Abstract

We address the problem of tomogram reconstruction in frequency-domain optical-coherence tomography. We propose a new technique for suppressing the autocorrelation artifacts that are commonly encountered with the conventional Fourier-transform-based approach. The technique is based on the assumptions that the scattering function is causal and that the intensity of the light reflected from the object is smaller than that of the reference. The technique is noniterative, nonlinear, and yields an exact solution in the absence of noise. Results on synthesized data and experimental measurements show that the technique offers superior quality reconstruction and is computationally more efficient than the iterative technique reported in the literature.

© 2008 Optical Society of America

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    [CrossRef]

2007 (1)

A. Rosset, J. Heuberger, and O. Ratib, “OsiriX imaging software” (2007), http://www.osirix-viewer.com/.

2006 (4)

2005 (2)

A. Szkulmowska, M. Wojtkowski, I. Gorczynska, T. Bajraszewski, M. Szkulmowski, P. Targowski, A. Kowalczyk, and J. J. Kaluzny, “Coherent noise-free ophthalmic imaging by spectral optical coherence tomography,” J. Phys. D 38, 2606-2611 (2005).
[CrossRef]

U. Schmidt-Erfurth, R. A. Leitgeb, S. Michels, B. Povazay, S. Sacu, B. Hermann, C. Ahlers, H. Sattmann, C. Scholda, A. F. Fercher, and W. Drexler, “Three-dimensional ultrahigh-resolution optical-coherence tomography of macular diseases,” Invest. Ophthalmol. Visual Sci. 46, 3393-3402 (2005).
[CrossRef]

2004 (4)

2003 (2)

2002 (4)

H. H. Bauschke, P. L. Combettes, and D. Luke, “Phase retrieval, error reduction algorithm, and Fienup variants: a view from convex optimization,” J. Opt. Soc. Am. A 19, 1334-1345 (2002).
[CrossRef]

M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, and A. Fercher, “In vivo human retinal imaging by Fourier-domain optical-coherence tomography,” J. Biomed. Opt. 7, 457-463 (2002).
[CrossRef] [PubMed]

M. Wojtkowski, A. Kowalczyk, R. A. Leitgeb, and A. Fercher, “Full-range complex spectral optical-coherence tomography technique in eye imaging,” Opt. Lett. 27, 1415-1417 (2002).
[CrossRef]

B. E. Bouma and G. J. Tearney, eds., Handbook of Optical Coherence Tomography (Marcel Dekker, 2002).

2000 (1)

E. Cuche, “Numerical reconstruction of digital holograms: application to phase-contrast imaging and microscopy,” Ph.D. dissertation (Ecole polytechnique fédérale de Lausanne (EPFL), 2000).

1999 (3)

M. Unser, “Splines: a perfect fit for signal and image processing,” IEEE Signal Process. Mag. 16, 22-38 (1999).
[CrossRef]

A. F. Fercher, R. A. Leitgeb, C. K. Hitzenberger, H. Sattmann, and M. Wojtkowski, “Complex spectral interferometry OCT,” Proc. SPIE 3564, 173-178 (1999).
[CrossRef]

A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing, 2nd ed. (Prentice-Hall, 1999).

1998 (1)

G. Hausler and M. W. Lindner, “Coherence radar and spectral radar±new tools for dermatological analysis,” J. Biomed. Opt. 3, 21-31 (1998).
[CrossRef]

1995 (1)

A. F. Fercher, C. K. Hitzenberger, G. Kamp, and S. Y. El-Zaiat, “Measurement of intraocular distances by backscattering spectral interferometry,” Opt. Commun. 117, 43-48 (1995).
[CrossRef]

1988 (1)

1982 (1)

1981 (1)

T. F. Quatieri, Jr. and A. V. Oppenheim, “Iterative techniques for minimum phase signal reconstruction from phase or magnitude,” IEEE Trans. Acoust., Speech, Signal Process. 29, 1187-1193 (1981).
[CrossRef]

1977 (1)

D. G. Childers, D. P. Skinner, and R. C. Kemerait, “The cepstrum: a guide to processing,” Proc. IEEE 65, 1428-1443 (1977).
[CrossRef]

1969 (1)

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153-156 (1969).
[CrossRef]

Ahlers, C.

