Towards automatic calibration of Fourier-Domain OCT for robot-assisted vitreoretinal surgery

Xuan Liu; Marcin Balicki; Russell H. Taylor; Jin U. Kang

doi:10.1364/OE.18.024331

Towards automatic calibration of Fourier-Domain OCT for robot-assisted vitreoretinal surgery

Xuan Liu, Marcin Balicki, Russell H. Taylor, and Jin U. Kang

Optics Express
Vol. 18,
Issue 23,
pp. 24331-24343
(2010)
•https://doi.org/10.1364/OE.18.024331

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Xuan Liu, Marcin Balicki, Russell H. Taylor, and Jin U. Kang, "Towards automatic calibration of Fourier-Domain OCT for robot-assisted vitreoretinal surgery," Opt. Express 18, 24331-24343 (2010)

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Related Topics
Table of Contents Category
- Medical Optics and Biotechnology
Optics & Photonics Topics
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The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Medical and biological imaging (170.3880)
- Optical coherence tomography (170.4500)

About this Article
History
- Original Manuscript: September 1, 2010
- Revised Manuscript: October 25, 2010
- Manuscript Accepted: October 27, 2010
- Published: November 5, 2010
Virtual Issues
Virtual Journal for Biomedical Optics Vol. 6, Iss. 1

Accessible

Open Access

Abstract

We present a new real-time automatic spectral calibration (ASC) method for Fourier domain optical coherence tomography (FD OCT) that can be automatically performed by the system. The ASC method proposed can be performed during OCT scanning operation and does not require an external calibrating light source or a commercial optical spectrum analyzer. Spectral data used for calibration can be interferograms obtained from an arbitrary sample which may have complicated internal structures, such as ones found in biological tissue. Moreover, our ASC method incorporates known robot motion to calibrate physical pixel spacing of the A-scan in static or dynamic environments. Experimental results show that our ASC method can provide high-performance calibration for FD OCT in terms of axial resolution and ranging accuracy without increasing hardware complexity.

