Abstract

Quite recently, a method has been presented to reconstruct X-ray scattering tensors from projections obtained in a grating interferometry setup. The original publications present a rather specialised approach, for instance by suggesting a single SART-based solver. In this work, we propose a novel approach to solving the inverse problem, allowing the use of other algorithms than SART (like conjugate gradient), a faster tensor recovery, and an intuitive visualisation. Furthermore, we introduce constraint enforcement for X-ray tensor tomography (cXTT) and demonstrate that this yields visually smoother results in comparison to the state-of-art approach, similar to regularisation.

© 2015 Optical Society of America

Full Article  |  PDF Article
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References

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    [Crossref]
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  20. Q. Li and J. G. Griffiths, “Least squares ellipsoid specific fitting,” Proc. Geom. Model. Process.335–340 (2004).
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  23. C. M. Bishop, Pattern Recognition and Machine Learning (Springer, 2006), chap. 12.1.
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    [Crossref]
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  27. C. R. Tench, P. S. Morgan, M. Wilson, and L. D. Blumhardt, “White matter mapping using diffusion tensor MRI,” Magn. Reson. Med. 47, 967–972 (2002).
    [Crossref] [PubMed]
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2014 (3)

F. L. Bayer, S. Hu, A. Maier, T. Weber, G. Anton, T. Michel, and C. P. Riess, “Reconstruction of scalar and vectorial components in X-ray dark-field tomography,” Proc. Natl. Acad. Sci. U.S.A. 111, 12699–12704 (2014).
[PubMed]

A. Fehringer, T. Lasser, I. Zanette, P. B. Noël, and F. Pfeiffer, “A versatile tomographic forward- and backprojection approach on multi-GPUs,” Proc. SPIE 9034, 90344F (2014).

H. H. B. Sørensen and P. C. Hansen, “Multicore performance of block algebraic iterative reconstruction methods,” SIAM J. Sci. Comp. 36, C524–C546 (2014).
[Crossref]

2013 (2)

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, e61268 (2013).
[Crossref] [PubMed]

A. Yaroshenko, F. G. Meinel, M. Bech, A. Tapfer, A. Velroyen, S. Schleede, S. Auweter, A. Bohla, A. Ö. Yildirim, K. Nikolaou, F. Bamberg, O. Eickelberg, M. F. Reiser, and F. Pfeiffer, “Pulmonary emphysema diagnosis with a preclinical small-animal X-ray dark-field scatter-contrast scanner,” Radiol. 269, 427–433 (2013).

2012 (1)

V. Revol, C. Kottler, R. Kaufmann, A. Neels, and A. Dommann, “Orientation-selective X-ray dark field imaging of ordered systems,” J. Appl. Phys. 112, 114903 (2012).
[Crossref]

2011 (2)

V. Revol, I. Jerjen, C. Kottler, P. Schütz, R. Kaufmann, T. Lüthi, U. Sennhauser, U. Straumann, and C. Urban, “Sub-pixel porosity revealed by X-ray scatter dark field imaging,” J. Appl. Phys. 110, 044912 (2011).

S. Sidhu, G. Falzon, S. A. Hart, J. G. Fox, R. A. Lewis, and K. K. W. Siu, “Classification of breast tissue using a laboratory system for small-angle x-ray scattering (SAXS),” Phys. Med. Biol. 56, 6779–6791 (2011).
[PubMed]

2010 (2)

F. Arfelli, L. Rigon, and R. H. Menk, “Microbubbles as X-ray scattering contrast agents using analyzer-based imaging,” Phys. Med. Biol. 55, 1643–1658 (2010).
[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]

2009 (1)

A. Filler, “Magnetic resonance neurography and diffusion tensor imaging: Origins, history, and clinical impact of the first 50 000 cases with an assessment of efficacy and utility in a prospective 5000-patient study group,” Neurosurg. 65, A29–A43 (2009).

