Abstract

The analysis of spectroscopic optical coherence tomography (SOCT) signals suffers the trade-off between time resolution and frequency resolution. Various joint time–frequency distributions (TFDs) can optimize this trade-off. Synthesized signals were generated and experimentally acquired data were obtained to compare and validate several different TFDs under different SOCT imaging schemes. Specific criteria were designed to quantify the TFD performance. We found that different SOCT imaging schemes require different optimal TFDs. Cohen’s class TFDs generate the most compact time–frequency (TF) analysis, while linear TFDs offer the most reliable TF analysis. In both cases, if some prior information is known, model-based TF analysis can improve the performance.

© 2005 Optical Society of America

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  1. D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
    [CrossRef] [PubMed]
  2. J. G. Fujimoto, M. E. Brezinski, G. J. Tearney, S. A. Boppart, B. E. Bouma, M. R. Hee, J. F. Southern, E. A. Swanson, “Biomedical imaging and optical biopsy using optical coherence tomography,” Nat. Med. 1, 970–972 (1995).
    [CrossRef] [PubMed]
  3. J. F. de Boer, T. E. Milner, M. J. C. van Germert, J. S. Nelson, “Two-dimensional birefringence imaging in biological tissue by polarization sensitive optical coherence tomography,” Opt. Lett. 22, 934–936 (1997).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  5. S. A. Boppart, B. E. Bouma, C. Pitris, G. J. Tearney, J. F. Southern, M. E. Brezinski, J. G. Fujimoto, “Intraoperative assessment of microsurgery with three dimensional optical coherence tomography,” Radiology 208, 81–86 (1998).
    [PubMed]
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    [CrossRef]
  7. R. Leitgeb, M. Wojtkowski, A. Kowalczyk, C. K. Hitzenberger, M. Sticker, A. F. Fercher, “Spectral measurement of absorption by spectroscopic frequency-domain optical coherence tomography,” Opt. Lett. 25, 820–822 (2000).
    [CrossRef]
  8. S. A. Boppart, W. Drexler, U. Morgner, F. X. Kartner, J. G. Fujimoto, “Ultrahigh resolution and spectroscopic optical coherence tomography imaging of cellular morphology and function,” presented at the Inter-Institute Workshop on In Vivo Optical Imaging at the National Institutes of Health, Bethesda, Md., 16–17 September 1999.
  9. B. Hermann, K. Bizheva, A. Unterhuber, B. Povazay, H. Sattmann, L. Schmetterer, A. F. Fercher, W. Drexler, “Precision of extracting absorption profiles from weakly scattering media with spectroscopic time-domain optical coherence tomography,” Opt. Express12, 1677–1688 (2004), http://www.opticsexpress.org .
    [CrossRef]
  10. C. Xu, J. Ye, D. L. Marks, S. A. Boppart, “Near-infrared dyes as contrast-enhancing agents for spectroscopic optical coherence tomography,” Opt. Lett. 29, 1647–1649 (2004).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  15. R. Carmona, W. Hwang, B. Torresani, Practical Time–Frequency Analysis (Academic, San Diego, Calif., 1998).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]

2004 (2)

2003 (2)

2000 (2)

1998 (2)

S. A. Boppart, B. E. Bouma, C. Pitris, F. F. Southern, M. E. Brezinski, J. G. Fujimoto, “In vivo cellular optical coherence tomography imaging,” Nat. Med. 4, 861–864 (1998).
[CrossRef] [PubMed]

S. A. Boppart, B. E. Bouma, C. Pitris, G. J. Tearney, J. F. Southern, M. E. Brezinski, J. G. Fujimoto, “Intraoperative assessment of microsurgery with three dimensional optical coherence tomography,” Radiology 208, 81–86 (1998).
[PubMed]

1997 (1)

1995 (1)

J. G. Fujimoto, M. E. Brezinski, G. J. Tearney, S. A. Boppart, B. E. Bouma, M. R. Hee, J. F. Southern, E. A. Swanson, “Biomedical imaging and optical biopsy using optical coherence tomography,” Nat. Med. 1, 970–972 (1995).
[CrossRef] [PubMed]

1991 (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

1990 (2)

I. Daubechies, “The wavelet transform, time–frequency localization and signal analysis,” IEEE Trans. Inf. Theory 36, 961–1005 (1990).
[CrossRef]

D. L. Jones, T. W. Parks, “A high resolution data-adaptive time-frequency representation,” IEEE Trans. Acoust. Speech Signal Process. 38, 2127–2135 (1990).
[CrossRef]

1989 (1)

L. Cohen, “Time–frequency distributions—a review,” Proc. IEEE 77, 941–981 (1989).
[CrossRef]

Aalders, M. C. G.

