Abstract

For the first time we applied the maximum entropy method (MEM) to spectral domain optical coherence tomography to enhance axial resolution (AR). The MEM estimates the power spectrum by fitting. For an onion with optimization of M=70, the AR of 18.8μm by discrete Fourier transform (DFT) was improved three times compared with peak widths. The calculation time by the MEM with M=70 was 20 times longer than that of DFT. However, further studies are needed for practical applications, because the validity of the MEM depends on the sample structures.

© 2007 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, and J. G. Fujimoto, "Optical coherence tomograpy," Science 254, 1178-1181 (1991).
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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2006

A. D. Aguirre, N. Nishizawa, and J. G. Fujimoto, "Continuum generation in a novel photonic crystal fiber for ultrahigh resolution optical coherence tomography at 800 nm and 1300 nm," Opt. Express 14, 1145-1160 (2006).
[CrossRef] [PubMed]

D. Merino, C. Dainty, A. Bradu, and A. Gh. Podoleanu, "Adaptive optics enhanced simultaneous en-face optical coherence tomography and scanning laser opthalmoscopy," Opt. Express 14, 3345-3353 (2006).
[CrossRef] [PubMed]

R. Huber, M. Wojtkowski, and J. G. Fujimoto, "Fourier domain mode locking (FDML): a new laser operating regime and applications for optical coherence tomography," Opt. Express 14, 3225-3236 (2006).
[CrossRef] [PubMed]

A. Ozcan, O. T. Inan, N. I. Maluf, G. T. A. Kovacs, M. J. F. Digonnet, and G. S. Kino, "Alternative data processing for frequency-domain optical coherence tomography," presented at the Conference on Lasers and Electro-Optics, CLEO2006 (Optical Society of America, 2006), paper CMZ3.

Y. Takahashi, Y. Watanabe, and M. Sato, "Application of the maximum entropy method to Fourier domain optical coherence tomography," presented at the Conference on Lasers and Electro-Optics, CLEO2006 (Optical Society of America, 2006), paper JTuD53.

2005

M. Szkulmowski, M. Wajtkowski, T. Bajraszewski, I. Gorczynska, P. Targowski, W. Wasilewski, A. Kowalczyk, and C. Randzewicz, "Quality improvement for high resolution in vivo images by spectral domain optical coherence tomography with supercontinuum source," Opt. Commun. 246, 569-578 (2005).
[CrossRef]

2004

2003

2002

2001

E. D. J. Smith, N. Wada, W. Chujo, and D. D. Sampson, "High resolution OCDR using 1.55 μm supercontinuum source and quadrature spectral detection," Electron. Lett. 37, 1305-1307 (2001).
[CrossRef]

1998

M. Bashkansky, B. D. Duncan, J. Reintjes, and P. R. Battle, "Signal processing for improving field cross-correlation function in optical coherence tomography," Appl. Opt. 37, 8137-8138 (1998).

G. Hausler and M. W. Lindner, "Coherence rader and spectral rader--new tools for dermatological diagnostics," J. Biomed. Opt. 3, 21-31 (1998).
[CrossRef]

1993

S. A. Teukolsky, W. H. Press, and W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, 1993).

1991

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, and J. G. Fujimoto, "Optical coherence tomograpy," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

1987

S. L. Marple, Jr., Digital Spectral Analysis (Prentice Hall, 1987).

1983

1982

R. Kuc, Introduction to Digital Signal Processing (McGraw-Hill, 1982).

Aguirre, A. D.

Apolonski, A.

Bajraszewki, T.

Bajraszewski, T.

M. Szkulmowski, M. Wajtkowski, T. Bajraszewski, I. Gorczynska, P. Targowski, W. Wasilewski, A. Kowalczyk, and C. Randzewicz, "Quality improvement for high resolution in vivo images by spectral domain optical coherence tomography with supercontinuum source," Opt. Commun. 246, 569-578 (2005).
[CrossRef]

Bashkansky, M.

Battle, P. R.

Bizheva, K

Bizheva, K.

Boppart, S. A.

