## Reference spectrum extraction and fixed-pattern noise removal in optical coherence tomography |

Optics Express, Vol. 18, Issue 24, pp. 24395-24404 (2010)

http://dx.doi.org/10.1364/OE.18.024395

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### Abstract

We present a new signal processing method that extracts the reference spectrum information from an acquired optical coherence tomography (OCT) image without a separate calibration step of reference spectrum measurement. The reference spectrum is used to remove the fixed-pattern noise that is a characteristic artifact of Fourier-domain OCT schemes. It was found that the conventional approach based on an averaged spectrum, or mean spectrum, is prone to be influenced by the high-amplitude data points whose statistical distribution is hardly randomized. Thus, the conventional mean-spectrum subtraction method cannot completely eliminate the artifact but may leave residual horizontal lines in the final image. This problem was avoided by utilizing an advanced statistical analysis tool of the median A-line. The reference A-line was obtained by taking a complex median of each horizontal-line data. As an optional method of high-speed calculation, we also propose a minimum-variance mean A-line that can be calculated from an image by a collection of mean A-line values taken from a horizontal segment whose complex variance of the data points is the minimum. By comparing the images processed by those methods, it was found that our new processing schemes of the median-line subtraction and the minimum-variance mean-line subtraction successfully suppressed the fixed-pattern noise. The inverse Fourier transform of the obtained reference A-line well matched the reference spectrum obtained by a physical measurement as well.

© 2010 OSA

## 1. Introduction

1. R. Leitgeb, C. K. Hitzenberger, and A. F. Fercher, “Performance of fourier domain vs. time domain optical coherence tomography,” Opt. Express **11**(8), 889–894 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-8-889. [CrossRef] [PubMed]

4. S. Yun, G. Tearney, J. de Boer, N. Iftimia, and B. Bouma, “High-speed optical frequency-domain imaging,” Opt. Express **11**(22), 2953–2963 (2003), http://www.opticsinfobase.org/abstract.cfm?URI=oe-11-22-2953. [CrossRef] [PubMed]

4. S. Yun, G. Tearney, J. de Boer, N. Iftimia, and B. Bouma, “High-speed optical frequency-domain imaging,” Opt. Express **11**(22), 2953–2963 (2003), http://www.opticsinfobase.org/abstract.cfm?URI=oe-11-22-2953. [CrossRef] [PubMed]

6. 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), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-3-367. [CrossRef] [PubMed]

*mean-spectrum subtraction*method is popular for both SD-OCT and SS-OCT due to the practical implementation simplicity and inherent immunity to the variation of the reference. One can utilize a slightly modified method of numerically dividing a detected spectrum by a reference spectrum of the mean spectrum instead of the more common subtraction processing [3,6

6. 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), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-3-367. [CrossRef] [PubMed]

## 2. Fixed-pattern noise and mean-spectrum subtraction

**, and the desired reflection signal from a sample is expressed by a sum of complex sample fields**

*E*_{r}**’s, the measured spectrum,**

*E*_{n}

*G*(*k*

**)**, is given as a function of frequency,

*k*, bywhere

*n*and

*m*are running integer indices for summation. In the right-hand side of the equation, the first two terms represent the reference spectrum and the power sum of the sample fields, respectively. The third term contains the desired information of OCT reflectance in its interferogram. And the fourth term corresponds to the mutual interference between the sample fields, which is known as the autocorrelation noise of FD-OCT. This autocorrelation signal of the fourth term is usually much weaker than the

**⋅**

*E*_{r}**products of the third term because of the relatively high amplitude of the reference field**

*E*_{n}**compared to the sample field**

*E*_{r}**. The first term is usually referred to the reference spectrum since its field of**

*E*_{n}**is the probing field that forms the optical interference of cross-correlation in the third term (**

*E*_{r}**⋅**

*E*_{r}**). This reference spectrum can be directly measured by a calibration process with no sample in place. On the other hand, the reference signal can be alternatively defined by a sum of the first two terms of Eq. (1) as they together form the nearly static pattern in the final A-line. Due to the relatively low power of the second term (sample reflection power), the difference between those two definitions may be ignored.**

*E*_{n}*N*) of A-line spectra, the mean spectrum [denoted by

**〈**, here] effectively becomes the reference spectrum of the first two terms in Eq. (1) aswhere

*G*〉**is a measured spectrum and**

*G*_{l}**〈⋅⋅⋅〉**operator stands for taking a mean value. Note that the averaging is done with spectrum data points of the same frequency (

*k*), not along the frequency. The reference spectrum obtained by Eq. (2) is subtracted from each A-line spectrum (

**−**

*G*_{l}**〈**to remove the static components in the mean-spectrum subtraction method.

*G*〉)*N*= 1,024. As shown in Fig. 1(c), most of the fixed pattern was removed by this method. But some horizontal lines were still visible as a noisy pattern. The axial positions of those lines coincided with the positions of the horizontal tissue-air interfaces of strong reflectance.

