2014).
[Crossref] [PubMed]

H. Wen, H. Miao, E. E. Bennett, N. M. Adamo, and L. Chen, “Flexible Retrospective Phase Stepping in X-ray Scatter Correction and Phase Contrast Imaging Using Structured Illumination,” PLOS ONE 8, e78276 (2013).
[Crossref] [PubMed]

Radiology (2)

A. A. Fingerle, M. Willner, J. Herzen, D. Münzel, D. Hahn, E. J. Rummeny, P. B. Noël, and F. Pfeiffer, “Simulated Cystic Renal Lesions: Quantitative X-ray Phase-Contrast CT – An in Vitro Phantom Study,” Radiology 272, 739–748 (2014).
[Crossref] [PubMed]

H. Hetterich, M. Willner, S. Fill, J. Herzen, F. Bamberg, A. Hipp, U. Schüller, S. Adam-Neumair, S. Wirth, M. Reiser, F. Pfeiffer, and T. Saam, “Phase-contrast CT: qualitative and quantitative evaluation of atherosclerotic carotid artery plaque,” Radiology 271, 870–878 (2014).
[Crossref] [PubMed]

Rev. Sci. Instrum. (1)

V. Revol, C. Kottler, R. Kaufmann, U. Straumann, and C. Urban, “Noise analysis of grating-based x-ray differential phase contrast imaging,” Rev. Sci. Instrum. 81, 073709 (2010).
[Crossref] [PubMed]

Sci. Rep. (3)

M. Marschner, M. Willner, G. Potdevin, A. Fehringer, P. B. Noël, F. Pfeiffer, and J. Herzen, “Helical X-ray phase-contrast computed tomography without phase stepping,” Sci. Rep. 6, 23953 (2016).
[Crossref] [PubMed]

L. Birnbacher, M. Willner, A. Velroyen, M. Marschner, A. Hipp, J. Meiser, F. Koch, T. Schröter, D. Kunka, J. Mohr, F. Pfeiffer, and J. Herzen, “Experimental Realisation of High-sensitivity Laboratory X-ray Grating-based Phase-contrast Computed Tomography,” Sci. Rep. 6, 24022 (2016).
[Crossref] [PubMed]

M. Strobl, “General solution for quantitative dark-field contrast imaging with grating interferometers,” Sci. Rep. 4, 7243 (2014).
[Crossref] [PubMed]

Other (1)

T. Souders, D. Flach, C. Hagwood, and G. Yang, “The effects of timing jitter in sampling systems,” in “6th IEEE Conference Record., Instrumentation and Measurement Technology Conference,” (IEEE, 1989), pp. 199–203.

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Figures (6)

Fig. 1
Fig. 1 (a): Effect of absorption, refraction and small-angle scattering on downstream intensity patterns in the Talbot-Lau interferometer. (b): The intensity patterns are resolved indirectly by recording the detector signals I for different lateral shifts x of either of the three gratings (of period p).
Fig. 2
Fig. 2 Correlation of stepping curve phase Φ 1 s (a) and residual oscillations in projections of transmittance T (b), differential-phase φ (c) and dark-field D (d) of a porcine cartilage sample in saline solution. A PMMA rod used for energy calibration is also visible. The phase-wrapping pattern in (a) correlates with the pattern of fringe artifacts in all three modalities. The artifacts in (c) and (d) evidently have twice the spatial frequency of the pattern in (b). Figs. 2(b)2(d) were retrieved from a set of 11 phase-stepping images over one grating period (and another 11 flat-field phase-stepping images). Exposure time per image was 3 s. Acquisition was performed at the setup in [13], at 40 kVp (Mo anode) and a current of 70 mA.
Fig. 3
Fig. 3 Intensities are measured at sampling positions x k t [abscissa in (a)], which, due to limited positioning accuracy, deviate from intended, equidistant stepping points xk [abscissa in (b)]. Misattribution of the measured values to the stepping points xk lead to a deviation between true and fitted curve parameters (dashed and dotted lines).
Fig. 4
Fig. 4 Simulation of two non-equidistant stepping curves, followed by signal extraction with the assumption of equidistant sampling leads to errors in the three sample modalities, depending on the stepping curve phase (solid line). Knowing the exact stepping positions, the errors can be approximated with the results in Table 1. The step deviations ξk were independently drawn from a normal distribution with mean zero and standard deviation σ = 20% × 2π/n, where n = 7 is the total number of steps. It was assumed that Vs,t = Vf,t.
Fig. 5
Fig. 5 Effect of the fringe artifact removal algorithm on projection modalities. Left and right columns: Before and after application of the correction. (a), (b): transmittance, (c), (d): differential-phase (values in radians), (e), (f): dark-field of an example projection. Narrow windowing was used to highlight artifacts. This is one of 801 projections comprising the gbPC-CT scan shown in Fig. 6.
Fig. 6
Fig. 6 Effect of the fringe artifact removal algorithm on tomograms of refractive index decrement δ of a liquid phantom submerged in a water bath [(a), (b): without correction, (c), (d): with correction, (e), (f): difference]. The phantom consists of seven plastic tubes containing different liquids (center: water, from bottom left clockwise: 5% / 10% NaCl in water; 25% / 50% / 75% / 100% glycerol in ethanol). The gbPC-CT scan consists of 801 phase-stepping projections, each composed of 11 individual phase-stepping images and an exposure time of 3 s per step. The acquisition was performed at the setup from [13], at 40 kVp (Mo anode) and a current of 70 mA. To show the image’s dynamic range, but also highlight reconstruction artifacts, each image is shown with two different gray-value scales (top and bottom row). Simultaneous integration and filtered backprojection of differential-phase data was performed using an imaginary Hilbert filter [21, 22]. Since samples are immersed in a water bath, all flat-fields are acquired with the water bath in the beam. Therefore, the refractive index decrement δ is determined relative to water.

