Selection of convolution kernel in non-uniform fast Fourier transform for Fourier domain optical coherence tomography

Kenny K.H. Chan; Shuo Tang

doi:10.1364/OE.19.026891

Selection of convolution kernel in non-uniform fast Fourier transform for Fourier domain optical coherence tomography

Kenny K.H. Chan and Shuo Tang

Optics Express
Vol. 19,
Issue 27,
pp. 26891-26904
(2011)
•https://doi.org/10.1364/OE.19.026891

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Kenny K.H. Chan and Shuo Tang, "Selection of convolution kernel in non-uniform fast Fourier transform for Fourier domain optical coherence tomography," Opt. Express 19, 26891-26904 (2011)

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Related Topics
Table of Contents Category
- Medical Optics and Biotechnology
Optics & Photonics Topics
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The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Discrete optical signal processing (070.2025)
- Image reconstruction techniques (110.3010)
- Optical coherence tomography (170.4500)

About this Article
History
- Original Manuscript: October 14, 2011
- Revised Manuscript: December 3, 2011
- Manuscript Accepted: December 5, 2011
- Published: December 16, 2011
Virtual Issues
Virtual Journal for Biomedical Optics Vol. 7, Iss. 2

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Open Access

Abstract

Gridding based non-uniform fast Fourier transform (NUFFT) has recently been shown as an efficient method of processing non-linearly sampled data from Fourier-domain optical coherence tomography (FD-OCT). This method requires selecting design parameters, such as kernel function type, oversampling ratio and kernel width, to balance between computational complexity and accuracy. The Kaiser-Bessel (KB) and Gaussian kernels have been used independently on the NUFFT algorithm for FD-OCT. This paper compares the reconstruction error and speed for the optimization of these design parameters and justifies particular kernel choice for FD-OCT applications. It is found that for on-the-fly computation of the kernel function, the simpler Gaussian function offers a better accuracy-speed tradeoff. The KB kernel, however, is a better choice in the pre-computed kernel mode of NUFFT, in which the processing speed is no longer dependent on the kernel function type. Finally, the algorithm is used to reconstruct in-vivo images of a human finger at a camera limited 50k A-line/s.

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References

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Publication

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[Crossref] [PubMed]
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[Crossref] [PubMed]
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[Crossref]
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[Crossref] [PubMed]
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[Crossref] [PubMed]
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[Crossref]
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[Crossref] [PubMed]
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[Crossref] [PubMed]
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[Crossref]
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[Crossref] [PubMed]
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[Crossref] [PubMed]
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[Crossref] [PubMed]
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[Crossref]
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[Crossref]
J. D. O’Sullivan, “A fast sinc function gridding algorithm for fourier inversion in computer tomography,” IEEE Trans. Med. Imaging 4(4), 200–207 (1985).
[Crossref] [PubMed]
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, 249–274.
M. Frigo and S. G. Johnson, “The design and implementation of FFTW3,” Proc. IEEE 93(2), 216–231 (2005).
[Crossref]
B. Liu, E. Azimi, and M. E. Brezinski, “True logarithmic amplification of frequency clock in SS-OCT for calibration,” Biomed. Opt. Express 2(6), 1769–1777 (2011).
[Crossref] [PubMed]
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).
[Crossref] [PubMed]
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).
[Crossref] [PubMed]
OpenMP Architecture Review Board, “The OpenMP API specification for parallel programming,” http://www.openmp.org/ .
T. S. Sorensen, T. Schaeffter, K. O. Noe, and M. S. Hansen, “Accelerating the nonequispaced fast Fourier transform on commodity graphics hardware,” IEEE Trans. Med. Imaging 27(4), 538–547 (2008).
[Crossref] [PubMed]

D. Hillmann, G. Huttmann, and P. Koch, “Using nonequispaced fast Fourier transformation to process optical coherence tomography signals,” Proc. SPIE 7372, 73720R (2009).
[Crossref]

2008 (3)

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).
[Crossref] [PubMed]

T. S. Sorensen, T. Schaeffter, K. O. Noe, and M. S. Hansen, “Accelerating the nonequispaced fast Fourier transform on commodity graphics hardware,” IEEE Trans. Med. Imaging 27(4), 538–547 (2008).
[Crossref] [PubMed]

