Abstract

We present a fast algorithm for fully 3D regularized X-ray tomography reconstruction for both traditional and differential phase contrast measurements. In many applications, it is critical to reduce the X-ray dose while producing high-quality reconstructions. Regularization is an excellent way to do this, but in the differential phase contrast case it is usually applied in a slice-by-slice manner. We propose using fully 3D regularization to improve the dose/quality trade-off beyond what is possible using slice-by-slice regularization. To make this computationally feasible, we show that the two computational bottlenecks of our iterative optimization process can be expressed as discrete convolutions; the resulting algorithms for computation of the X-ray adjoint and normal operator are fast and simple alternatives to regridding. We validate this algorithm on an analytical phantom as well as conventional CT and differential phase contrast measurements from two real objects. Compared to the slice-by-slice approach, our algorithm provides a more accurate reconstruction of the analytical phantom and better qualitative appearance on one of the two real datasets.

© 2016 Optical Society of America

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References

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2016 (1)

F. Arcadu, M. Nilchian, A. Studer, M. Stampanoni, and F. Marone, “A forward regridding method with minimal oversampling for accurate and efficient iterative tomographic algorithms,” IEEE Trans. Image Process. 25(3), 1207–1218 (2016).
[Crossref]

2015 (5)

M. McGaffin and J. Fessler, “Alternating dual updates algorithm for X-ray CT reconstruction on the GPU,” IEEE Trans. Comput. Imaging 1(3), 186–199 (2015).
[Crossref]

H. Nien and J. Fessler, “Fast X-ray CT image reconstruction using a linearized augmented lagrangian method with ordered subsets,” IEEE Trans. Med. Imaging 34(2), 388–399 (2015).
[Crossref]

F. Momey, L. Denis, C. Burnier, E. Thiebaut, J.-M. Becker, and L. Desbat, “Spline driven: high accuracy projectors for tomographic reconstruction from few projections,” IEEE Trans. Image Process. 24(12), 4715–4725 (2015).
[Crossref] [PubMed]

M. Nilchian, J. Ward, C. Vonesch, and M. Unser, “Optimized Kaiser-Bessel window functions for computed tomography,” IEEE Trans. Image Process. 24(11), 3826–3833 (2015).
[Crossref] [PubMed]

J. Fu, X. Hu, A. Velroyen, M. Bech, M. Jiang, and F. Pfeiffer, “3D algebraic iterative reconstruction for cone-beam X-ray differential phase-contrast computed tomography,” PLoS One 10(3), e0117502 (2015).
[Crossref] [PubMed]

2014 (1)

E. Maire and P. J. Withers, “Quantitative X-ray tomography,” Int. Mater. Rev. 59(1), 1–43 (2014).
[Crossref]

2013 (2)

S. Lefkimmiatis, J. P. Ward, and M. Unser, “Hessian schatten-norm regularization for linear inverse problems,” IEEE Trans. Image Process. 22(5), 1873–1888 (2013).
[Crossref] [PubMed]

M. Nilchian, C. Vonesch, P. Modregger, M. Stampanoni, and M. Unser, “Fast iterative reconstruction of differential phase contrast X-ray tomograms,” Opt. Express 21(5), 5511–5528 (2013).
[Crossref] [PubMed]

2012 (3)

Q. Xu, E. Y. Sidky, X. Pan, M. Stampanoni, P. Modregger, and M. A. Anastasio, “Investigation of discrete imaging models and iterative image reconstruction in differential X-ray phase-contrast tomography,” Opt. Express 20(10), 10724–10749 (2012).
[Crossref] [PubMed]

A. Entezari, M. Nilchian, and M. Unser, “A box spline calculus for the discretization of computed tomography reconstruction problems,” IEEE Trans. Med. Imaging 31(8), 1532–1541 (2012).
[Crossref] [PubMed]

M. Beister, D. Kolditz, and W. A. Kalender, “Iterative reconstruction methods in X-ray CT,” Physica Med. 28(2), 94–108 (2012).
[Crossref]

2011 (5)

K. Zhang, Y. Hong, P. Zhu, Q. Yuan, W. Huang, Z. Wang, S. Chu, S. A. McDonald, F. Marone, M. Stampanoni, and Z. Wu, “Study of OSEM with different subsets in grating-based X-ray differential phase-contrast imaging,” Anal. Bioanal. Chem. 401(3), 837–844 (2011).
[Crossref] [PubMed]

