## Optical coherence tomography by using frequency measurements in wavelength domain |

Optics Express, Vol. 19, Issue 2, pp. 1324-1334 (2011)

http://dx.doi.org/10.1364/OE.19.001324

Acrobat PDF (787 KB)

### Abstract

Optical coherence tomography (OCT) reconstruction by using frequency measurements in the wavelength domain is presented in this paper. The method directly recovers the axial scan by formulating the frequency domain OCT (FD-OCT) into an algebraic reconstruction problem. In this way, the need for interpolation is removed. Then by solving the problem with ℓ_{1} optimization, the computational load is significantly reduced. It is demonstrated by experiment and simulation that the proposed method can achieve high resolution and longer imaging depth compared to the FD-OCT method.

© 2011 Optical Society of America

## 1. Introduction

2. O. P. Bruno and J. Chaubell, “Inverse scattering problem for optical coherence tomography,” Opt. Lett. **28**, 2049–2051 (2003). [CrossRef] [PubMed]

3. P. E Andersen, L. Thrane, H. T Yura, A. Tycho, T. M. Jrgensen, and M. H Frosz, “Advanced modelling of optical coherence tomography systems,” Phys. Med. Biol. **49**, 1307–1327 (2004). [CrossRef] [PubMed]

1. D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science **254**, 1178–1181 (1991). [CrossRef] [PubMed]

4. G. Häusler and M. W. Lindner, “‘Coherence radar’ and ‘spectral radar’–new tools for dermatological diagnosis,” J. Biomed. Opt. **3**, 21–31 (1998). [CrossRef]

6. C. Dorrer, N. Belabas, J. Likforman, and M. Joffre, “Spectral resolution and sampling issues in Fourier-transform spectral interferometry,” J. Opt. Soc. Am. B **17**, 1795–1802 (2000). [CrossRef]

6. C. Dorrer, N. Belabas, J. Likforman, and M. Joffre, “Spectral resolution and sampling issues in Fourier-transform spectral interferometry,” J. Opt. Soc. Am. B **17**, 1795–1802 (2000). [CrossRef]

9. S. Vergnole, D. Lvesque, and G. Lamouche, “Experimental validation of an optimized signal processing method to handle non-linearity in swept-source optical coherence tomography,” Opt. Express **18**, 10446–10461 (2010). [CrossRef] [PubMed]

**Ax=b**, where

**b**denotes the frequency measurements in wavelength domain, vector

**x**denotes the discretized depth profile of the subsurface interfaces in an axial scan, and

**A**is the measurement matrix which also contains the information of the white light used. The basic principle of ART involves solving the algebraic inverse problem that is to reconstruct

**x**from

**b**. Compared to other OCT methods, the ART based OCT method gives directly the depth profile without using any resampling or interpolation. High resolution can be achieved by using small discretization interval in spatial coordinates, which results in a high dimension of vector

**x**. Another merit point of ART is that the imaging depth would be free from Shannon Nyquist sampling theorem by using advanced signal processing techniques such as the compressed sensing technique [

**?**]. However, ART method is not widely adopted because of the high computational load particularly when ones attempt to have high resolution and large imaging depth.

11. S. Vergnole, D. Levesque, G. Lamouche, M. Dufour, and B. Gauthier, “Characterization of thin layered structures using deconvolution techniques in time-domain and Fourier-domain optical coherence tomography,” Proc SPIE **6796**, 67961 (2007). [CrossRef]

12. S. A. Boppart, A. L. Oldenburg, C. Xu, and D. L. Marks, “Optical probes and techniques for molecular contrast enhancement in coherence imaging,” J. Biomed. Opt. **10**, 041208 (2005). [CrossRef]

**x**in the above mentioned algebraic representation is essentially sparse. Consequently, this presents an opportunity to use a greatly reduced dimension of

**b**in ART, and in turn, the computational load of ART can be greatly reduced. Recent advances in compressed sensing [?] has demonstrated the feasibility of solving sparse unknowns from measurements of reduced dimension. The ART method proposed in this paper is based on the ℓ

_{1}-optimization technique used in compressed sensing. It inherits the merits of ART in delivering high resolution of OCT, while overcoming the limitations in computational load.