Tables (2)

Tables Icon

Table 1 Dependence of stepping-jitter-induced artifacts on stepping curve phase Φ 1 t and complex amplitudes A j. These amplitudes are coefficients of the discrete Fourier transform of the jitter vector ξ , see Eq. (35). A distinction between flat-field and sample phase-stepping is therefore necessary for T, φ, and D.

Tables Icon

Table 2 Standard deviations of relative artifact amplitudes, in agreement with the values presented by Revol et al. in [20].

Equations (43)

Equations on this page are rendered with MathJax. Learn more.

I s ( x ) = a 0 s + a 1 s cos ( x Φ 1 s ) ,
I f ( x ) = a 0 f + a 1 f cos ( x Φ 1 f ) .
T = a 0 s a 0 f , φ = Φ 1 s Φ 1 f , D = V s V f = a 1 s / a 0 s a 1 f / a 0 f .
y k t = a 0 t + a 1 t cos ( x k 1 Φ 1 t ) , k = 1 , , n .
y k = a 0 + a 1 cos ( x k Φ 1 ) = β 1 + β 2 cos x k + β 3 sin x k , k = 1 , , n .
β 1 a 0 ,
β 2 , β 3 a 1 cos ( Φ 1 ) , a 1 sin ( Φ 1 ) ,
a 1 , Φ 1 ( β 2 2 + β 3 2 ) 1 / 2 , arg ( β 2 + i β 3 ) .
β 1 , β 2 , β 3 = argmin β 1 , β 2 , β 3 k = 1 n [ y k ( β 1 , β 2 , β 3 ) y k t ] 2 .
Y j = { y t } j = 1 n k = 1 n exp ( 2 π i j k / n ) y k t = 1 n k = 1 n exp ( i j x k ) y k t .
β 1 = Y 0 = 1 n k = 1 n y k t ,
β 2 = 2 Re ( Y 1 ) = 2 n k = 1 n y k t cos x k ,
β 3 = 2 Im ( Y 1 ) = 2 n k = 1 n y k t sin x k .
β 1 ( x t ) = 1 n k = 1 n [ a 0 t + a 1 t cos ( x k t Φ 1 t ) ] ,
β 2 ( x t ) = 2 n k = 1 n [ a 0 t + a 1 t cos ( x k t Φ 1 t ) ] cos x k ,
β 3 ( x t ) = 2 n k = 1 n [ a 0 t + a 1 t cos ( x k t Φ 1 t ) ] sin x k .
δ β j = β j ( x t ) β j ( x )
k = 1 n β j x k t | x t = x ( x k t x k ) = ξ k
δ β 1 a 1 t n k = 1 n ξ k sin ( x k Φ 1 t ) ,
δ β 2 2 a 1 t n k = 1 n ξ k sin ( x k Φ 1 t ) cos x k ,
δ β 3 2 a 1 t n k = 1 n ξ k sin ( x k Φ 1 t ) sin x k .
δ a 1 k = 1 n a 1 x k t | x ξ k = k = 1 n ( a 1 β 2 | x β 2 x k t | x + a 1 β 3 | x β 3 x k t | x ) ξ k
= a 1 β 2 | x k = 1 n β 2 x k t | x ξ k + a 1 β 3 | x k = 1 n β 3 x k t | x ξ k
( 18 ) a 1 β 2 | x δ β 2 + a 1 β 3 | x δ β 3 .
