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Virtual Journal for Biomedical Optics

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  • Editors: Andrew Dunn and Anthony Durkin
  • Vol. 5, Iss. 14 — Nov. 16, 2010
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Rapid volumetric OCT image acquisition using Compressive Sampling

Evgeniy Lebed, Paul J. Mackenzie, Marinko V. Sarunic, and Mirza Faisal Beg  »View Author Affiliations


Optics Express, Vol. 18, Issue 20, pp. 21003-21012 (2010)
http://dx.doi.org/10.1364/OE.18.021003


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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

Volumetric Optical Coherence Tomography (OCT) is emerging as a dominant medical imaging modality for diagnostic ophthalmology. Morphometric analysis and clinical applications require high resolution OCT images which necessitate dense sampling leading to high scan times. Clinical imaging with commercial OCT systems requires volumetric acquisitions lasting several seconds. During such an OCT scan session, the subject is required to stay still, fixating gaze on a point without blinking. Research applications and morphometric analysis often require higher resolution OCT images which necessitate denser sampling leading to even longer duration for volume acquisition. Long scan times increase chances of unavoidable motion artifacts that can corrupt the image. They also cause discomfort and aggravate soreness particularly for those with eye problems. Reducing the total scan time while preserving image quality is a much desired goal due to benefits of reduced discomfort, reduced motion artifacts and possible higher quality, higher resolution images.

Fig. 1 Proposed scan pattern consisting of randomly spaced horizontal B-scans and randomly spaced vertical B-scans. The doted lines represent the fly-back trajectory of the galvonometer mirrors.

2. Problem Formulation

Let 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]

]. Mathematically, this is 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.

Let 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 H is the synthesis operator transforming the coefficients back to the image domain. Given incomplete measurements 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 2 mean-squared error ‖yRS H x2 is small. From the many potential solutions to this problem, the particular chosen is the one with smallest 1 norm ‖1, representing a measure of sparsity. Hence, CS-based recovery of sparsely sampled OCT images will be achieved via solving the following optimization problem:

x˜=argminxx1subject toyRSHx2ε.
(1)

Our experiments show that shift invariant wavelets [4

4. S. Mallat, A Wavelet Tour of Signal Processing, Second Edition (Academic Press, New York, 1999).

] offer good compressibility of OCT images. Hence, they are a natural candidate for the sparse transform domain (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:

x˜=argminxx1+λyRSHx2,
(2)

where λ is a positive parameter that controls the trade-off between the sparsity imposed on the coefficients and the commensurate data mismatch that would result. The IST solver [9

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]

] is an attractive algorithm to use to solve this problem as it only consists of matrix-vector operations and converges to a solution in a timely manner. Using this approach, the iterative update xTλ(x+A T(yAy)), where A is defined to be RS H, 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 = S H . As a measure of recovery quality, signal to noise ratio (SNR), defined by −20log‖ff̃ ‖2/log‖f2 can be used with high numbers representing better recovery.

3. Numerical Results

A full 3D OCT original image volume (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.

]. The spectrometer based OCT system used operated at a line rate of 20kHz. The axial resolution of the system was nominally 4μm in tissue, and the lateral resolution was 25μm. 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.

Fig. 2 (a) One en-face slice of an optic nerve head volume. (b) The data as it would appear if acquired with the proposed sampling scheme (53% of the data missing). The red portions in the image indicate regions where no data was acquired. The four green lines represent recovered B-scans shown in Fig. 3. (c) En-face image generated from the CS recovered volume with 53% missing data. (d) Contrast-enhanced difference between (a) and (c).
Fig. 3 The B-scans in the first row are two typical vertical and two horizontal slices taken from the model, at locations shown by the green lines in Fig. 2(b). The images in the second row are the same slices taken from a volume with 53% missing data at locations corresponding to missing B-scans. The third row represents the slices that were recovered from the data by the CS interpolation method, while the fourth row represents B-scans that are the recovered by bilinear interpolation. The fifth row contains several zoomed sections of the CS-based and bilinear-based recovery. Note the presence of artifacts in recovered images with bilinear interpolation, and the standard good quality of CS-recovery compared to the original images.

The full 3D OCT image volume for each level of subsampling was recovered using our proposed CS based method employing IST solver (detail: using 6 inner and 40 outer loop iterations). SNR based on recovered images corresponding to 23, 35, 53, 61, and 75 percent missing data was found to be 14.9, 13.0, 10.6, 9.8, and 7.5 db respectively showing decreasing SNR with increasing degradation due to missing data. The SNR for bilinear interpolation results was found to be 11.5, 9.2, 6.1, 4.2 and 3.3 dB, respectively, for the above-described data volumes, showing a consistent 3-4 dB difference between the two interpolation schemes. The CS-recovery result for 53% missing data and the error |f| 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.