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).
[PubMed]

2006 (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).

2005 (1)

M. Ando, K. Yamasaki, F. Toyofuku, H. Sugiyama, C. Ohbayashi, G. Li, L. Pan, X. Jiang, W. Pattanasiriwisawa, D. Shimao, E. Hashimoto, T. Kimura, M. Tsuneyoshi, E. Ueno, K. Tokumori, A. Maksimenko, Y. Higashida, and M. Hirano, “Attempt at visualizing breast cancer with X-ray dark field imaging,” Jpn. J. Appl. Phys. 44, L528–L531 (2005).

2002 (1)

C. R. Tench, P. S. Morgan, M. Wilson, and L. D. Blumhardt, “White matter mapping using diffusion tensor MRI,” Magn. Reson. Med. 47, 967–972 (2002).
[Crossref] [PubMed]

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]

1984 (1)

A. H. Andersen and A. C. Kak, “Simultaneous algebraic reconstruction technique (SART): A superior implementation of the ART algorithm,” Ultrasonic Imaging 6, 81–94 (1984).
[Crossref] [PubMed]

1952 (1)

M. R. Hestenes and E. Stiefel, “Methods of conjugate gradients for solving linear systems,” J. Res. Nat. Bur. Stand. 49, 409–436 (1952).
[Crossref]

1933 (1)

H. Hotelling, “Analysis of a complex of statistical variables into principal components,” J. Educ. Psychol. 24, 417–441 (1933).
[Crossref]

1901 (2)

M. W. Kutta, “Beitrag zur näherungsweisen Integration totaler Differentialgleichungen [Contribution to the approximate integration of total differential equations],” Z. Math. Phys. 46, 435–453 (1901).

K. Pearson, “On lines and planes of closes fit to systems of points in space,” Philos. Mag. 2, 559–572 (1901).
[Crossref]

1895 (1)

C. Runge, “Über die numerische Auflösung von Differentialgleichungen [About numerically solving differential equations],” Math. Ann. 46, 167–178 (1895).
[Crossref]

Andersen, A. H.

A. H. Andersen and A. C. Kak, “Simultaneous algebraic reconstruction technique (SART): A superior implementation of the ART algorithm,” Ultrasonic Imaging 6, 81–94 (1984).
[Crossref] [PubMed]

Ando, M.

M. Ando, K. Yamasaki, F. Toyofuku, H. Sugiyama, C. Ohbayashi, G. Li, L. Pan, X. Jiang, W. Pattanasiriwisawa, D. Shimao, E. Hashimoto, T. Kimura, M. Tsuneyoshi, E. Ueno, K. Tokumori, A. Maksimenko, Y. Higashida, and M. Hirano, “Attempt at visualizing breast cancer with X-ray dark field imaging,” Jpn. J. Appl. Phys. 44, L528–L531 (2005).

Anton, G.

F. L. Bayer, S. Hu, A. Maier, T. Weber, G. Anton, T. Michel, and C. P. Riess, “Reconstruction of scalar and vectorial components in X-ray dark-field tomography,” Proc. Natl. Acad. Sci. U.S.A. 111, 12699–12704 (2014).
[PubMed]

Arfelli, F.

F. Arfelli, L. Rigon, and R. H. Menk, “Microbubbles as X-ray scattering contrast agents using analyzer-based imaging,” Phys. Med. Biol. 55, 1643–1658 (2010).
[PubMed]

Auweter, S.

A. Yaroshenko, F. G. Meinel, M. Bech, A. Tapfer, A. Velroyen, S. Schleede, S. Auweter, A. Bohla, A. Ö. Yildirim, K. Nikolaou, F. Bamberg, O. Eickelberg, M. F. Reiser, and F. Pfeiffer, “Pulmonary emphysema diagnosis with a preclinical small-animal X-ray dark-field scatter-contrast scanner,” Radiol. 269, 427–433 (2013).

Bamberg, F.