T. Khler, B. Brendel, and E. Roessl, “Iterative reconstruction for differential phase contrast imaging using spherically symmetric basis functions,” Med. Phys. 38(8), 4542–4545 (2011).
[Crossref]

M. Stampanoni, Z. Wang, T. Thring, C. David, E. Roessl, M. Trippel, R. A. Kubik-Huch, G. Singer, M. K. Hohl, and N. Hauser, “The first analysis and clinical evaluation of native breast tissue using differential phase-contrast mammography,” Invest. Radiol. 46(12), 801–806 (2011).
[Crossref] [PubMed]

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3(1), 1–122 (2011).
[Crossref]

I. Tosic and P. Frossard, “Dictionary learning,” IEEE Signal. Proc. Mag. 28(2), 27–38 (2011).
[Crossref]

2010 (1)

A. Sakdinawat and D. Attwood, “Nanoscale X-ray imaging,” Nat. Photonics 4(12), 840–848 (2010).
[Crossref]

2007 (3)

R. Fitzgerald, “Phase-sensitive X-ray imaging,” Phys. Today 53(7), 23–26 (2007).
[Crossref]

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., Sect. A 580(2), 925–928 (2007).
[Crossref]

J. A. Fessler, “On NUFFT-based gridding for non-Cartesian MRI,” J. Magn. Reson. 188(2), 191–195 (2007).
[Crossref] [PubMed]

2006 (1)

B. M. Weon, J. H. Je, Y. Hwu, and G. Margaritondo, “Phase contrast X-ray imaging,” Int. J. Nanotechnol. 3(2–3), 280–297 (2006).
[Crossref]

2005 (1)

2003 (1)

M. A. T. Figueiredo and R. D. Nowak, “An EM algorithm for wavelet-based image restoration,” IEEE Trans. Image Process. 12(8), 906–916 (2003).
[Crossref]

2002 (2)

J.-L. Starck, E. J. Candes, and D. L. Donoho, “The curvelet transform for image denoising,” IEEE Trans. Image Process. 11(6), 670–684 (2002).
[Crossref]

S. Horbelt, M. Liebling, and M. Unser, “Discretization of the radon transform and of its inverse by spline convolutions,” IEEE Trans. Med. Imaging 21(4), 363–376 (2002).
[Crossref] [PubMed]

1997 (1)

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmr, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced X-ray imaging,” Phys. Med. Biol. 42(11), 2015–2025 (1997).
[Crossref] [PubMed]

1996 (4)

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[Crossref] [PubMed]

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, “Phase-contrast imaging using polychromatic hard X-rays,” Nature 384(6607), 335–338 (1996).
[Crossref]

A. Momose, T. Takeda, Y. Itai, and K. Hirano, “Phase-contrast X-ray computed tomography for observing biological soft tissues,” Nat. Med. 2(4), 473–475 (1996).
[Crossref] [PubMed]

A. Delaney and Y. Bresler, “A fast and accurate Fourier algorithm for iterative parallel-beam tomography,” IEEE Trans. Image Process. 5(5), 740–753 (1996).
[Crossref] [PubMed]

1995 (3)

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of X-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66(12), 5486–5492 (1995).
[Crossref]

V. N. Ingal and E. A. Beliaevskaya, “X-ray plane-wave topography observation of the phase contrast from a non-crystalline object,” J. Phys. D: Appl. Phys. 28(11), 2314–2317 (1995).
[Crossref]

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature 373(6515), 595–598 (1995).
[Crossref]

1990 (1)

R. M. Lewitt, “Multidimensional digital image representations using generalized Kaiser-Bessel window functions,” J. Opt. Soc. Am. A. 7(10), 1834–1836 (1990).
[Crossref] [PubMed]

1980 (1)

L. A. Shepp, “Computerized tomography and nuclear magnetic resonance,” J. Comput. Assisted Tomogr. 4(1), 94–107 (1980).
[Crossref]

1965 (1)

U. Bonse and M. Hart, “An X-ray interferometer,” Appl. Phys. Lett. 6(8), 155–156 (1965).
[Crossref]

Anastasio, M. A.

