## Rapid volumetric OCT image acquisition using Compressive Sampling |

Optics Express, Vol. 18, Issue 20, pp. 21003-21012 (2010)

http://dx.doi.org/10.1364/OE.18.021003

Acrobat PDF (5379 KB)

### Abstract

Acquiring three dimensional image volumes with techniques such as Optical Coherence Tomography (OCT) relies on reconstructing the tissue layers based on reflection of light from tissue interfaces. One B-mode scan in an image is acquired by scanning and concatenating several A-mode scans, and several contiguous slices are acquired to assemble a full 3D image volume. In this work, we demonstrate how Compressive Sampling (CS) can be used to accurately reconstruct 3D OCT images with minimal quality degradation from a subset of the original image. The full 3D image is reconstructed from sparsely sampled data by exploiting the sparsity of the image in a carefully chosen transform domain. We use several sub-sampling schemes, recover the full 3D image using CS, and show that there is negligible effect on clinically relevant morphometric measurements of the optic nerve head in the recovered image. The potential outcome of this work is a significant reduction in OCT image acquisition time, with possible extensions to speeding up acquisition in other imaging modalities such as ultrasound and MRI.

© 2010 Optical Society of America

## 1. Introduction

1. B. Považay, B. Hofer, C. Torti, B. Hermann, A. R. Tumlinson, M. Esmaeelpour, C. A. Egan, A. C. Bird, and W. Drexler, “Impact of enhanced resolution, speed and penetration on three-dimensional retinal optical coherence tomography,” Opt. Express17, 4134–4150 (2009), http://www.opticsexpress.org/abstract.cfm?URI=oe-17-5-4134. [CrossRef] [PubMed]

2. T. Schmoll, C. Kolbitsch, and R. A. Leitgeb, “Ultra-high-speed volumetric tomography of human retinal blood flow,”, Opt. Express 17, 4166–4176 (2009), http://www.opticsexpress.org/abstract.cfm?URI=oe-17-5-4166.

1. B. Považay, B. Hofer, C. Torti, B. Hermann, A. R. Tumlinson, M. Esmaeelpour, C. A. Egan, A. C. Bird, and W. Drexler, “Impact of enhanced resolution, speed and penetration on three-dimensional retinal optical coherence tomography,” Opt. Express17, 4134–4150 (2009), http://www.opticsexpress.org/abstract.cfm?URI=oe-17-5-4134. [CrossRef] [PubMed]

2. T. Schmoll, C. Kolbitsch, and R. A. Leitgeb, “Ultra-high-speed volumetric tomography of human retinal blood flow,”, Opt. Express 17, 4166–4176 (2009), http://www.opticsexpress.org/abstract.cfm?URI=oe-17-5-4166.

3. M. K. K. Leung, A. Mariampillai, B. A. Standish, K. K. C. Lee, N. R. Munce, A. Vitkin, and V. X. D. Yang, “High-power wavelength-swept laser in Littman telescope-less polygon filter and dual-amplifier configuration for multichannel optical coherence tomography,” Opt. Lett.34, 2814–2816 (2009), http://ol.osa.org/abstract.cfm?URI=ol-34-18-2814 [CrossRef] [PubMed]

5. E. J. Candès and D. L. Donoho, “New tight frames of curvelets and optimal representations of objects with piecewise-C^{2} singularities,” Comm. Pure Appl. Math.57, 219–266 (2004), http://www.acm.caltech.edu/∼emmanuel/papers/CurveEdges.pdf. [CrossRef]

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

8. E. Candès and J. Romberg, “Sparsity and incoherence in compressive sampling,” Inverse Probl.23, 969–985 (2007), http://stacks.iop.org/0266-5611/23/969. [CrossRef]

9. I. Daubechies, M. Defrise, and C. De Mol “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Comm. Pure Appl. Math.11, 1413–1457 (2004), http://dx.doi.org/10.1002/cpa.20042. [CrossRef]

## 2. Problem Formulation

**f**denote the 3D tissue image (defined on a 3D Cartesian grid) that is to be imaged by OCT using a typical raster scanning geometry. The image

**f**is therefore a 3D matrix with values corresponding to the OCT image intensity at each point in the volume. Let

**R**be a restriction mask that denotes a subset of the samples of

**f**that are actually acquired. Therefore

**R**is also a 3D matrix, consisting of binary values corresponding to the scan pattern. The forward model of image acquisition describes the relationship with the acquired volume,