_{1}norm of

**x**is minimized to reconstruct the sparse depth profile including the positions of the subsurface interfaces and the reflectivity at the surfaces. By this way, the problem of axial scan reconstruction of the wavelength domain OCT is formulated into a convex optimization problem that can be solved by interior point based algorithm. The simulation and experiment results have shown that the proposed method delivers higher resolution and longer imaging depth compared to the FD-OCT method. The performance of the proposed method is also demonstrated by imaging an onion skin.

## 2. Principle of Algebraic OCT Based on ℓ_{1}-optimization

*z*-coordinate for axial scan as shown in Fig. 1 where the origin is set on the zero optical path difference with respect to the reference mirror,

*M*1. Assuming that absorption is weak or negligible, the frequency measurements captured from OSA are in the wavelength domain given as follows [4

4. G. Häusler and M. W. Lindner, “‘Coherence radar’ and ‘spectral radar’–new tools for dermatological diagnosis,” J. Biomed. Opt. **3**, 21–31 (1998). [CrossRef]

*G*(

*λ*) is the output optical spectrum of the low coherence light source,

*R*is the reflectivity of the reference mirror,

_{r}*a*(

*z*) denotes the depth profile which can also represent the reflectivity of the subsurface interfaces positioning at

*z*, and

*z*is the imaging depth.

_{max}*a*(

*z*) of the sample. The second term and the third term in Eq. (1) can be made non-overlapped in the reconstructed axial scan by positioning appropriately the reference mirror. The basic problem of OCT is to reconstruct

*a*(

*z*) from the known frequency measurements of

*S*(

*λ*) and

*G*(

*λ*).

4. G. Häusler and M. W. Lindner, “‘Coherence radar’ and ‘spectral radar’–new tools for dermatological diagnosis,” J. Biomed. Opt. **3**, 21–31 (1998). [CrossRef]

*κ*= 2

*π*/

*λ*, then they are interpolated and resampled on a uniformly spaced frequency grid with the same number of measurements. Thereafter, the depth profile is obtained from the fast Fourier transform (FFT) of the resampled measurements. This principle clearly shows that the resolution of the conventional FD-OCT method is determined by the coherence length of

*G*(

*λ*). In practice, FWHM (full width at half maximum) of

*G*(

*λ*) is used to characterize the resolution of the conventional FD-OCT method. Another consequence of the use of FFT is that the imaging depth of the conventional FD-OCT method is limited by the spectral resolution of the spectrum meter used to obtain frequency measurements [4

**3**, 21–31 (1998). [CrossRef]

**x**from the measurements

*S*(

*λ*) and

_{k}*G*(

*λ*) (

_{k}*k*= 1, 2,...,

*N*). It uses the frequency measurements obtained in wavelength domain, and it does not need the resampling process in the reconstruction. This is the same as in [7], but the proposed method is different from the one in [7] in that solving Eq. (2) is actually performing deconvolution simultaneously. Thus, the proposed method improves the resolution regardless of FWHM of

*G*(

*λ*). Moreover, the conventional ART method is to solve the equation by considering an optimization problem:

**Ax=b**where || · ||

_{2}denotes the 2-norm of a vector.

*a priori*knowledge on

**x**such as sparsity is available. This implies

*N*≥

*M*. The conventional ART method actually tries to minimize the total mean-square-error occurring to all the elements of

**x**. However, as stated above, the OCT measurements are actually coming from a layer structure or discrete scatterers or mixture of both. Thus,

**x**in Eq. (2) is essentially sparse. In this sense, ℓ

_{1}-optimization of

**x**offers an advantages of using much few number of measurements in

**b**, because one only needs to determine those non-zero elements in

**x**. This is feasible as shown by Restricted Isometry Property (RIP) condition in the compressed sensing theory [14

14. E. J. Candès and T. Tao, “Decoding by linear programming,” IEEE Trans. Inf. Theory **51**, 4203–4215 (2005). [CrossRef]

*N*≥ log(

*M*)

*μK*where

*K*is the sparsity of

**x**and

*μ*represents the coherence of

**A**. The method proposed in this paper is based on the ℓ

_{1}-optimization and it obtains the reconstruction of

**x**as follows: Considering noisy measurements in practice, the second implementation of the proposed method is given by the following mixed ℓ

_{1}/ℓ

_{2}-optimization: where

*ε*denotes the noise level.

