## Compressive SD-OCT: the application of compressed sensing in spectral domain optical coherence tomography |

Optics Express, Vol. 18, Issue 21, pp. 22010-22019 (2010)

http://dx.doi.org/10.1364/OE.18.022010

Acrobat PDF (1191 KB)

### Abstract

We applied compressed sensing (CS) to spectral domain optical coherence tomography
(SD OCT) and studied its effectiveness. We tested the CS reconstruction by randomly
undersampling the k-space SD OCT signal. We achieved this by applying pseudo-random
masks to sample 62.5%, 50%, and 37.5% of the CCD camera pixels. OCT images are
reconstructed by solving an optimization problem that minimizes the
*l*_{1} norm of a transformed image to enforce sparsity, subject to data
consistency constraints. CS could allow an array detector with fewer pixels to
reconstruct high resolution OCT images while reducing the total amount of data
required to process the images.

© 2010 OSA

## 1. Introduction

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

7. M. Choma, M. Sarunic, C. Yang, and J. Izatt, “Sensitivity advantage of swept source and
Fourier domain optical coherence tomography,” Opt.
Express **11**(18),
2183–2189 (2003), http://www.opticsinfobase.org/abstract.cfm?URI=oe-11-18-2183. [PubMed]

4. K. Zhang and J. U. Kang, “Real-time 4D signal processing and
visualization using graphics processing unit on a regular nonlinear-k
Fourier-domain OCT system,” Opt. Express **18**(11),
11772–11784 (2010), http://www.opticsinfobase.org/abstract.cfm?URI=oe-18-11-11772. [PubMed]

9. I. Grulkowski, M. Gora, M. Szkulmowski, I. Gorczynska, D. Szlag, S. Marcos, A. Kowalczyk, and M. Wojtkowski, “Anterior segment imaging with Spectral OCT
system using a high-speed CMOS camera,” Opt.
Express **17**(6),
4842–4858 (2009), http://www.opticsinfobase.org/abstract.cfm?URI=oe-17-6-4842. [PubMed]

11. K. Zhang, W. Wang, J. Han, and J. U. Kang, “A surface topology and motion compensation
system for microsurgery guidance and intervention based on common-path optical
coherence tomography,” IEEE Trans. Biomed.
Eng. **56**(9),
2318–2321 (2009). [PubMed]

14. M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application of compressed
sensing for rapid MR imaging,” Magn. Reson.
Med. **58**(6),
1182–1195 (2007). [PubMed]

*l*

_{1}norm of a transformed image to enforce sparsity, subject to data consistency constraints. To the best of our knowledge, this is the first time that CS has been applied to process an experimentally obtained OCT signal.

## 2. Common path SD OCT and conventional SD OCT signal processing

17. M. Mujat, B. H. Park, B. Cense, T. C. Chen, and J. F. de Boer, “Autocalibration of spectral-domain optical
coherence tomography spectrometers for in vivo quantitative retinal nerve fiber
layer birefringence determination,” J. Biomed.
Opt. **12**(4), 041205
(2007). [PubMed]

## 3. Compressed sensing for SD OCT: theory and image reconstruction algorithm

**x**can lead to accurate object reconstruction [12]. For OCT, the linear combinations are simply Fourier coefficients detected by the spectrometer (or k-space samples). Therefore, CS takes a small random subset of k-space data:In Eq. (5),

**F**indicates the matrix operator for incomplete Fourier measurement.

_{u}**x**has to have a sparse representation in a known transform domain,

**Φ**, where the image has only a few non-zero coefficients. Moreover, the k-space undersampling has to lead to incoherent interference in

**Φ**, i.e., the aliasing artifacts due to undersampling have to be noise-like instead of structural. Finally, CS uses a non-linear algorithm to reconstruct the image instead of a linear reconstruction.