D. Marks, P. S. Camey, and S. A. Boppart, "Adaptive spectral apodization for sidelobe reduction in optical coherence tomography images," J. Biomed. Opt. 9, 1281-1287 (2004).
[CrossRef] [PubMed]

Bouma, B. E.

Bradu, A.

Camey, P. S.

D. Marks, P. S. Camey, and S. A. Boppart, "Adaptive spectral apodization for sidelobe reduction in optical coherence tomography images," J. Biomed. Opt. 9, 1281-1287 (2004).
[CrossRef] [PubMed]

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, and J. G. Fujimoto, "Optical coherence tomograpy," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Choma, M. A.

Chujo, W.

E. D. J. Smith, N. Wada, W. Chujo, and D. D. Sampson, "High resolution OCDR using 1.55 μm supercontinuum source and quadrature spectral detection," Electron. Lett. 37, 1305-1307 (2001).
[CrossRef]

Dainty, C.

de Boer, J. F.

Digonnet, M. J. F.

A. Ozcan, O. T. Inan, N. I. Maluf, G. T. A. Kovacs, M. J. F. Digonnet, and G. S. Kino, "Alternative data processing for frequency-domain optical coherence tomography," presented at the Conference on Lasers and Electro-Optics, CLEO2006 (Optical Society of America, 2006), paper CMZ3.

Dolexler, W.

Drexler, W.

Duker, J. S.

Duncan, B. D.

Fercher, A. F.

Fijimoto, J. G.

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, and J. G. Fujimoto, "Optical coherence tomograpy," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Fujimoto, J. G.

Gorczynska, I.

M. Szkulmowski, M. Wajtkowski, T. Bajraszewski, I. Gorczynska, P. Targowski, W. Wasilewski, A. Kowalczyk, and C. Randzewicz, "Quality improvement for high resolution in vivo images by spectral domain optical coherence tomography with supercontinuum source," Opt. Commun. 246, 569-578 (2005).
[CrossRef]

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, and J. G. Fujimoto, "Optical coherence tomograpy," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Hausler, G.

G. Hausler and M. W. Lindner, "Coherence rader and spectral rader--new tools for dermatological diagnostics," J. Biomed. Opt. 3, 21-31 (1998).
[CrossRef]

Hee, M. R.

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, and J. G. Fujimoto, "Optical coherence tomograpy," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Hermann, B.

Hitzenberger, C. K.

Hoelzenbein, T.

Holzwarth, R.

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, and J. G. Fujimoto, "Optical coherence tomograpy," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Huber, R.

Inan, O. T.

A. Ozcan, O. T. Inan, N. I. Maluf, G. T. A. Kovacs, M. J. F. Digonnet, and G. S. Kino, "Alternative data processing for frequency-domain optical coherence tomography," presented at the Conference on Lasers and Electro-Optics, CLEO2006 (Optical Society of America, 2006), paper CMZ3.

Izatt, J. A.

Kawata, S.

Kino, G. S.

A. Ozcan, O. T. Inan, N. I. Maluf, G. T. A. Kovacs, M. J. F. Digonnet, and G. S. Kino, "Alternative data processing for frequency-domain optical coherence tomography," presented at the Conference on Lasers and Electro-Optics, CLEO2006 (Optical Society of America, 2006), paper CMZ3.

Knight, J. C.

Ko, T. H.

Kovacs, G. T. A.

A. Ozcan, O. T. Inan, N. I. Maluf, G. T. A. Kovacs, M. J. F. Digonnet, and G. S. Kino, "Alternative data processing for frequency-domain optical coherence tomography," presented at the Conference on Lasers and Electro-Optics, CLEO2006 (Optical Society of America, 2006), paper CMZ3.

Kowalczyk, A.

M. Szkulmowski, M. Wajtkowski, T. Bajraszewski, I. Gorczynska, P. Targowski, W. Wasilewski, A. Kowalczyk, and C. Randzewicz, "Quality improvement for high resolution in vivo images by spectral domain optical coherence tomography with supercontinuum source," Opt. Commun. 246, 569-578 (2005).
[CrossRef]

M. Wojtkowski, V. J. Srinivasan, T. H. Ko, J. G. Fijimoto, A. Kowalczyk, and J. S. Duker, "Ultrahigh-resolution, high-speed, Fourier domain optical coherence tomography and methods for dispersion compensation," Opt. Express 12, 2404-2422 (2004).
[CrossRef] [PubMed]

Kuc, R.

