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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 1 — Jan. 14, 2013
  • pp: 329–344
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Energy-guided learning approach to compressive FD-OCT

Shimon Schwartz, Chenyi Liu, Alexander Wong, David A. Clausi, Paul Fieguth, and Kostadinka Bizheva  »View Author Affiliations


Optics Express, Vol. 21, Issue 1, pp. 329-344 (2013)
http://dx.doi.org/10.1364/OE.21.000329


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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

1. Introduction

Optical Coherence Tomography (OCT) is a non-invasive optical imaging modality, that can provide cross-sectional and volumetric images of biological tissue with cellular level resolution and at depths of up to 2mm in biological tissues [1

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,” Science 254, 1178–1181 (1991). [CrossRef] [PubMed]

, 2

2. W. Drexler and J. G. Fujimoto, “Optical coherence tomography,” Springer BerlinHeidelberg (2008).

]. As such, OCT is well suited for non-invasive imaging of ocular tissue (retina and cornea) and over the past 15 years has emerged as one of the dominant ophthalmic diagnostic modalities [3

3. 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 Sci 52, 2425–2436 (2011). [CrossRef] [PubMed]

5

5. 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 Imaging 42(4), 15–27 (2011). [CrossRef]

]. Imaging large volumes of retinal or corneal tissue, while keeping the image quality high requires high density sampling of the imaged volume, which results in very large data sets. As a result, natural eye motion, such as fixational micro-saccades, that occur with frequency of 1Hz [6

6. S. Martinez-Conde, S. L. Macknik, and D. H. Hubel, “The role of fixational eye movements in visual perception,” Nat. Rev. Neurosci 5(3), 229–240 (2004). [CrossRef] [PubMed]

], can introduce motion artifacts in the imaged data, which can render partial or whole data sets unusable, and indirectly increase the patient examination time significantly. Different hardware or software approaches to dealing with eye motion have been proposed in the past. Low density scanning protocols are typical for commercial OCT systems however, in this approach the faster imaging acquisition comes at the expense of reduced image quality. Recently, a compressive sparse sampling algorithm was proposed for optimal OCT scanning in XY direction, that preserves high OCT image quality [7

7. 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. Express 2(9), 2690–2697 (2011). [CrossRef] [PubMed]

]. Use of FDML lasers has increased the OCT scanning rate from kilohertz to megahertz [8

8. 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. Express 19(4), 3044–3062 (2011). [CrossRef] [PubMed]

], however, currently OCT systems based on FDML technology are very complex, expensive and require high speed electronics. Combining eye tracking with OCT can correct most of the motion artefacts, however, this solution requires very complex hardware redesign and software algorithms. Furthermore, some clinical applications of OCT, for example, whole eye imaging, require very large scanning range, while maintaining the image quality (spatial resolution and SNR). In Fourier domain OCT (FD-OCT), the scanning range and the depth variation of the system SNR are dependent on the sampling density of the interferometric signal [9

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

, 10

10. M. A. Choma, M. V. Sarunic, C. Yang, and J. A. Izatt, “Sensitivity advantage of swept source and Fourier domain optical coherence tomography,” Opt. Express 11(18), 2183 (2003). [CrossRef] [PubMed]

]. In Fourier domain OCT, the number of sampling points is determined by the number of illuminated pixels in the camera interfaced to the spectrometer [9

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

]. In swept source OCT (SS-OCT), the number of sampling points is determined by the sweep rate of the tunable laser in relation to the digitizer rate [10

10. M. A. Choma, M. V. Sarunic, C. Yang, and J. A. Izatt, “Sensitivity advantage of swept source and Fourier domain optical coherence tomography,” Opt. Express 11(18), 2183 (2003). [CrossRef] [PubMed]

]. In both cases, the cost of the detection technology, CCD and CMOS cameras in the case of FD-OCT, and digitizers in the case of SS-OCT, increases monotonically and significantly with increasing the number of camera pixels or digitizing rate respectively.