Arcadu, F.

F. Arcadu, M. Nilchian, A. Studer, M. Stampanoni, and F. Marone, “A forward regridding method with minimal oversampling for accurate and efficient iterative tomographic algorithms,” IEEE Trans. Image Process. 25(3), 1207–1218 (2016).
[Crossref]

Arfelli, F.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmr, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced X-ray imaging,” Phys. Med. Biol. 42(11), 2015–2025 (1997).
[Crossref] [PubMed]

Attwood, D.

A. Sakdinawat and D. Attwood, “Nanoscale X-ray imaging,” Nat. Photonics 4(12), 840–848 (2010).
[Crossref]

Barnea, Z.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[Crossref] [PubMed]

Bech, M.

J. Fu, X. Hu, A. Velroyen, M. Bech, M. Jiang, and F. Pfeiffer, “3D algebraic iterative reconstruction for cone-beam X-ray differential phase-contrast computed tomography,” PLoS One 10(3), e0117502 (2015).
[Crossref] [PubMed]

Becker, J.-M.

F. Momey, L. Denis, C. Burnier, E. Thiebaut, J.-M. Becker, and L. Desbat, “Spline driven: high accuracy projectors for tomographic reconstruction from few projections,” IEEE Trans. Image Process. 24(12), 4715–4725 (2015).
[Crossref] [PubMed]

Beister, M.

M. Beister, D. Kolditz, and W. A. Kalender, “Iterative reconstruction methods in X-ray CT,” Physica Med. 28(2), 94–108 (2012).
[Crossref]

Beliaevskaya, E. A.

V. N. Ingal and E. A. Beliaevskaya, “X-ray plane-wave topography observation of the phase contrast from a non-crystalline object,” J. Phys. D: Appl. Phys. 28(11), 2314–2317 (1995).
[Crossref]

Bevins, N.

Z. Qi, J. Zambelli, N. Bevins, and G.-H. Chen, “A novel method to reduce data acquisition time in differential phase contrast: computed tomography using compressed sensing,” in SPIE Proceedings, vol. 7258, pp. 72,584A–72,584A–8 (Lake Buena Vista, Florida, United States, 2009).
[Crossref]

Bonse, U.

U. Bonse and M. Hart, “An X-ray interferometer,” Appl. Phys. Lett. 6(8), 155–156 (1965).
[Crossref]

Boyd, S.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3(1), 1–122 (2011).
[Crossref]

Brendel, B.

T. Khler, B. Brendel, and E. Roessl, “Iterative reconstruction for differential phase contrast imaging using spherically symmetric basis functions,” Med. Phys. 38(8), 4542–4545 (2011).
[Crossref]

Bresler, Y.

A. Delaney and Y. Bresler, “A fast and accurate Fourier algorithm for iterative parallel-beam tomography,” IEEE Trans. Image Process. 5(5), 740–753 (1996).
[Crossref] [PubMed]

Bunk, O.

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., Sect. A 580(2), 925–928 (2007).
[Crossref]

Burnier, C.

F. Momey, L. Denis, C. Burnier, E. Thiebaut, J.-M. Becker, and L. Desbat, “Spline driven: high accuracy projectors for tomographic reconstruction from few projections,” IEEE Trans. Image Process. 24(12), 4715–4725 (2015).
[Crossref] [PubMed]

Candes, E. J.

J.-L. Starck, E. J. Candes, and D. L. Donoho, “The curvelet transform for image denoising,” IEEE Trans. Image Process. 11(6), 670–684 (2002).
[Crossref]

Chapman, D.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmr, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced X-ray imaging,” Phys. Med. Biol. 42(11), 2015–2025 (1997).
[Crossref] [PubMed]

Chen, G.-H.

Z. Qi, J. Zambelli, N. Bevins, and G.-H. Chen, “A novel method to reduce data acquisition time in differential phase contrast: computed tomography using compressed sensing,” in SPIE Proceedings, vol. 7258, pp. 72,584A–72,584A–8 (Lake Buena Vista, Florida, United States, 2009).
[Crossref]

Chu, E.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3(1), 1–122 (2011).
[Crossref]

Chu, S.