18. Z. Jian, Z. Yu, L. Yu, B. Rao, Z. Chen, and B. J. Tromberg, “Speckle attenuation in optical coherence
tomography by curvelet shrinkage,” Opt.
Lett. **34**(10),
1516–1518 (2009). [PubMed]

**x**which has

**elements and**

*N***non-zero coefficients in the representation domain**

*T***Φ, x**can be reconstructed exactly with probability of at least

*1−O(*

*N**with*

^{-δ})**measurements in Fourier domain, and**

*K***has to satisfy the following inequality [12,13,16].In Eq. (6),**

*K**G*is a small quantity. According to experimental results, a

_{δ}**value that is 2–5 times**

*K***can offer satisfying signal through CS reconstruction, i.e.,**

*T***can be much smaller than required by Eq. (6) in practice [14**

*K*14. M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application of compressed
sensing for rapid MR imaging,” Magn. Reson.
Med. **58**(6),
1182–1195 (2007). [PubMed]

**Ψ**and representation domain

**Φ**has been elaboratively studied and is mathematically evaluated by the transform point-spread function (TPSF), defined as Eq. (7) [14

14. M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application of compressed
sensing for rapid MR imaging,” Magn. Reson.
Med. **58**(6),
1182–1195 (2007). [PubMed]

**e**denotes the i

_{i}^{th}vector in

**Φ**(i.e., having ‘1’ at the i

^{th}location and zeroes elsewhere), and

**e**the j

_{j}^{th}vector.

**W**is the sparsifying operator that transforms

**x**to the representation domain

**Φ**, such as a wavelet transformation operator.

**W**can also be an identity matrix

**I**if the image is sparse in pixel representation; in that case, TPSF becomes PSF. Please note that this PSF is fundamentally different from the PSF used to characterize the resolution of an imaging system.

**x**from undersampled measurements, as shown in Eq. (5), we cannot simply apply the inverse of

**F**to both sides of Eq. (5), because, due to undersampling,

_{u}**F**is usually ill-conditioned and the inverse of

_{u}**F**does not exist. Instead, CS recovers

_{u}**x**by solving the following constrained optimization problem:In Eq. (8), ε controls the data consistency; the objective function of this optimization problem is the

*l*

_{1}norm of the image in

**Φ**and the

*l*

_{1}norm is defined as ||α||

_{1}= Σ

_{i}|α

_{i}|. Minimizing the

*l*

_{1}norm of the image essentially promotes sparsity. Various methods that solve Eq. (8) have been developed; in this study we solve Eq. (8) iteratively, using non-linear conjugate gradients (CG) and backtracking line-search [19]. We re-write Eq. (8) as an unconstrained Lagrangian form in Eq. (9):To find out the solution of this optimization problem, we searched the space where

**x**is defined. In each iteration, the searching direction is calculated based on the gradient of the cost function f(

**x**), and is further adjusted using the Fletcher-Reeves (FR) formula. The searching step size is obtained by backtracking line search. More detailed description of this algorithm can be found in [14

**58**(6),
1182–1195 (2007). [PubMed]

## 4. Result

### 4.1 Sampling scheme and simulated PSF

### 4.2 Evaluation of CS SD OCT by imaging a mirror

**3**to the fully sampled and undersampled spectral data and show the results as M-mode images in Fig. 5 . Each A-scan in Fig. 5(a) has a single peak corresponding to the mirror surface. However, Figs. 5(b) and 5(c) exhibit strong undersampling artifacts. Figure 5(b) shows structural, coherent aliasing due to uniform density undersampling, while the random sampling results in noise-like interference in Fig. 5(c), consistent with the simulated result in Fig. 3. It is worth mentioning that Fig. 5(b) does not show undersampling aliasing as superposition of shifted replicas of the signal, which should be the case when k-space is undersampled uniformly, zero-filled, and processed by inverse Fourier transformation. The reason is that we do not directly sample the k-space uniformly, but sample the pixels of CCD uniformly.

_{10}[max(

**x**)

^{2}/

*var*],

*var*is the noise variance) compared to Fig. 5(a). We calculated the SNR of the first A-scan in Fig. 5(a) and Fig. 6(d) to be 42dB and 54dB, respectively. The improved SNR is due to the non-linear image recovery process which is inherently a denoising procedure [22].