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References

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[Crossref] [PubMed]
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K. Zhang, W. Wang, J. Han, and J. U. Kang, “A surface topology and motion compensation system for microsurgery guidance and intervention based on common-path optical coherence tomography,” IEEE Trans. Biomed. Eng. 56(9), 2318–2321 (2009).
[Crossref] [PubMed]
J. Han, M. Balicki, K. Zhang, X. Liu, J. Handa, R. Taylor, and J. U. Kang, “Common-path Fourier-domain Optical Coherence Tomography with a Fiber Optic Probe Integrated Into a Surgical Needle,” Proceedings of CLEO Conference (2009)
Y. K. Tao, J. P. Ehlers, C. A. Toth, and J. A. Izatt, “Intraoperative spectral domain optical coherence tomography for vitreoretinal surgery,” Opt. Lett. 35(20), 3315–3317 (2010).
[Crossref] [PubMed]
J. F. de Boer, B. Cense, B. H. Park, M. C. Pierce, G. J. Tearney, and B. E. Bouma, “Improved signal-to-noise ratio in spectral-domain compared with time-domain optical coherence tomography,” Opt. Lett. 28(21), 2067–2069 (2003).
[Crossref] [PubMed]
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[Crossref] [PubMed]
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[Crossref] [PubMed]
X. Li, J. H. Han, X. Liu, and J. U. Kang, “Signal-to-noise ratio analysis of all-fiber common-path optical coherence tomography,” Appl. Opt. 47(27), 4833–4840 (2008).
[Crossref] [PubMed]
U. Sharma, N. M. Fried, and J. U. Kang, “All-fiber Fizeau optical coherence tomography: sensitivity optimization and system analysis,” IEEE J. Quantum Electron. 11, 11799–11805 (2005).
X. Liu, X. Li, D. Kim, I. Ilev, and J. U. Kang, “Fiber-optic Fourier-domain common-path OCT,” Chin. Opt. Lett. 6(12), 899–901 (2008).
[Crossref]
A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography - principles and applications,” Rep. Prog. Phys. 66(2), 239–303 (2003).
[Crossref]
B. E. Bouma, and G. J. Tearney, Handbook of Optical Coherence Tomography, (Marcel Dekker, New York, 2002).
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[Crossref] [PubMed]
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[Crossref] [PubMed]
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[Crossref] [PubMed]
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[Crossref] [PubMed]
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[Crossref] [PubMed]
A. Uneri, M. Balicki, J. Handa, P. Gehlbach, R. Taylor, and I. Iordachita, “New Steady-Hand Eye Robot with Microforce Sensing for Vitreoretinal Surgery Research,” IEEE BioRob (2010).
M. Wojtkowski, V. Srinivasan, T. Ko, J. Fujimoto, A. Kowalczyk, and J. Duker, “Ultrahigh-resolution, high-speed, Fourier domain optical coherence tomography and methods for dispersion compensation,” Opt. Express 12(11), 2404–2422 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-11-2404 .
[Crossref] [PubMed]
Y. Yasuno, Y. Hong, S. Makita, M. Yamanari, M. Akiba, M. Miura, and T. Yatagai, “In vivo high-contrast imaging of deep posterior eye by 1-um swept source optical coherence tomography and scattering optical coherence angiography,” Opt. Express 15(10), 6121–6139 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-10-6121 .
[Crossref] [PubMed]
X. Liu, E. Meisne, J. Han, K. Zhang, P. Gehlbach, and R. Taylor, “Internal limiting membrane layer visualization and vitreoretinal surgery guidance using OCT integrated microsurgical tool,” Proc. SPIE 7550, 755003 (2010).
[Crossref]
J. U. Kang, J. Han, X. Liu, K. Zhang, C. Song, and P. Gehlbach, “Endoscopic Functional Fourier Domain Common Path Optical Coherence Tomography for Microsurgery,” IEEE J. Sel. Top. Quantum Electron. 16(4), 781–792 (2010), doi:.
[Crossref]
Z. Xu, L. Carrion, and R. Maciejko, “A zero-crossing detection method applied to Doppler OCT,” Opt. Express 16(7), 4394–4412 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-7-4394 .
[Crossref] [PubMed]
B. Cense, N. Nassif, T. Chen, M. Pierce, S. H. Yun, B. Park, B. Bouma, G. Tearney, and J. de Boer, “Ultrahigh-resolution high-speed retinal imaging using spectral-domain optical coherence tomography,” Opt. Express 12(11), 2435–2447 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-11-2435 .
[Crossref] [PubMed]

2010 (6)

C. Ding, P. Bu, X. Wang, and O. Sasaki, “A new spectral calibration method for Fourier domain optical coherence tomography,” Optik (Stuttg.) 121(11), 965–970 (2010).

E. Azimi, B. Liu, and M. E. Brezinski, “Real-time and high-performance calibration method for high-speed swept-source optical coherence tomography,” J. Biomed. Opt. 15(1), 016005 (2010).
[Crossref] [PubMed]

X. Liu, E. Meisne, J. Han, K. Zhang, P. Gehlbach, and R. Taylor, “Internal limiting membrane layer visualization and vitreoretinal surgery guidance using OCT integrated microsurgical tool,” Proc. SPIE 7550, 755003 (2010).
[Crossref]

J. U. Kang, J. Han, X. Liu, K. Zhang, C. Song, and P. Gehlbach, “Endoscopic Functional Fourier Domain Common Path Optical Coherence Tomography for Microsurgery,” IEEE J. Sel. Top. Quantum Electron. 16(4), 781–792 (2010), doi:.
[Crossref]

J. Xi, L. Huo, J. Li, and X. Li, “Generic real-time uniform K-space sampling method for high-speed swept-Source optical coherence tomography,” Opt. Express 18(9), 9511–9517 (2010), http://www.opticsinfobase.org/abstract.cfm?URI=oe-18-9-9511 .
[Crossref] [PubMed]

Y. K. Tao, J. P. Ehlers, C. A. Toth, and J. A. Izatt, “Intraoperative spectral domain optical coherence tomography for vitreoretinal surgery,” Opt. Lett. 35(20), 3315–3317 (2010).
[Crossref] [PubMed]

2009 (1)