Boppart, S. A.

C. Xu, J. Ye, D. L. Marks, S. A. Boppart, “Near-infrared dyes as contrast-enhancing agents for spectroscopic optical coherence tomography,” Opt. Lett. 29, 1647–1649 (2004).
[CrossRef] [PubMed]

D. L. Marks, A. L. Oldenburg, J. J. Reynolds, S. A. Boppart, “Digital algorithms for dispersion correction in optical coherence tomography for homogeneous and stratified media,” Appl. Opt. 42, 204–217 (2003).
[CrossRef] [PubMed]

S. A. Boppart, B. E. Bouma, C. Pitris, F. F. Southern, M. E. Brezinski, J. G. Fujimoto, “In vivo cellular optical coherence tomography imaging,” Nat. Med. 4, 861–864 (1998).
[CrossRef] [PubMed]

S. A. Boppart, B. E. Bouma, C. Pitris, G. J. Tearney, J. F. Southern, M. E. Brezinski, J. G. Fujimoto, “Intraoperative assessment of microsurgery with three dimensional optical coherence tomography,” Radiology 208, 81–86 (1998).
[PubMed]

J. G. Fujimoto, M. E. Brezinski, G. J. Tearney, S. A. Boppart, B. E. Bouma, M. R. Hee, J. F. Southern, E. A. Swanson, “Biomedical imaging and optical biopsy using optical coherence tomography,” Nat. Med. 1, 970–972 (1995).
[CrossRef] [PubMed]

S. A. Boppart, W. Drexler, U. Morgner, F. X. Kartner, J. G. Fujimoto, “Ultrahigh resolution and spectroscopic optical coherence tomography imaging of cellular morphology and function,” presented at the Inter-Institute Workshop on In Vivo Optical Imaging at the National Institutes of Health, Bethesda, Md., 16–17 September 1999.

Bouma, B. E.

S. A. Boppart, B. E. Bouma, C. Pitris, F. F. Southern, M. E. Brezinski, J. G. Fujimoto, “In vivo cellular optical coherence tomography imaging,” Nat. Med. 4, 861–864 (1998).
[CrossRef] [PubMed]

S. A. Boppart, B. E. Bouma, C. Pitris, G. J. Tearney, J. F. Southern, M. E. Brezinski, J. G. Fujimoto, “Intraoperative assessment of microsurgery with three dimensional optical coherence tomography,” Radiology 208, 81–86 (1998).
[PubMed]

J. G. Fujimoto, M. E. Brezinski, G. J. Tearney, S. A. Boppart, B. E. Bouma, M. R. Hee, J. F. Southern, E. A. Swanson, “Biomedical imaging and optical biopsy using optical coherence tomography,” Nat. Med. 1, 970–972 (1995).
[CrossRef] [PubMed]

Brezinski, M. E.

S. A. Boppart, B. E. Bouma, C. Pitris, G. J. Tearney, J. F. Southern, M. E. Brezinski, J. G. Fujimoto, “Intraoperative assessment of microsurgery with three dimensional optical coherence tomography,” Radiology 208, 81–86 (1998).
[PubMed]

S. A. Boppart, B. E. Bouma, C. Pitris, F. F. Southern, M. E. Brezinski, J. G. Fujimoto, “In vivo cellular optical coherence tomography imaging,” Nat. Med. 4, 861–864 (1998).
[CrossRef] [PubMed]

J. G. Fujimoto, M. E. Brezinski, G. J. Tearney, S. A. Boppart, B. E. Bouma, M. R. Hee, J. F. Southern, E. A. Swanson, “Biomedical imaging and optical biopsy using optical coherence tomography,” Nat. Med. 1, 970–972 (1995).
[CrossRef] [PubMed]

Carmona, R.

R. Carmona, W. Hwang, B. Torresani, Practical Time–Frequency Analysis (Academic, San Diego, Calif., 1998).

Chang, W.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Choma, M. A.

Cohen, L.

L. Cohen, “Time–frequency distributions—a review,” Proc. IEEE 77, 941–981 (1989).
[CrossRef]

Daubechies, I.

I. Daubechies, “The wavelet transform, time–frequency localization and signal analysis,” IEEE Trans. Inf. Theory 36, 961–1005 (1990).
[CrossRef]

de Boer, J. F.

Drexler, W.