A. Yaroshenko, F. G. Meinel, M. Bech, A. Tapfer, A. Velroyen, S. Schleede, S. Auweter, A. Bohla, A. Ö. Yildirim, K. Nikolaou, F. Bamberg, O. Eickelberg, M. F. Reiser, and F. Pfeiffer, “Pulmonary emphysema diagnosis with a preclinical small-animal X-ray dark-field scatter-contrast scanner,” Radiol. 269, 427–433 (2013).

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, e61268 (2013).
[Crossref] [PubMed]

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, e61268 (2013).
[Crossref] [PubMed]

Bayer, F. L.

F. L. Bayer, S. Hu, A. Maier, T. Weber, G. Anton, T. Michel, and C. P. Riess, “Reconstruction of scalar and vectorial components in X-ray dark-field tomography,” Proc. Natl. Acad. Sci. U.S.A. 111, 12699–12704 (2014).
[PubMed]

Bech, M.

A. Yaroshenko, F. G. Meinel, M. Bech, A. Tapfer, A. Velroyen, S. Schleede, S. Auweter, A. Bohla, A. Ö. Yildirim, K. Nikolaou, F. Bamberg, O. Eickelberg, M. F. Reiser, and F. Pfeiffer, “Pulmonary emphysema diagnosis with a preclinical small-animal X-ray dark-field scatter-contrast scanner,” Radiol. 269, 427–433 (2013).

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).
[PubMed]

Biernath, 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, e61268 (2013).
[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,” EPL105 (2014).

Bishop, C. M.

C. M. Bishop, Pattern Recognition and Machine Learning (Springer, 2006), chap. 12.1.

Blumhardt, L. D.

C. R. Tench, P. S. Morgan, M. Wilson, and L. D. Blumhardt, “White matter mapping using diffusion tensor MRI,” Magn. Reson. Med. 47, 967–972 (2002).
[Crossref] [PubMed]

Bohla, A.

A. Yaroshenko, F. G. Meinel, M. Bech, A. Tapfer, A. Velroyen, S. Schleede, S. Auweter, A. Bohla, A. Ö. Yildirim, K. Nikolaou, F. Bamberg, O. Eickelberg, M. F. Reiser, and F. Pfeiffer, “Pulmonary emphysema diagnosis with a preclinical small-animal X-ray dark-field scatter-contrast scanner,” Radiol. 269, 427–433 (2013).

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).
[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).
[PubMed]

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).

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).
[PubMed]

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).

Dommann, A.

V. Revol, C. Kottler, R. Kaufmann, A. Neels, and A. Dommann, “Orientation-selective X-ray dark field imaging of ordered systems,” J. Appl. Phys. 112, 114903 (2012).
[Crossref]

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]

Edwards, C. H.

C. H. Edwards and D. E. Penney, Differential Equations (Pearson, 2007), chap. 4.3.

Eggl, 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, e61268 (2013).
[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,” EPL105 (2014).

Eickelberg, O.

A. Yaroshenko, F. G. Meinel, M. Bech, A. Tapfer, A. Velroyen, S. Schleede, S. Auweter, A. Bohla, A. Ö. Yildirim, K. Nikolaou, F. Bamberg, O. Eickelberg, M. F. Reiser, and F. Pfeiffer, “Pulmonary emphysema diagnosis with a preclinical small-animal X-ray dark-field scatter-contrast scanner,” Radiol. 269, 427–433 (2013).

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).
[PubMed]

Falzon, G.

S. Sidhu, G. Falzon, S. A. Hart, J. G. Fox, R. A. Lewis, and K. K. W. Siu, “Classification of breast tissue using a laboratory system for small-angle x-ray scattering (SAXS),” Phys. Med. Biol. 56, 6779–6791 (2011).
[PubMed]

Fehringer, A.