K. Zhang, W. Wang, J. Han, and J. U. Kang, “A surface topology and motion compensation system for microsurgery guidance and intervention based on common-path optical coherence tomography,” IEEE Trans. Biomed. Eng. 56(9), 2318–2321 (2009).
[Crossref] [PubMed]

2008 (5)

S. Han, M. V. Sarunic, J. Wu, M. Humayun, and C. Yang, “Handheld forward-imaging needle endoscope for ophthalmic optical coherence tomography inspection,” J. Biomed. Opt. 13(2), 020505 (2008).
[Crossref] [PubMed]

Z. Xu, L. Carrion, and R. Maciejko, “A zero-crossing detection method applied to Doppler OCT,” Opt. Express 16(7), 4394–4412 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-7-4394 .
[Crossref] [PubMed]

N. V. Iftimia, D. X. Hammer, R. D. Ferguson, M. Mujat, D. Vu, and A. A. Ferrante, “Dual-beam Fourier domain optical Doppler tomography of zebrafish,” Opt. Express 16(18), 13624–13636 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-18-13624 .
[Crossref] [PubMed]

X. Li, J. H. Han, X. Liu, and J. U. Kang, “Signal-to-noise ratio analysis of all-fiber common-path optical coherence tomography,” Appl. Opt. 47(27), 4833–4840 (2008).
[Crossref] [PubMed]

X. Liu, X. Li, D. Kim, I. Ilev, and J. U. Kang, “Fiber-optic Fourier-domain common-path OCT,” Chin. Opt. Lett. 6(12), 899–901 (2008).
[Crossref]

2007 (2)

Y. Yasuno, Y. Hong, S. Makita, M. Yamanari, M. Akiba, M. Miura, and T. Yatagai, “In vivo high-contrast imaging of deep posterior eye by 1-um swept source optical coherence tomography and scattering optical coherence angiography,” Opt. Express 15(10), 6121–6139 (2007), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-10-6121 .
[Crossref] [PubMed]

M. Mujat, B. H. Park, B. Cense, T. C. Chen, and J. F. de Boer, “Autocalibration of spectral-domain optical coherence tomography spectrometers for in vivo quantitative retinal nerve fiber layer birefringence determination,” J. Biomed. Opt. 12(4), 041205 (2007).
[Crossref] [PubMed]

2005 (2)

U. Sharma, N. M. Fried, and J. U. Kang, “All-fiber Fizeau optical coherence tomography: sensitivity optimization and system analysis,” IEEE J. Quantum Electron. 11, 11799–11805 (2005).

Y. R. Chen, B. Sun, T. Han, Y. F. Kong, C. H. Xu, P. Zhou, X. F. Li, S. Y. Wang, Y. X. Zheng, and L. Y. Chen, “Densely folded spectral images of the CCD spectrometer working in the full 200-1000nm wavelength range with high resolution,” Opt. Express 13(25), 10049–10054 (2005), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-25-10049 .
[Crossref] [PubMed]

2004 (2)

M. Wojtkowski, V. Srinivasan, T. Ko, J. Fujimoto, A. Kowalczyk, and J. Duker, “Ultrahigh-resolution, high-speed, Fourier domain optical coherence tomography and methods for dispersion compensation,” Opt. Express 12(11), 2404–2422 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-11-2404 .
[Crossref] [PubMed]

B. Cense, N. Nassif, T. Chen, M. Pierce, S. H. Yun, B. Park, B. Bouma, G. Tearney, and J. de Boer, “Ultrahigh-resolution high-speed retinal imaging using spectral-domain optical coherence tomography,” Opt. Express 12(11), 2435–2447 (2004), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-11-2435 .
[Crossref] [PubMed]

2003 (4)

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography - principles and applications,” Rep. Prog. Phys. 66(2), 239–303 (2003).
[Crossref]

R. Leitgeb, C. Hitzenberger, and A. Fercher, “Performance of fourier domain vs. time domain optical coherence tomography,” Opt. Express 11(8), 889–894 (2003), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-8-889 .
[Crossref] [PubMed]

J. F. de Boer, B. Cense, B. H. Park, M. C. Pierce, G. J. Tearney, and B. E. Bouma, “Improved signal-to-noise ratio in spectral-domain compared with time-domain optical coherence tomography,” Opt. Lett. 28(21), 2067–2069 (2003).
[Crossref] [PubMed]

M. Choma, M. Sarunic, C. Yang, and J. Izatt, “Sensitivity advantage of swept source and Fourier domain optical coherence tomography,” Opt. Express 11(18), 2183–2189 (2003), http://www.opticsinfobase.org/abstract.cfm?URI=oe-11-18-2183 .
[Crossref] [PubMed]

Akiba, M.

