Abstract

The useful imaging range in spectral domain optical coherence tomography (SD-OCT) is often limited by the depth dependent sensitivity fall-off. Processing SD-OCT data with the non-uniform fast Fourier transform (NFFT) can improve the sensitivity fall-off at maximum depth by greater than 5dB concurrently with a 30 fold decrease in processing time compared to the fast Fourier transform with cubic spline interpolation method. NFFT can also improve local signal to noise ratio (SNR) and reduce image artifacts introduced in post-processing. Combined with parallel processing, NFFT is shown to have the ability to process up to 90k A-lines per second. High-speed SD-OCT imaging is demonstrated at camera-limited 100 frames per second on an ex-vivo squid eye.

© 2010 OSA

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References

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  1. Z. Hu and A. M. Rollins, “Fourier domain optical coherence tomography with a linear-in-wavenumber spectrometer,” Opt. Lett. 32(24), 3525–3527 (2007).
    [PubMed]
  2. Y. Zhang, X. Li, L. Wei, K. Wang, Z. Ding, and G. Shi, “Time-domain interpolation for Fourier-domain optical coherence tomography,” Opt. Lett. 34(12), 1849–1851 (2009).
    [PubMed]
  3. K. Wang, Z. Ding, T. Wu, C. Wang, J. Meng, M. Chen, and L. Xu, “Development of a non-uniform discrete Fourier transform based high speed spectral domain optical coherence tomography system,” Opt. Express 17(14), 12121–12131 (2009).
    [PubMed]
  4. G. Hausler and M. W. Lindner, “Coherence radar and spectral radar – new tools for dermatological diagnosis,” J. Biomed. Opt. 3(1), 21–31 (1998).
  5. N. Nassif, B. Cense, B. Park, M. Pierce, S. Yun, B. Bouma, G. Tearney, T. Chen, and J. de Boer, “In vivo high-resolution video-rate spectral-domain optical coherence tomography of the human retina and optic nerve,” Opt. Express 12(3), 367–376 (2004).
    [PubMed]
  6. G. Liu, J. Zhang, L. Yu, T. Xie, and Z. Chen, “Real-time polarization-sensitive optical coherence tomography data processing with parallel computing,” Appl. Opt. 48(32), 6365–6370 (2009).
    [PubMed]
  7. E. Maeland, “On the comparison of interpolation methods,” IEEE Trans. Med. Imaging 7(3), 213–217 (1988).
    [PubMed]
  8. H. Hou and H. C. Andrews, “Cubic splines for image interpolation and digital filtering,” IEEE Trans. Acoust. Speech Signal Process. 26(6), 508–516 (1978).
  9. G. E. Sarty, R. Bennett, and R. W. Cox, “Direct reconstruction of non-Cartesian k-space data using a nonuniform fast Fourier transform,” Magn. Reson. Med. 45(5), 908–915 (2001).
    [PubMed]
  10. S. De Francesco and A. M. F. da Silva, “Efficient NUFFT-based direct Fourier algorithm for fan beam CT reconstruction,” Proc. SPIE 5370, 666–677 (2004).
  11. M. M. Bronstein, A. M. Bronstein, M. Zibulevsky, and H. Azhari, “Reconstruction in diffraction ultrasound tomography using nonuniform FFT,” IEEE Trans. Med. Imaging 21(11), 1395–1401 (2002).
    [PubMed]
  12. A. Dutt and V. Rokhlin, “Fast Fourier transforms for nonequispaced data,” SIAM J. Sci. Comput. 14(6), 1368–1393 (1993).
  13. J. Lee and L. Greengard, “The type 3 nonuniform FFT and its application,” J. Comput. Phys. 206(iss. 1), 1–5 (2005).
  14. J. A. Fessler and B. P. Sutton, “Nonuniform fast Fourier transforms using min-max interpolation,” IEEE Trans. Signal Process. 51(2), 560–574 (2003).
  15. L. Greengard and J. Lee, “Accelerating the Nonuniform Fast Fourier Transform,” SIAM Rev. 46(3), 443–454 (2004).
  16. D. Potts, G. Steidl, and M. Tasche, “Fast Fourier transforms for nonequispaced data: a tutorial,” in Modern Sampling Theory: Mathematics and Applications, J.J.Benedetto and P.Ferreira, eds. (Springer, 2001), Chap. 12, pp. 249–274.
  17. A. J. W. Duijndam and M. A. Schonewille, “Nonuniform fast Fourier transform,” Geophys. 64, 539–551 (1999).
  18. Y. Rolain, J. Schoukens, and G. Vandersteen, ““Signal Reconstruction for Non-Equidistant Finite Length Sample Sets: A “KIS” Approach,” IEEE Trans. Instrum. Meas. 47(5), 1046–1052 (1998).
  19. C. Dorrer, N. Belabas, J. P. Likforman, and M. Joffre, “Spectral resolution and sampling issues in Fourier-transform spectral interferometry,” J. Opt. Soc. Am. B 17, 1795–1802 (2000).
  20. P. Thevenaz, T. Blu, and M. Unser, Handbook of Medical Imaging (Academic Press, 2000), Chap. 25.
  21. M. Choma, M. V. Sarunic, C. Yang, and J. Izatt, “Sensitivity advantage of swept source and Fourier domain optical coherence tomography,” Opt. Express 11(18), 2183–2189 (2003).
    [PubMed]
  22. M. Frigo, and S. G. Johnson, “FFTW: an adaptive software architecture for the FFT,” in Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing. (Institute of Electrical and Electronics Engineers, New York, 1988), pp. 1381–1384.
  23. 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).
    [PubMed]
  24. OpenMP Architecture Review Board, “The OpenMP API specification for parallel programming,” http://www.openmp.org/ .
  25. T. E. Ustun, N. V. Iftimia, R. D. Ferguson, and D. X. Hammer, “Real-time processing for Fourier domain optical coherence tomography using a field programmable gate array,” Rev. Sci. Instrum. 79(11), 114301 (2008).
    [PubMed]
  26. A. W. Schaefer, J. J. Reynolds, D. L. Marks, and S. A. Boppart, “Real-time digital signal processing-based optical coherence tomography and Doppler optical coherence tomography,” IEEE Trans. Biomed. Eng. 51(1), 186–190 (2004).
    [PubMed]