A. Fehringer, T. Lasser, I. Zanette, P. B. Noël, and F. Pfeiffer, “A versatile tomographic forward- and backprojection approach on multi-GPUs,” Proc. SPIE 9034, 90344F (2014).

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]

Filler, A.

A. Filler, “Magnetic resonance neurography and diffusion tensor imaging: Origins, history, and clinical impact of the first 50 000 cases with an assessment of efficacy and utility in a prospective 5000-patient study group,” Neurosurg. 65, A29–A43 (2009).

Fox, J. G.

S. Sidhu, G. Falzon, S. A. Hart, J. G. Fox, R. A. Lewis, and K. K. W. Siu, “Classification of breast tissue using a laboratory system for small-angle x-ray scattering (SAXS),” Phys. Med. Biol. 56, 6779–6791 (2011).
[PubMed]

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,” EPL105 (2014).

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, e61268 (2013).
[Crossref] [PubMed]

Griffiths, J. G.

Q. Li and J. G. Griffiths, “Least squares ellipsoid specific fitting,” Proc. Geom. Model. Process.335–340 (2004).

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).
[PubMed]

Hansen, P. C.

H. H. B. Sørensen and P. C. Hansen, “Multicore performance of block algebraic iterative reconstruction methods,” SIAM J. Sci. Comp. 36, C524–C546 (2014).
[Crossref]

Hart, S. A.

S. Sidhu, G. Falzon, S. A. Hart, J. G. Fox, R. A. Lewis, and K. K. W. Siu, “Classification of breast tissue using a laboratory system for small-angle x-ray scattering (SAXS),” Phys. Med. Biol. 56, 6779–6791 (2011).
[PubMed]

Hashimoto, E.

M. Ando, K. Yamasaki, F. Toyofuku, H. Sugiyama, C. Ohbayashi, G. Li, L. Pan, X. Jiang, W. Pattanasiriwisawa, D. Shimao, E. Hashimoto, T. Kimura, M. Tsuneyoshi, E. Ueno, K. Tokumori, A. Maksimenko, Y. Higashida, and M. Hirano, “Attempt at visualizing breast cancer with X-ray dark field imaging,” Jpn. J. Appl. Phys. 44, L528–L531 (2005).

Hestenes, M. R.

M. R. Hestenes and E. Stiefel, “Methods of conjugate gradients for solving linear systems,” J. Res. Nat. Bur. Stand. 49, 409–436 (1952).
[Crossref]

Higashida, Y.

M. Ando, K. Yamasaki, F. Toyofuku, H. Sugiyama, C. Ohbayashi, G. Li, L. Pan, X. Jiang, W. Pattanasiriwisawa, D. Shimao, E. Hashimoto, T. Kimura, M. Tsuneyoshi, E. Ueno, K. Tokumori, A. Maksimenko, Y. Higashida, and M. Hirano, “Attempt at visualizing breast cancer with X-ray dark field imaging,” Jpn. J. Appl. Phys. 44, L528–L531 (2005).

Hirano, M.

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Supplementary Material (3)

» Media 1: MOV (14563 KB)     
» Media 2: MOV (23374 KB)     
» Media 3: MOV (35264 KB)     

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Figures (16)