Azimi, E.

Bouma, B.

Bouma, B. E.

Brezinski, M. E.

Bu, P.

C. Ding, P. Bu, X. Wang, and O. Sasaki, “A new spectral calibration method for Fourier domain optical coherence tomography,” Optik (Stuttg.) 121(11), 965–970 (2010).

Carrion, L.

Cense, B.

Chen, L. Y.

Chen, T.

Chen, T. C.

Chen, Y. R.

Choma, M.

de Boer, J.

de Boer, J. F.

Ding, C.

C. Ding, P. Bu, X. Wang, and O. Sasaki, “A new spectral calibration method for Fourier domain optical coherence tomography,” Optik (Stuttg.) 121(11), 965–970 (2010).

Drexler, W.

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography - principles and applications,” Rep. Prog. Phys. 66(2), 239–303 (2003).
[Crossref]

Duker, J.

Ehlers, J. P.

Fercher, A.

Fercher, A. F.

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography - principles and applications,” Rep. Prog. Phys. 66(2), 239–303 (2003).
[Crossref]

Ferguson, R. D.

Ferrante, A. A.

Fried, N. M.

U. Sharma, N. M. Fried, and J. U. Kang, “All-fiber Fizeau optical coherence tomography: sensitivity optimization and system analysis,” IEEE J. Quantum Electron. 11, 11799–11805 (2005).

Fujimoto, J.

Gehlbach, P.

Hammer, D. X.

Han, J.

Han, J. H.

X. Li, J. H. Han, X. Liu, and J. U. Kang, “Signal-to-noise ratio analysis of all-fiber common-path optical coherence tomography,” Appl. Opt. 47(27), 4833–4840 (2008).
[Crossref] [PubMed]

Han, S.

Han, T.

Hitzenberger, C.

Hitzenberger, C. K.

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography - principles and applications,” Rep. Prog. Phys. 66(2), 239–303 (2003).
[Crossref]

Hong, Y.

Humayun, M.

Huo, L.

Iftimia, N. V.

Ilev, I.

X. Liu, X. Li, D. Kim, I. Ilev, and J. U. Kang, “Fiber-optic Fourier-domain common-path OCT,” Chin. Opt. Lett. 6(12), 899–901 (2008).
[Crossref]

Izatt, J.

Izatt, J. A.

Kang, J. U.

X. Liu, X. Li, D. Kim, I. Ilev, and J. U. Kang, “Fiber-optic Fourier-domain common-path OCT,” Chin. Opt. Lett. 6(12), 899–901 (2008).
[Crossref]

X. Li, J. H. Han, X. Liu, and J. U. Kang, “Signal-to-noise ratio analysis of all-fiber common-path optical coherence tomography,” Appl. Opt. 47(27), 4833–4840 (2008).
[Crossref] [PubMed]

U. Sharma, N. M. Fried, and J. U. Kang, “All-fiber Fizeau optical coherence tomography: sensitivity optimization and system analysis,” IEEE J. Quantum Electron. 11, 11799–11805 (2005).

Kim, D.

X. Liu, X. Li, D. Kim, I. Ilev, and J. U. Kang, “Fiber-optic Fourier-domain common-path OCT,” Chin. Opt. Lett. 6(12), 899–901 (2008).
[Crossref]

Ko, T.

Kong, Y. F.

Kowalczyk, A.

Lasser, T.

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography - principles and applications,” Rep. Prog. Phys. 66(2), 239–303 (2003).
[Crossref]

Leitgeb, R.

Li, J.

Li, X.

