## Homotopic, non-local sparse reconstruction of optical coherence tomography imagery |

Optics Express, Vol. 20, Issue 9, pp. 10200-10211 (2012)

http://dx.doi.org/10.1364/OE.20.010200

Acrobat PDF (3147 KB)

### Abstract

The resolution in optical coherence tomography imaging is an important parameter which determines the size of the smallest features that can be visualized. Sparse sampling approaches have shown considerable promise in producing high resolution OCT images with fewer camera pixels, reducing both the cost and the complexity of an imaging system. In this paper, we propose a non-local approach to the reconstruction of high resolution OCT images from sparsely sampled measurements. An iterative strategy is introduced for minimizing a homotopic, non-local regularized functional in the spatial domain, subject to data fidelity constraints in the *k*-space domain. The novel algorithm was tested on human retinal, corneal, and limbus images, acquired in-vivo, demonstrating the effectiveness of the proposed approach in generating high resolution reconstructions from a limited number of camera pixels.

© 2012 OSA

## 1. Introduction

1. S. A. Boppart, “Optical coherence tomography: technology and applications for neuroimaging,” Psychophysiology **40**, 529–541 (2003). [CrossRef] [PubMed]

2. 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]

5. W. Drexler and J. G. Fujimoto, *Optical Coherence Tomography* (Springer, 2008). [CrossRef]

6. M. Wojtkowski, A. Kowalczyk, R. Leitgeb, and A. F. Fercher, “Full range complex spectral optical coherence tomography technique in eye imaging,” Opt. Lett. **27**, 1415–1417 (2002). [CrossRef]

7. R. Leitgeb, W. Drexler, A. Unterhuber, B. Hermann, T. Bajraszewski, T. Le, A. Stingl, and A. Fercher, “Ultrahigh resolution Fourier domain optical coherence tomography,” Opt. Express **12**, 2156–2165 (2004). [CrossRef] [PubMed]

8. R. Leitgeb, C. Hitzenberger, and A. Fercher, “Performance of Fourier domain vs. time domain optical coherence tomography,” Opt. Express **11**, 889–894 (2003). [CrossRef] [PubMed]

9. A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography—principles and applications,” Rep. Prog. Phys. **66**, 239–303 (2003). [CrossRef]

10. E. Candës, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory **52**, 489–509 (2006). [CrossRef]

11. D. Donoho, “Compressive sensing,” IEEE Trans. Inf. Theory **52**, 1289–1306 (2006). [CrossRef]

*L*

_{1}based minimization method [12

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

*L*

_{0}minimization approach [13

13. J. Trzasko and A. Manduca, “Highly undersampled magnetic resonance image reconstruction via homotopic l0-minimization,” IEEE Trans. Med. Imag. **28**, 106–121 (2009). [CrossRef]

*L*

_{1}for spine and wrist MRI images, however the effectiveness of this approach may be limited, given its use of the finite differential transform, which is poorly suited for handling the fine details and variations found in other MRI images [14]. Liang et al. [15] explored the use of the non-local total variation regularization framework for MRI reconstruction, where regional characteristics between non-local sites is used as a penalty constraint, and found that such an approach allowed for improved handling of fine details and variations. More recently, the combination of a regional differential sparsifying transform and a homotopic

*L*

_{0}framework was proposed by Wong et al. [14], which was able to lower computational complexity while still maintaining structural fidelity.

16. 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). [CrossRef]

17. X. Liu and J. U. Kang, “Compressive SD-OCT: the application of compressed sensing in spectral domain optical coherence tomography,” Opt. Express **18**, 22010–22019 (2010). [CrossRef] [PubMed]

*L*

_{1}minimization methods to reconstruct OCT imagery with highly undersampled

*k*-space data. However, these early works have been evaluated using either simulated signals or objects such as onions, which do not necessarily reflect the characteristics of living tissue. One of the key applications of ophthalmic OCT is the imaging of retinal, corneal, and limbus tissues, which are characterized by complex structural characteristics that are important for clinical observation and diagnosis. Given the complexities of the human retina and limbus, the classical

*L*

_{1}based minimization approach may not provide an accurate reconstruction given sparser sampling, and may lead to blurring and loss of fine structure. As such, the integration of a non-local strategy for sparse OCT reconstruction is worth investigating for improved image quality.