### 4.3 Evaluation of CS SD OCT by imaging onion cells

**W**=

**I,**using 62.5%, 50%, and 37.5% of the spectral data. In addition, we sparsified the images by Symlets4 wavelet transformation and pursued sparsity in the wavelet domain. Figures 7(e) to 7(g) are the resultant images corresponding to 62.5%, 50%, and 37.5% sampling, respectively. All the figures reconstructed by CS show good consistency with the ground truth image Fig. 7(a), because both pixel representation and wavelet representation of the image is sparse.

**58**(6),
1182–1195 (2007). [PubMed]

11. K. Zhang, W. Wang, J. Han, and J. U. Kang, “A surface topology and motion compensation
system for microsurgery guidance and intervention based on common-path optical
coherence tomography,” IEEE Trans. Biomed.
Eng. **56**(9),
2318–2321 (2009). [PubMed]

**P**

_{CS}and

**P**

_{GT}, for CS images and ground truth image, respectively. We show

**P**

_{GT}versus lateral position in Fig. 8(a) . We also calculated Δ

**P**, the difference between

**P**

_{CS}and

**P**

_{GT}, and show the histograms of Δ

**P**corresponding to different sampling rates in Fig. 8(b). Seen from Fig. 8(b), most of the A-scans in CS OCT image allow us to extract the sample surface exactly the same as using ground truth OCT image and none of the A-scans lead to a Δ

**P**with an absolute value larger than 2. Results in Fig. 8(b) show that CS SD OCT can lead to reliable profiling of the sample surface.

## 5. Discussion

18. Z. Jian, Z. Yu, L. Yu, B. Rao, Z. Chen, and B. J. Tromberg, “Speckle attenuation in optical coherence
tomography by curvelet shrinkage,” Opt.
Lett. **34**(10),
1516–1518 (2009). [PubMed]

**Y**, which equals

_{u}**F**), and compared with the originally undersampled spectrum,

_{u}x**y**. The difference between

_{u}**Y**and

_{u}**y**,

_{u}**Δy = ||F**has to be small enough for us to stop the iterations. A large Δα may not induce a significant change in the signal amplitude and thus the appearance of the image; however it will result in

_{u}x- y_{u}||_{2},**Y**to be a shifted version of

_{u}**y**, which leads to a large

_{u}**Δy**and therefore violates the requirement of data consistency. Based on the above discussion, we come to a conclusion that CS OCT inherently preserve phase coherence.

## 6. Conclusion

## Acknowledgement

## References and links

1. | M. Brezinski, |

2. | B. E. Bouma, and G. J. Tearney, |

3. | U. Sharma, N. M. Fried, and J. U. Kang, “All-fiber Fizeau optical coherence
tomography: sensitivity optimization and system analysis,”
IEEE J. Sel. Top. Quantum Electron. |

4. | K. Zhang and J. U. Kang, “Real-time 4D signal processing and
visualization using graphics processing unit on a regular nonlinear-k
Fourier-domain OCT system,” Opt. Express |

5. | J. U. Kang, J. Han, X. Liu, K. Zhang, C. Song, and P. Gehlbach, “Endoscopic Functional Fourier Domain Common
Path Optical Coherence Tomography for Microsurgery,”
IEEE J. Sel. Top. Quantum Electron. |

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

7. | M. Choma, M. Sarunic, C. Yang, and J. Izatt, “Sensitivity advantage of swept source and
Fourier domain optical coherence tomography,” Opt.
Express |

8. | X. Liu, X. Li, D. Kim, I. Ilev, and J. U. Kang, “Fiber Optic Fourier-domain Common-path
OCT,” Chin. Opt. Lett. |

9. | I. Grulkowski, M. Gora, M. Szkulmowski, I. Gorczynska, D. Szlag, S. Marcos, A. Kowalczyk, and M. Wojtkowski, “Anterior segment imaging with Spectral OCT
system using a high-speed CMOS camera,” Opt.
Express |