Fig. 1
Fig. 1 Sketch of an X-ray grating interferometry setup. X-ray tube (T), source grating (G0), shifting interferometer grating (G1), specimen (S), static interferometer grating (G2), and detector (D).
Fig. 2
Fig. 2 Three components of a sample projection of a tooth: Absorption, phase contrast and dark field. The contrast of these images has been manually improved for better visibility, and the images have been cropped. The structure in the lower-left quadrant is the sample holder.
Fig. 3
Fig. 3 Expected X-ray scattering (blue ellipsoids) at fibre- or tube-like structures (grey). Vice versa, we will interpret the smallest half-axes of reconstructed scattering ellipsoids as fibre directions for visualisation.
Fig. 4
Fig. 4 Sampling directions, and ellipsoid fitting for a single voxel. Usually, we use K = 13 directions for reconstruction, but limit ourselves to K = 7 in this sketch, for clarity. They consist of the standard base vectors – red, green, blue in (a) for x, y, z, respectively – and diagonals – black. For a given voxel x, reconstruction yields a scattering coefficient ζk(x) for every sampling direction ε̂k. These are indicated by bold black marks in (b), and we mirror them along the negative sampling direction, yielding the small black marks. Finally, an ellipsoid can be fitted to that scaled, mirrored ‘bouquet’ afterwards, see (c).
Fig. 5
Fig. 5 Different variants of the same data, a downsampled detail from an actual result. Reconstruction is performed in terms of scaling the sampling directions (a), the scattering tensors are obtained by retrospectively fitting ellipsoids in a voxel-wise fashion (b). The colours indicate the directions of the smallest half-axes, see Fig. 8(a).
Fig. 6
Fig. 6 Viewing directions and Euler cradle. Only the perspectives marked by red lines are in the ‘normal’ viewing plane as used in standard CT applications. Blue lines mark additional off-plane perspectives. The dashed black line indicates the trajectory of the X-ray source-detector “camera” around the green cube, representing a specimen. The coverage gaps are due to limitations imposed by the Euler cradle.
Fig. 7
Fig. 7 Constraint enforcement. In both cases, the reconstructed coefficients are forced to be close to the manifold of valid ellipsoids while iterating. The hard constraint (a) projects back onto ellipsoids directly, where a – in this toy example exaggerated – scattering coefficient (dotted black line) is shortened appropriately (solid black line). The soft constraint smoothes the coefficients with respect to the other sampling directions of the same voxel respecting the angular relation, thus favouring ellipsoids in a relaxed sense. This is done using a Gaussian smoothing kernel based on the scalar product as shown in (b), ranging between 0 (blue) and 1 (yellow), before normalisation.
Fig. 8
Fig. 8 Colour coding of streamline visualisation. We use colours sampled from a sphere (a) to give visual cues about the orientation of the ‘fibres’. Note that the colour ball is symmetric with respect to the x-y-, x-z-, and y-z-planes. In a sample picture of the carbon knot (b), looking down from a slightly elevated perspective, vertically (green), horizontally (red) and obliquely (orange and yellow) oriented parts can be easily distinguished.
Fig. 9
Fig. 9 Photographies of the samples. Knot (a), branch (b), and tooth (c).
Fig. 10
Fig. 10 Wide-angle view of the actual setup. From left to right: X-ray source (T) with source grating (G0) directly in front of it, at the center the phase grating (G1), then the Euler cradle with sample (S), and adjacent to it the analyser grating (G2, in the cross-shaped holder); behind the latter the X-ray detector (D, hidden).
Fig. 11
Fig. 11 Behaviour of the algorithm for the knot. The plots show the normalised residual norm r(q) as defined in Eq. (21) over iterations q (left) and the normalised mean update Δ(q) as defined in Eq. (22) over iterations q (right). The unconstrained version yields smaller residuals, but the updates are noisy.
Fig. 12
Fig. 12 Comparison of attenuation reconstruction (a), Malecki’s tensor reconstruction (b), and our (unconstrained) tensor reconstruction (c). The first image is included for reference, to show that scattering data is of a considerably different nature than usual attenuation reconstructions. The two scattering reconstructions are largely equivalent, considering the different algorithms with different parameters.
Fig. 13
Fig. 13 Volume renderings of the knot’s scattering coefficients for sampling direction ε̂k = [0, 0, 1]T. The unconstrained reconstruction (a) shows strong streak artefacts, the two constrained versions (b) and (c) are much clearer.
Fig. 14
Fig. 14 Streamline visualisation of the knot’s scattering ellipsoids. (Also see Media 1.) The streamlines are supposed to follow the directions of the carbon fibres, see Fig. 9(a), but are not intended to accurately reconstruct individual fibres. The unconstrained version (a) shows noise, the two constrained versions (b) and (c) are visually considerably smoother. The ‘waves’ in the lower right appear to be additional scattering caused by the sample holder.
Fig. 15
Fig. 15 Streamline visualisation of the branch. (Also see Media 2.) Scattering is supposed to be caused by the tiny tubular vessels embedded in wood that transport water towards the leaves. The streamlines are thus supposed to run mainly in parallel to the individual branches. The unconstrained version (a) fails to recover useful scattering along the main branch. The lightly constrained versions (b) and (c) show more reasonable scattering there. The hard constraint overshoots and produces wavy patterns along the main branch, and introduces a ‘combing’ effect for the other branches. The latter can already be seen in (c).
Fig. 16
Fig. 16 Reconstruction results of the tooth. (Also see Media 3.) Volume rendering of X-ray attenuation (a) showing enamel (white) and dentine (gray). Equivalent attenuation volume rendering showing dentine only (b). Scattering streamlines obtained without constraint (c), with soft constraint with μ = 0.08 (d) and μ = 0.10 (e), and with hard constraint (f). Scattering is caused by dentinal tubules, tiny structures in radial direction, as indicated by the streamlines. The pulp chamber was obtained by masking.