*k*-space domain. This novel non-local sparse reconstruction is specifically applied to human retinal, corneal, and limbus OCT data, with results demonstrating that high-quality, high resolution OCT images can be reconstructed using the proposed method.

## 2. Methods

*L*

_{0}minimization framework for reconstructing OCT images from sparsely-sampled measurements acquired using a spectral domain OCT (SD-OCT) imaging system.

*k*-space. Denoting the sample as

*f*(

*x*), and the measurements in the

*k*-space domain as

*F*(

*k*), the relationship between

*f*(

*x*) and

*F*(

*k*) is formulated as where 𝔽

^{−1}is the inverse Fourier operator.

17. X. Liu and J. U. Kang, “Compressive SD-OCT: the application of compressed sensing in spectral domain optical coherence tomography,” Opt. Express **18**, 22010–22019 (2010). [CrossRef] [PubMed]

*K*in the

*k*-space domain should obey where

*x*is the maximum imaging depth, and Δ

_{max}*k*is the spectral bandwidth, which is inversely proportional to spatial resolution. A large value for

*K*is necessary to achieve a high imaging resolution and large imaging depth, which means that a large total number of CCD camera pixels is required. As such, strategies are desired that obtain high resolution OCT imagery without requiring a large number of camera pixels.

### 2.1. L_{p} minimization for sparse reconstruction

10. E. Candës, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory **52**, 489–509 (2006). [CrossRef]

11. D. Donoho, “Compressive sensing,” IEEE Trans. Inf. Theory **52**, 1289–1306 (2006). [CrossRef]

*f*(

*x*) can be obtained by maximizing the sparsity of the signal in the transformed domain and enforcing data fidelity in the

*k*-space domain. This can be formulated as a constrained

*L*

_{0}minimization problem, Two commonly used transformation operators are the finite differential transformation and the wavelet transformation [12

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

*L*

_{0}problem is essentially NP hard, intractable in practice [20

20. B. K. Natarajan, “Sparse approximate solutions to linear systems,” SIAM J. Comput. **24**, 227–234 (1995). [CrossRef]

10. E. Candës, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory **52**, 489–509 (2006). [CrossRef]

11. D. Donoho, “Compressive sensing,” IEEE Trans. Inf. Theory **52**, 1289–1306 (2006). [CrossRef]

*L*

_{0}and

*L*

_{1}minimization lead to the same solutions. Theoretically, solving the

*L*

_{1}problem can get exactly the same solution as solving the

*L*

_{0}problem, at the cost of a substantial increase in the number of measurements required [13

13. J. Trzasko and A. Manduca, “Highly undersampled magnetic resonance image reconstruction via homotopic l0-minimization,” IEEE Trans. Med. Imag. **28**, 106–121 (2009). [CrossRef]

*L*

_{1}minimization framework, which is known to have an edge-preservation effect: Both Mohan et al. [16

16. 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). [CrossRef]

17. X. Liu and J. U. Kang, “Compressive SD-OCT: the application of compressed sensing in spectral domain optical coherence tomography,” Opt. Express **18**, 22010–22019 (2010). [CrossRef] [PubMed]

*L*

_{1}minimization framework, which represents the state-of-the-art in sparse OCT reconstruction, and their results show that OCT imagery can be reconstructed in a meaningful manner using sparsely sampled measurements in

*k*-space.

### 2.2. Homotopic, non-local regularized reconstruction

*L*

_{1}problem can be noticeably higher than that for the

*L*

_{0}problem, therefore requiring a greater number of camera pixels in a SD-OCT system. Hence, an alternative reconstruction strategy that addresses these two problems is desired.

*L*

_{0}minimization framework, as it is more is well-suited for handling fine structural details. The proposed homotopic, non-local regularization framework is formulated as: where

*η*denotes the homotopic non-local regularization functional, and

*σ*controls the approximation degree: where Ω denotes the set of all pixels in the image,

*𝒩*(

*x*) denote the neighborhood search space for pixel

*x*,

**N**(

*x*) and

**N**(

*y*) denotes the neighborhoods similarity space around

*x*and

*y*,

*σ*controls the decay of the exponential function, and

*w*(

*x,y,σ*) is defined as To reduce the computation complexity, the neighborhood search space is limited to a search window around the pixel to be estimated, so that the regularization by

*η*(|Ψ

*f*(

*x*)|,

*σ*) efficiently suppresses erroneous artifacts without destroying the structure of the image. At the same time, we use a data fidelity term to ensure that the reconstructed image complies with the original signal while accounting for some level of noise.

