Abstract

High quality, large size volumetric imaging of biological tissue with optical coherence tomography (OCT) requires large number and high density of scans, which results in large data acquisition volume. This may lead to corruption of the data with motion artifacts related to natural motion of biological tissue, and could potentially cause conflicts with the maximum permissible exposure of biological tissue to optical radiation. Therefore, OCT can benefit greatly from different approaches to sparse or compressive sampling of the data where the signal is recovered from its sub-Nyquist measurements. In this paper, a new energy-guided compressive sensing approach is proposed for improving the quality of images acquired with Fourier domain OCT (FD-OCT) and reconstructed from sparse data sets. The proposed algorithm learns an optimized sampling probability density function based on the energy distribution of the training data set, which is then used for sparse sampling instead of the commonly used uniformly random sampling. It was demonstrated that the proposed energy-guided learning approach to compressive FD-OCT of retina images requires 45% fewer samples in comparison with the conventional uniform compressive sensing (CS) approach while achieving similar reconstruction performance. This novel approach to sparse sampling has the potential to significantly reduce data acquisition while maintaining image quality.

© 2013 OSA

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2012 (3)

2011 (8)

G. Puy, P. Vandergheynst, and Y. Wiaux, “On variable density compressive sampling,” IEEE Sig. Proc. Lett18, 595–598 (2011).
[CrossRef]

X. Liu and J.U. Kang, “sparse OCT: optimizing compressed sensing in spectral domain optical coherence tomography,” Proc. SPIE7904, 79041C (2011).
[CrossRef]

M.F. Duarte and Y.C. Elda, “Structured compressed sensing from theory to applications,” IEEE Tran. Sig. Proc. Lett59, 4053–4085 (2011).
[CrossRef]

M. Young, E. Lebed, Y. Jian, P. J. Mackenzie, M. F. Beg, and M. V. Sarunic, “Real-time high-speed volumetric imaging using compressive sampling optical coherence tomography,” Opt. Express2, 2690–2697 (2011)
[CrossRef]

ML Gabriele, G Wollstein, H Ishikawa, L Kagemann, J Xu, LS Folio, and JS Schuman, “Optical coherence tomography: history, current status, and laboratory work,” Invest Ophthalmol Vis Sci52, 2425–2436 (2011).
[CrossRef] [PubMed]

J. Wang, M. A. Shousha, V. L. Perez, C. L. Karp, S. H. Yoo, M. Shen, L. Cui, V. Hurmeriz, C. Du, D. Zhu, Q. Chen, and M. Li, “Ultra-high resolution optical coherence tomography for imaging the anterior segment of the eye,” Ophthalmic Surg Lasers Imaging42(4), 15–27 (2011).
[CrossRef]

M. Young, E. Lebed, Y. Jian, P. J. Mackenzie, M. F. Beg, and M. V. Sarunic, “The role of fixational eye movements in visual perception,” Biomed. Opt. Express2(9), 2690–2697 (2011).
[CrossRef] [PubMed]

T. Klein, W. Wieser, C. M. Eigenwillig, B. R. Biedermann, and R. Huber, “Megahertz OCT for ultrawide-field retinal imaging with a 1050 nm Fourier domain mode-locked laser,” Opt. Express19(4), 3044–3062 (2011).
[CrossRef] [PubMed]

2010 (5)

W. Geitzenauer, C. K. Hitzenberger, and U. M. Schmidt-Erfurth, “Retinal optical coherence tomography: past, present and future perspectives,” Br. J. Ophthalmol95(2), 171–177 (2010).
[CrossRef] [PubMed]

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).

N. Mohan, I. Stojanovic, W. C. Karl, B. E. A. Saleh, and M. C. Teich, “Compressed sensing in optical coherence tomography,” Proc. SPIE7570, 75700L–75700L-5 (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]

Z. Wang and G. Arce, “Variable density compressed image sampling,”IEEE Tran. on Image Proc.19, 264–270 (2010).
[CrossRef]

2009 (5)

M. Rudelson and R. Vershynin, “Sparse reconstruction by convex relaxation: Fourier and Gaussian measurements,” 40th An. Conf. Inf. Sc. Sys.207–210 (2009).

J. Trzasko and A. Manduca, “Highly undersampled magnetic resonance image reconstruction via homotopic l0-minimization,” IEEE Trans. Med. Image28, 106–121 (2009).
[CrossRef]

D. Liang, H. Wang, and L. Ying, “SENSE reconstruction with nonlocal TV regularization,” Proc. IEEE Eng. Med. Biol. Soc.1032–1035 (2009).

