Abstract

Optical tomography has recently witnessed a substantial increase in the size of the data sets used, mainly owing to the use of CCD cameras. Larger data sets render 3D reconstructions more robust, quantitative, and reproducible, but also significantly increase the computing time needed to generate the reconstructed data. Approaches working with spatial-frequencies instead of real space variables seem to be the method of choice in this case, and a direct inversion method that can produce three-dimensional images from very large detector numbers (>105) using either very large source numbers (>103) [Phys. Rev. E 64, 035601 (2001) ] or structured illumination [Opt. Lett. 34, 983 (2009) ] has been presented. However, most small animal imaging setups typically incur a practical upper limit of only 102 sources mainly due to imaging time constraints, and currently all relying on point source illumination. In this Letter, what we believe to be a new approach, which combines Fourier and real space functions, is shown, which fills the gap between traditional fiber-based small data sets that are solved in real space and the very large data sets solved entirely in spatial-frequency domain.

© 2010 Optical Society of America

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References

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  1. V. A. Markel and J. C. Schotland, Phys. Rev. E 64, 035601 (2001).
    [CrossRef]
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    [CrossRef] [PubMed]
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2009

2008

2004

V. A. Markel and J. C. Schotland, Phys. Rev. E 70, 056616 (2004).
[CrossRef]

2003

2001

V. A. Markel and J. C. Schotland, Phys. Rev. E 64, 035601 (2001).
[CrossRef]

V. Ntziachristos and R. Weissleder, Opt. Lett. 26, 893 (2001).
[CrossRef]

1999

Arridge, S.

S. Arridge and J. C. Schotland, Inverse Probl. 25 (2009).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Carminati, R.

Chance, B.

Culver, J. P.

Durduran, T.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968), Chap. 3.

Holboke, M. J.

Li, X. D.

Lukic, V.

Markel, V. A.

Nieto-Vesperinas, M.

Ntziachristos, V.

Panasyuk, G. Y.

Pattanayak, D. N.

Ripoll, J.

Schotland, J. C.

V. Lukic, V. A. Markel, and J. C. Schotland, Opt. Lett. 34, 983 (2009).
[CrossRef] [PubMed]

S. Arridge and J. C. Schotland, Inverse Probl. 25 (2009).
[CrossRef]

G. Y. Panasyuk, Z. M. Wang, J. C. Schotland, and V. A. Markel, Opt. Lett. 33, 1744 (2008).
[CrossRef] [PubMed]

V. A. Markel and J. C. Schotland, Phys. Rev. E 70, 056616 (2004).
[CrossRef]

V. A. Markel and J. C. Schotland, Phys. Rev. E 64, 035601 (2001).
[CrossRef]

Schulz, R. B.

Wang, Z. M.

Weissleder, R.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, 1999).

Yodh, A. G.

Zubkov, L.

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Figures (3)

Fig. 1
Fig. 1

The absolute value of the Fourier transform in log scale of a point source at z d = L for all K y > 0 values. Note how all the information is located in the center of the image (i.e., at the low frequencies). The insets show the real and imaginary parts of a 10 × 10 window, which corresponds to K max = 2 K res taking only the positive frequencies.

Fig. 2
Fig. 2

Reconstructed fluorescence concentration for four-point fluorophores located at z = 0.3   cm and z = 0.7   cm (see text for details). The insets depict slices through the z = 0.3 , 0.5, and 0.7 cm planes.

Fig. 3
Fig. 3

Effect of the maximum frequency cutoff for different values of K max and/or number of values taken on the 3D reconstruction shown in Fig. 2 for a slice at z = 0.7   cm . The numbers in parentheses indicates the size of the frequency window (see text for details).

Equations (4)

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U ̃ f l ( R s , z s ; K d , z d ) = V U ( e x c ) ( R s , z s ; R , z ) F ( R , z ) G ̃ ( R , z ; K d , z d ) d R d z .
U ̃ ( e x c ) ( R s , z s ; K d , z d ) = S 0 ( R s , z s ) G ̃ ( R s , z s ; K d , z d ) ,
U ̃ n ( R s , z s ; K d , z d ) = i = 1 N g ( R s , z s ; R i , z i ) G ̃ ( R i , z i ; K d , z d ) G ̃ ( R s , z s ; K d , z d ) F ( R i , z i ) Δ V ,
W ̃ ( R s , z s ; R i , z i ; K d , z d ) = g ( R s , z s ; R i , z i ) exp [ i q 0 ( K d ) ( z s z i ) ] exp [ i K d ( R i R s ) ] Δ V .

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