## 3. Performance Evaluation of the Algebraic OCT Method Based on ℓ_{1}-optimization

*μ*m. This choice corresponds to the output spectrum of the light source (SLED 1550, Denselight) used in experiments reported in the later sections. In the following experiments, the frequency measurements of

*S*(

*λ*) and

*G*(

*λ*) are captured by an optical spectrum analyzer (ANDO6317B, ANDO) with a tunable spectral resolution that can go as small as 0.01 nm. Without loss of generality, only one axial scan is considered in the following simulation and experiment studies in this section.

### 3.1. Imaging Depth

**3**, 21–31 (1998). [CrossRef]

**3**, 21–31 (1998). [CrossRef]

*λ*is the center wavelength,

*δλ*is the spectral resolution of the OSA, and

*n*is the effective refractive index of the sample.

_{1}-optimization to reconstruct sparse data. As shown in the recent development of compressed sensing theory [?], a sparse signal can still be recovered beyond the limit imposed by Shannon-Nyquist sampling theorem. This means that the imaging depth is no longer limited to Eq. (7).

*z*= 0 mm to

*z*= 5 mm in a step size of Δ

*z*= 1

*μ*m. Thus, we discretize

*a*(

*z*) over the axial range from 0 mm to 5 mm with discretization interval of Δ

*z*= 1

*μ*m. Referring to Eq.(2),

*M*= 5000. At the

*k*-th step of motion of the mirror, the position of the mirror is denoted by

*z*. That is,

_{k}*a*(

*z*) = 1 and

_{k}*a*(

*z*) = 0 for all

*z*≠

*z*. The frequency measurements at this step are taken at the wavelengths denoted by

_{k}*i*= 1, 2,..., 771 where

*N*= 150 samples among the 771 frequency measurements. Denote the corresponding wavelengths by

*λ*for (

_{i}*i*= 1, 2,...,

*N*= 150) for Eq. (3) and Eq. (4), without loss of generality. The reconstructed

**x**is obtained by performing ℓ

_{1}optimization in Eq. (5). The optimization can be done by many well-developed algorithms such as the CVX matlab module [15

15. M. Grant and S. Boyd, CVX: Matlab software for disciplined convex programming (web page and software). http://stanford.edu/$_{\mathaccent''0365\relax{~}}$boyd/cvx (2009).

**x**. After all motion steps are finished, the imaging depth is identified as the maximum value of the positions which are correctly recovered. We repeat the above process for other values of the spectral resolution of the OSA. The imaging depth is plotted against the spectral resolution in Fig. 2 which also includes the imaging depth for the conventional FD-OCT method as calculated by Eq. (7). This figure shows that the imaging depth of the proposed method is no longer limited by the spectral resolution of the spectrum meter used.

*a*(

*z*) by 5001 points over a range of [0 mm, 5 mm] in optical path, that is

*M*= 5001 and solve Eq. (6) with empirically determined

*ε*= 0.5 × 10

^{−3}. The frequency measurements are captured in experiment at 771 wavelengths from 1473 nm to 1627 nm for the conventional FD-OCT method, out of which 150 measurements are chosen evenly randomly from 1513 nm and 1573 nm for the proposed method. The reason for choosing the center portion of the optical spectrum is that the signal there has a higher SNR. For the conventional FD-OCT method, the whole set of 771 measurements with no zero padding is used.

*a*(

*z*) is plotted in logarithmic scale against the optical path difference recorded by the motion stage driving the sample mirror. Comparing the results obtained from the two methods, a clear drop trend of the magnitude is observed for both methods. This is attributed to the sensitivity roll off. However, experiment results show clearly that the mirror position reconstructed by the conventional FD-OCT method bounces back to less than 3 mm when the mirror goes actually beyond 3 mm in its third move marked by Step 3. In the Fig. 3(a), the reconstructed step profile at Step 3 has also a superimposed background or hump. This is similar to the effect observed after interpolation is performed on spectral interferogram with large time delay [6

6. C. Dorrer, N. Belabas, J. Likforman, and M. Joffre, “Spectral resolution and sampling issues in Fourier-transform spectral interferometry,” J. Opt. Soc. Am. B **17**, 1795–1802 (2000). [CrossRef]

### 3.2. Computational Load

**A**. In turn, the dimensions of matrix

**A**are determined by the values of

*N*and

*M*. They denote respectively the number of wavelengths at which the frequency measurements are obtained, and the number of points used to discretize

*a*(

*z*). They basically determine the computational complexity.