K. Zhang, Y. Hong, P. Zhu, Q. Yuan, W. Huang, Z. Wang, S. Chu, S. A. McDonald, F. Marone, M. Stampanoni, and Z. Wu, “Study of OSEM with different subsets in grating-based X-ray differential phase-contrast imaging,” Anal. Bioanal. Chem. 401(3), 837–844 (2011).
[Crossref] [PubMed]

Cloetens, P.

Cookson, D. F.

K. A. Nugent, T. E. Gureyev, D. F. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996).
[Crossref] [PubMed]

David, C.

M. Stampanoni, Z. Wang, T. Thring, C. David, E. Roessl, M. Trippel, R. A. Kubik-Huch, G. Singer, M. K. Hohl, and N. Hauser, “The first analysis and clinical evaluation of native breast tissue using differential phase-contrast mammography,” Invest. Radiol. 46(12), 801–806 (2011).
[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., Sect. A 580(2), 925–928 (2007).
[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(16), 6296–6304 (2005).
[Crossref] [PubMed]

Davis, T. J.

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, “Phase-contrast imaging of weakly absorbing materials using hard X-rays,” Nature 373(6515), 595–598 (1995).
[Crossref]

Delaney, A.

A. Delaney and Y. Bresler, “A fast and accurate Fourier algorithm for iterative parallel-beam tomography,” IEEE Trans. Image Process. 5(5), 740–753 (1996).
[Crossref] [PubMed]

Denis, L.

F. Momey, L. Denis, C. Burnier, E. Thiebaut, J.-M. Becker, and L. Desbat, “Spline driven: high accuracy projectors for tomographic reconstruction from few projections,” IEEE Trans. Image Process. 24(12), 4715–4725 (2015).
[Crossref] [PubMed]

Desbat, L.

F. Momey, L. Denis, C. Burnier, E. Thiebaut, J.-M. Becker, and L. Desbat, “Spline driven: high accuracy projectors for tomographic reconstruction from few projections,” IEEE Trans. Image Process. 24(12), 4715–4725 (2015).
[Crossref] [PubMed]

Diaz, A.

Donoho, D. L.

J.-L. Starck, E. J. Candes, and D. L. Donoho, “The curvelet transform for image denoising,” IEEE Trans. Image Process. 11(6), 670–684 (2002).
[Crossref]

Eckstein, J.

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Figures (13)

Fig. 1
Fig. 1

Geometry of X-ray projections. (a)–(c) A function of 3D space, f, (grey-green) and an X-ray projection plane spanned by the columns of P θ T (pink), shown from two orthogonal views and in perspective. (d) The X-ray projection of f onto the pictured plane plotted in the coordinate system specified by P θ T. In this geometry, x2 = y1.

Fig. 2
Fig. 2

High-level description of the proposed algorithm.

Fig. 3
Fig. 3

Accuracy (left) and computation time (right) vs problem size for our fast algorithm for HT g with upsamping rates of 1, 2, and 4 (left to right bars). On the right, K3 + C (dashed) and K2 logK + C (dotted) are plotted for comparison. Our method is a highly accurate and consistently faster than the exact implementation.

Fig. 4
Fig. 4

Accuracy (left) and computation time (right) as functions of input/output size for our fast algorithm for HT Hc with upsamping rates of 1, 2, and 4. On the right, K3 + C (dashed) and K2 logK + C (dotted) are plotted for comparison. Our method is highly accurate and is much faster and scales better than the exact implementation.

Fig. 5
Fig. 5

Example parameter sweep from low (a) to high (h) regularization strength using the physical phantom dataset.

Fig. 6
Fig. 6

Reconstruction quality versus number of views for the slice-by-slice and fully 3D reconstruction methods. Reducing the number of views negatively impacts both methods, but regularization mitigates the effect.

Fig. 7
Fig. 7

Representative slices of the fully 3D reconstruction of the Shepp3D dataset. (a) When the number of views is large, regularization only reduces noise in the reconstruction resulting from noise in the sinogram. (b) When the number of views is small, regularization also reduces the consequent line artifacts.

Fig. 8
Fig. 8

Representative slices of the slice-by-slice and fully 3D reconstruction of the Shepp3D dataset from 101 views with regularization. The fully 3D achieves superior results by enforcing consistency in all three dimensions.

Fig. 9
Fig. 9

Phantom dataset results.

