Abstract

Dispersion encoded full range (DEFR) optical coherence tomography (OCT) has become highly attractive as it is a simple way to increase the measurement range of OCT systems. Full range OCT is especially favorable as it does not only increase the measurement range but also shifts the highest sensitivity into the center of the measurement range. While the early versions of DEFR were highly computational expensive, new versions reduce the number of necessary Fourier transforms. Recently it has been shown that a GPU based algorithm can perform DEFR with more than 20,000 A-lines per second. We present a new version of the DEFR algorithm that requires only one Fourier transform per A-scan and uses convolution in z-space instead of multiplication in k-space, therefore reducing the computational effort considerably. While dispersion encoding has so far only been used to suppress mirror artifacts, we show that, with dispersion encoding and only one more Fourier transform, autocorrelation terms can be removed likewise. Since very high values of dispersion reduce the effective measurement range in dispersion encoded OCT, we present an estimate for a sufficient amount of dispersion for a successful image recovery, which is depending on the thickness of the scattering layers. Furthermore, we demonstrate the usability of ZnSe as a new dispersive material with a very high dispersion and describe a simple method to extract the dispersive phase from the measurement of a single reflex of a glass surface. Using a standard consumer PC, an artifact-free recovery of 1000 – 2000 A-scans per second with 2048 depth values including autocorrelation removal was achieved. The dynamic range (sensitivity) is not reduced and the suppression ratio of mirror artifacts and autocorrelation signals is more than 50dB using ZnSe.

© 2012 OSA

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2012 (2)

L. Wang, B. Hofer, J. A. Guggenheim, and B. Povazay, “Graphics processing unit-based dispersion encoded full-range frequency-domain optical coherence tomography,” J. Biomed. Opt.17, 077007 (2012).
[CrossRef] [PubMed]

C.-T. Wu, T.-T. Chi, Y.-W. Kiang, and C. C. Yang, “Computation time-saving mirror image suppression method in fourier-domain optical coherence tomography,” Opt. Express20, 8270–8283 (2012).
[CrossRef] [PubMed]

2011 (4)

P. D. Woolliams and P. H. Tomlins, “Estimating the resolution of a commercial optical coherence tomography system with limited spatial sampling,” Meas. Sci. Technol.22, 065502 (2011).
[CrossRef]

S. Marschall, B. Sander, M. Mogensen, T. Jørgensen, and P. Andersen, “Optical coherence tomography – current technology and applications in clinical and biomedical research,” Anal. Bioanal. Chem.400, 2699–2720 (2011).
[CrossRef] [PubMed]

J. Walther, M. Gaertner, P. Cimalla, A. Burkhardt, L. Kirsten, S. Meissner, and E. Koch, “Optical coherence tomography in biomedical research,” Anal. Bioanal. Chem.400, 2721–2743 (2011).
[CrossRef] [PubMed]

T. Wu, Z. Ding, C. Wang, and M. Chen, “Full-range swept source optical coherence tomography based on carrier frequency by transmissive dispersive optical delay line,” J. Biomed. Opt.16, 126008 (2011).
[CrossRef] [PubMed]

2010 (6)

2009 (5)

2008 (3)

2007 (1)

2006 (2)

Y. Yasuno, S. Makita, T. Endo, G. Aoki, M. Itoh, and T. Yatagai, “Simultaneous b-m-mode scanning method for real-time full-range fourier domain optical coherence tomography,” Appl. Opt.45, 1861–1865 (2006).
[CrossRef] [PubMed]

M. Duarte, M. Davenport, M. Wakin, and R. Baraniuk, “Sparse signal detection from incoherent projections,” in Proc. Int. Conf. on Acoustics, Speech and Signal Processing (ICASSP)3, III305–308 (2006).

2005 (2)

2004 (2)

2003 (3)

2002 (1)

2001 (1)

1991 (1)

D. Huang, E. Swanson, C. Lin, J. Schuman, W. Stinson, W. Chang, M. Hee, T. Flotte, K. Gregory, C. Puliafito, and “Optical coherence tomography,” Science254, 1178–1181 (1991).
[CrossRef] [PubMed]

An, L.

Andersen, P.

S. Marschall, B. Sander, M. Mogensen, T. Jørgensen, and P. Andersen, “Optical coherence tomography – current technology and applications in clinical and biomedical research,” Anal. Bioanal. Chem.400, 2699–2720 (2011).
[CrossRef] [PubMed]

Aoki, G.