### 2.3. Implementation

*k*-space domain to correct estimation errors. This two-step process is iteratively repeated, with the approximation degree

*σ*used to control the homotopic non-local regularization functional

*η*decreasing as the number of iterations increases, until convergence. Empirically, all experimental applications of this proposed method converged successfully. The pseudocode for the proposed homotopic, non-local sparse reconstruction algorithm is shown in Algorithm 1.

## 3. Results and discussions

22. P. Puvanathasan, P. Forbes, Z. Ren, D. Malchow, S. Boyd, and K. Bizheva, “High-speed, high-resolution Fourier-domain optical coherence tomography system for retinal imaging in the 1060 nm wavelength region,” Opt. Lett. **33**, 2479–2481 (2008). [PubMed]

*λ*-c = 1020nm,

*δλ*= 110nm,

*P*= 10mW) and a 47kHz InGaAs linear array, 1024 pixel camera (SUI, Goodrich) interfaced with a high performance spectrometer. The SD-OCT system provides 3

_{out}*μ*m axial and 15

*μ*m lateral resolution in the human cornea and limbus, and 6

*μ*m axial resolution in the human retina, and 100dB SNR for 1.3mW of optical power incident on the sample. The OCT images were acquired from healthy subjects using an imaging procedure carried out in accordance with University of Waterloo research ethics regulations, at 1000 A-scans. To reconstruct OCT imagery from only a percentage of camera pixels, SD-OCT spectral data is randomly sampled from the camera array using a uniformly distributed pseudo-random mask. The obtained spectral data was then used to populate the

*k*-space grid according to the known functional dependency of wavenumber on pixel index [17

**18**, 22010–22019 (2010). [CrossRef] [PubMed]

*II*X3 3.10 GHz machine with 3.25GB of RAM. For comparison purposes, the proposed method was compared with two other sparse reconstruction techniques: the baseline

*L*

_{2}minimization approach described in Section 2.1, and the

*L*

_{1}minimization approach with total variation, as proposed by Mohan et al. [16

16. 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). [CrossRef]

**18**, 22010–22019 (2010). [CrossRef] [PubMed]

*f*(

*x*) is original image,

*f*̂(

*x*) is reconstructed image, and

*N*is the number of pixels in each image.

*𝒩*) to 7 × 7, and the size of a neighborhood (

**N**) to 3×3. If

*N*is the number of pixels of the image, then the final computational complexity of the algorithm is 9 × 49 ×

*N*.

*L*

_{2}method leads to considerable blur and artifacts, making it difficult to see any of the underlying structure and detail in the reconstructed retinal image except for the very highly reflective (black) lines corresponding to the inner and outer photoreceptor junctions and the retinal pigmented epithelium. The rest of the retinal layers, along with the inner retina vasculature, cannot be vizualized because of the algorithm induced blur. The

*L*

_{1}method results in noticeably better image quality as compared to that produced using the

*L*

_{2}method, although the contrast of the individual retinal layers is not as good as in the original image. The proposed novel NLR algorithm results in overall higher contrast of the individual retinal layers as compared to

*L*

_{1}, with features such as the thin, highly reflective retinal layers and the cross-sections of the retinal capillaries appearing sharp and are distinctly visible (Fig. 2), appearing similar to the reconstructed retinal image from 100% of the acquired samples.

*L*

_{1}method results in an image where most of the layers and some of the keratocytes are still visible, however, the overall contrast of the image is drastically lower as compared to the image reconstructed from 100% of the acquired samples. The NLR approach result in significantly better reconstruction of the corneal morphological details, as well as higher image contrast as compared to the

*L*

_{2}and

*L*

_{1}methods. Once again, the image reconstructed using the NLR approach is closer to the image reconstructed from 100% of the acquired samples.