C. M. Eigenwillig, B. R. Biedermann, G. Palte, and R. Huber, “K-space linear Fourier domain mode locked laser and applications for optical coherence tomography,” Opt. Express 16(12), 8916–8937 (2008).
[Crossref] [PubMed]

2007 (2)

Z. Hu and A. M. Rollins, “Fourier domain optical coherence tomography with a linear-in-wavenumber spectrometer,” Opt. Lett. 32(24), 3525–3527 (2007).
[Crossref] [PubMed]

D. C. Adler, Y. Chen, R. Huber, J. Schmitt, J. Connolly, and J. G. Fujimoto, “Three-dimensional endomicroscopy using optical coherence tomography,” Nat. Photonics 1(12), 709–716 (2007).
[Crossref]

2005 (2)

M. Frigo and S. G. Johnson, “The design and implementation of FFTW3,” Proc. IEEE 93(2), 216–231 (2005).
[Crossref]

P. J. Beatty, D. G. Nishimura, and J. M. Pauly, “Rapid gridding reconstruction with a minimal oversampling ratio,” IEEE Trans. Med. Imaging 24(6), 799–808 (2005).
[Crossref] [PubMed]

2004 (3)

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

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).
[Crossref] [PubMed]

M. Wojtkowski, V. J. Srinivasan, T. H. Ko, J. G. Fujimoto, A. Kowalczyk, and J. S. Duker, “Ultrahigh-resolution, high-speed, Fourier domain optical coherence tomography and methods for dispersion compensation,” Opt. Express 12(11), 2404–2422 (2004).
[Crossref] [PubMed]

2003 (2)

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(18), 2183–2189 (2003).
[Crossref] [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).
[Crossref]

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).
[Crossref] [PubMed]

1999 (1)

A. J. W. Duijndam and M. A. Schonewille, “Nonuniform fast Fourier transform,” Geophysics 64(2), 539–551 (1999).
[Crossref]

1993 (1)

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

1991 (1)

J. I. Jackson, C. H. Meyer, D. G. Nishimura, and A. Macovski, “Selection of a convolution function for Fourier inversion using gridding [computerised tomography application],” IEEE Trans. Med. Imaging 10(3), 473–478 (1991).
[Crossref] [PubMed]

1985 (1)

J. D. O’Sullivan, “A fast sinc function gridding algorithm for fourier inversion in computer tomography,” IEEE Trans. Med. Imaging 4(4), 200–207 (1985).
[Crossref] [PubMed]

Adler, D. C.

Azimi, E.

B. Liu, E. Azimi, and M. E. Brezinski, “True logarithmic amplification of frequency clock in SS-OCT for calibration,” Biomed. Opt. Express 2(6), 1769–1777 (2011).
[Crossref] [PubMed]

Beatty, P. J.

P. J. Beatty, D. G. Nishimura, and J. M. Pauly, “Rapid gridding reconstruction with a minimal oversampling ratio,” IEEE Trans. Med. Imaging 24(6), 799–808 (2005).
[Crossref] [PubMed]

Bennett, R.

Biedermann, B. R.

Boppart, S. A.

Brezinski, M. E.

B. Liu, E. Azimi, and M. E. Brezinski, “True logarithmic amplification of frequency clock in SS-OCT for calibration,” Biomed. Opt. Express 2(6), 1769–1777 (2011).
[Crossref] [PubMed]

Chan, K. K. H.

K. K. H. Chan and S. Tang, “High-speed spectral domain optical coherence tomography using non-uniform fast Fourier transform,” Biomed. Opt. Express 1(5), 1309–1319 (2010).
[Crossref] [PubMed]

Chen, M.

Chen, Y.

Chen, Z.

Choma, M. A.

Connolly, J.

Cox, R. W.

Ding, Z.

Duijndam, A. J. W.

A. J. W. Duijndam and M. A. Schonewille, “Nonuniform fast Fourier transform,” Geophysics 64(2), 539–551 (1999).
[Crossref]

Duker, J. S.

Dutt, A.

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

Eigenwillig, C. M.

Ferguson, R. D.

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).
[Crossref]

Frigo, M.

M. Frigo and S. G. Johnson, “The design and implementation of FFTW3,” Proc. IEEE 93(2), 216–231 (2005).
[Crossref]

Fujimoto, J. G.

Greengard, L.

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

Hammer, D. X.

Hansen, M. S.