D. Donoho and J. Tanner, “Counting faces of randomly-projected polytopes when the projection radically lowers dimension,” J. AMS22, 1–5 (2009).

W. Guo and F. Huang, “Adaptive total variation based filtering for MRI images with spatially inhomogeneous noise and artifacts,” Int. Sym. Biomed Imag101–104 (2009).

2008 (5)

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. Lett33, 2479–2481 (2008).
[PubMed]

R. Robucci, L.K. Chiu, J. Gray, J. Romberg, P. Hasler, and D. Anderson, “Compressive sensing on a CMOS separable transform image sensor,”IEEE Int. Conf. Ac. Speech Sig. Proc.5125–5128 (2008).
[CrossRef]

S. Mendelson, A. Pajor, and N. Tomczak-Jaegermann, “Uniform uncertainty principle for Bernoulli and subgaussian ensembles,” Constructive Approximation28, 277–289 (2008).
[CrossRef]

R. Baraniuk, M. Davenport, R. DeVore, and M. Wakin, “A simple proof of the restricted isometry property for random matrices,” Constructive Approximation28, 253–263 (2008).
[CrossRef]

E. J. Candes, “Restricted isometry property and its implications for compressed sensing,” Comptes rendus - Mathematique386, 589–592 (2008).
[CrossRef]

2007 (2)

E. J. Candes and J. Romberg, “Sparsity and incoherence in compressive sampling,” Inverse Probl23, 969–985 (2007).
[CrossRef]

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

2006 (4)

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]

J. A. Tropp, “Just relax: convex programming methods for identifying sparse signals in noise,” IEEE Trans. Inf. Theory51, 1030–1051 (2006).
[CrossRef]

E. Candes, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,”Communications on Pure and Applied Mathematics59, 1207–1221 (2006).
[CrossRef]

2004 (4)

J. A. Tropp, “Greed is good: algorithmic results for sparse approximation,” IEEE Trans. Inf. Theory50, 2231–2242 (2004).
[CrossRef]

E.J. Candes and T. Tao, “Near-optimal signal recovery from random projections: universal encoding strategies,” IEEE Trans. Inf. Theory52, 5406–5425 (2004).
[CrossRef]

S. Martinez-Conde, S. L. Macknik, and D. H. Hubel, “The role of fixational eye movements in visual perception,” Nat. Rev. Neurosci5(3), 229–240 (2004).
[CrossRef] [PubMed]

R. Leitgeb, W. Drexler, A. Unterhuber, B. Hermann, T. Bajraszewski, T. Le, A. Stingl, and A. Fercher, “Ultrahigh resolutionFourier domain optical coherence tomography,” Opt. Express12, 2156–2165 (2004).
[CrossRef] [PubMed]

2003 (3)

M. A. Choma, M. V. Sarunic, C. Yang, and J. A. Izatt, “Sensitivity advantage of swept source and Fourier domain optical coherence tomography,” Opt. Express11(18), 2183 (2003).
[CrossRef] [PubMed]

R. Gribonval and M. Nielsen, “Sparse representations in unions of bases,” IEEE Trans. Inf. Theory49, 3320–3325 (2003).
[CrossRef]

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. SPIE4975, 115 (2003).
[CrossRef]

2002 (1)

M. Elad and A.M. Bruckstein, “A generalized uncertainty principle and sparse representation in pairs of bases,” IEEE Trans. Inf. Theory48, 2558–2567 (2002).
[CrossRef]

2001 (1)

D.L. Donoho and X. Huo, “Uncertainty principles and ideal atom decomposition,” IEEE Trans. Inf. Theory47, 2845–2862 (2001).
[CrossRef]

1997 (1)

S.M. Potter, A. Mart, and J. Pine, “High-speed CCD movie camera with random pixel selection for neurobiology research,” Proc. SPIE2869, 243253 (1997).

1996 (1)

B. Dierickx, D. Scheffer, G. Meynants, W. Ogiers, and J. Vlummens, “Random addressable active pixel image sensors,” Proc. SPIE2950, 2–7 (1996).
[CrossRef]

1995 (1)

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

1991 (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,” Science254, 1178–1181 (1991).
[CrossRef] [PubMed]

Anderson, D.

R. Robucci, L.K. Chiu, J. Gray, J. Romberg, P. Hasler, and D. Anderson, “Compressive sensing on a CMOS separable transform image sensor,”IEEE Int. Conf. Ac. Speech Sig. Proc.5125–5128 (2008).
[CrossRef]

Arce, G.

Z. Wang and G. Arce, “Variable density compressed image sampling,”IEEE Tran. on Image Proc.19, 264–270 (2010).
[CrossRef]

Bajraszewski, T.

