Abstract

We develop and implement a compressive reconstruction method for tomographic recovery of refractive index distribution for weakly attenuating objects in a microfocus x-ray system. This is achieved through the development of a discretized operator modeling both the transport of intensity equation and the x-ray transform that is suitable for iterative reconstruction techniques.

© 2013 Optical Society of America

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2013

2012

2011

2010

2008

2007

J. P. Guigay, M. Langer, R. Boistel, and P. Cloetens, Opt. Lett. 32, 1617 (2007).
[CrossRef]

J. M. Bioucas-Dias and M. A. T. Figueiredo, IEEE Trans. Image Process. 16, 2992 (2007).
[CrossRef]

2006

V. V. Nagarkar, I. Shestakova, V. Gaysinskiy, B. Singh, B. W. Miller, and H. Bradford Barber, Nucl. Instr. Meth. 563, 45 (2006).
[CrossRef]

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, Nat. Phys. 2, 258 (2006).
[CrossRef]

A. Groso, R. Abela, and M. Stampanoni, Opt. Express 14, 8103 (2006).
[CrossRef]

T. E. Gureyev, G. R. Myers, Y. I. Nesterets, D. M. Paganin, K. M. Pavlov, and S. W. Wilkins, Proc. SPIE 6318, 63180V (2006).

E. Candès, J. Romberg, and T. Tao, IEEE Trans. Inf. Theory 52, 489 (2006).
[CrossRef]

2004

S. Matej, J. A. Fessler, and I. G. Kazantsev, IEEE Trans. Med. Imaging 23, 401 (2004).
[CrossRef]

2003

J. Fessler and B. Sutton, IEEE Trans. Signal Process. 51, 560 (2003).
[CrossRef]

2002

1998

1996

A. Momose, T. Takeda, Y. Itai, and K. Hirano, Nat. Med. 2, 473 (1996).
[CrossRef]

1995

T. Davis, D. Gao, T. Gureyev, A. Stevenson, and S. Wilkins, Nature 373, 595 (1995).
[CrossRef]

Abela, R.

Anastasio, M. A.

Baillie, T. W.

Barbastathis, G.

J. Petruccelli, L. Tian, and G. Barbastathis, in Imaging and Applied Optics, J. Christou and D. Miller, eds., OSA Technical Digest (online) (Optical Society of America, 2013), paper CTu2C.3.

Batenburg, K. J.

Bioucas-Dias, J. M.

J. M. Bioucas-Dias and M. A. T. Figueiredo, IEEE Trans. Image Process. 16, 2992 (2007).
[CrossRef]

Boistel, R.

Bradford Barber, H.

V. V. Nagarkar, I. Shestakova, V. Gaysinskiy, B. Singh, B. W. Miller, and H. Bradford Barber, Nucl. Instr. Meth. 563, 45 (2006).
[CrossRef]

Brady, D.

Bronnikov, A. V.

Bunk, O.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, Nat. Phys. 2, 258 (2006).
[CrossRef]

Burvall, A.

Candès, E.

E. Candès, J. Romberg, and T. Tao, IEEE Trans. Inf. Theory 52, 489 (2006).
[CrossRef]

Chawla, A.

Chen, R. C.

Cloetens, P.

David, C.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, Nat. Phys. 2, 258 (2006).
[CrossRef]

Davis, T.

T. Davis, D. Gao, T. Gureyev, A. Stevenson, and S. Wilkins, Nature 373, 595 (1995).
[CrossRef]

Fessler, J.

J. Fessler and B. Sutton, IEEE Trans. Signal Process. 51, 560 (2003).
[CrossRef]

Fessler, J. A.

S. Matej, J. A. Fessler, and I. G. Kazantsev, IEEE Trans. Med. Imaging 23, 401 (2004).
[CrossRef]

Figueiredo, M. A. T.

J. M. Bioucas-Dias and M. A. T. Figueiredo, IEEE Trans. Image Process. 16, 2992 (2007).
[CrossRef]

Gao, D.

T. Davis, D. Gao, T. Gureyev, A. Stevenson, and S. Wilkins, Nature 373, 595 (1995).
[CrossRef]

Gaysinskiy, V.

V. V. Nagarkar, I. Shestakova, V. Gaysinskiy, B. Singh, B. W. Miller, and H. Bradford Barber, Nucl. Instr. Meth. 563, 45 (2006).
[CrossRef]

Groso, A.

Guigay, J. P.

Gureyev, T.