Currently, OCT image generation algorithms operate under the assumption that high quality reconstruction of the imaging data needs to comply with the Nyquist limit, which requires a minimum of twice the sampling points per period of the recorded interferogram. However, recent studies in compressive sensing [11

11. 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]

, 12

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

] have shown that certain signals can be reconstructed with a high level of accuracy when sampled below the Nyquist rate, given that they can be sparsely represented in some transformed domain. As long as they are not pure noise, signals can be sparse in their original domain, or some transformed domain. Here we define sparsity as the number of coefficients to represent the original signal is close to 0. The ground work done by Donoho et al. [12

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

, 13

13. D. Donoho and J. Tanner, “Counting faces of randomly-projected polytopes when the projection radically lowers dimension,” J. AMS 22, 1–5 (2009).

] and Candes et al. [11

11. 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]

, 14

14. E.J. Candes and T. Tao, “Near-optimal signal recovery from random projections: universal encoding strategies,” IEEE Trans. Inf. Theory 52, 5406–5425 (2004). [CrossRef]

] proved that excellent reconstruction results can be achieved by using L0 minimization. This concept has been widely used in clinical image reconstruction,and specifically in magnetic resonance imaging (MRI) [15

15. 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]

18

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

]. The prior work explored L1 minimization [11

11. 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]

], homotopic L0 minimization [16

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

], non-local total variation minimization [18

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

] and regional differential sparsifying transformation of the original domain [17

17. 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).

], to best account for the characteristics of different images.

The research on compressive sensing (CS) is very active and its core research can be grouped into three major research areas,
  • Sparse representation basis: Sufficiently sparse representation of images provides an unique solution for the reconstruction problem. Sparse representation basis and its implications in CS are well researched [23

    23. 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]

    27

    27. E. Candes, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,”Communications on Pure and Applied Mathematics 59, 1207–1221 (2006). [CrossRef]

    ]
  • Image reconstruction: Efficient reconstruction methods have been used to reconstruct the original image. Common reconstruction methods, such as: greedy methods [28

    28. J. A. Tropp, “Greed is good: algorithmic results for sparse approximation,” IEEE Trans. Inf. Theory 50, 2231–2242 (2004). [CrossRef]

    , 29

    29. J. A. Tropp, “Just relax: convex programming methods for identifying sparse signals in noise,” IEEE Trans. Inf. Theory 51, 1030–1051 (2006). [CrossRef]

    ], L1 minimization [11

    11. 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]

    , 30

    30. 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.

    , 31

    31. 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).

    ], non-local homotopic sparse reconstruction [22

    22. C. Liu, A. Wong, K. Bizheva, P. Fieguth, and H. Bie, “Homotopic, non-local sparse reconstruction of optical coherence tomography imagery,” Opt. Express 20, 10200–10211 (2012). [CrossRef] [PubMed]

    ] and other have been proven efficient in finding the sparsest solution for CS [32

    32. E. J. Candes and J. Romberg, “Sparsity and incoherence in compressive sampling,” Inverse Probl 23, 969–985 (2007). [CrossRef]

    ]. Image reconstruction methods have been well studied.
  • Generalization of the sampling procedure: Common sampling procedure in the CS research community is sampling the entire image randomly and uniformly based on Gaussian distributions and discrete Bernoulli distributions [33

    33. S. Mendelson, A. Pajor, and N. Tomczak-Jaegermann, “Uniform uncertainty principle for Bernoulli and subgaussian ensembles,” Constructive Approximation 28, 277–289 (2008). [CrossRef]

    ]. Those sensing distributions satisfy the restricted isometry property (RIP) with a controlled probability [13

    13. D. Donoho and J. Tanner, “Counting faces of randomly-projected polytopes when the projection radically lowers dimension,” J. AMS 22, 1–5 (2009).