2009 (3)

2008 (1)

T. E. Ustun, N. V. Iftimia, R. D. Ferguson, and D. X. Hammer, “Real-time processing for Fourier domain optical coherence tomography using a field programmable gate array,” Rev. Sci. Instrum. 79(11), 114301 (2008).
[PubMed]

2007 (1)

2005 (1)

J. Lee and L. Greengard, “The type 3 nonuniform FFT and its application,” J. Comput. Phys. 206(iss. 1), 1–5 (2005).

2004 (5)

L. Greengard and J. Lee, “Accelerating the Nonuniform Fast Fourier Transform,” SIAM Rev. 46(3), 443–454 (2004).

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).
[PubMed]

N. Nassif, B. Cense, B. Park, M. Pierce, S. Yun, B. Bouma, G. Tearney, T. Chen, and J. de Boer, “In vivo high-resolution video-rate spectral-domain optical coherence tomography of the human retina and optic nerve,” Opt. Express 12(3), 367–376 (2004).
[PubMed]

S. De Francesco and A. M. F. da Silva, “Efficient NUFFT-based direct Fourier algorithm for fan beam CT reconstruction,” Proc. SPIE 5370, 666–677 (2004).

A. W. Schaefer, J. J. Reynolds, D. L. Marks, and S. A. Boppart, “Real-time digital signal processing-based optical coherence tomography and Doppler optical coherence tomography,” IEEE Trans. Biomed. Eng. 51(1), 186–190 (2004).
[PubMed]

2003 (2)

M. Choma, M. V. Sarunic, C. Yang, and J. Izatt, “Sensitivity advantage of swept source and Fourier domain optical coherence tomography,” Opt. Express 11(18), 2183–2189 (2003).
[PubMed]

J. A. Fessler and B. P. Sutton, “Nonuniform fast Fourier transforms using min-max interpolation,” IEEE Trans. Signal Process. 51(2), 560–574 (2003).

2002 (1)

M. M. Bronstein, A. M. Bronstein, M. Zibulevsky, and H. Azhari, “Reconstruction in diffraction ultrasound tomography using nonuniform FFT,” IEEE Trans. Med. Imaging 21(11), 1395–1401 (2002).
[PubMed]

2001 (1)

G. E. Sarty, R. Bennett, and R. W. Cox, “Direct reconstruction of non-Cartesian k-space data using a nonuniform fast Fourier transform,” Magn. Reson. Med. 45(5), 908–915 (2001).
[PubMed]

2000 (1)

1999 (1)

A. J. W. Duijndam and M. A. Schonewille, “Nonuniform fast Fourier transform,” Geophys. 64, 539–551 (1999).

1998 (2)

Y. Rolain, J. Schoukens, and G. Vandersteen, ““Signal Reconstruction for Non-Equidistant Finite Length Sample Sets: A “KIS” Approach,” IEEE Trans. Instrum. Meas. 47(5), 1046–1052 (1998).