Baclayon, M.

Bajraszewski, T.

Baraniuk, R.

M. Duarte, M. Davenport, M. Wakin, and R. Baraniuk, “Sparse signal detection from incoherent projections,” in Proc. Int. Conf. on Acoustics, Speech and Signal Processing (ICASSP)3, III305–308 (2006).

Biedermann, B. R.

Bouma, B.

Bouma, B. E.

Burkhardt, A.

J. Walther, M. Gaertner, P. Cimalla, A. Burkhardt, L. Kirsten, S. Meissner, and E. Koch, “Optical coherence tomography in biomedical research,” Anal. Bioanal. Chem.400, 2721–2743 (2011).
[CrossRef] [PubMed]

Cense, B.

Chang, S.

Chang, W.

D. Huang, E. Swanson, C. Lin, J. Schuman, W. Stinson, W. Chang, M. Hee, T. Flotte, K. Gregory, C. Puliafito, and “Optical coherence tomography,” Science254, 1178–1181 (1991).
[CrossRef] [PubMed]

Chen, M.

T. Wu, Z. Ding, C. Wang, and M. Chen, “Full-range swept source optical coherence tomography based on carrier frequency by transmissive dispersive optical delay line,” J. Biomed. Opt.16, 126008 (2011).
[CrossRef] [PubMed]

Chen, T.

Chen, Y.-P.

L. Wang, B. Hofer, Y.-P. Chen, J. A. Guggenheim, W. Drexler, and B. Povazay, “Highly reproducible swept-source, dispersion-encoded full-range biometry and imaging of the mouse eye,” J. Biomed. Opt.15, 046004 (2010).
[CrossRef] [PubMed]

Cheng, H.-C.

H.-C. Cheng, J.-F. Huang, and Y.-H. Hsieh, “Numerical analysis of one-shot full-range fd-oct system based on orthogonally polarized light,” Opt. Commun.282, 3040–3045 (2009).
[CrossRef]

Chi, T.-T.

Choma, M.

Cimalla, P.

J. Walther, M. Gaertner, P. Cimalla, A. Burkhardt, L. Kirsten, S. Meissner, and E. Koch, “Optical coherence tomography in biomedical research,” Anal. Bioanal. Chem.400, 2721–2743 (2011).
[CrossRef] [PubMed]

P. Cimalla, J. Walther, M. Mehner, M. Cuevas, and E. Koch, “Simultaneous dual-band optical coherence tomography in the spectral domain for high resolution in vivo imaging,” Opt. Express17, 19486–19500 (2009).
[CrossRef] [PubMed]

Cuevas, M.

Davenport, M.

M. Duarte, M. Davenport, M. Wakin, and R. Baraniuk, “Sparse signal detection from incoherent projections,” in Proc. Int. Conf. on Acoustics, Speech and Signal Processing (ICASSP)3, III305–308 (2006).

de Boer, J.

de Boer, J. F.

Ding, Z.

T. Wu, Z. Ding, C. Wang, and M. Chen, “Full-range swept source optical coherence tomography based on carrier frequency by transmissive dispersive optical delay line,” J. Biomed. Opt.16, 126008 (2011).
[CrossRef] [PubMed]

Drexler, W.

Duarte, M.

M. Duarte, M. Davenport, M. Wakin, and R. Baraniuk, “Sparse signal detection from incoherent projections,” in Proc. Int. Conf. on Acoustics, Speech and Signal Processing (ICASSP)3, III305–308 (2006).

Duker, J.

Eigenwillig, C. M.

Endo, T.

Fabritius, T.

Fercher, A.

Fercher, A. F.

Flotte, T.

D. Huang, E. Swanson, C. Lin, J. Schuman, W. Stinson, W. Chang, M. Hee, T. Flotte, K. Gregory, C. Puliafito, and “Optical coherence tomography,” Science254, 1178–1181 (1991).
[CrossRef] [PubMed]

Flueraru, C.

Fujimoto, J.

Gaertner, M.

J. Walther, M. Gaertner, P. Cimalla, A. Burkhardt, L. Kirsten, S. Meissner, and E. Koch, “Optical coherence tomography in biomedical research,” Anal. Bioanal. Chem.400, 2721–2743 (2011).
[CrossRef] [PubMed]

Götzinger, E.