**y**, which is a product of the actual volume data modified by the sampling pattern (restriction mask) [10

10. F. J. Herrmann and G. Hennenfent, “Non-parametric seismic data recovery with curvelet frames,” Geophys. J. Int. **173**, 233–248 (2008). [CrossRef]

**y**=

**RMf**+

**n**, where

**y**is the observed incomplete signal containing a fraction of the full 3D image measurements from

**f**and

**n**is system noise. Here, the matrix

**M**contains the basis of the domain in which

**f**is acquired. Since OCT images are typically acquired in 3D space with Cartesian geometry,

**M**is the Dirac basis; so

**M**:=

**I**.

**S**be a transform domain offering good compressibility i. e. a small number of coefficients

**x = Sf**can describe the 3D image

**f**in that domain. Given the coefficients

**x**, the signal

**f**can be reconstructed as

**f**=

**S**

^{H}**x**where

**S**

*is the synthesis operator transforming the coefficients back to the image domain. Given incomplete measurements*

^{H}**y**of the image

**f**, instead of recovering the unknown image

**f**directly, the key idea from CS is instead to find the unknown coefficients

**x**parameterizing

**f**such that the reconstructed signal

**S**

^{H}**x**followed by known masking

**R**is close to the observed incomplete data

**y**i.e. the

*ℓ*

_{2}mean-squared error ‖

**y**–

**RS**

^{H}**x**‖

_{2}is small. From the many potential solutions to this problem, the particular

**x̃**chosen is the one with smallest

*ℓ*

_{1}norm ‖

**x̃**‖

_{1}, representing a measure of sparsity. Hence, CS-based recovery of sparsely sampled OCT images will be achieved via solving the following optimization problem:

**S**) required to solve the above CS-based recovery. The constrained optimization problem stated above is converted to an unconstrained optimization problem and rewritten as:

9. I. Daubechies, M. Defrise, and C. De Mol “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Comm. Pure Appl. Math.11, 1413–1457 (2004), http://dx.doi.org/10.1002/cpa.20042. [CrossRef]

**x**→

*T*(

_{λ}**x**+

**A**

*(*

^{T}**y**−

**Ay**)), where

**A**is defined to be

**RS**

*, converges to the solution for a particular value of λ as the number of iterations goes to infinity. In practice, only a finite number of iterations are required to achieve convergence. The final recovered image is given by*

^{H}**f̃**=

**S**

^{H}**x̃**. As a measure of recovery quality, signal to noise ratio (SNR), defined by −20log‖

*f*−

*f*̃ ‖

_{2}/log‖

*f*‖

_{2}can be used with high numbers representing better recovery.

## 3. Numerical Results

**f**) was acquired by raster-scanning the optic nerve head of a healthy male subject using OCT [13

13. M. Young, S. Lee, E. Gibson, K. Hsu, M. F. Beg, P. J. Mackenzie, and M. V. Sarunic, “Morphometric analysis of the optic nerve head with optical coherence tomography.” In proceedings of OCT and Coherence Domain Optical Methods in Biomedicine XIV7554, (2004), http://link.aip.org/link/?PSI/7554/75542L/1.

*μ*m in tissue, and the lateral resolution was 25

*μ*m. The cross sectional B-scans consisted of 512 A-scans, and the volume consisted of 400 B-scan elevations over an area of 3.7mm × 1.8mm. An example of a transversal slice, or C-scan, extracted from the OCT volume is shown in Fig. 2(a). To simulate the sparsely sampled OCT image volume (

**y**) that would be acquired by our proposed method of skipping horizontal and vertical raster-scan lines, we apply to this image a binary restriction mask

**R**. This mask consists of vertical and horizontal lines with irregular (but fixed) placing generated through a random number generator in software. Five levels of restriction mask giving 23, 35, 53, 61, and 75 percent missing data were created, and were applied to the original image volume to generate five different sub-sampled volumes

**y**. An example of mask that discards 53% of the original samples is shown in Fig. 2(b) where the red lines denote data that was discarded.

**f**–

**f̃**| image for this cross-section is shown in Fig. 2(c) and 2(d), respectively. Fig. 3 shows two vertical and horizontal B-scans recovered from a volume with 53% missing data compared with bilinear interpolation of the same B-scans. One can observe a distinct, undesirable “staircase” pattern in the bilinear interpolation results. Figure 4 shows the same typical slice that was recovered from volumes with 23, 35, 61 and 75 percent missing data, from a region where data was not sampled. One can see that the overall image quality begins to deteriorate as more data is discarded, but never the less, the fidelity is still acceptable even for a recovered volume from 75% missing data.

