## High-speed spectral domain optical coherence tomography using non-uniform fast Fourier transform |

Biomedical Optics Express, Vol. 1, Issue 5, pp. 1309-1319 (2010)

http://dx.doi.org/10.1364/BOE.1.001309

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### 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

## 1. Introduction

*k*domain to axial depth

*z*domain. DFT can be computed using the fast Fourier transform (FFT) algorithm if the data is uniformly sampled. However, diffraction gratings in SD-OCT systems separate spectral components almost linearly in wavelength λ. The data becomes unevenly sampled in

*k*domain due to the inverse relationship,

*k*in order to use FFT. The accuracy of the resampling method is important to the image reconstruction. Traditional resampling methods include linear and cubic spline interpolations. Although relatively fast, linear interpolation introduces a large amount of interpolation error. Alternatively, cubic spline interpolation can be used to reduce this error, but this method requires a long processing time. The performance of these traditional interpolation algorithms degrades as the signal frequency approaches Nyquist sampling rate. This causes the sensitivity to decrease for signals originating at greater depths which correspond to a higher oscillation frequency in the interference fringes.

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]

## 2. SD-OCT processing principles

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]

*s(k)*is the spectral intensity distribution of the light source. R

_{r}is the reflectivity of the reference arm mirror. R

_{i}and R

_{j}are the reflectivity in the i

^{th}and j

^{th}layers of the sample; z

_{i}is the optical path length difference of the i

^{th}layer compared to the reference arm and similarly z

_{ij}is the path length difference between the i

^{th}and j

^{th}sample layers. The third term in Eq. (1) encapsulates the axial depth information in the sample which appears as interferences of light waves. The axial reflectivity profile of the sample can be retrieved by performing a discrete Fourier transform from

*k*to axial depth z domain, resulting in the following equation:

### 2.1 Traditional software reconstruction methods

*k*domain to

*z*domain. In order to separate the spectral contents of the signal, most SD-OCT systems use a grating based spectrometer, which disperses the light evenly with respect to λ. The inverse relationship

*k*value. A simple method for resampling, linear interpolation, is used in high-speed SD-OCT systems [6

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]

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]

*k*using a Vendermode matrix [3

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]

*O(M*where

^{2}),*M*is the number of samples. Although NDFT is one of the more successful algorithms in alleviating the sensitivity fall-off problem [3

**17**(14), 12121–12131 (2009). [PubMed]

### 2.2 Non-uniform fast Fourier transform (NFFT)

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]

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]

*O(MlogM)*[12]. There are three types of NFFT which are distinguished by its inputs and outputs. Type I NFFT transforms data from a non-uniform grid to a uniform grid, type II NFFT goes from uniform sampling to non-uniform sampling and type III NFFT starts on a non-uniform grid that results in another non-uniform grid [13]. This paper will focus on the used of type I NFFT, specifically transforming data non-uniformly sampled in

*k*-domain to axial reflectivity information in the uniform z-domain.

*z*is the axial depth location,

_{m}*ΔK*is the wavenumber range,

*m*is the index for samples in the axial depth

*z*domain,

*I(k*) is the interference signal sampled at non-uniform

_{n}*k*spacing and

*M*is the number of sample points. Equation (3) cannot be computed using existing FFT algorithm because

*k*are not evenly spaced. NFFT, however, will resample the signal to an evenly spaced grid via a convolution based interpolation as illustrated in Fig. 2 . The signal can be interpolated using an user defined interpolation kernel G

_{n}_{τ}(

*k*) [15]. The interpolated signal is then resampled on a uniform grid. In the following calculation, an Gaussian interpolation kernel is selected which is defined aswhere

*M*is the number of sample points,

*R*is the upsampling ratio

*M*

_{r}/

*M*where

*M*

_{r}is the length of the upsampled signal, and

*M*

_{sp}is the kernel width which denotes the number of grid points on each side of the original data point to which the Gaussian kernel is accounted for in calculation. An infinite length Gaussian would produce the most accurate results, but the value of

*M*

_{sp}is often set to a small finite value in consideration of computational efficiency. The use of finite

*M*

_{sp}value introduces a truncation error [16] because the tail of the Gaussian is not used. Another type of error introduced in NFFT is aliasing. By resampling the interpolated signal in the

*k*domain onto a uniform grid, aliasing would occur in the

*z*domain [17]. Increasing the upsampling ratio

*R*would decrease the amount of aliasing and hence increase the accuracy of NFFT. The truncation and aliasing errors account for the small deviation between the results of NFFT and NDFT. Readers should refer to [12,15,16] for a detailed derivation of the computational errors and the method of choosing τ. To balance the processing time and the accuracy, we used

*M*

_{sp}of three and

*R*of two. Theoretically this combination of

*M*

_{sp}and

*R*would result in an error of less than 1.9 × 10

^{−3}when compared to NDFT [15].

*G*) with

_{τ}(k*I(k)*gives the intermediate function

*I*that can be defined as,

_{τ}(k)*I*is resampled in an evenly spaced grid with

_{τ}(k)*M*

_{r}samples. In the discrete from,

*M*

_{r}points.