_{1}optimization [16

16. W. Qiu and E. Skafidas, “Robust estimation of GCD with sparse coefficients,” Signal Processing **90**, 972–976 (2010). [CrossRef]

*O*(

*M*

^{3/2}

*log*

^{2}(

*M*)

*μK*). For the conventional ART method, SVD algorithms are usually employed for fast computational which gives the computational load of 6

*M*

^{3}[17]. It is easy to verify that

*O*(

*M*

^{3/2}

*log*

^{2}(

*M*)

*μK*) << 6

*M*

^{3}. With this, we anticipate that the computational load of the proposed method is greatly reduced compared to the conventional ART method. In order to verify this, we perform numerical simulations to compare the computational time of the proposed method and the conventional ART method on a Pentium-4 computer with 3.0 GHz processor and 1 GB RAM, and the software used is MATLAB 7.4 under Microsoft Windows XP 32 bit.

*z*= 300

_{k}*μ*m. This implies that

*a*(

*z*) = 0 except when

*z*=

*z*We let the spectrum

_{k}*G*(

*λ*) take a rectangular shape with wavelength range [600 nm, 1600 nm] and simulate

*S*(

*λ*) (

_{i}*i*= 1, 2...2000) using Eq. (1). We chose a broader wavelength range here just in order to make sure of

*rank*(

**A**) =

*M*. This is in favor of the conventional ART method in a sense that it would yield the exact and correct

*a*(

*z*). For the conventional ART method, we let

*M*= 500 and find the value of corresponding

*N*such that the condition

*rank*(

**A**) =

*M*is satisfied. Then we solve the inverse problem

**Ax**=

**b**using the pseudo inverse function in MATLAB and record the computational time. For the proposed method, we let

*M*= 500 and

*N*= 20. As a reference, the computational time for the conventional FD-OCT method is also evaluated. This consists of the time needed for MATLAB to spline interpolate all the simulated

*G*(

*λ*) to

*G*(

*κ*) and the time of performing the FFT algorithm.

### 3.3. Signal to Noise Ratio (SNR)

**Ax**–

**b**||

_{2}<

*ε*where

*ε*is a measure of the noise power in the measurements and the value of

*ε*is usually determined empirically. In this section, simulation studies are carried out by using Eq. (6) for the proposed method. The reconstruction error is chosen as the performance indicator, and it is examined by varying the number of measurements

*N*under different SNR conditions.

*R*= 2.5, [

_{r}*z*

_{1}= 1

*μ*m,

*z*

_{2}= 2

*μ*m, ...,

*z*= 300

_{k}*μ*m, ...,

*z*= 1000

_{M}*μ*m],

*M*= 1000, and

*a*(

*z*=

*z*

_{3}00) = 1. The spectrum is simulated over the range of wavelengths from 1473 nm to 1627 nm at 771 wavelengths. The spectrum of the light source is assumed as a Gaussian over the range. Then additive Gaussian noise with zero mean and variance of

*x*||

_{2}/ ||

*n*||

_{2}) with

*n*denoting a realization of the Gaussian noise. For each SNR value,

*N*samples are taken from the noisy frequency measurements randomly in an even distribution. Then the ℓ

_{2}norm of the difference between the reconstructed

*x*and their true values from

*a*(

*z*) is calculated and it is denoted as the reconstruction error. Figure 4 plots the reconstruction error against SNR for different number of frequency measurements used in Eq. (6). It is quite straightforward to see that reconstruction errors become bigger in the presence of a big noise level, given the number of frequency measurements. On the other hand, for a given SNR level, it shows that the reconstruction can be made more robust to noise by using more frequency measurements. And the dependence of the number of frequency measurements becomes lesser when a big enough value of

*N*is chosen.

## 4. Onion cell imaging

*μ*m is used to cover the onion sample. All glass are made from BK7. Such a packaging is to remove the reflection from the bottom surface of covering glass slide, and the top surface of the covering glass slide can be used as a reference plane which does not contribute too strong reflection.

*M*1 is removed and the collimator is replaced by a microscope objective lens (Numerical aperature=0.1). The interference is formed by the light reflected respectively from the subsurface features in the onion and the top surface of the covering glass slide. There are several reasons of using common path interferometer here. First, the reference light from the top surface of the covering glass slide is much weaker than a reference mirror, a better contrast is obtained because the non-varying component (DC) of the reflected spectrum in Eq. (1) is reduced [18

18. M. S. Muller and J. M. Fraser, “Contrast improvement in Fourier-domain optical coherence tomography through time gating,” J. Opt. Soc. Am. A **26**, 969–976 (2009). [CrossRef]

*μ*m over a range of 1 mm by 1 mm. For each axial scan at a lateral position, a set of raw frequency measurements

*S*(

*λ*) at wavelengths over a span of 154 nm centered at 1550 nm are obtained by the OSA with its spectral resolution set at 0.4 nm. The corresponding wavelengths are denoted by

*λ*= 0.4 nm and

*N*′ = 386.