U. Schmidt-Erfurth, R. A. Leitgeb, S. Michels, B. Povazay, S. Sacu, B. Hermann, C. Ahlers, H. Sattmann, C. Scholda, A. F. Fercher, and W. Drexler, “Three-dimensional ultrahigh-resolution optical-coherence tomography of macular diseases,” Invest. Ophthalmol. Visual Sci. 46, 3393-3402 (2005).
[CrossRef]

Aoki, G.

Bachmann, A. H.

A. H. Bachmann, R. A. Leitgeb, and T. Lasser, “Complex ultrahigh resolution Fourier-domain optical-coherence tomography,” Proc. SPIE 6079, 60790 (2006).
[CrossRef]

R. A. Leitgeb, M. L. Villiger, A. H. Bachmann, L. Steinmann, and T. Lasser, “Extended focus depth for Fourier-domain optical-coherence tomography,” Opt. Lett. 31, 2450-2452 (2006).
[CrossRef] [PubMed]

Bajraszewski, T.

A. Szkulmowska, M. Wojtkowski, I. Gorczynska, T. Bajraszewski, M. Szkulmowski, P. Targowski, A. Kowalczyk, and J. J. Kaluzny, “Coherent noise-free ophthalmic imaging by spectral optical coherence tomography,” J. Phys. D 38, 2606-2611 (2005).
[CrossRef]

M. Wojtkowski, T. Bajraszewski, I. Gorczynska, P. Targowski, A. Kowalczyk, W. Wasilewski, and C. Radzewicz, “Ophthalmic imaging by spectral optical-coherence tomography,” Am. J. Ophthalmol. 138, 412-419 (2004).
[CrossRef] [PubMed]

R. A. Leitgeb, C. K. Hitzenberger, A. F. Fercher, and T. Bajraszewski, “Phase-shifting algorithm to achieve high-speed long-depth-range probing by frequency-domain optical-coherence tomography,” Opt. Lett. 28, 2201-2203 (2003).
[CrossRef] [PubMed]

M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, and A. Fercher, “In vivo human retinal imaging by Fourier-domain optical-coherence tomography,” J. Biomed. Opt. 7, 457-463 (2002).
[CrossRef] [PubMed]

Bauschke, H. H.

Blu, T.

M. Liebling, T. Blu, and M. Unser, “Complex-wave retrieval from a single off-axis hologram,” J. Opt. Soc. Am. A 27, 367-377 (2004).
[CrossRef]

Bouma, B. E.

Cai, X.

Cense, B.

Chang, S.

Chen, T. C.

Childers, D. G.

D. G. Childers, D. P. Skinner, and R. C. Kemerait, “The cepstrum: a guide to processing,” Proc. IEEE 65, 1428-1443 (1977).
[CrossRef]

Choma, M. A.

Combettes, P. L.

Cuche, E.

E. Cuche, “Numerical reconstruction of digital holograms: application to phase-contrast imaging and microscopy,” Ph.D. dissertation (Ecole polytechnique fédérale de Lausanne (EPFL), 2000).

de Boer, J. F.

Digonnet, M. J. F.

Drexler, W.

U. Schmidt-Erfurth, R. A. Leitgeb, S. Michels, B. Povazay, S. Sacu, B. Hermann, C. Ahlers, H. Sattmann, C. Scholda, A. F. Fercher, and W. Drexler, “Three-dimensional ultrahigh-resolution optical-coherence tomography of macular diseases,” Invest. Ophthalmol. Visual Sci. 46, 3393-3402 (2005).
[CrossRef]

El-Zaiat, S. Y.

A. F. Fercher, C. K. Hitzenberger, G. Kamp, and S. Y. El-Zaiat, “Measurement of intraocular distances by backscattering spectral interferometry,” Opt. Commun. 117, 43-48 (1995).
[CrossRef]

Endo, T.

Fercher, A.