Gregory, K.

D. Huang, E. Swanson, C. Lin, J. Schuman, W. Stinson, W. Chang, M. Hee, T. Flotte, K. Gregory, C. Puliafito, and “Optical coherence tomography,” Science254, 1178–1181 (1991).
[CrossRef] [PubMed]

Groot, M. L.

Guggenheim, J. A.

L. Wang, B. Hofer, J. A. Guggenheim, and B. Povazay, “Graphics processing unit-based dispersion encoded full-range frequency-domain optical coherence tomography,” J. Biomed. Opt.17, 077007 (2012).
[CrossRef] [PubMed]

L. Wang, B. Hofer, Y.-P. Chen, J. A. Guggenheim, W. Drexler, and B. Povazay, “Highly reproducible swept-source, dispersion-encoded full-range biometry and imaging of the mouse eye,” J. Biomed. Opt.15, 046004 (2010).
[CrossRef] [PubMed]

Hagen-Eggert, M.

M. Hagen-Eggert, P. Koch, and G. Hüttmann, “Analysis of the signal fall-off in spectral domain optical coherence tomography systems,” in Optical Coherence Tomography and Coherence Domain Optical Methods in Biomedicine XVI, Joseph A. Izatt, James G. Fujimoto, and Valery V. Tuchin, eds., Proc. SPIE 8213, 82131K (2012).

Hassler, K.

Hee, M.

D. Huang, E. Swanson, C. Lin, J. Schuman, W. Stinson, W. Chang, M. Hee, T. Flotte, K. Gregory, C. Puliafito, and “Optical coherence tomography,” Science254, 1178–1181 (1991).
[CrossRef] [PubMed]

Hermann, B.

Hitzenberger, C.

Hofer, B.

Hsieh, Y.-H.

H.-C. Cheng, J.-F. Huang, and Y.-H. Hsieh, “Numerical analysis of one-shot full-range fd-oct system based on orthogonally polarized light,” Opt. Commun.282, 3040–3045 (2009).
[CrossRef]

Huang, D.

D. Huang, E. Swanson, C. Lin, J. Schuman, W. Stinson, W. Chang, M. Hee, T. Flotte, K. Gregory, C. Puliafito, and “Optical coherence tomography,” Science254, 1178–1181 (1991).
[CrossRef] [PubMed]

Huang, J.-F.

H.-C. Cheng, J.-F. Huang, and Y.-H. Hsieh, “Numerical analysis of one-shot full-range fd-oct system based on orthogonally polarized light,” Opt. Commun.282, 3040–3045 (2009).
[CrossRef]

Huber, R.

Hüttmann, G.

M. Hagen-Eggert, P. Koch, and G. Hüttmann, “Analysis of the signal fall-off in spectral domain optical coherence tomography systems,” in Optical Coherence Tomography and Coherence Domain Optical Methods in Biomedicine XVI, Joseph A. Izatt, James G. Fujimoto, and Valery V. Tuchin, eds., Proc. SPIE 8213, 82131K (2012).

Itoh, M.

Izatt, J.

Jaillon, F.

Jørgensen, T.

S. Marschall, B. Sander, M. Mogensen, T. Jørgensen, and P. Andersen, “Optical coherence tomography – current technology and applications in clinical and biomedical research,” Anal. Bioanal. Chem.400, 2699–2720 (2011).
[CrossRef] [PubMed]

Karamata, B.

Kiang, Y.-W.

Kim, D. Y.

D. Y. Kim, J. S. Werner, and R. J. Zawadzki, “Comparison of phase-shifting techniques for in vivo full-range, high-speed fourier-domain optical coherence tomography,” J. Biomed. Opt.15, 056011 (2010).
[CrossRef] [PubMed]

Kirsten, L.

J. Walther, M. Gaertner, P. Cimalla, A. Burkhardt, L. Kirsten, S. Meissner, and E. Koch, “Optical coherence tomography in biomedical research,” Anal. Bioanal. Chem.400, 2721–2743 (2011).
[CrossRef] [PubMed]

Klein, T.

Ko, T.

Koch, E.