Baraniuk, R.

R. Baraniuk, M. Davenport, R. DeVore, and M. Wakin, “A simple proof of the restricted isometry property for random matrices,” Constructive Approximation28, 253–263 (2008).
[CrossRef]

Beg, M.

B. Wu, E. Lebed, M. Sarunic, and M. Beg, “Quantitative evaluation of transform domains for compressive sampling-based recovery of sparsely sampled volumetric OCT images,” IEEE Trans. Bio. Eng.1–9 (2012).
[CrossRef]

Beg, M. F.

M. Young, E. Lebed, Y. Jian, P. J. Mackenzie, M. F. Beg, and M. V. Sarunic, “Real-time high-speed volumetric imaging using compressive sampling optical coherence tomography,” Opt. Express2, 2690–2697 (2011)
[CrossRef]

M. Young, E. Lebed, Y. Jian, P. J. Mackenzie, M. F. Beg, and M. V. Sarunic, “The role of fixational eye movements in visual perception,” Biomed. Opt. Express2(9), 2690–2697 (2011).
[CrossRef] [PubMed]

Bie, H.

Biedermann, B. R.

Bizheva, K.

C. Liu, A. Wong, K. Bizheva, P. Fieguth, and H. Bie, “Homotopic, non-local sparse reconstruction of optical coherence tomography imagery,” Opt. Express20, 10200–10211 (2012).
[CrossRef] [PubMed]

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. Lett33, 2479–2481 (2008).
[PubMed]

Boyd, S.

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. Lett33, 2479–2481 (2008).
[PubMed]

S. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “A method for large-scale L1-regularized least squares problems with applications in signal processing and statistics,” Manuscript, (2007).

Bruckstein, A.M.

M. Elad and A.M. Bruckstein, “A generalized uncertainty principle and sparse representation in pairs of bases,” IEEE Trans. Inf. Theory48, 2558–2567 (2002).
[CrossRef]

Candes, E.

E. Candes, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,”Communications on Pure and Applied Mathematics59, 1207–1221 (2006).
[CrossRef]

Candes, E. J.

E. J. Candes, “Restricted isometry property and its implications for compressed sensing,” Comptes rendus - Mathematique386, 589–592 (2008).
[CrossRef]

E. J. Candes and J. Romberg, “Sparsity and incoherence in compressive sampling,” Inverse Probl23, 969–985 (2007).
[CrossRef]

Candes, E.J.

E.J. Candes and T. Tao, “Near-optimal signal recovery from random projections: universal encoding strategies,” IEEE Trans. Inf. Theory52, 5406–5425 (2004).
[CrossRef]

Candës, E.

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]

Casella, G.

C. P. Robert and G. Casella, “Stable signal recovery from incomplete and inaccurate measurements,”Monte Carlo Statistical Methods, New York: Springer-Verlag (1999).

Chang, T.

L. He, T. Chang, S. Osher, T. Fang, and P. Speier, “MR image reconstruction by using the iterative refinement method and nonlinear inverse scale space methods,” UCLA CAM Reports6–35, 2006.

Chang, W.

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]

Chen, H.

H. Chen, Tutorial on monte carlo sampling (The Ohio state university, department of chemcal and biomolecular engineering, technical report, 2005).

Chen, Q.

J. Wang, M. A. Shousha, V. L. Perez, C. L. Karp, S. H. Yoo, M. Shen, L. Cui, V. Hurmeriz, C. Du, D. Zhu, Q. Chen, and M. Li, “Ultra-high resolution optical coherence tomography for imaging the anterior segment of the eye,” Ophthalmic Surg Lasers Imaging42(4), 15–27 (2011).
[CrossRef]

Chiu, L.K.

R. Robucci, L.K. Chiu, J. Gray, J. Romberg, P. Hasler, and D. Anderson, “Compressive sensing on a CMOS separable transform image sensor,”IEEE Int. Conf. Ac. Speech Sig. Proc.5125–5128 (2008).
[CrossRef]

Choma, M. A.

Clausi, D.

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).

Clausi, D. A.

S. Schwartz, A. Wong, and D. A. Clausi, “Compressive fluorescence microscopy using saliency-guided sparse reconstruction ensemble fusion,” Opt. Express20, 17281–17296 (2012).
[CrossRef] [PubMed]

S. Schwartz, A. Wong, and D. A. Clausi, “Saliency-guided compressive sensing approach to efficient laser range measurement,” Journal of Visual Communication and Image Representation (2012).
[CrossRef]

Cui, L.