M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, and A. Fercher, “In vivo human retinal imaging by Fourier-domain optical-coherence tomography,” J. Biomed. Opt. 7, 457-463 (2002).
[CrossRef] [PubMed]

M. Wojtkowski, A. Kowalczyk, R. A. Leitgeb, and A. Fercher, “Full-range complex spectral optical-coherence tomography technique in eye imaging,” Opt. Lett. 27, 1415-1417 (2002).
[CrossRef]

Fercher, A. F.

U. Schmidt-Erfurth, R. A. Leitgeb, S. Michels, B. Povazay, S. Sacu, B. Hermann, C. Ahlers, H. Sattmann, C. Scholda, A. F. Fercher, and W. Drexler, “Three-dimensional ultrahigh-resolution optical-coherence tomography of macular diseases,” Invest. Ophthalmol. Visual Sci. 46, 3393-3402 (2005).
[CrossRef]

R. A. Leitgeb, C. K. Hitzenberger, A. F. Fercher, and T. Bajraszewski, “Phase-shifting algorithm to achieve high-speed long-depth-range probing by frequency-domain optical-coherence tomography,” Opt. Lett. 28, 2201-2203 (2003).
[CrossRef] [PubMed]

A. F. Fercher, R. A. Leitgeb, C. K. Hitzenberger, H. Sattmann, and M. Wojtkowski, “Complex spectral interferometry OCT,” Proc. SPIE 3564, 173-178 (1999).
[CrossRef]

A. F. Fercher, C. K. Hitzenberger, G. Kamp, and S. Y. El-Zaiat, “Measurement of intraocular distances by backscattering spectral interferometry,” Opt. Commun. 117, 43-48 (1995).
[CrossRef]

Fienup, J. R.

Flueraru, C.

Gorczynska, I.

A. Szkulmowska, M. Wojtkowski, I. Gorczynska, T. Bajraszewski, M. Szkulmowski, P. Targowski, A. Kowalczyk, and J. J. Kaluzny, “Coherent noise-free ophthalmic imaging by spectral optical coherence tomography,” J. Phys. D 38, 2606-2611 (2005).
[CrossRef]

M. Wojtkowski, T. Bajraszewski, I. Gorczynska, P. Targowski, A. Kowalczyk, W. Wasilewski, and C. Radzewicz, “Ophthalmic imaging by spectral optical-coherence tomography,” Am. J. Ophthalmol. 138, 412-419 (2004).
[CrossRef] [PubMed]

Hausler, G.

G. Hausler and M. W. Lindner, “Coherence radar and spectral radar±new tools for dermatological analysis,” J. Biomed. Opt. 3, 21-31 (1998).
[CrossRef]

Hermann, B.

U. Schmidt-Erfurth, R. A. Leitgeb, S. Michels, B. Povazay, S. Sacu, B. Hermann, C. Ahlers, H. Sattmann, C. Scholda, A. F. Fercher, and W. Drexler, “Three-dimensional ultrahigh-resolution optical-coherence tomography of macular diseases,” Invest. Ophthalmol. Visual Sci. 46, 3393-3402 (2005).
[CrossRef]

Heuberger, J.

A. Rosset, J. Heuberger, and O. Ratib, “OsiriX imaging software” (2007), http://www.osirix-viewer.com/.

Hitzenberger, C. K.

R. A. Leitgeb, C. K. Hitzenberger, A. F. Fercher, and T. Bajraszewski, “Phase-shifting algorithm to achieve high-speed long-depth-range probing by frequency-domain optical-coherence tomography,” Opt. Lett. 28, 2201-2203 (2003).
[CrossRef] [PubMed]

A. F. Fercher, R. A. Leitgeb, C. K. Hitzenberger, H. Sattmann, and M. Wojtkowski, “Complex spectral interferometry OCT,” Proc. SPIE 3564, 173-178 (1999).
[CrossRef]

A. F. Fercher, C. K. Hitzenberger, G. Kamp, and S. Y. El-Zaiat, “Measurement of intraocular distances by backscattering spectral interferometry,” Opt. Commun. 117, 43-48 (1995).
[CrossRef]

Itoh, M.

Izatt, J. A.

Kaluzny, J. J.