T. Davis, D. Gao, T. Gureyev, A. Stevenson, and S. Wilkins, Nature 373, 595 (1995).
[CrossRef]

Gureyev, T. E.

Hertz, H. M.

Hirano, K.

A. Momose, T. Takeda, Y. Itai, and K. Hirano, Nat. Med. 2, 473 (1996).
[CrossRef]

Itai, Y.

A. Momose, T. Takeda, Y. Itai, and K. Hirano, Nat. Med. 2, 473 (1996).
[CrossRef]

Kazantsev, I. G.

S. Matej, J. A. Fessler, and I. G. Kazantsev, IEEE Trans. Med. Imaging 23, 401 (2004).
[CrossRef]

King, A.

Kostenko, A.

Krishnamurthy, K.

Langer, M.

Larsson, D. H.

Longo, R.

Lundström, U.

MacCabe, K.

Marks, D.

Matej, S.

S. Matej, J. A. Fessler, and I. G. Kazantsev, IEEE Trans. Med. Imaging 23, 401 (2004).
[CrossRef]

Mayo, S. C.

Miller, B. W.

V. V. Nagarkar, I. Shestakova, V. Gaysinskiy, B. Singh, B. W. Miller, and H. Bradford Barber, Nucl. Instr. Meth. 563, 45 (2006).
[CrossRef]

Momose, A.

A. Momose, T. Takeda, Y. Itai, and K. Hirano, Nat. Med. 2, 473 (1996).
[CrossRef]

Myers, G. R.

G. R. Myers, D. M. Paganin, T. E. Gureyev, and S. C. Mayo, Opt. Express 16, 908 (2008).
[CrossRef]

T. E. Gureyev, G. R. Myers, Y. I. Nesterets, D. M. Paganin, K. M. Pavlov, and S. W. Wilkins, Proc. SPIE 6318, 63180V (2006).

Nagarkar, V. V.

V. V. Nagarkar, I. Shestakova, V. Gaysinskiy, B. Singh, B. W. Miller, and H. Bradford Barber, Nucl. Instr. Meth. 563, 45 (2006).
[CrossRef]

Nesterets, Y. I.

T. E. Gureyev, G. R. Myers, Y. I. Nesterets, D. M. Paganin, K. M. Pavlov, and S. W. Wilkins, Proc. SPIE 6318, 63180V (2006).

Offerman, S. E.

Paganin, D. M.

G. R. Myers, D. M. Paganin, T. E. Gureyev, and S. C. Mayo, Opt. Express 16, 908 (2008).
[CrossRef]

T. E. Gureyev, G. R. Myers, Y. I. Nesterets, D. M. Paganin, K. M. Pavlov, and S. W. Wilkins, Proc. SPIE 6318, 63180V (2006).

Pan, X.

Pavlov, K. M.

T. W. Baillie, T. E. Gureyev, J. A. Schmalz, and K. M. Pavlov, Opt. Express 20, 16913 (2012).
[CrossRef]

T. E. Gureyev, G. R. Myers, Y. I. Nesterets, D. M. Paganin, K. M. Pavlov, and S. W. Wilkins, Proc. SPIE 6318, 63180V (2006).

Petruccelli, J.

J. Petruccelli, L. Tian, and G. Barbastathis, in Imaging and Applied Optics, J. Christou and D. Miller, eds., OSA Technical Digest (online) (Optical Society of America, 2013), paper CTu2C.3.

Pfeiffer, F.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, Nat. Phys. 2, 258 (2006).
[CrossRef]

Rigon, L.

Romberg, J.

E. Candès, J. Romberg, and T. Tao, IEEE Trans. Inf. Theory 52, 489 (2006).
[CrossRef]

Samei, E.

Schmalz, J. A.

Shestakova, I.

V. V. Nagarkar, I. Shestakova, V. Gaysinskiy, B. Singh, B. W. Miller, and H. Bradford Barber, Nucl. Instr. Meth. 563, 45 (2006).
[CrossRef]

Sidky, E. Y.

Singh, B.

V. V. Nagarkar, I. Shestakova, V. Gaysinskiy, B. Singh, B. W. Miller, and H. Bradford Barber, Nucl. Instr. Meth. 563, 45 (2006).
[CrossRef]

Stampanoni, M.

Stevenson, A.

T. Davis, D. Gao, T. Gureyev, A. Stevenson, and S. Wilkins, Nature 373, 595 (1995).
[CrossRef]

Sutton, B.

J. Fessler and B. Sutton, IEEE Trans. Signal Process. 51, 560 (2003).
[CrossRef]

Takeda, T.