Hillmann, D.

D. Hillmann, G. Huttmann, and P. Koch, “Using nonequispaced fast Fourier transformation to process optical coherence tomography signals,” Proc. SPIE 7372, 73720R (2009).
[Crossref]

Hu, Z.

Z. Hu and A. M. Rollins, “Fourier domain optical coherence tomography with a linear-in-wavenumber spectrometer,” Opt. Lett. 32(24), 3525–3527 (2007).
[Crossref] [PubMed]

Huber, R.

Huo, L.

Huttmann, G.

D. Hillmann, G. Huttmann, and P. Koch, “Using nonequispaced fast Fourier transformation to process optical coherence tomography signals,” Proc. SPIE 7372, 73720R (2009).
[Crossref]

Iftimia, N. V.

Izatt, J. A.

Jackson, J. I.

Johnson, S. G.

M. Frigo and S. G. Johnson, “The design and implementation of FFTW3,” Proc. IEEE 93(2), 216–231 (2005).
[Crossref]

Kang, J. U.

Ko, T. H.

Koch, P.

D. Hillmann, G. Huttmann, and P. Koch, “Using nonequispaced fast Fourier transformation to process optical coherence tomography signals,” Proc. SPIE 7372, 73720R (2009).
[Crossref]

Kowalczyk, A.

Lamouche, G.

Lee, J.

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

Lévesque, D.

Li, J.

Li, X.

Liu, B.

B. Liu, E. Azimi, and M. E. Brezinski, “True logarithmic amplification of frequency clock in SS-OCT for calibration,” Biomed. Opt. Express 2(6), 1769–1777 (2011).
[Crossref] [PubMed]

Liu, G.

Macovski, A.

Marks, D. L.

Meng, J.

Meyer, C. H.

Nishimura, D. G.

P. J. Beatty, D. G. Nishimura, and J. M. Pauly, “Rapid gridding reconstruction with a minimal oversampling ratio,” IEEE Trans. Med. Imaging 24(6), 799–808 (2005).
[Crossref] [PubMed]

Noe, K. O.

O’Sullivan, J. D.

J. D. O’Sullivan, “A fast sinc function gridding algorithm for fourier inversion in computer tomography,” IEEE Trans. Med. Imaging 4(4), 200–207 (1985).
[Crossref] [PubMed]

Palte, G.

Pauly, J. M.

P. J. Beatty, D. G. Nishimura, and J. M. Pauly, “Rapid gridding reconstruction with a minimal oversampling ratio,” IEEE Trans. Med. Imaging 24(6), 799–808 (2005).
[Crossref] [PubMed]

Reynolds, J. J.

Rokhlin, V.

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

Rollins, A. M.

Z. Hu and A. M. Rollins, “Fourier domain optical coherence tomography with a linear-in-wavenumber spectrometer,” Opt. Lett. 32(24), 3525–3527 (2007).
[Crossref] [PubMed]

Sarty, G. E.

Sarunic, M. V.

Schaefer, A. W.

Schaeffter, T.

Schmitt, J.

Schonewille, M. A.

A. J. W. Duijndam and M. A. Schonewille, “Nonuniform fast Fourier transform,” Geophysics 64(2), 539–551 (1999).
[Crossref]

Sorensen, T. S.

Srinivasan, V. J.

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).
[Crossref]

Tang, S.

K. K. H. Chan and S. Tang, “High-speed spectral domain optical coherence tomography using non-uniform fast Fourier transform,” Biomed. Opt. Express 1(5), 1309–1319 (2010).
[Crossref] [PubMed]

Ustun, T. E.

Vergnole, S.

Wang, C.

Wang, K.

Wojtkowski, M.

Wu, T.

Xi, J.

Xie, T.

Xu, L.

Yang, C.

Yu, L.

Zhang, J.

Zhang, K.