U. Morgner, W. Drexler, F. C. Kartner, X. D. Li, C. Pitris, E. P. Ippen, J. G. Fujimoto, “Spectroscopic optical coherence tomography,” Opt. Lett. 25, 111–113 (2000).
[CrossRef]

S. A. Boppart, W. Drexler, U. Morgner, F. X. Kartner, J. G. Fujimoto, “Ultrahigh resolution and spectroscopic optical coherence tomography imaging of cellular morphology and function,” presented at the Inter-Institute Workshop on In Vivo Optical Imaging at the National Institutes of Health, Bethesda, Md., 16–17 September 1999.

Faber, D. J.

Fercher, A. F.

Flotte, T.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Fujimoto, J. G.

U. Morgner, W. Drexler, F. C. Kartner, X. D. Li, C. Pitris, E. P. Ippen, J. G. Fujimoto, “Spectroscopic optical coherence tomography,” Opt. Lett. 25, 111–113 (2000).
[CrossRef]

S. A. Boppart, B. E. Bouma, C. Pitris, F. F. Southern, M. E. Brezinski, J. G. Fujimoto, “In vivo cellular optical coherence tomography imaging,” Nat. Med. 4, 861–864 (1998).
[CrossRef] [PubMed]

S. A. Boppart, B. E. Bouma, C. Pitris, G. J. Tearney, J. F. Southern, M. E. Brezinski, J. G. Fujimoto, “Intraoperative assessment of microsurgery with three dimensional optical coherence tomography,” Radiology 208, 81–86 (1998).
[PubMed]

J. G. Fujimoto, M. E. Brezinski, G. J. Tearney, S. A. Boppart, B. E. Bouma, M. R. Hee, J. F. Southern, E. A. Swanson, “Biomedical imaging and optical biopsy using optical coherence tomography,” Nat. Med. 1, 970–972 (1995).
[CrossRef] [PubMed]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

S. A. Boppart, W. Drexler, U. Morgner, F. X. Kartner, J. G. Fujimoto, “Ultrahigh resolution and spectroscopic optical coherence tomography imaging of cellular morphology and function,” presented at the Inter-Institute Workshop on In Vivo Optical Imaging at the National Institutes of Health, Bethesda, Md., 16–17 September 1999.

Gregory, K.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Hee, M. R.

J. G. Fujimoto, M. E. Brezinski, G. J. Tearney, S. A. Boppart, B. E. Bouma, M. R. Hee, J. F. Southern, E. A. Swanson, “Biomedical imaging and optical biopsy using optical coherence tomography,” Nat. Med. 1, 970–972 (1995).
[CrossRef] [PubMed]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Hitzenberger, C. K.

Huang, D.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Hwang, W.

R. Carmona, W. Hwang, B. Torresani, Practical Time–Frequency Analysis (Academic, San Diego, Calif., 1998).

Ippen, E. P.

Izatt, J. A.

Jones, D. L.

D. L. Jones, T. W. Parks, “A high resolution data-adaptive time-frequency representation,” IEEE Trans. Acoust. Speech Signal Process. 38, 2127–2135 (1990).
[CrossRef]

Kartner, F. C.

Kartner, F. X.

S. A. Boppart, W. Drexler, U. Morgner, F. X. Kartner, J. G. Fujimoto, “Ultrahigh resolution and spectroscopic optical coherence tomography imaging of cellular morphology and function,” presented at the Inter-Institute Workshop on In Vivo Optical Imaging at the National Institutes of Health, Bethesda, Md., 16–17 September 1999.

Kowalczyk, A.

Lamb, L. E.

Leitgeb, R.

Li, X. D.

Lin, C. P.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Marks, D. L.

Mik, E. G.

Milner, T. E.

Morgner, U.

U. Morgner, W. Drexler, F. C. Kartner, X. D. Li, C. Pitris, E. P. Ippen, J. G. Fujimoto, “Spectroscopic optical coherence tomography,” Opt. Lett. 25, 111–113 (2000).
[CrossRef]

S. A. Boppart, W. Drexler, U. Morgner, F. X. Kartner, J. G. Fujimoto, “Ultrahigh resolution and spectroscopic optical coherence tomography imaging of cellular morphology and function,” presented at the Inter-Institute Workshop on In Vivo Optical Imaging at the National Institutes of Health, Bethesda, Md., 16–17 September 1999.

Nelson, J. S.

Oldenburg, A. L.

Papandreou-Suppappola, A.