*L*

_{2}method generates artifacts and results in overall poor contrast and visibility of the tissue morphology. The

*L*

_{1}and the NLR approaches generate images of significantly higher quality than

*L*

_{2}with fairly good contrast and preservation of the morphological details. As expected, the overall image contrast of the image reconstructed using the NLR method is better than that produced using the

*L*

_{1}method, and closer to the quality of the image reconstructed from 100% of the acquired samples.

## 4. Conclusions

*L*

_{2}and

*L*

_{1}total variation reconstruction methods. Results show that the proposed approach is able to achieve a significantly higher signal-to-noise ratio and visual quality using a lower number of camera pixels, thus illustrating the potential for obtaining high resolution images with lower equipment costs and reduced imaging times. In future work, a comprehensive analysis of different sparsifying transforms (wavelets, curvelets, etc.) will be investigated on different types of OCT imagery to identify the optimal transform for reconstructing OCT imagery from sparse measurements.

## Acknowledgments

## References and links

1. | S. A. Boppart, “Optical coherence tomography: technology and applications for neuroimaging,” Psychophysiology |

2. | 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 |

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

4. | M. Wojtkowski, V. Srinivasan, J. G. Fujimoto, T. Ko, J. S. Schuman, A. Kowalczyk, and J. S. Duker, “Three-dimensional retinal imaging with high-speed ultrahigh-resolution optical coherence tomography,” Ophthalmology |

5. | W. Drexler and J. G. Fujimoto, |

6. | M. Wojtkowski, A. Kowalczyk, R. Leitgeb, and A. F. Fercher, “Full range complex spectral optical coherence tomography technique in eye imaging,” Opt. Lett. |

7. | R. Leitgeb, W. Drexler, A. Unterhuber, B. Hermann, T. Bajraszewski, T. Le, A. Stingl, and A. Fercher, “Ultrahigh resolution Fourier domain optical coherence tomography,” Opt. Express |

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

9. | A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography—principles and applications,” Rep. Prog. Phys. |

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

11. | D. Donoho, “Compressive sensing,” IEEE Trans. Inf. Theory |

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

13. | J. Trzasko and A. Manduca, “Highly undersampled magnetic resonance image reconstruction via homotopic l0-minimization,” IEEE Trans. Med. Imag. |

14. | A. Wong, A. Mishra, D. Clausi, and P. Fieguth, “Sparse reconstruction of breast MRI using homotopic L0 minimization in a regional sparsified domain,” |

15. | D. Liang, H. Wang, and L. Ying, “SENSE reconstruction with nonlocal TV regularization,” |

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. | X. Liu and J. U. Kang, “Compressive SD-OCT: the application of compressed sensing in spectral domain optical coherence tomography,” Opt. Express |

18. | G. Gilboa and S. Osher, “Nonlocal operators with applications to image processing,” Tech. Rep. CAM Report 07-23, Univ. California, Los Angeles, 2007. |

19. | P. Fieguth, |

20. | B. K. Natarajan, “Sparse approximate solutions to linear systems,” SIAM J. Comput. |

21. | W. Guo and F. Huang, “Adaptive total variation based filtering for MRI images with spatially inhomogeneous noise and artifacts,” Int. Sym. Biomed Imag , 101–104 (2009). |

22. | P. Puvanathasan, P. Forbes, Z. Ren, D. Malchow, S. Boyd, and K. Bizheva, “High-speed, high-resolution Fourier-domain optical coherence tomography system for retinal imaging in the 1060 nm wavelength region,” Opt. Lett. |

**OCIS Codes**

(100.0100) Image processing : Image processing

(100.3010) Image processing : Image reconstruction techniques

(110.4500) Imaging systems : Optical coherence tomography

(170.4470) Medical optics and biotechnology : Ophthalmology

**ToC Category:**

Image Processing

**History**

Original Manuscript: March 1, 2012

Revised Manuscript: April 5, 2012

Manuscript Accepted: April 11, 2012

Published: April 19, 2012

**Virtual Issues**

Vol. 7, Iss. 6 *Virtual Journal for Biomedical Optics*

**Citation**

Chenyi Liu, Alexander Wong, Kostadinka Bizheva, Paul Fieguth, and Hongxia Bie, "Homotopic, non-local sparse reconstruction of optical coherence tomography imagery," Opt. Express **20**, 10200-10211 (2012)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-20-9-10200