J. Walther, M. Gaertner, P. Cimalla, A. Burkhardt, L. Kirsten, S. Meissner, and E. Koch, “Optical coherence tomography in biomedical research,” Anal. Bioanal. Chem.400, 2721–2743 (2011).
[CrossRef] [PubMed]

P. Cimalla, J. Walther, M. Mehner, M. Cuevas, and E. Koch, “Simultaneous dual-band optical coherence tomography in the spectral domain for high resolution in vivo imaging,” Opt. Express17, 19486–19500 (2009).
[CrossRef] [PubMed]

Koch, P.

M. Hagen-Eggert, P. Koch, and G. Hüttmann, “Analysis of the signal fall-off in spectral domain optical coherence tomography systems,” in Optical Coherence Tomography and Coherence Domain Optical Methods in Biomedicine XVI, Joseph A. Izatt, James G. Fujimoto, and Valery V. Tuchin, eds., Proc. SPIE 8213, 82131K (2012).

Kowalczyk, A.

Lasser, T.

Laubscher, M.

Lee, K.-S.

Leitgeb, R.

Lin, C.

D. Huang, E. Swanson, C. Lin, J. Schuman, W. Stinson, W. Chang, M. Hee, T. Flotte, K. Gregory, C. Puliafito, and “Optical coherence tomography,” Science254, 1178–1181 (1991).
[CrossRef] [PubMed]

Makita, S.

Mansvelder, H. D.

Mao, Y.

Marschall, S.

S. Marschall, B. Sander, M. Mogensen, T. Jørgensen, and P. Andersen, “Optical coherence tomography – current technology and applications in clinical and biomedical research,” Anal. Bioanal. Chem.400, 2699–2720 (2011).
[CrossRef] [PubMed]

Matz, G.

Meemon, P.

Mehner, M.

Meier, C.

Meissner, S.

J. Walther, M. Gaertner, P. Cimalla, A. Burkhardt, L. Kirsten, S. Meissner, and E. Koch, “Optical coherence tomography in biomedical research,” Anal. Bioanal. Chem.400, 2721–2743 (2011).
[CrossRef] [PubMed]

Mogensen, M.

S. Marschall, B. Sander, M. Mogensen, T. Jørgensen, and P. Andersen, “Optical coherence tomography – current technology and applications in clinical and biomedical research,” Anal. Bioanal. Chem.400, 2699–2720 (2011).
[CrossRef] [PubMed]

Nassif, N.

Park, B.

Park, B. H.

Peterman, E. J.

Pierce, M.

Pierce, M. C.

Pircher, M.

Povazay, B.

L. Wang, B. Hofer, J. A. Guggenheim, and B. Povazay, “Graphics processing unit-based dispersion encoded full-range frequency-domain optical coherence tomography,” J. Biomed. Opt.17, 077007 (2012).
[CrossRef] [PubMed]

L. Wang, B. Hofer, Y.-P. Chen, J. A. Guggenheim, W. Drexler, and B. Povazay, “Highly reproducible swept-source, dispersion-encoded full-range biometry and imaging of the mouse eye,” J. Biomed. Opt.15, 046004 (2010).
[CrossRef] [PubMed]

Považay, B.

Puliafito, C.

D. Huang, E. Swanson, C. Lin, J. Schuman, W. Stinson, W. Chang, M. Hee, T. Flotte, K. Gregory, C. Puliafito, and “Optical coherence tomography,” Science254, 1178–1181 (1991).
[CrossRef] [PubMed]

Rey, S.

Rolland, J. P.

Sander, B.

S. Marschall, B. Sander, M. Mogensen, T. Jørgensen, and P. Andersen, “Optical coherence tomography – current technology and applications in clinical and biomedical research,” Anal. Bioanal. Chem.400, 2699–2720 (2011).
[CrossRef] [PubMed]

Sarunic, M.

Schuman, J.

D. Huang, E. Swanson, C. Lin, J. Schuman, W. Stinson, W. Chang, M. Hee, T. Flotte, K. Gregory, C. Puliafito, and “Optical coherence tomography,” Science254, 1178–1181 (1991).
[CrossRef] [PubMed]

Sherif, S.

Srinivasan, V.

Sticker, M.

Stinson, W.

D. Huang, E. Swanson, C. Lin, J. Schuman, W. Stinson, W. Chang, M. Hee, T. Flotte, K. Gregory, C. Puliafito, and “Optical coherence tomography,” Science254, 1178–1181 (1991).
[CrossRef] [PubMed]

Swanson, E.