A. Szkulmowska, M. Wojtkowski, I. Gorczynska, T. Bajraszewski, M. Szkulmowski, P. Targowski, A. Kowalczyk, and J. J. Kaluzny, “Coherent noise-free ophthalmic imaging by spectral optical coherence tomography,” J. Phys. D 38, 2606-2611 (2005).
[CrossRef]

Kamp, G.

A. F. Fercher, C. K. Hitzenberger, G. Kamp, and S. Y. El-Zaiat, “Measurement of intraocular distances by backscattering spectral interferometry,” Opt. Commun. 117, 43-48 (1995).
[CrossRef]

Kemerait, R. C.

D. G. Childers, D. P. Skinner, and R. C. Kemerait, “The cepstrum: a guide to processing,” Proc. IEEE 65, 1428-1443 (1977).
[CrossRef]

Kino, G. S.

Kowalczyk, A.

A. Szkulmowska, M. Wojtkowski, I. Gorczynska, T. Bajraszewski, M. Szkulmowski, P. Targowski, A. Kowalczyk, and J. J. Kaluzny, “Coherent noise-free ophthalmic imaging by spectral optical coherence tomography,” J. Phys. D 38, 2606-2611 (2005).
[CrossRef]

M. Wojtkowski, T. Bajraszewski, I. Gorczynska, P. Targowski, A. Kowalczyk, W. Wasilewski, and C. Radzewicz, “Ophthalmic imaging by spectral optical-coherence tomography,” Am. J. Ophthalmol. 138, 412-419 (2004).
[CrossRef] [PubMed]

M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, and A. Fercher, “In vivo human retinal imaging by Fourier-domain optical-coherence tomography,” J. Biomed. Opt. 7, 457-463 (2002).
[CrossRef] [PubMed]

M. Wojtkowski, A. Kowalczyk, R. A. Leitgeb, and A. Fercher, “Full-range complex spectral optical-coherence tomography technique in eye imaging,” Opt. Lett. 27, 1415-1417 (2002).
[CrossRef]

Lasser, T.

A. H. Bachmann, R. A. Leitgeb, and T. Lasser, “Complex ultrahigh resolution Fourier-domain optical-coherence tomography,” Proc. SPIE 6079, 60790 (2006).
[CrossRef]

R. A. Leitgeb, M. L. Villiger, A. H. Bachmann, L. Steinmann, and T. Lasser, “Extended focus depth for Fourier-domain optical-coherence tomography,” Opt. Lett. 31, 2450-2452 (2006).
[CrossRef] [PubMed]

Leitgeb, R.

M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, and A. Fercher, “In vivo human retinal imaging by Fourier-domain optical-coherence tomography,” J. Biomed. Opt. 7, 457-463 (2002).
[CrossRef] [PubMed]

Leitgeb, R. A.

A. H. Bachmann, R. A. Leitgeb, and T. Lasser, “Complex ultrahigh resolution Fourier-domain optical-coherence tomography,” Proc. SPIE 6079, 60790 (2006).
[CrossRef]

R. A. Leitgeb, M. L. Villiger, A. H. Bachmann, L. Steinmann, and T. Lasser, “Extended focus depth for Fourier-domain optical-coherence tomography,” Opt. Lett. 31, 2450-2452 (2006).
[CrossRef] [PubMed]

U. Schmidt-Erfurth, R. A. Leitgeb, S. Michels, B. Povazay, S. Sacu, B. Hermann, C. Ahlers, H. Sattmann, C. Scholda, A. F. Fercher, and W. Drexler, “Three-dimensional ultrahigh-resolution optical-coherence tomography of macular diseases,” Invest. Ophthalmol. Visual Sci. 46, 3393-3402 (2005).
[CrossRef]

R. A. Leitgeb, C. K. Hitzenberger, A. F. Fercher, and T. Bajraszewski, “Phase-shifting algorithm to achieve high-speed long-depth-range probing by frequency-domain optical-coherence tomography,” Opt. Lett. 28, 2201-2203 (2003).
[CrossRef] [PubMed]

M. Wojtkowski, A. Kowalczyk, R. A. Leitgeb, and A. Fercher, “Full-range complex spectral optical-coherence tomography technique in eye imaging,” Opt. Lett. 27, 1415-1417 (2002).
[CrossRef]

A. F. Fercher, R. A. Leitgeb, C. K. Hitzenberger, H. Sattmann, and M. Wojtkowski, “Complex spectral interferometry OCT,” Proc. SPIE 3564, 173-178 (1999).
[CrossRef]

Liebling, M.