    ,14

    14. E.J. Candes and T. Tao, “Near-optimal signal recovery from random projections: universal encoding strategies,” IEEE Trans. Inf. Theory 52, 5406–5425 (2004). [CrossRef]

    ,34

    34. R. Baraniuk, M. Davenport, R. DeVore, and M. Wakin, “A simple proof of the restricted isometry property for random matrices,” Constructive Approximation 28, 253–263 (2008). [CrossRef]

    36

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

    ]. Improvements in this area have been minimal when compared to the other two areas. In recent years there has been interest in optimizing the sampling probability density. Representations of MRI images have nonrandom structures since most of the image energy is concentrated close to the representation domain origin [15

    15. 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]

    ]. Therefore, it was proposed [15

    15. 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]

    ,37

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

    ,38

    38. G. Puy, P. Vandergheynst, and Y. Wiaux, “On variable density compressive sampling,” IEEE Sig. Proc. Lett 18, 595–598 (2011). [CrossRef]

    ] to consider a variable density random under-sampling which would sample closer to the origin and farther in the periphery of the representation domain. The proposed sampling function is expected to adjust the probability density according to the power of distance from the origin [15

    15. 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]

    ] in the Fourier domain. It was also proposed [37

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

    ] to use a similar method of variable density random under-sampling for the images domain. Based on the same concept, it was proposed to apply this variable density random under-sampling approach to OCT [39

    39. X. Liu and J.U. Kang, “sparse OCT: optimizing compressed sensing in spectral domain optical coherence tomography,” Proc. SPIE 7904, 79041C (2011). [CrossRef]

    ] though a static sampling pattern. Those non-uniform sensing approaches are static, considering general representation domains properties and hardware constraints [40

    40. M.F. Duarte and Y.C. Elda, “Structured compressed sensing from theory to applications,” IEEE Tran. Sig. Proc. Lett 59, 4053–4085 (2011). [CrossRef]

    , 41

    41. 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]

    ] in order to improve reconstruction performance.

While the main focus of the CS research community is image reconstruction, an optimal and adaptive sampling procedure has been less studied for practical applications such as compressive FD-OCT. The design of a data adaptive sampling procedure can have a significant impact on CS performance for practical applications such FD-OCT, where the objects of interest have structured characteristics in the frequency domain, thus making the sampling procedure worth investigating. Existing CS based systems employ a sampling scheme that samples the entire scene in the same manner regardless of the underlying data. However, such an approach is limiting for many practical applications, which involve distinct regions of interest in some basis, since it does not consider data importance. In many cases such region of interest are of greater interest for analysis purposes, one is motivated to obtain higher quality reconstructions for those regions than the background regions.

In a recent study, a saliency-guided sparse measurement model [42

42. 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]

,43

43. S. Schwartz, A. Wong, and D. A. Clausi, “Compressive fluorescence microscopy using saliency-guided sparse reconstruction ensemble fusion,” Opt. Express 20, 17281–17296 (2012). [CrossRef] [PubMed]

] has been proposed for a significant CS reconstruction improvement. This method optimizes the sampling probability density function according to salient regions in the spatial domain, where high saliency is sampled with high probability and low saliency is sampled at lower probability. It has been shown that this approach achieves greater reconstruction performance in comparison to the common uniform sampling distribution or matching reconstruction performance with much fewer samples. While signal acquisition in most current FD-OCT systems is based on static pixel array, multiple studies [20

20. 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]

, 44

44. S.M. Potter, A. Mart, and J. Pine, “High-speed CCD movie camera with random pixel selection for neurobiology research,” Proc. SPIE 2869, 243253 (1997).

46

46. B. Dierickx, D. Scheffer, G. Meynants, W. Ogiers, and J. Vlummens, “Random addressable active pixel image sensors,” Proc. SPIE 2950, 2–7 (1996). [CrossRef]

] have explored in detail the development of CCD cameras with randomly addressable pixels sensors that are being designed for efficient imaging acquisition. As with all existing literature in compressive FD-OCT, the novel methodology being developed here acts as the theoretical foundation for advanced research in efficient FD-OCT systems that are optimized for the integration of such cameras once they become available for FD-OCT systems, with the aim to acquire much fewer samples while maintaining image quality.