## 3. Statistical analysis of the fixed-pattern noise

**F**{

**⋅⋅⋅**} in this report), the Fourier transform of an averaged spectrum is equal to the average of the transformed A-line spectra:where

*N*is the number of A-lines involved with the averaging. Thus, the mean-spectrum subtraction can be replaced by its transformed version. The

*mean A-line*or

*mean line*,

**〈**, can be defined in the same way as Eq. (2) after Fourier transform:where

*g*〉**(**

*g*_{l}*z*) is an A-line that consists of complex-reflectance data points along the axial coordinate

*z*. In the

*mean-line subtraction*method, a final A-line is obtained by subtracting the mean line from each A-line (

**−**

*g*_{l}**〈**to remove the fixed-pattern noise. This method produces a mathematically equal result to that of the mean-spectrum subtraction as Eq. (3) suggests. Only the mean-spectrum subtraction has been used in the OCT signal processing so far, probably because of its conceptual simplicity and computational ease of processing real numbers. Even though the mean-line subtraction is mathematically equivalent to the mean-spectrum subtraction, this approach gives more insight on the characteristic of the fixed-pattern noise. It becomes easier to isolate the cause of the error found in those methods after the transform.

*g*〉)**+ i**

*x***) obtained from the horizontal line of Fig. 1(a) at the axial position of observation.**

*y**i.e*., the mean complex number). In this method, each dot is shifted back by this mean vector for the final distribution to be relocated around the origin in an even and symmetric distribution manner. However, the mean vector may not coincide with the effective center of the distribution because of the statistical oddness. As observed in Figs. 2(a) and (d), some distinguished points have distinctly high amplitudes while the others are densely located around the origin with low amplitudes. These high-amplitude points are distributed in an odd and asymmetric manner so that their distribution does not look purely random. These were generated by a different reflectance source from those of the central area and obeyed a distinguished distribution function as a consequence. The high-amplitude points came from the high-contrast interface of the tissue and the air shown by a bright horizontal curve in Fig. 1. Their amplitudes were far higher than the others in one or two orders of magnitude,

*i.e*., 20 to 40 dB above the reflectivities of the other ordinary points. Thus, the average of the high-amplitude points has a chance to produce a statistical fluctuation that is stronger than the low amplitudes of the ordinary data points.

**>0,**

*x***<0) with a certain average phase. It was obvious that the smooth tissue-air interface of the sample could make a certain correlation of those phases. In Fig. 2(d), the black dot represents the mean complex number which was averaged over the whole data for the 162nd line. It was offset to the lower right with respect to the effective center of the distribution as anticipated by the odd phase distribution of the high-amplitude points.**

*y**M*, their mean produces a sampling error of a standard deviation to be (1/

*M*)

^{1/2}of the initial deviation of the samples which roughly equals the mean amplitude. In a rough estimation for the given case of Fig. 2, the sampling error in the mean of the high-amplitude points must have been larger than 1/10 of their average amplitude for

*M*<100. This error is still higher than the low amplitudes of a majority of data points located around the center. Therefore, even under the most favorable assumption of the random phase distribution, a mean can produce an error in finding the distribution center when a part of the data points have extraordinarily high amplitudes beyond the magnitude expected by a monotonic normal distribution.

## 4. Median-line subtraction

*median-line subtraction*to avoid the observed drawback of the conventional mean-spectrum subtraction method. A median is defined by the halfway value in a set of data when they are sorted in an ascending or descending order. In this research, a median for complex numbers is defined by a sum of the median of the real terms and that of the imaginary terms for simplicity. For a complex-numbered A-line data

*g**, the median line,*

_{l}**is obtained bywhere**

*g′***Re**{} represents taking a real term,

**Im**{} represents taking an imaginary term, and

**med**{} represents a median value operation of a set defined by an integer index

*l*, respectively. Thus, the mean line consists of complex median values taken for each set of a horizontal line.

*k*-domain) before Fourier transform. A median of a Fourier transform is not equal to a Fourier transform of a median unlike the linear property of a mean operation suggested by Eq. (3). The effect of a high-amplitude reflection point spreads over the full range of the spectrum before the transform, not to be removed in the spectral domain with ease. Thus, our median-line subtraction should be processed with the A-line data after the domain conversion.

7. R. Tripathi, N. Nassif, J. S. Nelson, B. H. Park, and J. F. de Boer, “Spectral shaping for non-Gaussian source spectra in optical coherence tomography,” Opt. Lett. **27**(6), 406–408 (2002), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-27-6-406. [CrossRef]

## 5. Minimum-variance mean-line subtraction

*N*data points, an efficient sorting algorithm requires an order of

*N*⋅log

_{2}

*N*computational complexity like a fast Fourier transform (FFT). We observed that a sorting task for 1,024 complex numbers (two separate sorting tasks in real numbers) takes eight times longer than a computation time used in an FFT task for 1,024 real numbers in Matlab

^{TM}7.0. This comparison was made by using the internal functions of

*sort*() and

*fft*() for randomly generated numbers. In most cases, it is unnecessary to extract a reference for every OCT frame. The variation of a reference spectrum is usually so slow that it just needs to be refreshed with a long interval such as tens of seconds. However, some applications, such as an OCT endoscope, may demand a more frequent reference extraction because of the relatively unstable imaging conditions.