D. Huang, E. Swanson, C. Lin, J. Schuman, W. Stinson, W. Chang, M. Hee, T. Flotte, K. Gregory, C. Puliafito, and “Optical coherence tomography,” Science254, 1178–1181 (1991).
[CrossRef] [PubMed]

Szkulmowska, A.

Szkulmowski, M.

Tearney, G.

Tearney, G. J.

Tomlins, P. H.

P. D. Woolliams and P. H. Tomlins, “Estimating the resolution of a commercial optical coherence tomography system with limited spatial sampling,” Meas. Sci. Technol.22, 065502 (2011).
[CrossRef]

Toonen, R. F.

Unterhuber, A.

Wakin, M.

M. Duarte, M. Davenport, M. Wakin, and R. Baraniuk, “Sparse signal detection from incoherent projections,” in Proc. Int. Conf. on Acoustics, Speech and Signal Processing (ICASSP)3, III305–308 (2006).

Walther, J.

J. Walther, M. Gaertner, P. Cimalla, A. Burkhardt, L. Kirsten, S. Meissner, and E. Koch, “Optical coherence tomography in biomedical research,” Anal. Bioanal. Chem.400, 2721–2743 (2011).
[CrossRef] [PubMed]

P. Cimalla, J. Walther, M. Mehner, M. Cuevas, and E. Koch, “Simultaneous dual-band optical coherence tomography in the spectral domain for high resolution in vivo imaging,” Opt. Express17, 19486–19500 (2009).
[CrossRef] [PubMed]

Wang, C.

T. Wu, Z. Ding, C. Wang, and M. Chen, “Full-range swept source optical coherence tomography based on carrier frequency by transmissive dispersive optical delay line,” J. Biomed. Opt.16, 126008 (2011).
[CrossRef] [PubMed]

Wang, L.

L. Wang, B. Hofer, J. A. Guggenheim, and B. Povazay, “Graphics processing unit-based dispersion encoded full-range frequency-domain optical coherence tomography,” J. Biomed. Opt.17, 077007 (2012).
[CrossRef] [PubMed]

L. Wang, B. Hofer, Y.-P. Chen, J. A. Guggenheim, W. Drexler, and B. Povazay, “Highly reproducible swept-source, dispersion-encoded full-range biometry and imaging of the mouse eye,” J. Biomed. Opt.15, 046004 (2010).
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Figures (10)

Fig. 1
Fig. 1

Iteration procedure in the DEFR algorithm. A peak detector finds the strongest signal component in z-space (Fig. 1(a)) and removes it from the spectrum. In order to remove its mirror artifacts, twice the dispersive phase is applied on the signal in k-space (Fig. 1(b)), which is calculated via Fourier transform. The resulting signal after the nonlinear phase shift is depicted in Fig. 1(c). The mirror component, which is just one peak in z-space after inverse Fourier transform (Fig. 1(d)), can be finally removed via subtraction of the peak corresponding to the true signal component found in Fig. 1(a) (these two signals are symmetric around the zero delay position). Again, the nonlinear phase shift in k-space is applied after a Fourier transform and the ’original’ spectrum in z-space is retrieved via inverse Fourier transform.

Fig. 2
Fig. 2

Simple implementation of the DEFR algorithm without autocorrelation removal. Independent from the actual processing steps, the inverse Fourier transform (IFFT) of twice the inverse dispersive phase function e−i2ϕ(kn) is calculated (denoted as p 2 ( n )) for the purpose of complex conjugate artifact removal (step (1)). After performing the numerical dispersion compensation in k-space, the depth signal in z-space is calculated as usual for each A-scan via inverse Fourier transform (step (2); the detector signal, resampled in frequency and background corrected, and the initial depth signal in z-space are denoted as Pint (kn) and c 1 0 ( n ), respectively). From now on, all processing steps will be executed directly in z-space without the need of any additional Fourier transforms. The iterations are comprised of step (3) and step (4). In step (3) a peak detector finds the strongest signal component c 1 i ( n 1 i ), whose mirror component is subtracted directly in z-space using the information provided by Eq. (23), term (b) and Eq. (23), term (c). The true signal component c 1 i ( n 1 i ) is added to the output d 1 i ( n ) and its complex value is also subtracted from the spectrum. The iteration finishes if a maximum number of iterations is reached or the residual spectrum c 1 i ( n ) contains only noise. Optionally, the residual spectrum c 1 i ( n ) can be added to the output d 1 i ( n ).