Fig. 4 Representative typical slices recovered with CS in volumes with 23%, 35%, 61% and 75% missing data. Note that even at 75% missing data the image quality remains comparable to the original B-scan.

3.1. Validation

We performed experiments to quantify how the degradation in recovered images affected the measurement of anatomically meaningful parameters such as the optic nerve cup shape. Morphometric analysis of changes in the shape of the optic nerve head have been proposed as a possible early diagnostic indicator of glaucomatous change. This was done by manually segmenting the clinically relevant Inner Limiting Membrane (ILM) surface for the reconstructed image volume for each of the five levels of degradation using an internal protocol developed in collaboration with clinical author (PJM). These segmentations were performed on each B-scan slice of every volume. The extracted ILM surface from one of the segmentations of the original 3D image is shown in Fig. 5(a) where the color map represents the vertical height (to better present the 3D nature of the cup). The ILM surface extracted from the recovered volume with 53% missing data, shown in Fig. 5(b), closely resembles that of the original ILM.

Fig. 5 (a) Extracted internal limiting membrane (ILM) of the optic nerve head from the original image. Color map represents the height function. (b) Extracted ILM from the recovered volume with 53% missing data. (c) Standard deviation of distance of the three segmentations of the original image volume, computed to measure the natural variability in segmentation of ILM from original image.

For each point on the ILM surface segmented from the reconstructed data, the relative position to the average ILM surface (the baseline) was found. This process generates the so-called Topographical Change Analysis (TCA) map, [14

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. 41, 775–782 (2000). [PubMed]

], representing the variability in position of the ILM surface from a known baseline/ground truth. Points on recovered ILM surface lying more than one standard deviation away from the average baseline surface are shown as the TCA overlaid on top of the summed voxels projection image of the original volume in Fig. 6. One can see that the quality of the summed voxel projection images begins to deteriorate at the 61% missing data volume, but is still acceptible even at 75% missing data. The red and green colors represent posterior and anterior change in position, respectively. For the volume that was recovered from 23% missing data, nowhere did the segmentations fall outside the one standard deviation regions. For the other cases with more missing data, most points in the image were reconstructed faithfully so that the errors in position exceeded one standard deviation in only a few locations. These were typically in slices with missing data that were located along the steeply sloping part of the ILM surface. Given that only a few locations in recovered surfaces deviated by more than one standard deviation for each of the five cases tested (as an example, <8% points on the recovered ILM surface were farther than one standard deviation in the case of 75% missing data), clinical measurements such as cup volume and area (see Table 1) did not significantly differ from the measurements made on the original volume. The cup volume was measured as the space enclosed by a reference plane near the top of the volume and the manual segmentation of the ILM in the cup. The surface area of the ILM closed by the reference plane and the cup was also calculated. Changes in surface area and volume of the nerve cup from recovered images were assessed to be clinically negligible by the clinical author (PJM).

Fig. 6 Topographical Change Analysis (TCA) maps showing position of ILM surface overlaid on top of summed-voxels-projection images of recovered volumes at (a) 23%, (b) 35%, (c)53%, (d) 61% and (e) 75% missing data, respectively. The relative surface position relative to standard deviation is shown with red representing posterior surface location, and green representing anterior surface location in the recovered image.

Table 1. Relative change in surface area (ΔS.A.) and volume (ΔVOL) in CS-based recovery

table-icon
View This Table

The reduction in scan time, presented as a function of percent of missing data is shown in Fig. 7. This function is computed by counting the number of B-scans required for a particular level of subsampling and dividing this by the number of B-scans required in a full raster scan. The fly-back time of the galvanometer mirrors is implicitly taken into account into the time it takes to perform one B-scan. The benefits in reduced scan time of the proposed scanning pattern begin to be observable when one discards approximately 25% of the data, or more. Using the subsampling corresponding to 53% or 61% missing data roughly halved the acquisition time.

Fig. 7 Percent reduction in scan time by the proposed scanning pattern, as a function of the percent of missing data. The diagonal line shows the asymptotic behavior of the percentage of reduction in scan time as a function of percentage of missing data.

4. Discussion

Compressive Sampling (CS), pioneered by works like [9

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]

, 11

11. M. Holschneider, R. Kronland-Martinet, J. Morlet, and P. Tchamitchian, Wavelets, Time-Frequency Methods and Phase Space (Springer-Verlag, Berlin, 1989).