Tables (1)

Tables Icon

Alg. 1 Generic tomographic X-ray tensor reconstruction. A denotes the system matrix describing the imaging process, Dk a scaling matrix containing weighting factors as defined in Eq. (2). m is the measurement vector, and η k ( q ) a vector containing the qth iterate of the voxel-wise squared scattering coefficients corresponding to sampling direction k. approximate is a function running a single iteration of an arbitrary iterative linear solver.

Equations (22)

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d j = exp [ L j k | I ^ j × ε ^ k | ( ζ k ( x ) ε ^ k ) , t ^ j 2 d x ]
v k j : = ( | I ^ j × ε ^ k | ε ^ k , t ^ j ) 2
ln d j = L j k v k j η k ( x ) d x
= k v k j L j η k ( x ) d x
m j = ln d j = k v k j a j , η k = k v k j a j T η k
m = ( v 11 a 1 T v 12 a 2 T v 1 J a J T ) η 1 + ( v 21 a 1 T v 22 a 2 T v 2 J a J T ) η 2 + + ( v K 1 a 1 T v K 2 a 2 T v K J a J T ) η K
= ( v 11 v 12 v 1 J ) ( a 1 T a 2 T a J T ) η 1 + ( v 21 v 22 v 2 J ) ( a 1 T a 2 T a J T ) η 2 +
= D 1 A η 1 + D 2 A η 2 + + D K A η K = k D k A η k
= ( D 1 A , D 2 A , , D K A ) ( η 1 η 2 η K )
H s
( D k A ) t k ( q ) = m ˜ k ( q 1 )
m ˜ k ( q 1 ) = m l k D l A η l ( q 1 )
η k ( q ) = K 1 K η k ( q 1 ) + 1 K t k ( q ) .
ζ k ( x ) = | η k ( x ) |
S i : = { ± | η 1 ( x i ) | ε ^ 1 , ± | η 2 ( x i ) | ε ^ 2 , }
= { ± ζ 1 ( x 1 ) ε ^ 1 , ± ζ 2 ( x i ) ε ^ 2 , }
V i Λ i = C i V i
σ i , k 2 [ ( x i , k r i , 1 ) 2 + ( y i , k r i , 2 ) 2 + ( z i , k r i , 3 ) 2 ] = 1 .
η i , k ( q ) = σ i , k 2
η i , k ( q ) = g k T , [ η i , 1 ( q ) , , η i , K ( q ) ] .
r ( q ) : = m k D k A η k ( q ) 2 / m 2
Δ ( q ) : = mean k η k ( q ) η k ( q 1 ) 2 / η k ( q ) 2 .

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