The main contribution of this paper is the improvement of compressive FD-OCT reconstruction performance by dynamically adapting the sampling model according to energy-guided statistical learning approach based on the underlying data in the frequency domain. By doing so, the proposed method optimizes the sampling pattern in an automatic fashion in order to reduce the sampling rate and improve reconstruction quality.

The organization of this paper is as follows. Section 2 describes the theory and implementation of the proposed method. Experimental results using OCT retinal, corneal, and fingertip data are presented and discussed in Section 3, with conclusions and future work drawn in Section 4.

2. Methodology

2.1. FD-OCT

In FD-OCT, spatial locations in depth, f(x) within the imaged object are correlated with information in the spectral domain F(k) through the inverse Fourier transform 𝔽−1:
f(x)=𝔽1{F(k)}
(1)

According to the Nyquist criterion [20

20. 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]

], the number of samples K in the k-space domain should obey
K2xmaxΔk/π,
(2)
where xmax is the maximum scanning range, and Δk is the spectral bandwidth, which is inversely proportional to the OCT axial resolution. A large value for K is necessary to achieve a long scanning range in OCT, which means that CCD or CMOS linear array cameras with large number of pixels, or digitizers with high sampling rates are required for developing long scanning range FD-OCT and SS-OCT systems respectively.

2.2. Compressive FD-OCT

Ideally, it is desirable to exactly reconstruct an OCT image f, from only a small subset of sampled signal in k-space domain. Let an undersampled reconstruction fu be expressed as
fu(x)=𝔽1{ΦF(k)}
(3)
where Φ is a measurement operator defining which of the spectral sites are measured (non-measured k-space indices will be assigned to 0 by Φ). Our goal is to reconstruct f(x) from a sparse sampling of F(k). With only few measurements made, which is essentially an ill-posed inverse problem [47

47. P. Fieguth, Statistical image processing and multidimensional modeling (SpringerNew York, 2010).

], and multi solutions exist if there is no regularization.

According to the emerging theory of compressive sensing [11

11. 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]

, 12

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

], a better estimate of 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 L0 minimization problem. Unfortunately, solving the L0 problem is essentially NP hard, intractable in practice [48

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

]. To address this problem, pioneering work was done by Candes et al. [11

11. 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]

] and Donoho [12

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

], showing that under certain conditions, L0 and L1 minimization lead to the same solutions. Theoretically, solving the L1 problem can get exactly the same solution as solving the L0 problem, at the cost of a substantial increase in the number of measurements required [16

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

].

2.3. Energy guided CS model

Sampling method can have a significant impact on FD-OCT data acquisition efficiency in terms of amount of required samples while maintaining high reconstruction quality. The traditional CS approach, the entire sampling scene is sampled uniformly, where all sampling locations are considered equally [34

34. R. Baraniuk, M. Davenport, R. DeVore, and M. Wakin, “A simple proof of the restricted isometry property for random matrices,” Constructive Approximation 28, 253–263 (2008). [CrossRef]

, 35

35. E. J. Candes, “Restricted isometry property and its implications for compressed sensing,” Comptes rendus - Mathematique 386, 589–592 (2008). [CrossRef]

]. In some cases [15

15. 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]

] a static non-uniform sampling approach was implemented in the frequency domain for MRI measurements. In this case a static and generic function (one over power distance from the origin) is used, not considering the underlying data. However, in practical situations such as FD-OCT, the energy of coefficients in the frequency domain, usually are concentrated more in some frequency bands and sparse in others frequency locations across the spectrum. In other words, the energy spectral density (ESD) has structural characteristics. The conventional uniform CS sampling approach is limited in the capability of preserving high energy spectral density.