R. Kuc, Introduction to Digital Signal Processing (McGraw-Hill, 1982).

Le, T.

Leitgeb, R.

Lgitgeb, R. A.

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, and J. G. Fujimoto, "Optical coherence tomograpy," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Lindner, M. W.

G. Hausler and M. W. Lindner, "Coherence rader and spectral rader--new tools for dermatological diagnostics," J. Biomed. Opt. 3, 21-31 (1998).
[CrossRef]

Maluf, N. I.

A. Ozcan, O. T. Inan, N. I. Maluf, G. T. A. Kovacs, M. J. F. Digonnet, and G. S. Kino, "Alternative data processing for frequency-domain optical coherence tomography," presented at the Conference on Lasers and Electro-Optics, CLEO2006 (Optical Society of America, 2006), paper CMZ3.

Marks, D.

D. Marks, P. S. Camey, and S. A. Boppart, "Adaptive spectral apodization for sidelobe reduction in optical coherence tomography images," J. Biomed. Opt. 9, 1281-1287 (2004).
[CrossRef] [PubMed]

Marple, S. L.

S. L. Marple, Jr., Digital Spectral Analysis (Prentice Hall, 1987).

Mei, M.

Merino, D.

Minami, K.

Minami, S.

Nassif, N.

Nelson, J. S.

Nishizawa, N.

Ozcan, A.

A. Ozcan, O. T. Inan, N. I. Maluf, G. T. A. Kovacs, M. J. F. Digonnet, and G. S. Kino, "Alternative data processing for frequency-domain optical coherence tomography," presented at the Conference on Lasers and Electro-Optics, CLEO2006 (Optical Society of America, 2006), paper CMZ3.

Park, B. H.

Pehamberger, H.

Podoleanu, A. Gh.

Povazay, B.

Press, W. H.

S. A. Teukolsky, W. H. Press, and W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, 1993).

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, and J. G. Fujimoto, "Optical coherence tomograpy," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Randzewicz, C.

M. Szkulmowski, M. Wajtkowski, T. Bajraszewski, I. Gorczynska, P. Targowski, W. Wasilewski, A. Kowalczyk, and C. Randzewicz, "Quality improvement for high resolution in vivo images by spectral domain optical coherence tomography with supercontinuum source," Opt. Commun. 246, 569-578 (2005).
[CrossRef]

Reintjes, J.

Russell, P. St. J.

Sampson, D. D.

E. D. J. Smith, N. Wada, W. Chujo, and D. D. Sampson, "High resolution OCDR using 1.55 μm supercontinuum source and quadrature spectral detection," Electron. Lett. 37, 1305-1307 (2001).
[CrossRef]

Sarunic, M. V.

Sato, M.

Y. Takahashi, Y. Watanabe, and M. Sato, "Application of the maximum entropy method to Fourier domain optical coherence tomography," presented at the Conference on Lasers and Electro-Optics, CLEO2006 (Optical Society of America, 2006), paper JTuD53.

Sattmann, H.

Scherzer, E.

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, and J. G. Fujimoto, "Optical coherence tomograpy," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Smith, E. D. J.

E. D. J. Smith, N. Wada, W. Chujo, and D. D. Sampson, "High resolution OCDR using 1.55 μm supercontinuum source and quadrature spectral detection," Electron. Lett. 37, 1305-1307 (2001).
[CrossRef]

Srinivasan, V. J.

Sting, A.

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, and J. G. Fujimoto, "Optical coherence tomograpy," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Swanson, E. 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, and J. G. Fujimoto, "Optical coherence tomograpy," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Szkulmowski, M.