A. Papandreou-Suppappola, Applications in Time–Frequency Signal Processing (CRC Press, Boca Raton, Fla., 2002).
[CrossRef]

Parks, T. W.

D. L. Jones, T. W. Parks, “A high resolution data-adaptive time-frequency representation,” IEEE Trans. Acoust. Speech Signal Process. 38, 2127–2135 (1990).
[CrossRef]

Pitris, C.

U. Morgner, W. Drexler, F. C. Kartner, X. D. Li, C. Pitris, E. P. Ippen, J. G. Fujimoto, “Spectroscopic optical coherence tomography,” Opt. Lett. 25, 111–113 (2000).
[CrossRef]

S. A. Boppart, B. E. Bouma, C. Pitris, F. F. Southern, M. E. Brezinski, J. G. Fujimoto, “In vivo cellular optical coherence tomography imaging,” Nat. Med. 4, 861–864 (1998).
[CrossRef] [PubMed]

S. A. Boppart, B. E. Bouma, C. Pitris, G. J. Tearney, J. F. Southern, M. E. Brezinski, J. G. Fujimoto, “Intraoperative assessment of microsurgery with three dimensional optical coherence tomography,” Radiology 208, 81–86 (1998).
[PubMed]

Puliafito, C. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Reynolds, J. J.

Schuman, J. S.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Simon, J. D.

Southern, F. F.

S. A. Boppart, B. E. Bouma, C. Pitris, F. F. Southern, M. E. Brezinski, J. G. Fujimoto, “In vivo cellular optical coherence tomography imaging,” Nat. Med. 4, 861–864 (1998).
[CrossRef] [PubMed]

Southern, J. F.

S. A. Boppart, B. E. Bouma, C. Pitris, G. J. Tearney, J. F. Southern, M. E. Brezinski, J. G. Fujimoto, “Intraoperative assessment of microsurgery with three dimensional optical coherence tomography,” Radiology 208, 81–86 (1998).
[PubMed]

J. G. Fujimoto, M. E. Brezinski, G. J. Tearney, S. A. Boppart, B. E. Bouma, M. R. Hee, J. F. Southern, E. A. Swanson, “Biomedical imaging and optical biopsy using optical coherence tomography,” Nat. Med. 1, 970–972 (1995).
[CrossRef] [PubMed]

Sticker, M.

Stinson, W. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Swanson, E. A.

J. G. Fujimoto, M. E. Brezinski, G. J. Tearney, S. A. Boppart, B. E. Bouma, M. R. Hee, J. F. Southern, E. A. Swanson, “Biomedical imaging and optical biopsy using optical coherence tomography,” Nat. Med. 1, 970–972 (1995).
[CrossRef] [PubMed]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Tearney, G. J.

S. A. Boppart, B. E. Bouma, C. Pitris, G. J. Tearney, J. F. Southern, M. E. Brezinski, J. G. Fujimoto, “Intraoperative assessment of microsurgery with three dimensional optical coherence tomography,” Radiology 208, 81–86 (1998).
[PubMed]

J. G. Fujimoto, M. E. Brezinski, G. J. Tearney, S. A. Boppart, B. E. Bouma, M. R. Hee, J. F. Southern, E. A. Swanson, “Biomedical imaging and optical biopsy using optical coherence tomography,” Nat. Med. 1, 970–972 (1995).
[CrossRef] [PubMed]

Torresani, B.

R. Carmona, W. Hwang, B. Torresani, Practical Time–Frequency Analysis (Academic, San Diego, Calif., 1998).

van Germert, M. J. C.

van Leeuwen, T. G.

Wojtkowski, M.

Xu, C.

Yang, C.

Ye, J.

Appl. Opt. (1)

IEEE Trans. Acoust. Speech Signal Process. (1)

D. L. Jones, T. W. Parks, “A high resolution data-adaptive time-frequency representation,” IEEE Trans. Acoust. Speech Signal Process. 38, 2127–2135 (1990).
[CrossRef]

IEEE Trans. Inf. Theory (1)

I. Daubechies, “The wavelet transform, time–frequency localization and signal analysis,” IEEE Trans. Inf. Theory 36, 961–1005 (1990).
[CrossRef]

Nat. Med. (2)

J. G. Fujimoto, M. E. Brezinski, G. J. Tearney, S. A. Boppart, B. E. Bouma, M. R. Hee, J. F. Southern, E. A. Swanson, “Biomedical imaging and optical biopsy using optical coherence tomography,” Nat. Med. 1, 970–972 (1995).
[CrossRef] [PubMed]