The proposed energy-guided learning approach to compressive FD-OCT is addressing this important aspect, preserving high energy spectral density to improve CS-OCT performance. Consider the scene being measured using a measurement system to contain K sampling frequency locations denoted by ΩK, with the measured value at each sampling location:
ΩK={k|k=1,,K},
(4)
Subsequently, ΩK is partitioned into two complementary sets ΩT and ΩTc such that
ΩK=ΩTΩTc,withΩTΩTc,
(5)
whose cardinality of ΩT is T and the cardinality of ΩTc is (KT). The subset ΩT denotes sampled frequency locations and ΩTc denote unsampled frequency locations. Subset ΩT is generated based on a probability function pS(k) that provides a quantitative measure of ESD at frequency sampling location k
pS(k)[0,1]k
(6)

Let {φm}m=1M denote a collection of MK discrete sampling functions. The linear measurements of F (Eq. (1)) can be generally written as:
ym=F,φm+εm=k=1Kφm(k)+εm
(7)
where m = 1, 2,...,M and εm denotes measurement noise.

In a more compact a matrix form:
y¯=ΦTF+ε¯
(8)
where ȳ and ε̄ is M × 1 vector and ΦT is M × K sampling matrix.

Since M < K in the energy-guided learning model, the sampling basis is modified to account for the lack of observations at frequency locations in ΩTc:
φT,m(k)={φm(k),if(k)ΩT,0,if(k)ΩTc.
(9)
Consequently, the energy-guided learning approach to compressive FD-OCT model can be expressed as
ym=F,φT,m+εm,m=1,2,,M
(10)
where φT,m values at location k are realizations of a random variable k whose probability is defined in Eq. (11):
pT(k|π)=(1πpS(k))δ(k)+πpS(k)
(11)
The probability of random variable k is defined with weight π ∈ [0, 1] which sets the under sampling level (for example, if 50% compression rate is required, π is set to 0.5). Here the probability density function (PDF) of δ(k) has zero mean with zero variance, while pS is based on learned spectral measured coefficients PDF. Thus, φmT(k) probability given by Eq. (11) is pT (k) = 0 with probability (1 − πpS(k)).

2.4. Implementation

The implementation of the proposed energy-guided learning approach to compressive FD-OCT includes two steps: learning step and energy-guided compressive sensing step (Fig. 1).

Fig. 1 Energy-guided compressive sensing implementation.

Assume that FD-OCT measured coefficients, can be quantified based energy spectral density. Given that high energy measured coefficients within the frequency domain scene can have structural characteristics, it would be useful to quantify such characteristics using measured coefficients probability density function (PDF). Therefore, in the proposed implementation, spectral measured coefficients PDF is learned in step one. Let define pS to be the measured coefficients PDF (Eq. (12)) that will be used to implement the sparse sampling probability density pT (Eq. (11)):
pS(k)=i|Fi(k)|ki|Fi(k)|
(12)
where Fi represents the ith reflectivity profile (A-scan), contains spatial information at the ith location. The absolute operator |Fi(k)| is selected based on experiments. In the energy-guided learning approach, the measurement sparsity is guided by the measurements PDF model. Therefore a sampling method, such as been used in Monte Carlo approximation [51

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

], can be used for sampling from the learned PDF. Since pS is an univariant distribution, Inverse Transform Sampling method can be used [51