Appl. Opt. (1)

Biomed. Opt. Express (3)

K. K. H. Chan and S. Tang, “High-speed spectral domain optical coherence tomography using non-uniform fast Fourier transform,” Biomed. Opt. Express 1(5), 1309–1319 (2010).
[Crossref] [PubMed]

B. Liu, E. Azimi, and M. E. Brezinski, “True logarithmic amplification of frequency clock in SS-OCT for calibration,” Biomed. Opt. Express 2(6), 1769–1777 (2011).
[Crossref] [PubMed]

Geophysics (1)

A. J. W. Duijndam and M. A. Schonewille, “Nonuniform fast Fourier transform,” Geophysics 64(2), 539–551 (1999).
[Crossref]

IEEE Trans. Biomed. Eng. (1)

IEEE Trans. Med. Imaging (4)

P. J. Beatty, D. G. Nishimura, and J. M. Pauly, “Rapid gridding reconstruction with a minimal oversampling ratio,” IEEE Trans. Med. Imaging 24(6), 799–808 (2005).
[Crossref] [PubMed]

J. D. O’Sullivan, “A fast sinc function gridding algorithm for fourier inversion in computer tomography,” IEEE Trans. Med. Imaging 4(4), 200–207 (1985).
[Crossref] [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).
[Crossref]

Magn. Reson. Med. (1)

Nat. Photonics (1)

Opt. Express (7)

Opt. Lett. (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).
[Crossref] [PubMed]

Proc. IEEE (1)

M. Frigo and S. G. Johnson, “The design and implementation of FFTW3,” Proc. IEEE 93(2), 216–231 (2005).
[Crossref]

Proc. SPIE (1)

D. Hillmann, G. Huttmann, and P. Koch, “Using nonequispaced fast Fourier transformation to process optical coherence tomography signals,” Proc. SPIE 7372, 73720R (2009).
[Crossref]

Rev. Sci. Instrum. (1)

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).
[Crossref]

SIAM Rev. (1)

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

Other (2)

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, 249–274.

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

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Fig. 1 Illustration of the resampling of data with Gaussian kernel into equally spaced grid with a width of 6 [9]. 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 by the crosses. The new evenly sampled data value is the summation of the values of all the crosses on each grid line.

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Fig. 2 The overview of the NUFFT algorithm in the spectral domain (k space) and the spatial domain (z space) (color online).

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Fig. 3 The effect on the kernel response by changing the kernel width W (left) and oversampling ratio R (right) of the Kasier-Bessel kernel function.

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Fig. 4 (left) Plot of kernel functions with R = 2, W = 3 and (right) Fourier transform of the kernel function a) Gaussian b) Kaiser-Bessel c) Two term cosine d) Three term cosine.

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Fig. 5 Relative L₂ error with varying oversampling ratio and kernel width for different kernel functions. a) Gaussian, b) Kaiser-Bessel, c) Two term cosine, d) Three term cosine.

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Fig. 6 Average absolute error with varying oversampling ratio and kernel width for different kernel functions. a) Gaussian, b) Kaiser-Bessel, c) Two term cosine, d) Three term cosine.

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Fig. 7 Computation time–error curves of the NUFFT (R = 2) in pre-computed mode (a, c) and in on-the-fly mode (b, d) for kernel width W = 2, 3, 4, 5, 6. The top row shows the L₂ error and the bottom row shows the average absolute error. Increasing the kernel width increases the processing time as expected. The arrow points to the most efficient kernel function: Kaiser-Bessel for the pre-computed mode and the Gaussian for the on-the-fly mode.

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Fig. 8 Images of the palmar surface of a finger reconstructed using NUFFT with different kernel functions and their differences from NDFT reconstructed image. From top to bottom the rows are Gaussian, Kaiser-Bessel, two term cosine, and three term cosine, respectively. The field of view is 2.5 mm wide by 1.7 mm deep. The scale bars are in dB.

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Tables (2)

Table 1 The parameters for cosine functions with an oversampling ratio of 2

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Table 2 NUFFT computation time (μs) for a 1024 pixels A-line and A-scan rate (lines/s)

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Equations (19)

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

f_{N D F T} (z_{m}) = \frac{1}{N} \sum_{n = 0}^{N - 1} F (k_{n}) \exp (- i \frac{2 π}{Δ K} m (k_{n} - k_{o})), m = 0, \dots, N - 1

F_{r} (k_{l}) = \sum_{n = 0}^{N - 1} F (k_{n}) C (k_{l} - k_{n}), l = 0, \dots, N_{r} - 1

f_{r} (z_{m}) \approx \frac{1}{N_{r}} \sum_{l = 0}^{N_{r} - 1} F_{r} (k_{l}) e^{- i 2 π \frac{m l}{N_{r}}}, m = 0, 1, \dots, N_{r} - 1

f_{N U F F T} (z_{m}) = \frac{f {}_{r}{(z_{m})}}{c (z_{m})}

e (z_{m}) = | f_{N U F F T} (z_{m}) - f_{N D F T} (z_{m}) |

= | \frac{1}{c (z_{m})} \sum_{i = - \infty}^{\infty} [f_{N D F T} (z_{m + i N_{r}}) c (z_{m + i N_{r}})] - f_{N D F T} (z_{m}) |