### 3.1. Validation

## 4. Discussion

9. I. Daubechies, M. Defrise, and C. De Mol “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Comm. Pure Appl. Math.11, 1413–1457 (2004), http://dx.doi.org/10.1002/cpa.20042. [CrossRef]

12. M. Elad, J. L. Starck, P. Querre, and D.L. Donoho, “Simultaneous cartoon and texture image inpainting using morphological component analysis (MCA),” Appl. Comput. Harmon. Anal.19, 340–358 (2005), http://www.sciencedirect.com/science/article/B6WB3-4GWC29F-2/2/61d7afc314d50b27968d84ff4a16acce. [CrossRef]

6. E. J. Candès, 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]

## 5. Conclusions

2. T. Schmoll, C. Kolbitsch, and R. A. Leitgeb, “Ultra-high-speed volumetric tomography of human retinal blood flow,”, Opt. Express 17, 4166–4176 (2009), http://www.opticsexpress.org/abstract.cfm?URI=oe-17-5-4166.

## Acknowledgments

## References and links

1. | B. Považay, B. Hofer, C. Torti, B. Hermann, A. R. Tumlinson, M. Esmaeelpour, C. A. Egan, A. C. Bird, and W. Drexler, “Impact of enhanced resolution, speed and penetration on three-dimensional retinal optical coherence tomography,” Opt. Express17, 4134–4150 (2009), http://www.opticsexpress.org/abstract.cfm?URI=oe-17-5-4134. [CrossRef] [PubMed] |

2. | T. Schmoll, C. Kolbitsch, and R. A. Leitgeb, “Ultra-high-speed volumetric tomography of human retinal blood flow,”, Opt. Express 17, 4166–4176 (2009), http://www.opticsexpress.org/abstract.cfm?URI=oe-17-5-4166. |

3. | M. K. K. Leung, A. Mariampillai, B. A. Standish, K. K. C. Lee, N. R. Munce, A. Vitkin, and V. X. D. Yang, “High-power wavelength-swept laser in Littman telescope-less polygon filter and dual-amplifier configuration for multichannel optical coherence tomography,” Opt. Lett.34, 2814–2816 (2009), http://ol.osa.org/abstract.cfm?URI=ol-34-18-2814 [CrossRef] [PubMed] |

4. | S. Mallat, |

5. | E. J. Candès and D. L. Donoho, “New tight frames of curvelets and optimal representations of objects with piecewise-C |

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

7. | D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory |

8. | E. Candès and J. Romberg, “Sparsity and incoherence in compressive sampling,” Inverse Probl.23, 969–985 (2007), http://stacks.iop.org/0266-5611/23/969. [CrossRef] |

9. | I. Daubechies, M. Defrise, and C. De Mol “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Comm. Pure Appl. Math.11, 1413–1457 (2004), http://dx.doi.org/10.1002/cpa.20042. [CrossRef] |

10. | F. J. Herrmann and G. Hennenfent, “Non-parametric seismic data recovery with curvelet frames,” Geophys. J. Int. |

11. | M. Holschneider, R. Kronland-Martinet, J. Morlet, and P. Tchamitchian, |

12. | M. Elad, J. L. Starck, P. Querre, and D.L. Donoho, “Simultaneous cartoon and texture image inpainting using morphological component analysis (MCA),” Appl. Comput. Harmon. Anal.19, 340–358 (2005), http://www.sciencedirect.com/science/article/B6WB3-4GWC29F-2/2/61d7afc314d50b27968d84ff4a16acce. [CrossRef] |

13. | M. Young, S. Lee, E. Gibson, K. Hsu, M. F. Beg, P. J. Mackenzie, and M. V. Sarunic, “Morphometric analysis of the optic nerve head with optical coherence tomography.” In proceedings of OCT and Coherence Domain Optical Methods in Biomedicine XIV7554, (2004), http://link.aip.org/link/?PSI/7554/75542L/1. |

14. | B. C. Chauhan, J. W. Blanchard, D. C. Hamilton, and R. P. LeBlanc, “Technique for Detecting Serial Topographic Changes in the Optic Disc and Peripapillary Retina Using Scanning Laser Tomography,”, Invest. Ophthalmol. Vis. Sci. |