Fig. 10
Fig. 10

Representative slices of the slice-by-slice reconstruction of the phantom dataset with regularization. When the number of views is low, there is an inconsistency between the slices apparent in the x1x2 cross section (d).

Fig. 11
Fig. 11

Representative slices of the fully 3D reconstruction of the phantom dataset with regularization. The reconstruction is of reasonable quality even with only 13 views.

Fig. 12
Fig. 12

Rat brain dataset results.

Fig. 13
Fig. 13

Representative x1x2 slices of the slice-by-slice (top row) and fully 3D (bottom row) reconstructions of the rat brain DPC dataset. For this complex sample with low-noise measurements, regularizing in full 3D does not visually improve the reconstruction quality.

Equations (27)

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𝒫 θ { f } ( y ) = f ( t θ + P θ T y ) d t ,
𝒫 θ { α f + β g } ( y ) = α 𝒫 θ { f } ( y ) + β 𝒫 θ { g } ( y ) ,
𝒫 θ { f ( x 0 ) } ( y ) = 𝒫 θ { f } ( y P θ x 0 ) .
f * = arg min f g H f + Ψ ( f ) ,
𝒟 θ , u { f } ( y ) = u , 𝒫 θ { f } ( y ) ,
f ( x ) = k 𝒦 c [ k ] φ ( x Λ x k ) ,
𝒫 θ { f } ( y ) = k 𝒦 c [ k ] 𝒫 θ { φ } ( y P θ Λ x k ) .
Hc [ m ] = g [ m ] = 𝒫 θ { f } ( Λ y m ) = k 𝒦 c [ k ] 𝒫 θ { φ } ( Λ y m P θ Λ x k ) ,
c * = arg min c g Hc + Ψ ( c ) ,
c * = arg min c g Hc + λ 1 Ψ 1 ( c ) + λ 2 Ψ 2 ( u ) , subject to u = Lc .
( c , u , α ) = 1 2 g Hc 2 + λ 1 Ψ 1 ( c ) + λ 2 Ψ 2 ( u ) + α T ( Lc u ) + μ 2 Lc u 2 ,
c k + 1 arg min c ( c , u k , α k )
u k + 1 arg min u ( c k + 1 , u , α k )
α k + 1 α k + μ ( Lc k + 1 u k + 1 ) .
( c , u k , α k ) = ( H T H + μ L T L + λ 1 Ψ 1 ) c ( H T g + μ L T ( u k α k μ ) ) .
H T g , c = g , Hc
= m g [ m ] k 𝒦 c [ k ] 𝒫 θ { φ } ( Λ y m P θ Λ x k )
= k 𝒦 c [ k ] m g [ m ] 𝒫 θ { φ } ( Λ y m P θ Λ x k )
( H T g ) [ k ] = m g [ m ] 𝒫 θ { φ } ( Λ y m P θ Λ x k ) .
( H T Hc ) [ k ] = m k 𝒦 c [ k ] 𝒫 θ { φ } ( Λ y m P θ Λ x k ) 𝒫 θ { φ } ( Λ y m P θ Λ x k ) .
( H T Hc ) [ k ] = k 𝒦 c [ k ] m 𝒫 θ { φ } ( Λ y m P θ Λ x k ) 𝒫 θ { φ } ( Λ y m P θ Λ x k ) .
m d f ( Λ m ) g ( Λ m ) = 1 det ( Λ ) d f ( x ) g ( x ) d x ,
( 21 ) = ( a ) k 𝒦 c [ k ] det ( Λ y ) d 1 𝒫 θ { φ } ( y P θ Λ x k ) 𝒫 θ { φ } ( y P θ Λ x k ) d y
= ( b ) k 𝒦 c [ k ] det ( Λ y ) d 1 𝒫 θ { φ } ( y P θ Λ x ( k k ) ) 𝒫 θ { φ } ( y ) d y
= k 𝒦 c [ k ] ( 𝒫 θ { φ } * 𝒫 θ { φ } ( ) ) ( P θ Λ x ( k k ) )
= ( c ) 1 det ( Λ y ) ( c * r ) [ k ]
r [ k ] = ( 𝒫 θ { φ } * 𝒫 θ { φ } ( ) ) ( P θ Λ x k ) ;

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