G. Hausler and M. W. Lindner, “Coherence radar and spectral radar – new tools for dermatological diagnosis,” J. Biomed. Opt. 3(1), 21–31 (1998).

1993 (1)

A. Dutt and V. Rokhlin, “Fast Fourier transforms for nonequispaced data,” SIAM J. Sci. Comput. 14(6), 1368–1393 (1993).

1988 (1)

E. Maeland, “On the comparison of interpolation methods,” IEEE Trans. Med. Imaging 7(3), 213–217 (1988).
[PubMed]

1978 (1)

H. Hou and H. C. Andrews, “Cubic splines for image interpolation and digital filtering,” IEEE Trans. Acoust. Speech Signal Process. 26(6), 508–516 (1978).

Andrews, H. C.

H. Hou and H. C. Andrews, “Cubic splines for image interpolation and digital filtering,” IEEE Trans. Acoust. Speech Signal Process. 26(6), 508–516 (1978).

Azhari, H.

M. M. Bronstein, A. M. Bronstein, M. Zibulevsky, and H. Azhari, “Reconstruction in diffraction ultrasound tomography using nonuniform FFT,” IEEE Trans. Med. Imaging 21(11), 1395–1401 (2002).
[PubMed]

Belabas, N.

Bennett, R.

G. E. Sarty, R. Bennett, and R. W. Cox, “Direct reconstruction of non-Cartesian k-space data using a nonuniform fast Fourier transform,” Magn. Reson. Med. 45(5), 908–915 (2001).
[PubMed]

Boppart, S. A.

A. W. Schaefer, J. J. Reynolds, D. L. Marks, and S. A. Boppart, “Real-time digital signal processing-based optical coherence tomography and Doppler optical coherence tomography,” IEEE Trans. Biomed. Eng. 51(1), 186–190 (2004).
[PubMed]

Bouma, B.

Bronstein, A. M.

M. M. Bronstein, A. M. Bronstein, M. Zibulevsky, and H. Azhari, “Reconstruction in diffraction ultrasound tomography using nonuniform FFT,” IEEE Trans. Med. Imaging 21(11), 1395–1401 (2002).
[PubMed]

Bronstein, M. M.

M. M. Bronstein, A. M. Bronstein, M. Zibulevsky, and H. Azhari, “Reconstruction in diffraction ultrasound tomography using nonuniform FFT,” IEEE Trans. Med. Imaging 21(11), 1395–1401 (2002).
[PubMed]

Cense, B.

Chen, M.

Chen, T.

Chen, Z.

Choma, M.

Cox, R. W.

G. E. Sarty, R. Bennett, and R. W. Cox, “Direct reconstruction of non-Cartesian k-space data using a nonuniform fast Fourier transform,” Magn. Reson. Med. 45(5), 908–915 (2001).
[PubMed]

da Silva, A. M. F.

S. De Francesco and A. M. F. da Silva, “Efficient NUFFT-based direct Fourier algorithm for fan beam CT reconstruction,” Proc. SPIE 5370, 666–677 (2004).

de Boer, J.

De Francesco, S.

S. De Francesco and A. M. F. da Silva, “Efficient NUFFT-based direct Fourier algorithm for fan beam CT reconstruction,” Proc. SPIE 5370, 666–677 (2004).

Ding, Z.

Dorrer, C.

Duijndam, A. J. W.

A. J. W. Duijndam and M. A. Schonewille, “Nonuniform fast Fourier transform,” Geophys. 64, 539–551 (1999).

Dutt, A.

A. Dutt and V. Rokhlin, “Fast Fourier transforms for nonequispaced data,” SIAM J. Sci. Comput. 14(6), 1368–1393 (1993).

Ferguson, R. D.

T. E. Ustun, N. V. Iftimia, R. D. Ferguson, and D. X. Hammer, “Real-time processing for Fourier domain optical coherence tomography using a field programmable gate array,” Rev. Sci. Instrum. 79(11), 114301 (2008).
[PubMed]

Fessler, J. A.