M. Liebling, T. Blu, and M. Unser, “Complex-wave retrieval from a single off-axis hologram,” J. Opt. Soc. Am. A 27, 367-377 (2004).
[CrossRef]

Lindner, M. W.

G. Hausler and M. W. Lindner, “Coherence radar and spectral radar±new tools for dermatological analysis,” J. Biomed. Opt. 3, 21-31 (1998).
[CrossRef]

Luke, D.

Makita, S.

Michels, S.

U. Schmidt-Erfurth, R. A. Leitgeb, S. Michels, B. Povazay, S. Sacu, B. Hermann, C. Ahlers, H. Sattmann, C. Scholda, A. F. Fercher, and W. Drexler, “Three-dimensional ultrahigh-resolution optical-coherence tomography of macular diseases,” Invest. Ophthalmol. Visual Sci. 46, 3393-3402 (2005).
[CrossRef]

Nakajima, N.

Nassif, N. A.

Oppenheim, A. V.

A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing, 2nd ed. (Prentice-Hall, 1999).

T. F. Quatieri, Jr. and A. V. Oppenheim, “Iterative techniques for minimum phase signal reconstruction from phase or magnitude,” IEEE Trans. Acoust., Speech, Signal Process. 29, 1187-1193 (1981).
[CrossRef]

Ozcan, A.

Park, B. H.

Pierce, M. C.

Povazay, B.

U. Schmidt-Erfurth, R. A. Leitgeb, S. Michels, B. Povazay, S. Sacu, B. Hermann, C. Ahlers, H. Sattmann, C. Scholda, A. F. Fercher, and W. Drexler, “Three-dimensional ultrahigh-resolution optical-coherence tomography of macular diseases,” Invest. Ophthalmol. Visual Sci. 46, 3393-3402 (2005).
[CrossRef]

Quatieri, T. F.

T. F. Quatieri, Jr. and A. V. Oppenheim, “Iterative techniques for minimum phase signal reconstruction from phase or magnitude,” IEEE Trans. Acoust., Speech, Signal Process. 29, 1187-1193 (1981).
[CrossRef]

Radzewicz, C.

M. Wojtkowski, T. Bajraszewski, I. Gorczynska, P. Targowski, A. Kowalczyk, W. Wasilewski, and C. Radzewicz, “Ophthalmic imaging by spectral optical-coherence tomography,” Am. J. Ophthalmol. 138, 412-419 (2004).
[CrossRef] [PubMed]

Ratib, O.

A. Rosset, J. Heuberger, and O. Ratib, “OsiriX imaging software” (2007), http://www.osirix-viewer.com/.

Rosset, A.

A. Rosset, J. Heuberger, and O. Ratib, “OsiriX imaging software” (2007), http://www.osirix-viewer.com/.

Sacu, S.

U. Schmidt-Erfurth, R. A. Leitgeb, S. Michels, B. Povazay, S. Sacu, B. Hermann, C. Ahlers, H. Sattmann, C. Scholda, A. F. Fercher, and W. Drexler, “Three-dimensional ultrahigh-resolution optical-coherence tomography of macular diseases,” Invest. Ophthalmol. Visual Sci. 46, 3393-3402 (2005).
[CrossRef]

Sattmann, H.

U. Schmidt-Erfurth, R. A. Leitgeb, S. Michels, B. Povazay, S. Sacu, B. Hermann, C. Ahlers, H. Sattmann, C. Scholda, A. F. Fercher, and W. Drexler, “Three-dimensional ultrahigh-resolution optical-coherence tomography of macular diseases,” Invest. Ophthalmol. Visual Sci. 46, 3393-3402 (2005).
[CrossRef]

A. F. Fercher, R. A. Leitgeb, C. K. Hitzenberger, H. Sattmann, and M. Wojtkowski, “Complex spectral interferometry OCT,” Proc. SPIE 3564, 173-178 (1999).
[CrossRef]

Schafer, R. W.