, 12

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]

] to name a few, provides a theoretical basis for the recovery of the full 3D imaging volume from only a subset of the acquired data. Given a subsampled volume where random data is missing and using the idea of transform sparsity, there is usually enough information in the available data to reconstruct the original image volume with an accuracy that is at least as good as the best compressed representation of the object [6

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]

].

In this work we have shown that the theory of CS can be gainfully applied to reconstruct a full 3D OCT image with high fidelity from only a subset of acquired OCT samples. This type of recovery has clear advantages to using the standard bilinear interpolation, as shown by results in Fig. 3. An obvious “staircase” pattern is visible in the bilinear interpolation result (fifth row from the top in Fig. 3), which is undesirable as it corrupts the inherently smooth nature of the image, while CS-based recovery preserves the smoothness.

The sub-sampling scheme we have chosen is physically motivated by OCT raster-scanning, and is one where the imaging volume is acquired at specific vertical and horizontal B-scans only, thereby shortening the image acquisition time. The ILM surfaces obtained from manual segmentations of the recovered images do not differ markedly from the surfaces obtained from segmentations of the original image. The differences primarily occur in regions where the ILM cup has the steepest topographical changes. The small number of points and the localized nature of the deviations of the recovered ILM surface from the original surface are found to have negligible effect on downstream morphometric measurements. The change in metrics such as surface area and volume calculated from ‘recovered’ images was found to be insignificant for the clinical assessment of diseases such as glaucoma. Acquisition of fewer image samples allows the reduction in total OCT scan time, which in turn leads to lower chances of motion artifacts and less subject discomfort. For scan patterns corresponding to 53% or 61% missing data, for which the recovery is found to of very high fidelity, the acquisition time is roughly halved providing valuable saving in scan time. Data acquisition sparsity was investigated through the selective acquisition of B-scan images. The number of lines acquired in a B-scan could also be reduced through the acquisition of randomly spaced A-scans. However, due to the limited response time of the galvanometer mirrors, and complexity of coordinating the camera capture with settled mirror position, capturing an entire line is more economical. Other sub-sampling patterns, which may reduce scan time by an even greater ratio without degrading the overall image quality, are a topic of further study.

5. Conclusions

We have described a subsampling scan pattern consisting of a sequence of irregularly spaced horizontal and vertical B-scans that works well with CS-based recovery. Fewer total B-scans are required for CS-based recovery of the OCT volume, reducing the acquisition time with minimal degradation in image quality. The reduction in the OCT scan time limits the discomfort of a subject during the scan and reduces the amount of motion artifact. Image reconstruction with CS preserved the smoothness of sharp features in the optic cup, whereas standard bilinear interpolation resulted in staircase effect and loss of detail. The Compressed Sensing scan pattern presented in this report is compatible with existing OCT acquisition hardware, and has the potential to reduce the acquisition time in applications such as retinal imaging where the A-scan rate is limited by the maximum permitted optical exposure [2

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.

]. The degradation in the CS-recovered image quality is small, even at 75% missing data, and does not significantly affect clinical measurements like cup surface area and volume.

Acknowledgments

Funding for this work is generously provided by Michael Smith Foundation for Health Research (MSFHR), Natural Sciences and Engineering Research Council of Canada (NSERC) and Canadian Institutes of Health Research (CIHR).

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, A Wavelet Tour of Signal Processing, Second Edition (Academic Press, New York, 1999).

5.

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]

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]

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]

10.

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

11.

M. Holschneider, R. Kronland-Martinet, J. Morlet, and P. Tchamitchian, Wavelets, Time-Frequency Methods and Phase Space (Springer-Verlag, Berlin, 1989).

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. 41, 775–782 (2000). [PubMed]

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/vjbo/abstract.cfm?URI=oe-18-20-21003


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References

  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. Express 17, 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, A Wavelet Tour of Signal Processing, Second Edition (Academic Press, New York, 1999).
  5. 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]
  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]
  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]
  10. F. J. Herrmann, and G. Hennenfent, “Non-parametric seismic data recovery with curvelet frames,” Geophys. J. Int. 173, 233–248 (2008). [CrossRef]
  11. M. Holschneider, R. Kronland-Martinet, J. Morlet, and P. Tchamitchian, Wavelets, Time-Frequency Methods and Phase Space (Springer-Verlag, Berlin, 1989).
  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 XIV 7554, (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. 41, 775–782 (2000). [PubMed]

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