X. Li, J. H. Han, X. Liu, and J. U. Kang, “Signal-to-noise ratio analysis of all-fiber common-path optical coherence tomography,” Appl. Opt. 47(27), 4833–4840 (2008).
[Crossref] [PubMed]

X. Liu, X. Li, D. Kim, I. Ilev, and J. U. Kang, “Fiber-optic Fourier-domain common-path OCT,” Chin. Opt. Lett. 6(12), 899–901 (2008).
[Crossref]

Li, X. F.

Liu, B.

Liu, X.

X. Li, J. H. Han, X. Liu, and J. U. Kang, “Signal-to-noise ratio analysis of all-fiber common-path optical coherence tomography,” Appl. Opt. 47(27), 4833–4840 (2008).
[Crossref] [PubMed]

X. Liu, X. Li, D. Kim, I. Ilev, and J. U. Kang, “Fiber-optic Fourier-domain common-path OCT,” Chin. Opt. Lett. 6(12), 899–901 (2008).
[Crossref]

Maciejko, R.

Makita, S.

Meisne, E.

Miura, M.

Mujat, M.

Nassif, N.

Park, B.

Park, B. H.

Pierce, M.

Pierce, M. C.

Sarunic, M.

Sarunic, M. V.

Sasaki, O.

C. Ding, P. Bu, X. Wang, and O. Sasaki, “A new spectral calibration method for Fourier domain optical coherence tomography,” Optik (Stuttg.) 121(11), 965–970 (2010).

Sharma, U.

U. Sharma, N. M. Fried, and J. U. Kang, “All-fiber Fizeau optical coherence tomography: sensitivity optimization and system analysis,” IEEE J. Quantum Electron. 11, 11799–11805 (2005).

Song, C.

Srinivasan, V.

Sun, B.

Tao, Y. K.

Taylor, R.

Tearney, G.

Tearney, G. J.

Toth, C. A.

Vu, D.

Wang, S. Y.

Wang, W.

Wang, X.

C. Ding, P. Bu, X. Wang, and O. Sasaki, “A new spectral calibration method for Fourier domain optical coherence tomography,” Optik (Stuttg.) 121(11), 965–970 (2010).

Wojtkowski, M.

Wu, J.

Xi, J.

Xu, C. H.

Xu, Z.

Yamanari, M.

Yang, C.

Yasuno, Y.

Yatagai, T.

Yun, S. H.

Zhang, K.

Zheng, Y. X.

Zhou, P.

Appl. Opt. (1)

X. Li, J. H. Han, X. Liu, and J. U. Kang, “Signal-to-noise ratio analysis of all-fiber common-path optical coherence tomography,” Appl. Opt. 47(27), 4833–4840 (2008).
[Crossref] [PubMed]

Chin. Opt. Lett. (1)

X. Liu, X. Li, D. Kim, I. Ilev, and J. U. Kang, “Fiber-optic Fourier-domain common-path OCT,” Chin. Opt. Lett. 6(12), 899–901 (2008).
[Crossref]

IEEE J. Quantum Electron. (1)

U. Sharma, N. M. Fried, and J. U. Kang, “All-fiber Fizeau optical coherence tomography: sensitivity optimization and system analysis,” IEEE J. Quantum Electron. 11, 11799–11805 (2005).

IEEE J. Sel. Top. Quantum Electron. (1)

IEEE Trans. Biomed. Eng. (1)

J. Biomed. Opt. (3)

Opt. Express (9)

Opt. Lett. (2)

Optik (Stuttg.) (1)

C. Ding, P. Bu, X. Wang, and O. Sasaki, “A new spectral calibration method for Fourier domain optical coherence tomography,” Optik (Stuttg.) 121(11), 965–970 (2010).

Proc. SPIE (1)

Rep. Prog. Phys. (1)

A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography - principles and applications,” Rep. Prog. Phys. 66(2), 239–303 (2003).
[Crossref]

Other (4)

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

A. Uneri, M. Balicki, J. Handa, P. Gehlbach, R. Taylor, and I. Iordachita, “New Steady-Hand Eye Robot with Microforce Sensing for Vitreoretinal Surgery Research,” IEEE BioRob (2010).