Fig. 3
Fig. 3

Advanced implementation of the DEFR algorithm with autocorrelation removal. While the basic idea is similar to the simple version presented in Sect. 2.2, the advanced DEFR algorithm with autocorrelation removal exploits the information provided by both the non dispersion compensated and the dispersion compensated spectrum in z-space c 1 i ( n ) and c 2 i ( n ), respectively, which are retrieved from the detector signal after resampling in frequency and background correction (step (2)). Again, the required inverse Fourier transforms (IFFT) of the dispersive phase functions arising in Eq. (23) and Eq. (25) (denoted as p 1 + ( n ), p 1 ( n ) and p 2 ( n )) are calculated separately from the actual processing steps in step (1). Two peak detectors determine simultaneously the strongest signal components in both spectra and provide their values and positions for the next processing steps (step (3)). Depending on whether the amplitude of c 1 i ( n 1 i ) is higher than the amplitude of c 2 i ( n 2 i ) or vice versa, this component is a true signal component or an autocorrelation artifact. In both cases the subtraction of the corresponding signals is performed directly in z-space and the located signals are added to the output. The iteration finishes if a maximum number of iterations is reached or if the residual spectrum c 1 i ( n ) contains only noise. Optionally, the residual spectrum c 1 i ( n ) can be added to the output d 1 i ( n ).

Fig. 4
Fig. 4

Procedure of dispersion measurement. The dispersive phase ϕ(k) can be directly extracted from the measured interference signal of a sharp reflector, i.e. the surface of a glass plate (Fig. 4(a)). Note that after resampling to 4096 spectral points in k-space and inverse Fourier transform only the middle half of the depth signal in z-space, comprising 2048 complex values, is displayed and the range of the normalized position covers these 2048 points. The dispersion broadened peak in z-space is filtered (Fig. 4(b)) and transformed into k-space via Fourier transform. The phase of the complex signal in k-space is retrieved and unwrapped (Fig. 4(c)). The dispersive phase is finally obtained after subtracting the linear part of the nonlinear phase (Fig. 4(d)).

Fig. 5
Fig. 5

Measured dispersive phases and shape of the corresponding complex conjugate mirror components. The dispersive phases were extracted from the signal phase in k-space, which was averaged over 50 A-scans, and fitted to a fourth order polynomial regression (depicted in Fig. 5(a)). The corresponding inverse Fourier transforms of twice the inverse dispersive phase functions, belonging to the complex conjugate mirror components of a true signal after numerical dispersion compensation, are depicted in Fig. 5(b).

Fig. 6
Fig. 6

Nanoparticle target; 6.35mm SF6; 45dB dynamic range; 3.89mm × 4.09mm (h × w); (a) – original image without dispersion compensation; (b) – dispersion compensated image; (c) – image without autocorrelation removal (simple DEFR processing); (d) – image with autocorrelation removal (advanced DEFR processing)

Fig. 7
Fig. 7

Capillary; 12.70mm SF6; 45dB dynamic range; 3.89mm × 0.54mm (h × w); (a) – original image without dispersion compensation; (b) – dispersion compensated image; (c) – image without autocorrelation removal (simple DEFR processing); (d) – image with autocorrelation removal (advanced DEFR processing)

Fig. 8
Fig. 8

Capillary; 6.25mm ZnSe; 45dB dynamic range; 3.89mm × 0.54mm (h × w); (a) – original image without dispersion compensation; (b) – dispersion compensated image; (c) – image without autocorrelation removal (simple DEFR processing); (d) – image with autocorrelation removal (advanced DEFR processing)

Fig. 9
Fig. 9

Optical filter; 12.70mm SF6; 45dB dynamic range; 3.89mm×1.79mm (h×w); (a) – original image without dispersion compensation; (b) – dispersion compensated image; (c) – image without autocorrelation removal (simple DEFR processing); (d) – image with autocorrelation removal (advanced DEFR processing)

Fig. 10
Fig. 10

Excised human eardrum ex vivo; 6.25mm ZnSe; 45dB dynamic range; 3.89mm× 8.18mm (h × w); (a) – original image without dispersion compensation; (b) – dispersion compensated image; (c) – image without autocorrelation removal (simple DEFR processing); (d) – image with autocorrelation removal (advanced DEFR processing)

Equations (53)