A. Momose, T. Takeda, Y. Itai, and K. Hirano, Nat. Med. 2, 473 (1996).
[CrossRef]

Takman, P. A. C.

Tao, T.

E. Candès, J. Romberg, and T. Tao, IEEE Trans. Inf. Theory 52, 489 (2006).
[CrossRef]

Tian, L.

J. Petruccelli, L. Tian, and G. Barbastathis, in Imaging and Applied Optics, J. Christou and D. Miller, eds., OSA Technical Digest (online) (Optical Society of America, 2013), paper CTu2C.3.

van Vliet, L. J.

Weitkamp, T.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, Nat. Phys. 2, 258 (2006).
[CrossRef]

Wilkins, S.

T. Davis, D. Gao, T. Gureyev, A. Stevenson, and S. Wilkins, Nature 373, 595 (1995).
[CrossRef]

Wilkins, S. W.

T. E. Gureyev, G. R. Myers, Y. I. Nesterets, D. M. Paganin, K. M. Pavlov, and S. W. Wilkins, Proc. SPIE 6318, 63180V (2006).

T. E. Gureyev and S. W. Wilkins, J. Opt. Soc. Am. A 15, 579 (1998).
[CrossRef]

IEEE Trans. Image Process.

J. M. Bioucas-Dias and M. A. T. Figueiredo, IEEE Trans. Image Process. 16, 2992 (2007).
[CrossRef]

IEEE Trans. Inf. Theory

E. Candès, J. Romberg, and T. Tao, IEEE Trans. Inf. Theory 52, 489 (2006).
[CrossRef]

IEEE Trans. Med. Imaging

S. Matej, J. A. Fessler, and I. G. Kazantsev, IEEE Trans. Med. Imaging 23, 401 (2004).
[CrossRef]

IEEE Trans. Signal Process.

J. Fessler and B. Sutton, IEEE Trans. Signal Process. 51, 560 (2003).
[CrossRef]

J. Opt. Soc. Am. A

Nat. Med.

A. Momose, T. Takeda, Y. Itai, and K. Hirano, Nat. Med. 2, 473 (1996).
[CrossRef]

Nat. Phys.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, Nat. Phys. 2, 258 (2006).
[CrossRef]

Nature

T. Davis, D. Gao, T. Gureyev, A. Stevenson, and S. Wilkins, Nature 373, 595 (1995).
[CrossRef]

Nucl. Instr. Meth.

V. V. Nagarkar, I. Shestakova, V. Gaysinskiy, B. Singh, B. W. Miller, and H. Bradford Barber, Nucl. Instr. Meth. 563, 45 (2006).
[CrossRef]

Opt. Express

Opt. Lett.

Proc. SPIE

T. E. Gureyev, G. R. Myers, Y. I. Nesterets, D. M. Paganin, K. M. Pavlov, and S. W. Wilkins, Proc. SPIE 6318, 63180V (2006).

Other

J. Petruccelli, L. Tian, and G. Barbastathis, in Imaging and Applied Optics, J. Christou and D. Miller, eds., OSA Technical Digest (online) (Optical Society of America, 2013), paper CTu2C.3.

Supplementary Material (2)

» Media 1: AVI (1667 KB)     
» Media 2: AVI (1629 KB)     

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Figures (4)

Fig. 1.
Fig. 1.

Imaging geometry for TIE tomography.

Fig. 2.
Fig. 2.

Illustration of the construction of (a) discrete projection operator R and (b) discrete TIE operator P.

Fig. 3.
Fig. 3.

Single projected measurement of I.

Fig. 4.
Fig. 4.

Reconstruction results for the refractive index deviation from air (1n). (a) Fourier-based TIE solver+ FBP, (b) compressive reconstruction, (c) plot along the dashed line illustrated in (a), and (d) plot along the dashed line illustrated in (b). The three cross sections are taken from the three orthogonal planes going through the center of the sample. (e), (f) Two perspectives of a 3D rendering of the reconstruction in (b). [Online: movies show the rotation of the renderings in (e) (Media 1) and (f) (Media 2) about their respective vertical axes.]

Equations (6)

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

g(x,y;θ)2πλd[1I(Msx,Msy)Ii(Msx,Msy)]=xy2ϕ(x,y;θ),
ϕ(x,y;θ)=n(x,ys,zs)δ(yyscosθ+zssinθ)dysdzs.
g=PRnAn,
n^=argminnnTVsuch thatg=An,
P=xy2=Fx,y1HFx,y,
A=Fx,y1HFxFNU.

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