= | \frac{1}{c (z_{m})} \sum_{i \neq 0} f_{N D F T} (z_{m + i N_{r}}) c (z_{m + i N_{r}}) |

e_{2} = \frac{{({\sum_{m} | f_{N U F F T} (z_{m}) - f_{N D F T} (z_{m}) |}^{2})}^{\frac{1}{2}}}{{(\sum_{m} {| f_{N D F T} (z_{m}) |}^{2})}^{\frac{1}{2}}}

e_{a b s} = \frac{1}{N} \sum_{m} | 20 \log_{10} [f_{N U F F T} (z_{m})] - 20 \log_{10} [f_{N D F T} (z_{m})] |

C (κ) = e^{\frac{- κ^{2}}{4 τ}}

C (κ) = \frac{1}{W} I_{o} {β {[1 - {(2 κ / W)}^{2}]}^{\frac{1}{2}}}

C (κ) = α + (1 - α) \cos (\frac{2 π}{W} κ)

C (κ) = α + β \cos (\frac{2 π}{W} κ) + (1 - α - β) \cos (\frac{4 π}{W} κ)

κ = | \frac{k_{n} - k_{l}}{δ k} | \leq \frac{W}{2}, δ k = \frac{k_{\max} - k_{\min}}{N_{r}}

τ = \frac{W}{2 N^{2}} \frac{π}{R (R - 0.5)}

β = π {[\frac{W^{2}}{R^{2}} {(R - \frac{1}{2})}^{2} - 0.8]}^{\frac{1}{2}}

c (z_{m}) = {(2 τ)}^{\frac{1}{2}} \exp (- z_{m}^{2} τ), m = 0, \dots, \frac{N}{2} - 1

c (z_{m}) = \frac{\sin ({[{(m π W / N_{r})}^{2} - β^{2}]}^{\frac{1}{2}})}{{[{(m π W / N_{r})}^{2} - β^{2}]}^{\frac{1}{2}}}

O [p W N + R N \log_{2} (R N) + N]

Kernel Width	Two term cosine	Three term cosine
Kernel Width	α	α	β
2	0.5273	0.4149	0.4990
3	0.5076	0.3954	0.4977
4		0.3850	0.5000
5		0.3823	0.5000

Kernel function	Pre-computed mode (R=2, W=3)		On-the-fly mode (R=2, W=3)		On-the-fly mode (R=2, W=5)
Kernel function	Processing time	A-scan rate	Processing time	A-scan rate	Processing time	A-scan rate
Gaussian	10.51	95.14k	18.01	55.52k	23.77	40.27k
Kaiser-Bessel			39.72	25.17k	59.42	16.83k
Two term cos			20.01	49.96k	24.95	40.08k
Three term cos			25.56	39.13k	36.15	27.66k
NDFT	1470	0.6803k	-	-	-	-

Kernel Width	Two term cosine	Three term cosine
Kernel Width	α	α	β
2	0.5273	0.4149	0.4990
3	0.5076	0.3954	0.4977
4		0.3850	0.5000
5		0.3823	0.5000

Kernel function	Pre-computed mode (R=2, W=3)		On-the-fly mode (R=2, W=3)		On-the-fly mode (R=2, W=5)
Kernel function	Processing time	A-scan rate	Processing time	A-scan rate	Processing time	A-scan rate
Gaussian	10.51	95.14k	18.01	55.52k	23.77	40.27k
Kaiser-Bessel			39.72	25.17k	59.42	16.83k
Two term cos			20.01	49.96k	24.95	40.08k
Three term cos			25.56	39.13k	36.15	27.66k
NDFT	1470	0.6803k	-	-	-	-

Abstract

References

2011 (2)

2010 (4)

2009 (3)

2008 (3)

2007 (2)

2005 (2)

2004 (3)

2003 (2)

2001 (1)

1999 (1)

1993 (1)

1991 (1)

1985 (1)

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