J. A. Fessler and B. P. Sutton, “Nonuniform fast Fourier transforms using min-max interpolation,” IEEE Trans. Signal Process. 51(2), 560–574 (2003).

Greengard, L.

J. Lee and L. Greengard, “The type 3 nonuniform FFT and its application,” J. Comput. Phys. 206(iss. 1), 1–5 (2005).

L. Greengard and J. Lee, “Accelerating the Nonuniform Fast Fourier Transform,” SIAM Rev. 46(3), 443–454 (2004).

Hammer, D. X.

T. E. Ustun, N. V. Iftimia, R. D. Ferguson, and D. X. Hammer, “Real-time processing for Fourier domain optical coherence tomography using a field programmable gate array,” Rev. Sci. Instrum. 79(11), 114301 (2008).
[PubMed]

Hausler, G.

G. Hausler and M. W. Lindner, “Coherence radar and spectral radar – new tools for dermatological diagnosis,” J. Biomed. Opt. 3(1), 21–31 (1998).

Hou, H.

H. Hou and H. C. Andrews, “Cubic splines for image interpolation and digital filtering,” IEEE Trans. Acoust. Speech Signal Process. 26(6), 508–516 (1978).

Hu, Z.

Iftimia, N. V.

T. E. Ustun, N. V. Iftimia, R. D. Ferguson, and D. X. Hammer, “Real-time processing for Fourier domain optical coherence tomography using a field programmable gate array,” Rev. Sci. Instrum. 79(11), 114301 (2008).
[PubMed]

Izatt, J.

Joffre, M.

Lee, J.

J. Lee and L. Greengard, “The type 3 nonuniform FFT and its application,” J. Comput. Phys. 206(iss. 1), 1–5 (2005).

L. Greengard and J. Lee, “Accelerating the Nonuniform Fast Fourier Transform,” SIAM Rev. 46(3), 443–454 (2004).

Li, X.

Likforman, J. P.

Lindner, M. W.

G. Hausler and M. W. Lindner, “Coherence radar and spectral radar – new tools for dermatological diagnosis,” J. Biomed. Opt. 3(1), 21–31 (1998).

Liu, G.

Maeland, E.

E. Maeland, “On the comparison of interpolation methods,” IEEE Trans. Med. Imaging 7(3), 213–217 (1988).
[PubMed]

Marks, D. L.

A. W. Schaefer, J. J. Reynolds, D. L. Marks, and S. A. Boppart, “Real-time digital signal processing-based optical coherence tomography and Doppler optical coherence tomography,” IEEE Trans. Biomed. Eng. 51(1), 186–190 (2004).
[PubMed]

Meng, J.

Nassif, N.

Park, B.

Pierce, M.

Reynolds, J. J.

A. W. Schaefer, J. J. Reynolds, D. L. Marks, and S. A. Boppart, “Real-time digital signal processing-based optical coherence tomography and Doppler optical coherence tomography,” IEEE Trans. Biomed. Eng. 51(1), 186–190 (2004).
[PubMed]

Rokhlin, V.

A. Dutt and V. Rokhlin, “Fast Fourier transforms for nonequispaced data,” SIAM J. Sci. Comput. 14(6), 1368–1393 (1993).

Rolain, Y.

Y. Rolain, J. Schoukens, and G. Vandersteen, ““Signal Reconstruction for Non-Equidistant Finite Length Sample Sets: A “KIS” Approach,” IEEE Trans. Instrum. Meas. 47(5), 1046–1052 (1998).

Rollins, A. M.

Sarty, G. E.

G. E. Sarty, R. Bennett, and R. W. Cox, “Direct reconstruction of non-Cartesian k-space data using a nonuniform fast Fourier transform,” Magn. Reson. Med. 45(5), 908–915 (2001).
[PubMed]

Sarunic, M. V.

Schaefer, A. W.

A. W. Schaefer, J. J. Reynolds, D. L. Marks, and S. A. Boppart, “Real-time digital signal processing-based optical coherence tomography and Doppler optical coherence tomography,” IEEE Trans. Biomed. Eng. 51(1), 186–190 (2004).
[PubMed]

Schonewille, M. A.

A. J. W. Duijndam and M. A. Schonewille, “Nonuniform fast Fourier transform,” Geophys. 64, 539–551 (1999).

Schoukens, J.