M. Szkulmowski, M. Wajtkowski, T. Bajraszewski, I. Gorczynska, P. Targowski, W. Wasilewski, A. Kowalczyk, and C. Randzewicz, "Quality improvement for high resolution in vivo images by spectral domain optical coherence tomography with supercontinuum source," Opt. Commun. 246, 569-578 (2005).
[CrossRef]

Takahashi, Y.

Y. Takahashi, Y. Watanabe, and M. Sato, "Application of the maximum entropy method to Fourier domain optical coherence tomography," presented at the Conference on Lasers and Electro-Optics, CLEO2006 (Optical Society of America, 2006), paper JTuD53.

Targowski, P.

M. Szkulmowski, M. Wajtkowski, T. Bajraszewski, I. Gorczynska, P. Targowski, W. Wasilewski, A. Kowalczyk, and C. Randzewicz, "Quality improvement for high resolution in vivo images by spectral domain optical coherence tomography with supercontinuum source," Opt. Commun. 246, 569-578 (2005).
[CrossRef]

Tearney, G. J.

Teukolsky, S. A.

S. A. Teukolsky, W. H. Press, and W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, 1993).

Tripathi, R.

Unterhuber, A.

Vetterlein, M.

Vetterling, W. T.

S. A. Teukolsky, W. H. Press, and W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, 1993).

Wacheck, V.

Wada, N.

E. D. J. Smith, N. Wada, W. Chujo, and D. D. Sampson, "High resolution OCDR using 1.55 μm supercontinuum source and quadrature spectral detection," Electron. Lett. 37, 1305-1307 (2001).
[CrossRef]

Wadsworth, W. J.

Wajtkowski, M.

M. Szkulmowski, M. Wajtkowski, T. Bajraszewski, I. Gorczynska, P. Targowski, W. Wasilewski, A. Kowalczyk, and C. Randzewicz, "Quality improvement for high resolution in vivo images by spectral domain optical coherence tomography with supercontinuum source," Opt. Commun. 246, 569-578 (2005).
[CrossRef]

Wasilewski, W.

M. Szkulmowski, M. Wajtkowski, T. Bajraszewski, I. Gorczynska, P. Targowski, W. Wasilewski, A. Kowalczyk, and C. Randzewicz, "Quality improvement for high resolution in vivo images by spectral domain optical coherence tomography with supercontinuum source," Opt. Commun. 246, 569-578 (2005).
[CrossRef]

Watanabe, Y.

Y. Takahashi, Y. Watanabe, and M. Sato, "Application of the maximum entropy method to Fourier domain optical coherence tomography," presented at the Conference on Lasers and Electro-Optics, CLEO2006 (Optical Society of America, 2006), paper JTuD53.

Wojtkowski, M.

Yang, C.

Yun, S. H.

Appl. Opt.

Electron. Lett.

E. D. J. Smith, N. Wada, W. Chujo, and D. D. Sampson, "High resolution OCDR using 1.55 μm supercontinuum source and quadrature spectral detection," Electron. Lett. 37, 1305-1307 (2001).
[CrossRef]

J. Biomed. Opt.

D. Marks, P. S. Camey, and S. A. Boppart, "Adaptive spectral apodization for sidelobe reduction in optical coherence tomography images," J. Biomed. Opt. 9, 1281-1287 (2004).
[CrossRef] [PubMed]

G. Hausler and M. W. Lindner, "Coherence rader and spectral rader--new tools for dermatological diagnostics," J. Biomed. Opt. 3, 21-31 (1998).
[CrossRef]

Opt. Commun.

M. Szkulmowski, M. Wajtkowski, T. Bajraszewski, I. Gorczynska, P. Targowski, W. Wasilewski, A. Kowalczyk, and C. Randzewicz, "Quality improvement for high resolution in vivo images by spectral domain optical coherence tomography with supercontinuum source," Opt. Commun. 246, 569-578 (2005).
[CrossRef]

Opt. Express

D. Merino, C. Dainty, A. Bradu, and A. Gh. Podoleanu, "Adaptive optics enhanced simultaneous en-face optical coherence tomography and scanning laser opthalmoscopy," Opt. Express 14, 3345-3353 (2006).
[CrossRef] [PubMed]