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

] which is define [52

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

]: Let C(x) be the cumulative density function (CDF) of random variable x, if random variable y comes from uniform (0,1) distribution, then random variable z = C−1(y) comes from the distribution of x. C−1 can be realized from the learned PDF (Eq. (12)). Let define random variable ω with a given probability distribution function (Eq. (12)) and let define I(k) as the inverse CDF of the learned PDF (Eq. (12)):
I(k)=P(ωk)
(13)
where P denotes probability that the random variable ω value is less or equal to x. For illustration purposes, learned distribution of different type of tissues are demonstrated in Fig. 2. Retinal and corneal PDF are presented at Fig. 2(b), Fig. 2(c). PDF of different type of tissue is examined as well - fingertip, Fig. 2(d). In addition an example of cornea background light is demonstrated in Fig. 2(a). In the case of sampling PDF learned from retina data acquisitions, it can be seen that the energy is concentrated at two ranges, highest energy range at samples 600–800 and secondary energy peak at samples 250–350. For sampling PDF learned from cornea data acquisitions, it can be seen that the energy is concentrated at two ranges, highest energy range at samples 550–650 and secondary energy peak at samples 250–300. The fingertip PDF is more uniformly distributed in comparison to retinal and corneal PDF. Sampling PDF learned from fingertip data acquisitions. It can be seen that the energy is concentrated at two ranges, highest energy range at samples 350–750 and secondary energy peak at samples 100–150

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.

For this practical realization, based on the learned PDF, the measurement sparsity is ESD-guided for each k frequency location of F(k). The first learning step is concluded once the sampling probability (Eq. (11)) as well as subset ΩT are determined. Based on comprehensive testing using OCT datasets of different types of tissues, it was found that the energy-guided compressive FD-OCT without the DC term has no notable impact on performance.

In the second step energy-guided compressive sensing step, F is sampled by ΦT (Eq. (8)) with probability pT (Eq. (11)) in order to measure higher energy coefficients at higher accuracy. The samples from subset ΩT are used for creating the sampling basis φT,k that create sampling matrix ΦT (Eq. (8)). The acquired samples used to reconstruct the OCT images at a higher reconstruction quality.

3. Results and discussions

As with the proposed energy-guided approach (denoted as EGCS), the non-local sparse reconstruction framework [22

22. C. Liu, A. Wong, K. Bizheva, P. Fieguth, and H. Bie, “Homotopic, non-local sparse reconstruction of optical coherence tomography imagery,” Opt. Express 20, 10200–10211 (2012). [CrossRef] [PubMed]

] was used for reconstructing the sampled data. To evaluate the effectiveness of the proposed method, a series of OCT images acquired in-vivo from the human retinal (Fig. 3(a), Fig. 4(a)), cornea (Fig. 5(a), Fig. 6(a)), and fingertip (Fig. 7(a), Fig. 8(a)) were processed with the novel proposed algorithm. For comparison purposes, the conventional uniform CS sampling approach was also evaluated as a baseline reference where sparse random sampling locations are distributed uniformly. In this reference case, pS (Eq. (11)) is adjusted pSCS to represent uniform distribution for the conventional CS approach. Therefore the sampling pattern φT,m (Eq. (10)) values for the conventional CS approach at location k are realizations of a random variable k whose probability is adjusted as well and is defined in Eq. (15):
pTCS(k|π)=(1πpSCS(k))δ(k)+πpSCS(k)
(15)
Both the sampling matrices used in conventional CS reconstruction and the proposed energy-guided compressive sensing reconstruction are random matrices. The difference between the matrices is that the conventional CS sparse sampling matrix contains zeros at random locations which are distributed uniformly, representing non-sampled locations, while the proposed energy-guided compressive sensing matrix includes zeros at random locations which are distributed based on underlying data.

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 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 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 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 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 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.

The images were acquired with a research grade, high-speed, FD-OCT system [53

53. 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]

], operating at 1060nm wavelength, that utilizes a super-luminescent diode (λc = 1020nm, δλ = 110nm, Pout = 10mW) and a 47kHz InGaAs linear array, 1024 pixel camera (SUI, Goodrich) interfaced with a high performance spectrometer (P&P Optica). The FD-OCT system provides 3μm axial and 15μm lateral resolution in the human corneal and fingertip tissue, and 6μm axial resolution in the human retina. The OCT images were acquired from healthy subjects using an imaging procedure carried out in accordance with the University of Waterloo research ethics regulations. To reconstruct OCT images from only a percentage of the camera pixels, the original data is sampled in the spectral domain using pseudo-random mask which its distribution is based on underlying data. The obtained spectral data is then used to populate the k-space grid according to the known functional dependency of wavenumber on the pixel index [20