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### References

- S. A. Boppart, “Optical coherence tomography: technology and applications for neuroimaging,” Psychophysiology40, 529–541 (2003). [CrossRef] [PubMed]
- 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,” Science254, 1178–1181 (1991). [CrossRef] [PubMed]
- B. E. Bouma and G. J. Tearney, Handbook of Optical Coherence Tomography (Informa Healthcare, 2001).
- M. Wojtkowski, V. Srinivasan, J. G. Fujimoto, T. Ko, J. S. Schuman, A. Kowalczyk, and J. S. Duker, “Three-dimensional retinal imaging with high-speed ultrahigh-resolution optical coherence tomography,” Ophthalmology112, 1734–1746 (2005). [CrossRef] [PubMed]
- W. Drexler and J. G. Fujimoto, Optical Coherence Tomography (Springer, 2008). [CrossRef]
- M. Wojtkowski, A. Kowalczyk, R. Leitgeb, and A. F. Fercher, “Full range complex spectral optical coherence tomography technique in eye imaging,” Opt. Lett.27, 1415–1417 (2002). [CrossRef]
- R. Leitgeb, W. Drexler, A. Unterhuber, B. Hermann, T. Bajraszewski, T. Le, A. Stingl, and A. Fercher, “Ultrahigh resolution Fourier domain optical coherence tomography,” Opt. Express12, 2156–2165 (2004). [CrossRef] [PubMed]
- R. Leitgeb, C. Hitzenberger, and A. Fercher, “Performance of Fourier domain vs. time domain optical coherence tomography,” Opt. Express11, 889–894 (2003). [CrossRef] [PubMed]
- A. F. Fercher, W. Drexler, C. K. Hitzenberger, and T. Lasser, “Optical coherence tomography—principles and applications,” Rep. Prog. Phys.66, 239–303 (2003). [CrossRef]
- E. Candës, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory52, 489–509 (2006). [CrossRef]
- D. Donoho, “Compressive sensing,” IEEE Trans. Inf. Theory52, 1289–1306 (2006). [CrossRef]
- M. Lustig, D. Donoho, and J. Pauly, “Sparse MRI: the application of compressed sesing for rapid MR imaging,” Magn. Reson. Med.58, 1182–1195 (2007). [CrossRef] [PubMed]
- J. Trzasko and A. Manduca, “Highly undersampled magnetic resonance image reconstruction via homotopic l0-minimization,” IEEE Trans. Med. Imag.28, 106–121 (2009). [CrossRef]
- A. Wong, A. Mishra, D. Clausi, and P. Fieguth, “Sparse reconstruction of breast MRI using homotopic L0 minimization in a regional sparsified domain,” Biomed. Eng. IEEE Trans, 1–10 (2010).
- D. Liang, H. Wang, and L. Ying, “SENSE reconstruction with nonlocal TV regularization,” Proc. IEEE Eng. Med. Biol. Soc., 1032–1035 (2009).
- N. Mohan, I. Stojanovic, W. C. Karl, B. E. A. Saleh, and M. C. Teich, “Compressed sensing in optical coherence tomography,” Proc. SPIE7570, 75700L (2010). [CrossRef]
- X. Liu and J. U. Kang, “Compressive SD-OCT: the application of compressed sensing in spectral domain optical coherence tomography,” Opt. Express18, 22010–22019 (2010). [CrossRef] [PubMed]
- G. Gilboa and S. Osher, “Nonlocal operators with applications to image processing,” Tech. Rep. CAM Report 07-23, Univ. California, Los Angeles, 2007.
- P. Fieguth, Statistical Image Processing and Multidimensional Modeling (Springer, 2010).
- B. K. Natarajan, “Sparse approximate solutions to linear systems,” SIAM J. Comput.24, 227–234 (1995). [CrossRef]
- W. Guo and F. Huang, “Adaptive total variation based filtering for MRI images with spatially inhomogeneous noise and artifacts,” Int. Sym. Biomed Imag, 101–104 (2009).
- P. Puvanathasan, P. Forbes, Z. Ren, D. Malchow, S. Boyd, and K. Bizheva, “High-speed, high-resolution Fourier-domain optical coherence tomography system for retinal imaging in the 1060 nm wavelength region,” Opt. Lett.33, 2479–2481 (2008). [PubMed]

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