Y. Rolain, J. Schoukens, and G. Vandersteen, ““Signal Reconstruction for Non-Equidistant Finite Length Sample Sets: A “KIS” Approach,” IEEE Trans. Instrum. Meas. 47(5), 1046–1052 (1998).

Shi, G.

Sutton, B. P.

J. A. Fessler and B. P. Sutton, “Nonuniform fast Fourier transforms using min-max interpolation,” IEEE Trans. Signal Process. 51(2), 560–574 (2003).

Tearney, G.

Ustun, T. E.

T. E. Ustun, N. V. Iftimia, R. D. Ferguson, and D. X. Hammer, “Real-time processing for Fourier domain optical coherence tomography using a field programmable gate array,” Rev. Sci. Instrum. 79(11), 114301 (2008).
[PubMed]

Vandersteen, G.

Y. Rolain, J. Schoukens, and G. Vandersteen, ““Signal Reconstruction for Non-Equidistant Finite Length Sample Sets: A “KIS” Approach,” IEEE Trans. Instrum. Meas. 47(5), 1046–1052 (1998).

Wang, C.

Wang, K.

Wei, L.

Wu, T.

Xie, T.

Xu, L.

Yang, C.

Yu, L.

Yun, S.

Yun, S. H.

Zhang, J.

Zhang, Y.

Zibulevsky, M.

M. M. Bronstein, A. M. Bronstein, M. Zibulevsky, and H. Azhari, “Reconstruction in diffraction ultrasound tomography using nonuniform FFT,” IEEE Trans. Med. Imaging 21(11), 1395–1401 (2002).
[PubMed]

Appl. Opt. (1)

Geophys. (1)

A. J. W. Duijndam and M. A. Schonewille, “Nonuniform fast Fourier transform,” Geophys. 64, 539–551 (1999).

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

H. Hou and H. C. Andrews, “Cubic splines for image interpolation and digital filtering,” IEEE Trans. Acoust. Speech Signal Process. 26(6), 508–516 (1978).

IEEE Trans. Biomed. Eng. (1)

A. W. Schaefer, J. J. Reynolds, D. L. Marks, and S. A. Boppart, “Real-time digital signal processing-based optical coherence tomography and Doppler optical coherence tomography,” IEEE Trans. Biomed. Eng. 51(1), 186–190 (2004).
[PubMed]

IEEE Trans. Instrum. Meas. (1)

Y. Rolain, J. Schoukens, and G. Vandersteen, ““Signal Reconstruction for Non-Equidistant Finite Length Sample Sets: A “KIS” Approach,” IEEE Trans. Instrum. Meas. 47(5), 1046–1052 (1998).

IEEE Trans. Med. Imaging (2)

M. M. Bronstein, A. M. Bronstein, M. Zibulevsky, and H. Azhari, “Reconstruction in diffraction ultrasound tomography using nonuniform FFT,” IEEE Trans. Med. Imaging 21(11), 1395–1401 (2002).
[PubMed]

E. Maeland, “On the comparison of interpolation methods,” IEEE Trans. Med. Imaging 7(3), 213–217 (1988).
[PubMed]

IEEE Trans. Signal Process. (1)

J. A. Fessler and B. P. Sutton, “Nonuniform fast Fourier transforms using min-max interpolation,” IEEE Trans. Signal Process. 51(2), 560–574 (2003).

J. Biomed. Opt. (1)

G. Hausler and M. W. Lindner, “Coherence radar and spectral radar – new tools for dermatological diagnosis,” J. Biomed. Opt. 3(1), 21–31 (1998).

J. Comput. Phys. (1)

J. Lee and L. Greengard, “The type 3 nonuniform FFT and its application,” J. Comput. Phys. 206(iss. 1), 1–5 (2005).

J. Opt. Soc. Am. B (1)

Magn. Reson. Med. (1)

G. E. Sarty, R. Bennett, and R. W. Cox, “Direct reconstruction of non-Cartesian k-space data using a nonuniform fast Fourier transform,” Magn. Reson. Med. 45(5), 908–915 (2001).
[PubMed]

Opt. Express (4)

Opt. Lett. (2)

Proc. SPIE (1)

S. De Francesco and A. M. F. da Silva, “Efficient NUFFT-based direct Fourier algorithm for fan beam CT reconstruction,” Proc. SPIE 5370, 666–677 (2004).