20. 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]

]. The testing methodology in this work, which employs pseudo-random masks [20

20. 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]

], is consistent with the common approach used for evaluating compressive FD-OCT frameworks, where the underlying rationale is that the use of random sparse sampling can indeed be practically realized in an FD-OCT based on the research and development of CCD cameras with randomly addressable pixels for efficient imaging acquisition [20

20. 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]

, 44

44. S.M. Potter, A. Mart, and J. Pine, “High-speed CCD movie camera with random pixel selection for neurobiology research,” Proc. SPIE 2869, 243253 (1997).

46

46. B. Dierickx, D. Scheffer, G. Meynants, W. Ogiers, and J. Vlummens, “Random addressable active pixel image sensors,” Proc. SPIE 2950, 2–7 (1996). [CrossRef]

].

For illustrative purposes, the PSNR was measured for retinal, corneal and fingertip measurements reconstructed across the range of 25% and 70% of the camera pixels.

From the PSNR vs. sampling rate plot shown in Fig. 9, it can be observed that the PSNR achieved using the Energy-guided compressive FD-OCT outperforms conventional uniform CS sampling approach through the entire tested sampling rate and for the different OCT images. The average difference between the Energy-guided compressive FD-OCT and conventional uniform CS sampling approach for the tested sampling rate of 25%–70% is corneal: 2.3dB, fingertip: 0.8dB and retinal: 3.1dB. Through the experiments, the training datasets were separated from the testing datasets. The learning dataset and the tested dataset are different datasets that were obtained from the same volumetric data. In other words, one plane of the volumetric dataset is dedicated for training and other planes are used for testing. In order to validate the concept even further, additional processing was carried out, where the learning dataset is based on a different retinal eye tissue which means one volumetric dataset was used for training and another volumetric dataset for testing. The average PSNR for this case is 3.0dB, which is very similar to the experiment where training dataset is obtained from the same volumetric data with average PSNR of 3.1dB. Furthermore, a qualitative visual assessment was performed on the reconstructed data to investigate the reconstruction performance and the preservation of details achieved using the tested methods at 50% sampling rate.

Fig. 9 PSNR vs. sampling rate for cornea, retina and fingertip measurements.

As a complementary performance evaluation, the effect on point-spread functions of the system is presented in Fig. 10. A point spread function is examined from axial and lateral direction. The analysis is performed for a bright point at cornea measurement for original 100% sampling image, conventional CS reconstruction at 40% sampling rate and the energy-guided compressive sensing reconstruction at 40% sampling rate. The plot in Fig. 10 presents normalized intensity vs. pixel location. From the point spread function analysis, it can be seen (Fig. 10) that the energy-guided compressive sensing reconstruction at 40% sampling rate is very close to the 100% sampling case while the conventional CS has much spread intensity at this location at the two directions (axial and lateral). This mean that the energy-guided compressive sensing provides better resolution compared to the conventional CS methods at the same sampling rate and the resolution is close to the 100% sampling even with much fewer sampling. Figures 3, 5, and 7 show examples for each of the three types of in-vivo human OCT imaging data, each reconstructed using the two reconstruction methods from spectral data acquired using 50% of the camera pixels. To visualize the improvements obtained from using the proposed method, regions of interest (ROI) extracted from the reconstructed images (shown as colored boxes in Figs. 3, 5, and 7 are shown for better visual comparison in Figs. 4, 6, and 8. The human retina (Fig. 3) contains a number of morphological details such as individual cellular and plexiform layers, cross-sections of retinal capillaries (circular black features in the retina), as well as large choroidal blood vessels (pale circular or elongated features below the retinal pigmented epithelium — the last thick black line). We can see very clearly that the conventional uniform CS sampling 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 visualized because of the algorithm induced blur. The energy-guided compressive FD-OCT method results in noticeably better image quality as compared to that produced using the conventional uniform CS sampling method, although the contrast of the individual retinal layers is not as good as in the original image. It can be seen that the image reconstructed using the energy-guided compressive FD-OCT approach is closer to the image reconstructed from 100% of the acquired samples.