*a*(

*z*) with the discretization interval of 4

*μ*m over the axial range from 0 mm to 2 mm. That is,

*z*

_{1}= 0

*μ*m,

*z*

_{2}= 4

*μ*m, ...,

*z*= 2000

_{M}*μ*m, and

*M*= 501. We measure the output spectrum

*G*(

*λ*) of the light source at the wavelengths

*N*′ frequency measurements both for

*S*(

*λ*) and for

*G*(

*λ*), 280 frequency measurements are taken at the wavelengths which are chosen randomly in an even distribution, that is

*N*=280. Using these measurements, we obtain

**A**and

**b**as in Eq. (3) and Eq. (4). Then we reconstruct each axial scan through solving the optimization problem given in Eq. (6). The image obtained by the proposed method is given in Fig. 5(a) where the vertical represents the depth but in logarithmic scale.

19. 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**, 12121–12131 (2009). [CrossRef] [PubMed]

*N*′ = 386 wavelengths given at

_{2}optimization is performed. The reconstructed cross-section images of using these method are shown respectively in Fig. 5(b), Fig. 5(c), and Fig. 5(d). To compare the imaging depth, we also put an arrow to depict the imaging depth of 1.5 mm as determined by Eq. (7) based on OSA resolution of 0.4 nm. But for all the images in Fig. 5, we crop the same top part without any meaningful data, and show only a part with depth range of 1 mm.

## 5. Conclusions

_{1}-optimization. Inheriting the merit of ART in achieving high resolution, the proposed method overcomes the bottleneck of ART method by greatly reducing its computational load. Simulation and experiment studies have demonstrated the merit performance of the proposed method. The proposed method is expected to be further develop as a fast effective method for high resolution and longer imaging depth subsurface imaging.

## References and links

1. | D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science |

2. | O. P. Bruno and J. Chaubell, “Inverse scattering problem for optical coherence tomography,” Opt. Lett. |

3. | P. E Andersen, L. Thrane, H. T Yura, A. Tycho, T. M. Jrgensen, and M. H Frosz, “Advanced modelling of optical coherence tomography systems,” Phys. Med. Biol. |

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

5. | M. E. Brezinski, |

6. | C. Dorrer, N. Belabas, J. Likforman, and M. Joffre, “Spectral resolution and sampling issues in Fourier-transform spectral interferometry,” J. Opt. Soc. Am. B |

7. | S. S. Sherif, C. Flueraru, Y. Mao, and S. Change, “Swept source optical coherence tomography with nonuniform frequency domain sampling,” in |

8. | D. Hillmann, G. Huttmann, and P. Koch, “Using Nonequispaced fast Fourier transformation to process optical coherence tomography signals,” Proc. SPIE |

9. | S. Vergnole, D. Lvesque, and G. Lamouche, “Experimental validation of an optimized signal processing method to handle non-linearity in swept-source optical coherence tomography,” Opt. Express |

10. | A. C. Kak and M. Slaney, “Algebraic reconstruction algorithms,” in |

11. | S. Vergnole, D. Levesque, G. Lamouche, M. Dufour, and B. Gauthier, “Characterization of thin layered structures using deconvolution techniques in time-domain and Fourier-domain optical coherence tomography,” Proc SPIE |

12. | S. A. Boppart, A. L. Oldenburg, C. Xu, and D. L. Marks, “Optical probes and techniques for molecular contrast enhancement in coherence imaging,” J. Biomed. Opt. |

13. | M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI,” IEEE Signal Processing Mag. |

14. | E. J. Candès and T. Tao, “Decoding by linear programming,” IEEE Trans. Inf. Theory |

15. | M. Grant and S. Boyd, CVX: Matlab software for disciplined convex programming (web page and software). http://stanford.edu/$_{\mathaccent''0365\relax{~}}$boyd/cvx (2009). |

16. | W. Qiu and E. Skafidas, “Robust estimation of GCD with sparse coefficients,” Signal Processing |