*minimum-variance mean-line subtraction*scheme, a segment of the minimum variance is selected for its mean value to be assigned as the mean-line value of that axial position. In other words, a mean-line value for

*z*is determined to be one of the segmental means,

*L*is the number of data points of Ω. Here,

*g**(*

_{l}*z*) is an A-line data point of the raw OCT image obtained by Fourier-transforming the measured spectrum. On the other hand, the segmental variance,

*v*

_{Ω}, is defined byfor a segment Ω. The minimum-variance mean is the segmental mean of a minimum-variance segment, whose variance of

*v*

_{Ω}is the minimum in a horizontal line for an axial position of

*z*.

## 6. Conclusion

## Acknowledgments

## References and links

1. | R. Leitgeb, C. K. Hitzenberger, and A. F. Fercher, “Performance of fourier domain vs. time domain optical coherence tomography,” Opt. Express |

2. | J. F. de Boer, B. Cense, B. H. Park, M. C. Pierce, G. J. Tearney, and B. E. Bouma, “Improved signal-to-noise ratio in spectral-domain compared with time-domain optical coherence tomography,” Opt. Lett. |

3. | J. F. de Boer, “Spectral/Fourier domain optical coherence tomography,” in |

4. | S. Yun, G. Tearney, J. de Boer, N. Iftimia, and B. Bouma, “High-speed optical frequency-domain imaging,” Opt. Express |

5. | R. A. Leitgeb, and M. Wojtkowski, “Complex and coherent noise free Fourier domain optical coherence tomography,” in |

6. | 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 |

7. | R. Tripathi, N. Nassif, J. S. Nelson, B. H. Park, and J. F. de Boer, “Spectral shaping for non-Gaussian source spectra in optical coherence tomography,” Opt. Lett. |

8. | C. C. Rosa and A. G. Podoleanu, “Limitation of the achievable signal-to-noise ratio in optical coherence tomography due to mismatch of the balanced receiver,” Appl. Opt. |

9. | R. Langley, |

**OCIS Codes**

(100.2000) Image processing : Digital image processing

(110.4500) Imaging systems : Optical coherence tomography

(170.4500) Medical optics and biotechnology : Optical coherence tomography

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: September 2, 2010

Revised Manuscript: October 15, 2010

Manuscript Accepted: October 21, 2010

Published: November 8, 2010

**Citation**

Sucbei Moon, Sang-Won Lee, and Zhongping Chen, "Reference spectrum extraction and fixed-pattern noise removal in optical coherence tomography," Opt. Express **18**, 24395-24404 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-24-24395

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### References

- R. Leitgeb, C. K. Hitzenberger, and A. F. Fercher, “Performance of fourier domain vs. time domain optical coherence tomography,” Opt. Express 11(8), 889–894 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-8-889 . [CrossRef] [PubMed]
- J. F. de Boer, B. Cense, B. H. Park, M. C. Pierce, G. J. Tearney, and B. E. Bouma, “Improved signal-to-noise ratio in spectral-domain compared with time-domain optical coherence tomography,” Opt. Lett. 28(21), 2067–2069 (2003), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-28-21-2067 . [CrossRef] [PubMed]
- J. F. de Boer, “Spectral/Fourier domain optical coherence tomography,” in Optical Coherence Tomography, Technology and Applications, Wolfgang Drexler, and James G. Fujimoto, eds. (Springer, 2008), pp. 147–175.
- S. Yun, G. Tearney, J. de Boer, N. Iftimia, and B. Bouma, “High-speed optical frequency-domain imaging,” Opt. Express 11(22), 2953–2963 (2003), http://www.opticsinfobase.org/abstract.cfm?URI=oe-11-22-2953 . [CrossRef] [PubMed]
- R. A. Leitgeb, and M. Wojtkowski, “Complex and coherent noise free Fourier domain optical coherence tomography,” in Optical Coherence Tomography, Technology and Applications, Wolfgang Drexler, and James G. Fujimoto, eds. (Springer, 2008), pp. 177–207.
- 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), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-12-3-367 . [CrossRef] [PubMed]
- R. Tripathi, N. Nassif, J. S. Nelson, B. H. Park, and J. F. de Boer, “Spectral shaping for non-Gaussian source spectra in optical coherence tomography,” Opt. Lett. 27(6), 406–408 (2002), http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-27-6-406 . [CrossRef]
- C. C. Rosa and A. G. Podoleanu, “Limitation of the achievable signal-to-noise ratio in optical coherence tomography due to mismatch of the balanced receiver,” Appl. Opt. 43(25), 4802–4815 (2004), http://www.opticsinfobase.org/ao/abstract.cfm?URI=ao-43-25-4802 . [CrossRef] [PubMed]
- R. Langley, Practical Statistics (Dover Publications, 1971).

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