S. A. Boppart, B. E. Bouma, C. Pitris, F. F. Southern, M. E. Brezinski, J. G. Fujimoto, “In vivo cellular optical coherence tomography imaging,” Nat. Med. 4, 861–864 (1998).
[CrossRef] [PubMed]

Opt. Lett. (6)

Proc. IEEE (1)

L. Cohen, “Time–frequency distributions—a review,” Proc. IEEE 77, 941–981 (1989).
[CrossRef]

Radiology (1)

S. A. Boppart, B. E. Bouma, C. Pitris, G. J. Tearney, J. F. Southern, M. E. Brezinski, J. G. Fujimoto, “Intraoperative assessment of microsurgery with three dimensional optical coherence tomography,” Radiology 208, 81–86 (1998).
[PubMed]

Science (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Other (4)

S. A. Boppart, W. Drexler, U. Morgner, F. X. Kartner, J. G. Fujimoto, “Ultrahigh resolution and spectroscopic optical coherence tomography imaging of cellular morphology and function,” presented at the Inter-Institute Workshop on In Vivo Optical Imaging at the National Institutes of Health, Bethesda, Md., 16–17 September 1999.

B. Hermann, K. Bizheva, A. Unterhuber, B. Povazay, H. Sattmann, L. Schmetterer, A. F. Fercher, W. Drexler, “Precision of extracting absorption profiles from weakly scattering media with spectroscopic time-domain optical coherence tomography,” Opt. Express12, 1677–1688 (2004), http://www.opticsexpress.org .
[CrossRef]

R. Carmona, W. Hwang, B. Torresani, Practical Time–Frequency Analysis (Academic, San Diego, Calif., 1998).

A. Papandreou-Suppappola, Applications in Time–Frequency Signal Processing (CRC Press, Boca Raton, Fla., 2002).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

SOCT synthetic signals: (a) synthetic signal 1, (b) ideal TFD of (a), (c) synthetic signal 2, (d) ideal TFD of (c), (e) synthetic signal 3, (f) ideal TFD of (e).

Fig. 2
Fig. 2

Sample construction for obtaining SOCT signals from two closely spaced interfaces. The geometrical dimensions are given in the text.

Fig. 3
Fig. 3

NIR dye properties. (a) Absorption spectra from NIR dye and emission spectrum of the Ti:sapphire laser used in this study. (b) Theoretical backscattered spectra for dye layer thicknesses from 0 to 10 μm.

Fig. 4
Fig. 4

STFT of SOCT signals backreflected from two glass interfaces separated by various distance with NIR dye in between: (a) SOCT signal for distance = 5 Δlc (coherence lengths), (b) STFT of signal in (a), (c) SOCT signal for distance = 2Δlc, (d) STFT of signal in (c), (e) SOCT signal for distance = Δlc, (f) STFT of signal in (e). Note that the x-axis scale is half for (e) and (f) to improve visualization of these two closely spaced pulses.

Fig. 5
Fig. 5

Different time–frequency representations of the signal in Fig. 4(e). (a) STFT with Hamming window of length corresponding to 1 μm. (b) Scalogram with the Morlet wavelet. The analyzing wavelet of half-length corresponds to 1 μm in air. (c) Choi–Williams distribution with time-smoothing window of length corresponding to 1 μm in air. (d) Model-based TFD as described in the text. Note the cross-term artifacts shown above the main signal in (b).

Fig. 6
Fig. 6

Comparison of extracted absorption spectra. Curves represent extracted absorption spectra from a dye-filled cuvette with different TFDs, compared with the mixture absorption spectra measured with a spectrometer. The spectral range of the laser (FWHM) is also shown, within which the absorption spectra of the media can be determined.

Tables (3)

Tables Icon

Table 1 Comparison of Time–Frequency Resolution of the TFDs on Synthetic Signal 1

Tables Icon

Table 2 Comparison of Minimal Resolving Distance of the TFDs on Synthetic Signal 2

Tables Icon

Table 3 Comparison of TFD Performance for Retrieving Absorption Spectra on Synthetic Signal 3

Equations (4)

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I ( λ ,     z ) = S ( λ ) H r ( λ ,     z ) H m ( λ , z ) H s ( λ ,     z ) .
C = TFD ( t , f ) 4 d t d f [ TFD ( t , f ) 2 d t d f ] 2 ,
error = frequency band A ( f ) - A ( f ) A ( f ) .
TFD = A × TFD M ( z ,     λ ) + TFD M ( z - z t ,     λ ) × exp [ - B ( λ ) ] ,

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