10. | M. Balicki, J. Han, I. Iordachita, P. Gehlbach, J. Handa, J. U. Kang, and R. Taylor, “Single Fiber Optical Coherence Tomography Microsurgical Instruments for Computer and Robot-Assisted Retinal Surgery,” Proceedings of the MICCAI Conference (2009) |

11. | K. Zhang, W. Wang, J. Han, and J. U. Kang, “A surface topology and motion compensation
system for microsurgery guidance and intervention based on common-path optical
coherence tomography,” IEEE Trans. Biomed.
Eng. |

12. | D. L. Donoho, “Compressed Sensing,”
IEEE Trans. Inf. Theory |

13. | E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal
reconstruction from highly incomplete frequency
information,” IEEE Trans. Inf. Theory |

14. | M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application of compressed
sensing for rapid MR imaging,” Magn. Reson.
Med. |

15. | Z. Guo, C. Li, L. Song, and L. V. Wang, “Compressed sensing in photoacoustic
tomography in vivo,” J. Biomed. Opt. |

16. | N. Mohan, I. Stojanovic, W. C. Karl, B. E. A. Saleh, and M. C. Teich, “Compressed sensing in optical coherence
tomography,” Proc. SPIE |

17. | M. Mujat, B. H. Park, B. Cense, T. C. Chen, and J. F. de Boer, “Autocalibration of spectral-domain optical
coherence tomography spectrometers for in vivo quantitative retinal nerve fiber
layer birefringence determination,” J. Biomed.
Opt. |

18. | Z. Jian, Z. Yu, L. Yu, B. Rao, Z. Chen, and B. J. Tromberg, “Speckle attenuation in optical coherence
tomography by curvelet shrinkage,” Opt.
Lett. |

19. | J. Shewchuk, “An introduction to the conjugate gradient method without the agonizing pain,” Technical Report CMUCS-TR-94–125, Carnegie Mellon University, (1994). |

20. | S. M. Potter, A. Mart, and J. Pine, “High-speed CCD movie camera with random
pixel selection for neurobiology research,” Proc.
SPIE |

21. | S. P. Monacos, R. K. Lam, A. A. Portillo, and G. G. Ortiz, “Design of an event-driven
random-access-windowing CCD-based camera,” Proc.
SPIE |

22. | D. Donoho and I. Johnstone, “Ideal spatial adaptation via wavelet
shrinkage,” Biometrika |

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

**OCIS Codes**

(100.3010) Image processing : Image reconstruction techniques

(170.4500) Medical optics and biotechnology : Optical coherence tomography

(110.3055) Imaging systems : Information theoretical analysis

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: August 9, 2010

Revised Manuscript: September 10, 2010

Manuscript Accepted: September 20, 2010

Published: October 1, 2010

**Virtual Issues**

Vol. 5, Iss. 14 *Virtual Journal for Biomedical Optics*

**Citation**

Xuan Liu and Jin U. Kang, "Compressive SD-OCT: the application of compressed sensing in spectral domain optical coherence tomography," Opt. Express **18**, 22010-22019 (2010)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-18-21-22010