J. Wang, M. A. Shousha, V. L. Perez, C. L. Karp, S. H. Yoo, M. Shen, L. Cui, V. Hurmeriz, C. Du, D. Zhu, Q. Chen, and M. Li, “Ultra-high resolution optical coherence tomography for imaging the anterior segment of the eye,” Ophthalmic Surg Lasers Imaging42(4), 15–27 (2011).
[CrossRef]

Davenport, M.

R. Baraniuk, M. Davenport, R. DeVore, and M. Wakin, “A simple proof of the restricted isometry property for random matrices,” Constructive Approximation28, 253–263 (2008).
[CrossRef]

DeVore, R.

R. Baraniuk, M. Davenport, R. DeVore, and M. Wakin, “A simple proof of the restricted isometry property for random matrices,” Constructive Approximation28, 253–263 (2008).
[CrossRef]

Dierickx, B.

B. Dierickx, D. Scheffer, G. Meynants, W. Ogiers, and J. Vlummens, “Random addressable active pixel image sensors,” Proc. SPIE2950, 2–7 (1996).
[CrossRef]

Donoho, D.

D. Donoho and J. Tanner, “Counting faces of randomly-projected polytopes when the projection radically lowers dimension,” J. AMS22, 1–5 (2009).

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

D. Donoho, “Compressive sensing,” IEEE Trans. Inf. Theory52, 1289–1306 (2006).
[CrossRef]

Donoho, D.L.

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Figures (10)

Fig. 1
Fig. 1

Energy-guided compressive sensing implementation.

Fig. 2
Fig. 2

Sampling data PDF obtained from energy-guided learning approach based on different type of tissues and background. All the PDF plots comply with the basic shape of background PDF, and have different characteristics according to different kind of tissues. For illustration purposes only, the function presented in Fig. 2 were smoothened. No filtering was used for smoothing Ps(k) (Eq. (12)) in the implementation.

Fig. 3
Fig. 3

Reconstruction results from 50% of the acquired human retinal fovea data. The colored boxes mark sections that are enlarged in Fig. 4. The reconstruction results using 100% of the samples are provided as a reference.

Fig. 4
Fig. 4

Zoomed-in regions from Fig. 3. The fine details and the structural detail of the individual layers are better maintained in the reconstructed results produced using EGCS compared to CS.

Fig. 5
Fig. 5

Reconstruction results from 50% of the acquired human corneal data. The colored boxes mark sections that are enlarged in Fig. 6. The reconstruction results using 100% of the samples are provided as a reference.

Fig. 6
Fig. 6

Zoomed-in regions from Fig. 5. The fine details and the structural detail of the individual layers are better maintained in the reconstructed results produced using EGCS compared to CS.

Fig. 7
Fig. 7

Reconstruction results from 50% of the acquired human fingertip data. The colored boxes mark sections that are enlarged in Fig. 8. The reconstruction results using 100% of the samples are provided as a reference.

Fig. 8
Fig. 8

Zoomed-in regions from Fig. 7. The fine details and the structural detail of the individual layers are better maintained in the reconstructed results produced using EGCS compared to CS.

Fig. 9
Fig. 9

PSNR vs. sampling rate for cornea, retina and fingertip measurements.

Fig. 10
Fig. 10

Effect on point-spread functions of the system through cornea measurements at 40% sampling rate.

Equations (17)

Equations on this page are rendered with MathJax. Learn more.

f ( x ) = 𝔽 1 { F ( k ) }
K 2 x max Δ k / π ,
f u ( x ) = 𝔽 1 { Φ F ( k ) }
Ω K = { k | k = 1 , , K } ,
Ω K = Ω T Ω T c , with Ω T Ω T c ,
p S ( k ) [ 0 , 1 ] k
y m = F , φ m + ε m = k = 1 K φ m ( k ) + ε m
y ¯ = Φ T F + ε ¯
φ T , m ( k ) = { φ m ( k ) , if ( k ) Ω T , 0 , if ( k ) Ω T c .
y m = F , φ T , m + ε m , m = 1 , 2 , , M
p T ( k | π ) = ( 1 π p S ( k ) ) δ ( k ) + π p S ( k )
p S ( k ) = i | F i ( k ) | k i | F i ( k ) |
I ( k ) = P ( ω k )
f ^ ( x ) = lim σ 0 argmin f ( x ) η ( | Ψ f ( x ) | , σ ) s . t . Φ F ^ ( k ) Φ F ( k ) 2 < ε
p T C S ( k | π ) = ( 1 π p S C S ( k ) ) δ ( k ) + π p S C S ( k )
P S N R = 10 log 10 ( max 2 ( f ( x ) ) M S E )
M S E = 1 N x Ω ( f ( x ) f ^ ( x ) ) 2

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