R. Leitgeb, C. K. Hitzenberger, and A. F. Fercher, "Performance of fourier domain vs. time domain optical coherence tomography," Opt. Express 11, 889-894 (2003).
[CrossRef] [PubMed]

M. A. Choma, M. V. Sarunic, C. Yang, and J. A. Izatt, "Sensitivity advantage of swept source and Fourier domain optical coherence tomography," Opt. Express 11, 2183-2189 (2003).
[CrossRef] [PubMed]

M. Wojtkowski, V. J. Srinivasan, T. H. Ko, J. G. Fijimoto, A. Kowalczyk, and J. S. Duker, "Ultrahigh-resolution, high-speed, Fourier domain optical coherence tomography and methods for dispersion compensation," Opt. Express 12, 2404-2422 (2004).
[CrossRef] [PubMed]

R. A. Lgitgeb, W. Dolexler, A. Unterhuber, B. Hermann, T. Bajraszewki, T. Le, A. Sting, and A. F. Fercher, "Ultrahigh resolution Fourier domain optical coherence tomography," Opt. Express 12, 2156-2165 (2004).
[CrossRef]

S. H. Yun, G. J. Tearney, B. E. Bouma, B. H. Park, and J. F. de Boer, "High-speed spectral-domain optical coherence tomography at 1.3 μm wavelength," Opt. Express 11, 3598-3604 (2003).
[CrossRef] [PubMed]

R. Huber, M. Wojtkowski, and J. G. Fujimoto, "Fourier domain mode locking (FDML): a new laser operating regime and applications for optical coherence tomography," Opt. Express 14, 3225-3236 (2006).
[CrossRef] [PubMed]

A. D. Aguirre, N. Nishizawa, and J. G. Fujimoto, "Continuum generation in a novel photonic crystal fiber for ultrahigh resolution optical coherence tomography at 800 nm and 1300 nm," Opt. Express 14, 1145-1160 (2006).
[CrossRef] [PubMed]

Opt. Lett.

Science

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, and J. G. Fujimoto, "Optical coherence tomograpy," Science 254, 1178-1181 (1991).
[CrossRef] [PubMed]

Other

Y. Takahashi, Y. Watanabe, and M. Sato, "Application of the maximum entropy method to Fourier domain optical coherence tomography," presented at the Conference on Lasers and Electro-Optics, CLEO2006 (Optical Society of America, 2006), paper JTuD53.

S. A. Teukolsky, W. H. Press, and W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, 1993).

S. L. Marple, Jr., Digital Spectral Analysis (Prentice Hall, 1987).

R. Kuc, Introduction to Digital Signal Processing (McGraw-Hill, 1982).

A. Ozcan, O. T. Inan, N. I. Maluf, G. T. A. Kovacs, M. J. F. Digonnet, and G. S. Kino, "Alternative data processing for frequency-domain optical coherence tomography," presented at the Conference on Lasers and Electro-Optics, CLEO2006 (Optical Society of America, 2006), paper CMZ3.

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Figures (14)

Fig. 1
Fig. 1

Schematic of the spectral domain OCT system: SLD, broad bandwidth light source; CCD, InGaAs line CCD camera; DG, diffraction grating; S, sample; BS, beam splitter; FG, frame grabber board; DAC, digital-to-analog converter; PC, computer.

Fig. 2
Fig. 2

OCT images of the mirror obtained from the same spectral data (a) calculated by the DFT algorithm, with a depth scale magnification of 4 × and (b) calculated by the MEM algorithm, with M = 20 , and a depth scale magnification of 4 × .

Fig. 3
Fig. 3

Profiles of the depth direction, using a mirror as the sample. The bottom profile is the DFT calculation; the center profile is the MEM(20) calculation; and the top profile is the MEM(40) calculation. All the curves were sampled by the center (dashed line) of the image of Fig. 2. Inset: The enlargement in peak regions with DFT and MEM(20).

Fig. 4
Fig. 4

Axial resolution and FPE measured as a function of processing by DFT or order number M of MEM, using a mirror as the sample.