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1 ( ( c ( z ) ) e ± i 2 ϕ ( k ) ) ( z ) ,
1 ( ( c ( z ) e ± i 2 ϕ ( k ) ) ( z ) = ( c * 1 ( e ± i 2 ϕ ( k ) ) ) ( z ) .
( c ˜ * 1 ( e ± i 2 ϕ ( k ) ) ) ( z ) = c ˜ × 1 ( e ± i 2 ϕ ( k ) ) ( z s ) ,
E i ( k , t ) = s ( k ) e i ( k z ω t ) e ρ ,
E r ( k , t ) = E i ( k , t ) r M e i 2 k z R ,
E r , dis ( k , t ) = E i ( k , t ) r M t G e i 2 ( k z F + k ( ω ) z G ) ,
k ( ω ) = l = 0 a l ( ω ω 0 ) l ,
a l = 1 l ! d l k ( ω ) d ω l | ω = ω 0 .
( ω ω 0 ) l = m = 0 l ( l m ) ω m ( ω 0 ) l m ,
k abs = l = 0 a l ( l 0 ) ω 0 ( ω 0 ) l 0 = l = 0 a l ( ω 0 ) l .
k lin = l = 1 a l ( l 1 ) ω 1 ( ω 0 ) l 1 = l = 1 a l l ω ( ω 0 ) l 1 .
z G , opt = z G c 0 k lin ω = z G c 0 l = 1 a l l ( ω 0 ) l 1 .
k dis = l = 2 a l m = 2 l ( l m ) ω m ( ω 0 ) l m ,
ϕ 0 = 2 k abs z G = 2 l = 0 a l ( ω 0 ) l z G ,
ϕ ( k ) = 2 k dis z G = 2 l = 2 a l m = 2 l ( l m ) ω m ( ω 0 ) l m z G ,
E r , dis ( k , t ) = E i ( k , t ) r M t G e i ϕ 0 e i 2 ( k z F + k lin z G ) e i ϕ ( k ) .
E s ( k , t ) = E i ( k , t ) l r S , l e i 2 k n ref z S , l .
E int ( k , t ) = E r , dis ( k , t ) + E s ( k , t ) = E i ( k , t ) ( r M t G e i ϕ 0 e i 2 ( k z F + k lin z G ) e i ϕ ( k ) + l r S , l e i 2 k n ref z S , l )
p ( k ) | E ( k , t ) | 2 t ,
P int ( k ) = P i ( k ) | r M t G e i ϕ 0 e i 2 ( k z F + k lin z G ) e i ϕ ( k ) + l r S , l e i 2 k n ref z S , l | 2 .
P int ( k ) = P i ( k ) [ r M 2 t G 2 + l r S , l 2 + ]
l r M t G r S , l e i ϕ 0 e i 2 ( k z F + k lin z G k n ref z S , l ) e i ϕ ( k ) +
l r M t G r S , l e i ϕ 0 e i 2 ( k z F + k lin z G k n ref z S , l ) e i ϕ ( k ) +
l m r S , l r S , m ( e i 2 k n ref ( z S , l z S , m ) + e i 2 k n ref ( z S , l z S , m ) ) ] .
P int ( k ) e i ϕ ( k ) = P i ( k ) [ ( r M 2 t G 2 + l r S , l 2 ) e i ϕ ( k ) +
l r M t G r S , l e i ϕ 0 e i 2 ( k z F + k lin z G k n ref z S , l ) +
l r M t G r S , l e i ϕ 0 e i 2 ( k z F + k lin z G k n ref z S , l ) e i 2 ϕ ( k ) +
( l m r S , l r S , m ( e i 2 k n ref ( z S , l z S , m ) + e i 2 k n ref ( z S , l z S , m ) ) ) e i ϕ ( k ) ]
1 ( P int ( k ) e i ϕ ( k ) ) ( z ) = { 1 ( P i ( k ) ) * [ ( r M 2 t G 2 + l r S , l 2 ) 1 ( e i ϕ ( k ) ) +
l r M t G r S , l e i ϕ 0 δ ( z 2 ( z F + k lin c 0 z G n ref z S , l ) ) +