A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing, 2nd ed. (Prentice-Hall, 1999).

Schmidt-Erfurth, U.

U. Schmidt-Erfurth, R. A. Leitgeb, S. Michels, B. Povazay, S. Sacu, B. Hermann, C. Ahlers, H. Sattmann, C. Scholda, A. F. Fercher, and W. Drexler, “Three-dimensional ultrahigh-resolution optical-coherence tomography of macular diseases,” Invest. Ophthalmol. Visual Sci. 46, 3393-3402 (2005).
[CrossRef]

Scholda, C.

U. Schmidt-Erfurth, R. A. Leitgeb, S. Michels, B. Povazay, S. Sacu, B. Hermann, C. Ahlers, H. Sattmann, C. Scholda, A. F. Fercher, and W. Drexler, “Three-dimensional ultrahigh-resolution optical-coherence tomography of macular diseases,” Invest. Ophthalmol. Visual Sci. 46, 3393-3402 (2005).
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A. Szkulmowska, M. Wojtkowski, I. Gorczynska, T. Bajraszewski, M. Szkulmowski, P. Targowski, A. Kowalczyk, and J. J. Kaluzny, “Coherent noise-free ophthalmic imaging by spectral optical coherence tomography,” J. Phys. D 38, 2606-2611 (2005).
[CrossRef]

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A. Szkulmowska, M. Wojtkowski, I. Gorczynska, T. Bajraszewski, M. Szkulmowski, P. Targowski, A. Kowalczyk, and J. J. Kaluzny, “Coherent noise-free ophthalmic imaging by spectral optical coherence tomography,” J. Phys. D 38, 2606-2611 (2005).
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[CrossRef] [PubMed]

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[CrossRef]

M. Wojtkowski, T. Bajraszewski, I. Gorczynska, P. Targowski, A. Kowalczyk, W. Wasilewski, and C. Radzewicz, “Ophthalmic imaging by spectral optical-coherence tomography,” Am. J. Ophthalmol. 138, 412-419 (2004).
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Am. J. Ophthalmol. (1)

M. Wojtkowski, T. Bajraszewski, I. Gorczynska, P. Targowski, A. Kowalczyk, W. Wasilewski, and C. Radzewicz, “Ophthalmic imaging by spectral optical-coherence tomography,” Am. J. Ophthalmol. 138, 412-419 (2004).
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Invest. Ophthalmol. Visual Sci. (1)

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[CrossRef]

J. Biomed. Opt. (2)

M. Wojtkowski, R. Leitgeb, A. Kowalczyk, T. Bajraszewski, and A. Fercher, “In vivo human retinal imaging by Fourier-domain optical-coherence tomography,” J. Biomed. Opt. 7, 457-463 (2002).
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Proc. IEEE (1)

D. G. Childers, D. P. Skinner, and R. C. Kemerait, “The cepstrum: a guide to processing,” Proc. IEEE 65, 1428-1443 (1977).
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A. H. Bachmann, R. A. Leitgeb, and T. Lasser, “Complex ultrahigh resolution Fourier-domain optical-coherence tomography,” Proc. SPIE 6079, 60790 (2006).
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Figures (10)

Fig. 1
Fig. 1

Schematic of the FDOCT setup.

Fig. 2
Fig. 2

Figure to illustrate the property that F 1 { log ( 1 + A ) } ( z ) and F 1 { log ( 1 + A * ) } ( z ) have nonoverlapping support.

Fig. 3
Fig. 3

Complex-function retrieval by application of Theorem 1. (a) and (d) are the real and imaginary parts of a ( z ) (ground truth); (b) and (e) are the estimates obtained by using the conventional Fourier-transform technique (note the autocorrelation artifacts in the dashed circles); (c) and (f) are the estimates obtained by applying Theorem 1.

Fig. 4
Fig. 4

(a) A homomorphic system and (b) its canonical representation.

Fig. 5
Fig. 5

(a) Characteristic system D { * , + } for convolution and (b) its inverse D 1 { + , * } .