Fig. 5
Fig. 5

Sensitivity as a function of processing by DFT or order number M of MEM, using a mirror as the sample.

Fig. 6
Fig. 6

Dependence of peak positions on the number of terms with MEM using a mirror as the sample. The peak position and peak width (dashed lines) were obtained with DFT by Gauss fitting.

Fig. 7
Fig. 7

Relation between signal light power and the peak signal intensity and peak width, using a mirror as the sample, that intensity was normalized with the signal intensity with the signal light power of 10   nW . DFT processing (intensity: triangles, peak width: dashed line); MEM ( 20 ) × 100 processing (intensity: rectangle, peak width: rectangles and solid curve).

Fig. 8
Fig. 8

OCT images of the double cover glasses computed from the same spectral data (a) calculated by the DFT algorithm and a depth scale magnification of 4 × and (b) calculated by the MEM algorithm, with M = 70 and a depth scale magnification of 4 × .

Fig. 9
Fig. 9

Profile of the depth direction, using double cover glasses as the sample. The bottom profile is the DFT calculation, and other profiles are MEM calculations by order number M. Inset: The enlargement in peak regions with DFT and MEM(70).

Fig. 10
Fig. 10

Axial resolution and FPE measured as a function of processing by DFT or order number M of MEM, using double cover glasses as samples.

Fig. 11
Fig. 11

OCT images of an onion computed from the same spectral data. (a) Calculated by the DFT algorithm with a depth scale magnification of 2 × . (b) Calculated by the MEM algorithm with M = 70 and a depth scale magnification of 2 × . (c) Calculated by the DFT algorithm and a depth scale magnification of 4 × , which was extracted from a portion of the enclosure in (a). (d) Calculated by the DFT algorithm, with a depth scale magnification of 4 × , which was extracted from a portion of the enclosure in (b).

Fig. 12
Fig. 12

Profile of the depth direction, using an onion as the sample. The bottom profile is DFT calculation, and the other profiles are MEM calculations arranged by order number M.

Fig. 13
Fig. 13

Axial resolution and FPE measured as a function of processing by DFT or order number M of MEM, using an onion as the sample.

Fig. 14
Fig. 14

Calculation time measured as a function of processing by FFT or order number M of MEM. FFT calculation time as a reference (dashed line). MEM calculation time (dotted and solid curves). Theoretical ratio between FFT and MEM of order number M (solid gray curve).

Equations (16)

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I ( k ) = S ( k ) FT [ a ^ ( z ) ] ,
IFT [ I ( k ) ] = IFT [ S ( k ) ] a ^ ( z ) ,
IFT [ I ( k ) ] = γ ( z ) a ^ ( z )
IFT [ I ( k ) ] a ^ ( z ) .
P ( z ) = FT [ R I I ] = P M Δ | 1 + l = 1 M a l   exp ( 2 π j z l Δ ) | 2 ,
P I ( m ) = DFT [ R I I ] = N 2 | γ ( m ) a ^ ( m ) | 2 ,
l c = 4   ln   2 π λ 0 2 Δ λ ,
I ( k ) = S ( k ) A ( k ) .
S ( k ) = FT [ γ ( z ) ] , A ( k ) = FT [ a ^ ( z ) ] ,
R I I ( n ) = l = 0 N 1 I ( l ) I ( n + l ) .
P I ( m ) = DFT [ R I I ( n ) ] ( m = 0 , 1 , 2 , , N 1 ) ,
I ( m ) = DFT [ I ( n ) ] ,
DFT [ S ( n ) ] = N γ ( m ) ,    DFT [ A ( n ) ] = N a ^ ( m ) .
DFT [ R I I ( n ) ] = R I I ( m ) = | I ( m ) | 2 = | DFT [ I ( n ) ] | 2 = | DFT [ S ( n ) A ( n ) ] | 2 = | DFT [ S ( n ) ] DFT [ A ( n ) ] | 2 , 
DFT [ R I I ( n ) ] = N 2 | γ ( m ) a ^ ( m ) | 2 .
P I ( m ) = DFT [ R I I ( n ) ] = N 2 | γ ( m ) a ^ ( m ) | 2 . 

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