M. Balicki, J. Han, I. Iordachita, P. Gehlbach, J. Handa, J. U. Kang, and R. Taylor, “Single Fiber Optical Coherence Tomography Microsurgical Instruments for Computer and Robot-Assisted Retinal Surgery,” Proceedings of the MICCAI Conference, London, pp. 108–115 (2009)

J. Han, M. Balicki, K. Zhang, X. Liu, J. Handa, R. Taylor, and J. U. Kang, “Common-path Fourier-domain Optical Coherence Tomography with a Fiber Optic Probe Integrated Into a Surgical Needle,” Proceedings of CLEO Conference (2009)

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Fig. 1 Schematic of CP FD OCT and robot system.

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Fig. 2 (a) Wavenumber versus pixel index; (b) simulated interferometric fringes in pixel space; (c) A-scan corresponding to spectrum in Fig. 2(b); differences between k(n) and

\tilde{k} (n)

, k(n) and p(n), k(n) and the 4th order polynomial fits of p(n) for data generated with R equals 0.1(d), 0.2(e), and 0.3(f).

Download Full Size | PPT Slide | PDF

Fig. 3 (a) Interferometric fringes detected at the center part of CCD (red, original spectrum; black, spectrum after bandpass filtering; blue circles, zero-crossing points); (b) initial wavenumber mapping using zero-crossing detection; (c) difference between the initial estimation of wavenumber mapping and the wavenumber mapping that maximizes A-scan sharpness; (d) interferometric fringes obtained from multilayered phantom, in pixel space (blue) and k-space (red). (e) A-scan corresponding to the spectrum in Fig. 3(d).

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Fig. 4 (a) M-scan obtained when scanning the OCT probe axially above a multilayered phantom; (b) robot motion derived from OCT signal in the unit of pixel (upper); commanded robot motion in the unit of mm(lower); (c) ranging error based on regression using Fig. 4(b); (d) M-scan obtained when the OCT probe was modulated sinusoidally and the sample was nonstatic; (e) peak indices of OCT signal (upper) and its frequency analysis result (lower); (f) robot motion derived from OCT signal in the unit of pixel (upper); commanded robot motion in mm (lower); (g) ranging error based on regression of Fig. 4(f).

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Fig. 5 (a) PSFs obtained without calibration; (b) PSFs obtained with calibration based on the initial estimation of the 4th order polynomial; (c) PSFs obtained with calibration based on the 4th order polynomial that maximizes the image sharpness; (d) FWHM resolution at different depths based on initial estimation of polynomial (blue) and the polynomial that maximizes the signal sharpness (red).

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Fig. 6 (a) ground truth spectral mapping obtained from a mirror: wavenumber versus pixel index in our spectrometer; (b) FWHM resolution at different depths using interferograms from a multilayered phantom by ASC (blue) and from a mirror by extracting the phase (red).

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Fig. 7 Image of multilayer phantom obtained with (a) and without (b) ASC.

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Fig. 8 (a) A-scans based on ground truth calibration and ASC; (b) OCT image (logarithmic scale) of human forearm skin, based on ground truth calibration(SC: stratum corneum; epidermis: EP; dermis: DE); (c) OCT image (logarithmic scale) of human forearm skin, based on our automatic calibration using sample interferogram.

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Equations (9)

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S_{n} = α S_{0} (k_{n}) \sum_{m} [R_{m} \cos (2 k_{n} l_{m})]

k_{n} = a_{4} n^{4} + a_{3} n^{3} + a_{2} n^{2} + a_{1} n + a_{0}

n_{m} = \frac{n_{1} S_{n_{1} + 1} - (n_{1} + 1) S_{n_{1}}}{S_{n_{1} + 1} - S_{n_{1}}}

k (n_{m}) = m δ k + k_{0}

\tilde{k} (n_{m}) = \frac{k (n_{m}) - k_{0}}{δ k}

Δ z= \frac{π}{η Δ k}

Z = i Δ {z+z}_{0}

Z_{0} \sin (ω t) + Z_{S} (t) = i (t) Δ {z+z}_{0}

Z_{0} \sin (ω t) = \hat{i} Δ z + {\hat{z}}_{0}