Fig. 6
Fig. 6

Spatial-domain aliasing as a function of the oversampling factor.

Fig. 7
Fig. 7

Signal-to-aliasing error ratio as a function of the oversampling factor.

Fig. 8
Fig. 8

Tomograms of a synthesized multilayer specimen: (a) ground truth, (b) standard Fourier reconstruction (notice the autocorrelation artifacts), (c) homomorphic reconstruction, and (d) iterative reconstruction.

Fig. 9
Fig. 9

Tomograms of a two-layered glass specimen obtained by the (a) conventional Fourier, (b) iterative, and (c) homomorphic techniques. The subplot (d) shows the standard deviation along scans as a function of depth for the three techniques. By comparing the standard deviation profiles in the dashed boxes, we note that the autocorrelation is significantly suppressed by the iterative and homomorphic techniques. The iterative technique has a performance that is superior to that of the conventional technique, but somewhat inferior to that of the homomorphic technique.

Fig. 10
Fig. 10

3D visualization of a segment of a mouse pancreas: (a) standard Fourier, (b) iterative, and (c) homomorphic reconstructions. The x step is 0.8 μ m and the y step is 1.5 μ m . The number of scans in the x and y directions is 1500 and 150 , respectively. The horizontal plane at the top corresponds to the zero-delay reference. The two horizontal lines are probably due to inaccuracies in estimating the background by averaging the scans. The tomogram reconstruction times, excluding the file access operations, are also indicated. The simulations are carried out in MATLAB 7.4 on a Macintosh 2.66 GHz dual-core Intel Xeon system.

Equations (20)

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I ( k ) = S ( k ) a R e j 2 k r + + a ( z ) e j 2 k [ r + n ( z ) z ] d z 2 .
I ( ω ) = S ( ω ) 1 + + a ( z ) e j ω z d z 2 ,
I ( ω ) = S ( ω ) ( 1 + + a ( z ) e j ω z d z + + a * ( z ) e j ω z d z + + + a ( z ) a * ( z ) e j ω ( z z ) d z d z ) .
I ( ω ) = S ( ω ) ( 1 + A ( ω ) ) ( 1 + A * ( ω ) ) .
log I ( ω ) log S ( ω ) = log ( 1 + A ( ω ) ) + log ( 1 + A * ( ω ) ) .
log ( 1 + A ( ω ) ) = n = 1 ( 1 ) n 1 A n ( ω ) n for A ( ω ) < 1 , ω .
log 1 + A ( ω ) 2 = log ( 1 + A ( ω ) ) + log ( 1 + A * ( ω ) ) .
c ( z ) = F 1 { log 1 + A 2 } ( z ) = F 1 { log ( 1 + A ) + log ( 1 + A * ) } ( z ) .
w ( z ) = { 0 , z < z 0 z 2 z 0 + 1 2 , z 0 z z 0 1 , z > z 0 .
F 1 { log ( 1 + A ) } ( z ) = F 1 { log 1 + A 2 } ( z ) w ( z ) a.e .
log ( 1 + A ( ω ) ) = F { F 1 { log 1 + A 2 } w } ( z ) ,
A ( ω ) = exp ( F { F 1 { log 1 + A 2 } ( z ) w } ( ω ) ) 1 ,
c ̂ f ( z ) = 1 2 π + log F ( ω ) e j ω z d ω .
c f ( z ) = 1 2 π + log F ( ω ) e j ω z d ω .
c f ( z ) = c ̂ f ( z ) + c ̂ f * ( z ) 2 .
T { f 1 f 2 } ( z ) = { T { f 1 } T { f 2 } } ( z ) ( linearity ) ,
T { c f 1 } ( z ) = { c T { f 1 } } ( z ) ( scaling property ) ,
I p ( k ) = n = + I ( k + n N ) ,
η = 10 log 10 ( l = 0 N 1 a 2 ( l Δ z ) l = 0 N 1 [ a ( l Δ z ) a ̂ ( l Δ z ) ] 2 ) ,
ξ = 10 log 10 ( 1 L l = 1 L Var { a l } Var { a l a ̂ l } ) ,

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