17. | G. H. Golub and C. F. Van Loan, |

18. | M. S. Muller and J. M. Fraser, “Contrast improvement in Fourier-domain optical coherence tomography through time gating,” J. Opt. Soc. Am. A |

19. | 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 |

**OCIS Codes**

(110.0110) Imaging systems : Imaging systems

(110.4500) Imaging systems : Optical coherence tomography

(350.5730) Other areas of optics : Resolution

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: April 19, 2010

Revised Manuscript: December 19, 2010

Manuscript Accepted: December 28, 2010

Published: January 12, 2011

**Virtual Issues**

Vol. 6, Iss. 2 *Virtual Journal for Biomedical Optics*

**Citation**

Hon Luen Seck, Ying Zhang, and Yeng Chai Soh, "Optical coherence tomography by using frequency measurements in wavelength
domain," Opt. Express **19**, 1324-1334 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-2-1324

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### References

- D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991). [CrossRef] [PubMed]
- O. P. Bruno, and J. Chaubell, “Inverse scattering problem for optical coherence tomography,” Opt. Lett. 28, 2049–2051 (2003). [CrossRef] [PubMed]
- P. E. Andersen, L. Thrane, H. T. Yura, A. Tycho, T. M. Jørgensen, and M. H. Frosz, “Advanced modelling of optical coherence tomography systems,” Phys. Med. Biol. 49, 1307–1327 (2004). [CrossRef] [PubMed]
- G. Häusler, and M. W. Lindner, “‘Coherence radar’ and ‘spectral radar’–new tools for dermatological diagnosis,” J. Biomed. Opt. 3, 21–31 (1998). [CrossRef]
- M. E. Brezinski, Optical Coherence Tomography: Principles and Applications (Academic Press,2006).
- C. Dorrer, N. Belabas, J. Likforman, and M. Joffre, “Spectral resolution and sampling issues in Fourier-transform spectral interferometry,” J. Opt. Soc. Am. B 17, 1795–1802 (2000). [CrossRef]
- S. S. Sherif, C. Flueraru, Y. Mao, and S. Change, “Swept source optical coherence tomography with nonuniform frequency domain sampling,” in Biomedical Optics, OSA Technical Digest (CD) (Optical Society of America, 2008), paper BMD86.
- D. Hillmann, G. Huttmann, and P. Koch, “Using nonequispaced fast Fourier transformation to process optical coherence tomography signals,” Proc. of SPIE-OSA Biomedical Optics 7372, 73730 (2009).
- S. Vergnole, D. Lvesque, and G. Lamouche, “Experimental validation of an optimized signal processing method to handle non-linearity in swept-source optical coherence tomography,” Opt. Express 18, 10446–10461 (2010). [CrossRef] [PubMed]
- A. C. Kak, and M. Slaney, “Algebraic reconstruction algorithms,” in Principles of Computerized Tomographic Imaging (IEEE Press, 1999).
- S. Vergnole, D. Levesque, G. Lamouche, M. Dufour, and B. Gauthier, “Characterization of thin layered structures using deconvolution techniques in time-domain and Fourier-domain optical coherence tomography,” Proc. SPIE 6796, 67961 (2007). [CrossRef]
- S. A. Boppart, A. L. Oldenburg, C. Xu, and D. L. Marks, “Optical probes and techniques for molecular contrast enhancement in coherence imaging,” J. Biomed. Opt. 10, 041208 (2005). [CrossRef]
- M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI,” IEEE Signal Process. Mag. 25, 72–82 (2008). [CrossRef]
- E. J. Candès, and T. Tao, “Decoding by linear programming,” IEEE Trans. Inf. Theory 51, 4203–4215 (2005). [CrossRef]
- M. Grant, and S. Boyd, CVX: Matlab software for disciplined convex programming (web page and software). http://stanford.edu/eboyd/cvx (2009).
- W. Qiu, and E. Skafidas, “Robust estimation of GCD with sparse coefficients,” Signal Process. 90, 972–976 (2010). [CrossRef]
- G. H. Golub, and C. F. Van Loan, Matrix computations (Johns Hopkins University Press, 1984).
- M. S. Muller, and J. M. Fraser, “Contrast improvement in Fourier-domain optical coherence tomography through time gating,” J. Opt. Soc. Am. A 26, 969–976 (2009). [CrossRef]
- 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, 12121–12131 (2009). [CrossRef] [PubMed]

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