δ Φ 1 Φ 1 β 2 | x δ β 2 + Φ 1 β 3 | x δ β 3 .
a 1 β 2 = β 2 ( β 2 2 + β 2 3 ) 1 / 2 = β 2 a 1 = cos Φ 1 , a 1 β 3 = β 3 ( β 2 2 + β 2 3 ) 1 / 2 = β 3 a 1 = sin Φ 1 ,
Φ 1 β 2 = β 3 β 2 2 + β 3 2 = β 3 a 1 2 = sin Φ 1 a 1 , Φ 1 β 3 = β 2 β 2 2 + β 3 2 = β 2 a 1 2 = cos Φ 1 a 1 .
a 1 β 2 | x = cos Φ 1 t , a 1 β 3 | x = sin Φ 1 t ,
Φ 1 β 2 | x = sin Φ 1 t a 1 t , Φ 1 β 3 | x = cos Φ 1 t a 1 t .
δ a 1 cos Φ 1 t × 2 a 1 t n k = 1 n ξ k sin ( x k Φ 1 t ) cos x k + sin Φ 1 t × 2 a 1 t n k = 1 n ξ k sin ( x k Φ 1 t ) sin x k = 2 a 1 t n k = 1 n ξ k sin ( x k Φ 1 t ) ( cos Φ 1 t cos x k + sin Φ 1 t sin x k ) = 2 a 1 t n k = 1 n ξ k sin ( x k Φ 1 t ) cos ( Φ 1 t x k ) δ a 1 a 1 t 1 n k = 1 n ξ k sin ( 2 Φ 1 t 2 x k ) .
δ Φ 1 = 1 n k = 1 n ξ k [ cos ( 2 Φ 1 t 2 x k ) 1 ] .
δ a 0 a 0 t a 1 t a 0 t Im [ A 1 exp ( i Φ 1 t ) ] = a 1 t a 0 t | A 1 | sin ( Φ 1 t + arg A 1 ) ,
δ a 1 a 1 t Im [ A 2 exp ( 2 i Φ 1 t ) ] = | A 2 | sin ( 2 Φ 1 t + arg A 2 ) ,
δ Φ 1 Re [ A 2 exp ( 2 i Φ 1 t ) A 0 ] = | A 2 | cos ( 2 Φ 1 t + arg A 2 ) A 0 ,
A j = 1 n k = 1 n ξ k exp ( i j x k ) = 1 n k = 1 n ξ k exp ( 2 π i j k / n ) .
δ φ = δ Φ 1 s δ Φ 1 f .
δ T = a 0 s , t + δ a 0 s a 0 f , t + δ a 0 f a 0 s a 0 f δ T T t δ a 0 s a 0 s , t δ a 0 f a 0 f , t ,
δ V = a 1 t + δ a 1 a 0 t + δ a 0 a 1 t a 0 t δ V V t δ a 1 a 1 t δ a 0 a 0 t ,
δ D = V s , t + δ V s V f , t + δ V f V s , t V f , t δ D D t δ V s V s , t δ V f V f , t δ a 1 s a 1 s , t δ a 1 f a 1 f , t + δ a 0 f a 0 f , t δ a 0 s a 0 s , t .
σ = σ mech 2 π p G ,
F [ s 0 , s 1 , c 1 , s 2 , c 2 ; Φ 1 ] = s 0 + s 1 sin ( Φ 1 ) + c 1 cos ( Φ 1 ) + s 2 sin ( 2 Φ 1 ) + c 2 cos ( 2 Φ 1 )
( s ^ 0 X , s ^ 1 X , c ^ 1 X , s ^ 2 X , c ^ 2 X ) = argmin s 0 , s 1 , c 1 , c 2 , c 2 p R { F [ s 0 , s 1 , c 1 , s 2 , c 2 ; Φ 1 ( p ) ] X ( p ) } 2
X corr ( p ) = X ( p ) F [ s ^ 0 , X , s ^ 1 X , c ^ 1 X , s ^ 2 X , c ^ 2 X ; Φ 1 ( p ) ] + s ^ 0 X p R 0

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