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### References

- M. Brezinski, Optical Coherence Tomography: Principles and Applications, (Academic Press, London, 2006).
- B. E. Bouma, and G. J. Tearney, Handbook of Optical Coherence Tomography, (Informa Healthcare, New York, 2001).
- U. Sharma, N. M. Fried, and J. U. Kang, “All-fiber Fizeau optical coherence tomography: sensitivity optimization and system analysis,” IEEE J. Sel. Top. Quantum Electron. 11(4), 799–805 (2005).
- K. Zhang and J. U. Kang, “Real-time 4D signal processing and visualization using graphics processing unit on a regular nonlinear-k Fourier-domain OCT system,” Opt. Express 18(11), 11772–11784 (2010), http://www.opticsinfobase.org/abstract.cfm?URI=oe-18-11-11772 . [PubMed]
- J. U. Kang, J. Han, X. Liu, K. Zhang, C. Song, and P. Gehlbach, “Endoscopic Functional Fourier Domain Common Path Optical Coherence Tomography for Microsurgery,” IEEE J. Sel. Top. Quantum Electron. 16(4), 781–792 (2010).
- R. Leitgeb, C. Hitzenberger, and A. Fercher, “Performance of fourier domain vs. time domain optical coherence tomography,” Opt. Express 11(8), 889–894 (2003), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-8-889 . [PubMed]
- M. Choma, M. Sarunic, C. Yang, and J. Izatt, “Sensitivity advantage of swept source and Fourier domain optical coherence tomography,” Opt. Express 11(18), 2183–2189 (2003), http://www.opticsinfobase.org/abstract.cfm?URI=oe-11-18-2183 . [PubMed]
- X. Liu, X. Li, D. Kim, I. Ilev, and J. U. Kang, “Fiber Optic Fourier-domain Common-path OCT,” Chin. Opt. Lett. 6(12), 899–903 (2008).
- I. Grulkowski, M. Gora, M. Szkulmowski, I. Gorczynska, D. Szlag, S. Marcos, A. Kowalczyk, and M. Wojtkowski, “Anterior segment imaging with Spectral OCT system using a high-speed CMOS camera,” Opt. Express 17(6), 4842–4858 (2009), http://www.opticsinfobase.org/abstract.cfm?URI=oe-17-6-4842 . [PubMed]
- M. Balicki, J. Han, I. Iordachita, P. Gehlbach, J. Handa, J. U. Kang, and R. Taylor, “Single Fiber Optical Coherence Tomography Microsurgical Instruments for Computer and Robot-Assisted Retinal Surgery,” Proceedings of the MICCAI Conference (2009)
- K. Zhang, W. Wang, J. Han, and J. U. Kang, “A surface topology and motion compensation system for microsurgery guidance and intervention based on common-path optical coherence tomography,” IEEE Trans. Biomed. Eng. 56(9), 2318–2321 (2009). [PubMed]
- D. L. Donoho, “Compressed Sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006).
- E. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
- M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: The application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58(6), 1182–1195 (2007). [PubMed]
- Z. Guo, C. Li, L. Song, and L. V. Wang, “Compressed sensing in photoacoustic tomography in vivo,” J. Biomed. Opt. 15(2), 021311 (2010). [PubMed]
- N. Mohan, I. Stojanovic, W. C. Karl, B. E. A. Saleh, and M. C. Teich, “Compressed sensing in optical coherence tomography,” Proc. SPIE 7570, 75700L (2010).
- M. Mujat, B. H. Park, B. Cense, T. C. Chen, and J. F. de Boer, “Autocalibration of spectral-domain optical coherence tomography spectrometers for in vivo quantitative retinal nerve fiber layer birefringence determination,” J. Biomed. Opt. 12(4), 041205 (2007). [PubMed]
- Z. Jian, Z. Yu, L. Yu, B. Rao, Z. Chen, and B. J. Tromberg, “Speckle attenuation in optical coherence tomography by curvelet shrinkage,” Opt. Lett. 34(10), 1516–1518 (2009). [PubMed]
- J. Shewchuk, “An introduction to the conjugate gradient method without the agonizing pain,” Technical Report CMUCS-TR-94–125, Carnegie Mellon University, (1994).
- S. M. Potter, A. Mart, and J. Pine, “High-speed CCD movie camera with random pixel selection for neurobiology research,” Proc. SPIE 2869, 243–253 (1997).
- S. P. Monacos, R. K. Lam, A. A. Portillo, and G. G. Ortiz, “Design of an event-driven random-access-windowing CCD-based camera,” Proc. SPIE 4975, 115 (2003).
- D. Donoho and I. Johnstone, “Ideal spatial adaptation via wavelet shrinkage,” Biometrika 81(3), 425–455 (1994).
- J. A. Fessler and B. P. Sutton, “Nonuniform fast Fourier transforms using min–max interpolation,” IEEE Trans. Signal Process. 51(2), 560–574 (2003).

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