*a*has been calculated,

_{τ}(z_{m})*a(z*can be calculated by a deconvolution in

_{m})*k*space by

*G*or alternatively with a simple division by the Fourier transform of

_{τ}(k)*G*in z space. The Fourier transform of

_{τ}(k)*G*can be expressed as,

_{τ}(k)*M*

_{r}points, which is larger than the original

*M*points input because of upsampling. The points

*a(z*, where

_{m})*m*>

*M*, represents deeper locations in the sample in which the interference fringes were not captured by the spectrometer. Recall that the imaging depth

*a(z*) is determined by the original sampling rate at

_{m}*M*points. The extra points in the z domain contain artifacts, primarily introduced through aliasing in interpolation and resampling of the data. No additional physical information from the sample is contained and thus the extra points can be discarded. Hence, the vector of useful data will contain only

*M*points as expected.

## 3. System and experiments

^{2}pixel size. The data from the camera is transferred to a computer via a CameraLink frame grabber (National instrument) for further processing. The spectrometer was designed to realize a source limited axial resolution of 7 μm and minimized sensitivity fall-off. The theoretical spectral resolution is 0.101 nm and the total imaging depth is 1.73 mm.

### 3.1 Experiment for sensitivity fall-off and artifact reduction

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]

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]

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]

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]

**12**(3), 367–376 (2004). [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]

### 3.2 Computation Speed

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]

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]

### 3.3 Demonstration of high-speed Imaging on an ex-vivo squid eye

## 4. Conclusion

## Acknowledgment

## References and links

1. | Z. Hu and A. M. Rollins, “Fourier domain optical coherence tomography with a linear-in-wavenumber spectrometer,” Opt. Lett. |

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. |

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 |

4. | G. Hausler and M. W. Lindner, “Coherence radar and spectral radar – new tools for dermatological diagnosis,” J. Biomed. Opt. |

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 |

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. |

7. | E. Maeland, “On the comparison of interpolation methods,” IEEE Trans. Med. Imaging |

8. | H. Hou and H. C. Andrews, “Cubic splines for image interpolation and digital filtering,” IEEE Trans. Acoust. Speech Signal Process. |

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. |

10. | S. De Francesco and A. M. F. da Silva, “Efficient NUFFT-based direct Fourier algorithm for fan beam CT reconstruction,” Proc. SPIE |

11. | M. M. Bronstein, A. M. Bronstein, M. Zibulevsky, and H. Azhari, “Reconstruction in diffraction ultrasound tomography using nonuniform FFT,” IEEE Trans. Med. Imaging |

12. | A. Dutt and V. Rokhlin, “Fast Fourier transforms for nonequispaced data,” SIAM J. Sci. Comput. |

13. | J. Lee and L. Greengard, “The type 3 nonuniform FFT and its application,” J. Comput. Phys. |

14. | J. A. Fessler and B. P. Sutton, “Nonuniform fast Fourier transforms using min-max interpolation,” IEEE Trans. Signal Process. |

15. | L. Greengard and J. Lee, “Accelerating the Nonuniform Fast Fourier Transform,” SIAM Rev. |

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. |

18. | Y. Rolain, J. Schoukens, and G. Vandersteen, ““Signal Reconstruction for Non-Equidistant Finite Length Sample Sets: A “KIS” Approach,” IEEE Trans. Instrum. Meas. |

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 |

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 |

22. | M. Frigo, and S. G. Johnson, “FFTW: an adaptive software architecture for the FFT,” in |

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 |

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. |

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. |

**OCIS Codes**

(170.4500) Medical optics and biotechnology : Optical coherence tomography

(070.2025) Fourier optics and signal processing : Discrete optical signal processing

(110.3010) Imaging systems : Image reconstruction techniques

**ToC Category:**

Optical Coherence Tomography

**History**

Original Manuscript: September 13, 2010

Revised Manuscript: October 29, 2010

Manuscript Accepted: October 30, 2010

Published: November 4, 2010

**Citation**

Kenny K. H. Chan and Shuo Tang, "High-speed spectral domain optical coherence tomography using non-uniform fast Fourier transform," Biomed. Opt. Express **1**, 1309-1319 (2010)

http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-1-5-1309

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### References

- 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]
- 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]
- 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]
- G. Hausler and M. W. Lindner, “Coherence radar and spectral radar – new tools for dermatological diagnosis,” J. Biomed. Opt. 3(1), 21–31 (1998).
- 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]
- 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]
- E. Maeland, “On the comparison of interpolation methods,” IEEE Trans. Med. Imaging 7(3), 213–217 (1988). [PubMed]
- 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).
- 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]
- 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).
- 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]
- A. Dutt and V. Rokhlin, “Fast Fourier transforms for nonequispaced data,” SIAM J. Sci. Comput. 14(6), 1368–1393 (1993).
- J. Lee and L. Greengard, “The type 3 nonuniform FFT and its application,” J. Comput. Phys. 206(iss. 1), 1–5 (2005).
- J. A. Fessler and B. P. Sutton, “Nonuniform fast Fourier transforms using min-max interpolation,” IEEE Trans. Signal Process. 51(2), 560–574 (2003).
- L. Greengard and J. Lee, “Accelerating the Nonuniform Fast Fourier Transform,” SIAM Rev. 46(3), 443–454 (2004).
- 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.
- A. J. W. Duijndam and M. A. Schonewille, “Nonuniform fast Fourier transform,” Geophys. 64, 539–551 (1999).
- 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).
- 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).
- P. Thevenaz, T. Blu, and M. Unser, Handbook of Medical Imaging (Academic Press, 2000), Chap. 25.
- 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]
- 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.
- 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]
- OpenMP Architecture Review Board, “The OpenMP API specification for parallel programming,” http://www.openmp.org/ .
- 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]
- 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]

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