Rev. Sci. Instrum. (1)

T. E. Ustun, N. V. Iftimia, R. D. Ferguson, and D. X. Hammer, “Real-time processing for Fourier domain optical coherence tomography using a field programmable gate array,” Rev. Sci. Instrum. 79(11), 114301 (2008).
[PubMed]

SIAM J. Sci. Comput. (1)

A. Dutt and V. Rokhlin, “Fast Fourier transforms for nonequispaced data,” SIAM J. Sci. Comput. 14(6), 1368–1393 (1993).

SIAM Rev. (1)

L. Greengard and J. Lee, “Accelerating the Nonuniform Fast Fourier Transform,” SIAM Rev. 46(3), 443–454 (2004).

Other (4)

D. Potts, G. Steidl, and M. Tasche, “Fast Fourier transforms for nonequispaced data: a tutorial,” in Modern Sampling Theory: Mathematics and Applications, J.J.Benedetto and P.Ferreira, eds. (Springer, 2001), Chap. 12, pp. 249–274.

M. Frigo, and S. G. Johnson, “FFTW: an adaptive software architecture for the FFT,” in Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing. (Institute of Electrical and Electronics Engineers, New York, 1988), pp. 1381–1384.

OpenMP Architecture Review Board, “The OpenMP API specification for parallel programming,” http://www.openmp.org/ .

P. Thevenaz, T. Blu, and M. Unser, Handbook of Medical Imaging (Academic Press, 2000), Chap. 25.

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Figures (5)

Fig. 1
Fig. 1

Flow chart of the NFFT processing algorithm.

Fig. 2
Fig. 2

Illustration of the resampling of data with Gaussian interpolation kernel into equally spaced grid. The circles are the original unevenly sampled data and the vertical dashed lines are the new uniform grid. A Gaussian function is convolved with each original data point, spreading its power over a few adjacent grid points as shown in the crosses. The new evenly sampled data value is the summation of the values of the crosses on each grid line.

Fig. 3
Fig. 3

Schematic of the SD-OCT system. SLD, superluminescent diode; OI, optical isolator, FC, fiber coupler, NDF: neutral density filter.

Fig. 4
Fig. 4

Left: Sensitivity fall-off based on different reconstruction methods. LI, Linear interpolation; CSI, cubic spline interpolation; NDFT, non-unifrom discrete Fourier transform; NFFT, non-uniform fast Fourier transform. Right: Typical axial reflectivity profile with a single partial reflector showing shoulder artifacts of linear and cubic spline interpolation. The data for NDFT and NFFT overlaps each other, showing the accuracy of the approximation.

Fig. 5
Fig. 5

(a) Corneal images obtained from different processing techniques. The arrows indicate the location of the image artifacts. EP, epithelium; S, stroma; EN, endothelium. (b) Representative part of an A-line located at the solid line in the corneal image. NFFT produced peaks with higher intensity as a result of the improved sensitivity fall-off. LI, Linear interpolation; CSI, cubic spline interpolation; NDFT, non-unifrom discrete Fourier transform; NFFT, non-uniform fast Fourier transform.

Tables (1)

Tables Icon

Table 1 Computation time for one A-line (μs) and display frame rate (fps) of a 512 × 512 pixels image based on different processing methods

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

I ( k ) = s ( k ) [ R r + i R i + 2 R r i R i cos ( k z i ) + 2 i j i R i R j cos ( k z i j ) ]
a ( z ) = | F T k z 1 [ I ( k ) ] | = Γ ( z ) { R r δ ( 0 ) + i R i δ ( 0 ) + R r i R i δ ( z ± z i ) + 2 i j i R i R j δ ( z ± z i j ) }
a ( z m ) = 1 M n = 0 M 1 I ( k n ) e i 2 π Δ K m k n , m [ 0 , M 1 ]
G τ ( k ) = e k 2 4 τ
τ = 1 M 2 π R ( R 0.5 ) M s p
I τ ( k ) = I ( k ) G τ ( k ) = I ( y ) G τ ( k y ) d y
I τ ( l M r Δ K ) = n = 0 M 1 I ( k n ) G τ ( l M r Δ K k n ) , l [ 0 , M r 1 ]
a τ ( z m ) 1 M r l = 0 M r 1 I τ ( l M r Δ K ) e i 2 π m l M r , m [ 0 , M r 1 ]
g ( z m ) = F T 1 [ G τ ( k ) ] = 2 τ exp ( z m 2 τ )
a ( z m ) = 1 2 τ exp ( z m 2 τ ) a ( z m ) τ

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