**OCIS Codes**

(100.2000) Image processing : Digital image processing

(100.3010) Image processing : Image reconstruction techniques

**ToC Category:**

Image Processing

**History**

Original Manuscript: May 27, 2010

Revised Manuscript: July 2, 2010

Manuscript Accepted: July 9, 2010

Published: September 20, 2010

**Virtual Issues**

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

**Citation**

Evgeniy Lebed, Paul J. Mackenzie, Marinko V. Sarunic, and Faisal M. Beg, "Rapid Volumetric OCT Image Acquisition Using Compressive Sampling," Opt. Express **18**, 21003-21012 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-20-21003

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### References

- B. Považay, B. Hofer, C. Torti, B. Hermann, A. R. Tumlinson, M. Esmaeelpour, C. A. Egan, A. C. Bird, and W. Drexler, “Impact of enhanced resolution, speed and penetration on three-dimensional retinal optical coherence tomography,” Opt. Express 17, 4134–4150 (2009), http://www.opticsexpress.org/abstract.cfm?URI=oe-17-5-4134. [CrossRef] [PubMed]
- T. Schmoll, C. Kolbitsch, and R. A. Leitgeb, “Ultra-high-speed volumetric tomography of human retinal blood flow,” Opt. Express 17, 4166–4176 (2009), http://www.opticsexpress.org/abstract.cfm?URI=oe-17-5-4166.
- M. K. K. Leung, A. Mariampillai, B. A. Standish, K. K. C. Lee, N. R. Munce, A. Vitkin, and V. X. D. Yang, “High-power wavelength-swept laser in Littman telescope-less polygon filter and dual amplifier configuration for multichannel optical coherence tomography,” Opt. Lett. 34, 2814–2816 (2009), http://ol.osa.org/abstract.cfm?URI=ol-34-18-2814. [CrossRef] [PubMed]
- S. Mallat, A Wavelet Tour of Signal Processing, Second Edition (Academic Press, New York, 1999).
- E. J. Candès, and D. L. Donoho, “New tight frames of curvelets and optimal representations of objects with piecewise-C2 singularities,” Comm. Pure Appl. Math. 57, 219–266 (2004), http://www.acm.caltech.edu/emmanuel/papers/CurveEdges.pdf. [CrossRef]
- E. J. Candès, 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]
- D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006). [CrossRef]
- E. Candès, and J. Romberg, “Sparsity and incoherence in compressive sampling,” Inverse Probl. 23, 969–985 (2007), http://stacks.iop.org/0266-5611/23/969. [CrossRef]
- I. Daubechies, M. Defrise, and C. De Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Comm. Pure Appl. Math. 11, 1413–1457 (2004), http://dx.doi.org/10.1002/cpa.20042. [CrossRef]
- F. J. Herrmann, and G. Hennenfent, “Non-parametric seismic data recovery with curvelet frames,” Geophys. J. Int. 173, 233–248 (2008). [CrossRef]
- M. Holschneider, R. Kronland-Martinet, J. Morlet, and P. Tchamitchian, Wavelets, Time-Frequency Methods and Phase Space (Springer-Verlag, Berlin, 1989).
- M. Elad, J. L. Starck, P. Querre, and D. L. Donoho, “Simultaneous cartoon and texture image inpainting using morphological component analysis (MCA),” Appl. Comput. Harmon. Anal. 19, 340–358 (2005), http://www.sciencedirect.com/science/article/B6WB3-4GWC29F-2/2/61d7afc314d50b27968d84ff4a16acce. [CrossRef]
- M. Young, S. Lee, E. Gibson, K. Hsu, M. F. Beg, P. J. Mackenzie, and M. V. Sarunic, “Morphometric analysis of the optic nerve head with optical coherence tomography.” In proceedings of OCT and Coherence Domain Optical Methods in Biomedicine XIV 7554, (2004), http://link.aip.org/link/?PSI/7554/75542L/1.
- B. C. Chauhan, J. W. Blanchard, D. C. Hamilton, and R. P. LeBlanc, “Technique for Detecting Serial Topographic Changes in the Optic Disc and Peripapillary Retina Using Scanning Laser Tomography,” Invest. Ophthalmol. Vis. Sci. 41, 775–782 (2000). [PubMed]

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