l r M t G r S , l e i ϕ 0 δ ( z + 2 ( z F + k lin c 0 z G n ref z S , l ) ) * 1 ( e i 2 ϕ ( k ) ) +
l m r S , l r S , m δ ( z ± 2 n ref ( z S , l z S , m ) ) * 1 ( e i ϕ ( k ) ) ] } ( z ) ,
1 ( x ( k ) ) ( z ) = x ( k ) e i 2 π k z d k .
1 ( P int ( k ) ) ( z ) = { 1 ( P i ( k ) ) * [ ( r M 2 t G 2 + l r S , l 2 ) +
l r M t G r S , l e i ϕ 0 δ ( z 2 ( z F + k lin c 0 z G n ref z S , l ) ) * 1 ( e i ϕ ( k ) ) +
l r M t G r S , l e i ϕ 0 δ ( z + 2 ( z F + k lin c 0 z G n ref z S , l ) ) * 1 ( e i ϕ ( k ) ) +
m l r S , l r S , m δ ( z ± 2 n ref ( z S , l z S , m ) ) ] } ( z ) .
| x ( t ) | 2 d t = | X ( f ) | 2 d f , where X ( f ) = ( x ( t ) ) ( f ) ,
p 1 + ( n ) = 1 N j = 0 N 1 e + i ϕ ( k j ) e i 2 π n N j ,
p 1 ( n ) = 1 N j = 0 N 1 e i ϕ ( k j ) e i 2 π n N j ,
p 2 ( n ) = 1 N j = 0 N 1 e i 2 ϕ ( k j ) e i 2 π n N j ,
c 1 0 ( n ) = 1 N j = 0 N 1 P int ( k j ) e i ϕ ( k j ) e i 2 π n N j ( dispersion compensated   sprectrum ) ,
c 0 2 ( n ) = 1 N j = 0 N 1 P int ( k j ) e i 2 π n N j ( non dispersion compensated   spectrum ) ,
d 1 i + 1 ( n ) = d 1 i ( n ) + c 1 i ( n 1 i ) , d 2 i + 1 ( n ) = d 2 i ( n ) , c 1 i + 1 ( n ) = c 1 i ( n ) c 1 i ( n 1 i ) ( c 1 i ( n 1 i ) ) * p 2 ( n + n 1 i ) , c 2 i + 1 ( n ) = c 2 i ( n ) { c 1 i ( n 1 i ) p 1 + ( n n 1 i ) , if n 1 i N / 2 ( c 1 i ( n 1 i ) ) * p 1 ( n + n 1 i ) , if n 1 i > N / 2
d 1 i + 1 ( n ) = d 1 i ( n ) , d 2 i + 1 ( n ) = d 2 i ( n ) + c 2 i ( n 2 i ) , c 1 i + 1 ( n ) = c 1 i ( n ) c 2 i ( n 2 i ) p 1 ( n n 2 i ) ( c 2 i ( n 2 i ) ) * p 1 ( n + n 2 i ) , c 2 i + 1 ( n ) = c 2 i ( n ) c 2 i ( n 2 i ) ,
c 1 i ( n 1 i ) = c ^ 1 i ( n 1 i ) + ( c ^ 1 i ( n 1 i ) ) * p 2 ( 2 n i 1 ) ,
c ^ 1 i ( n 1 i ) = c 1 i ( n 1 i ) ( c 1 i ( n 1 i ) ) * p 2 ( t ) 1 | p 2 ( t ) | 2 .
n 2 ( λ ) = 1 + B 1 λ 2 λ 2 C 1 + B 2 λ 2 λ 2 C 2 + B 3 λ 2 λ 2 C 3 ,
c ¯ ( z ) = ( 1 ( P i ( k ) ) * r M t G r S , 0 e i ϕ 0 δ ( z 2 ( z F + k lin c 0 z G n ref z S , 0 ) ) * 1 ( e i ϕ ( k ) ) ) ( z ) ,
( c ¯ ( z ) ) ( k ) = P i ( k ) r M t G r S , 0 e i ϕ 0 e i 2 ( k z F + k lin z G k n ref z S , 0 ) e i ϕ ( k ) .
ϕ ( k ) = ϕ 0 + 2 ( k z F + k lin z G k n ref z S , 0 ) + ϕ ( k ) = φ lin ( k ) + ϕ ( k ) .
ϕ ( k ) = φ ( k ) φ lin ( k ) .
φ lin ( k ) = ϕ ( k a ) + ϕ ( k b ) ϕ ( k a ) k b k a ( k k a ) ,

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