Fig. 10 Effect on point-spread functions of the system through cornea measurements at 40% sampling rate.

The human cornea (Fig. 5) contains a number of morphological features of different size and optical properties. It has four distinct layers: the epithelium, Bowman’s membrane, stroma, and Descemet’s-endothelial complex, each of which are clearly visible in the image reconstructed from 100% of the acquired samples. The small black dots in the corneal stroma correspond to reflections from keratocyte cells. As observed in Fig. 5 and Fig. 6, the conventional uniform CS sampling 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 energy-guided compressive FD-OCT approach result in significantly better reconstruction of the corneal morphological details, as well as higher image contrast as compared to the conventional uniform CS sampling method. Once again, the image reconstructed using the energy-guided compressive FD-OCT approach is closer to the image reconstructed from 100% of the acquired samples.

The human fingertip contains spiral shaped sweat glands in the skin epithelial region, which are clearly visible in Fig. 7. When applied to OCT images of the human fingertip (Fig. 7 and Fig. 8) It was observed that the conventional uniform CS sampling method and energy-guided compressive FD-OCT method have almost similar reconstruction performance with only 1 dB PSNR difference Fig. 9(b). The reason is the energy probability distribution. The energy is more concentrated in certain frequencies at the retinal and corneal datasets (Fig. 2(a) and Fig. 2(b)) while the energy is more spread in the fingertip case (Fig. 2(c)). The energy-guided compressive FD-OCT method optimizes the sampling probability according to underline data. Since the energy is spread in the fingertip case, the optimal sampling probability is closer to uniform probability in comparison to retinal or corneal data. The proposed energy-guided compressive FD-OCT approach produces reconstructed OCT data with higher PSNR values for all levels of camera pixel under-sampling when compared to the conventional uniform CS sampling method. The energy-guided compressive FD-OCT method outperformed the conventional uniform CS sampling approach even for fingertip data, though not significantly as in the case of the retinal and corneal images. Finally, note that, based on experiments using OCT datasets of different types of tissues, the variable density sampling approach [39

39. X. Liu and J.U. Kang, “sparse OCT: optimizing compressed sensing in spectral domain optical coherence tomography,” Proc. SPIE 7904, 79041C (2011). [CrossRef]

] provides lower PSNR when compared to the proposed method.

4. Conclusions

In this paper, we proposed an energy-guided learning approach for improving the efficiency of compressive FD-OCT. The energy-guided learning approach to compressive FD-OCT optimizes the sampling probability density function in the frequency domain to underlying data rather the common uniform sampling used in conventional uniform CS sampling approaches. This system learns the energy spectral density to optimize the sampling probability. This proposed work provides a framework for optimizing sampling and sparse reconstruction integrated system. The performance of this work was demonstrated by the experimental on different OCT imagery tissues. For example, we show that 4.1dB PSNR improvement can be achieved for retinal data at 65% sampling rate compared with conventional CS sampling method. Furthermore the proposed design allows for substantial reduction in amount of samples needed to achieve similar performance compared to conventional uniform CS sampling approach. For example, we showed that to achieve retina reconstruction PSNR of 16 dB, the energy-guided learning approach achieves similar performance as the conventional uniform CS sampling approach using 45% fewer samples. In future, it will be interesting to investigate different methods for optimizing sampling probability density based on underlying data, as well as different sparse reconstruction algorithms to best fit the imagery characteristics. In addition, a study can be performed for researching optimal structures for the measured coefficients PDF (Eq. (12)). Finally, this energy-guided sampling and sparse reconstruction system can be easily extended to other medical image techniques as well.

Acknowledgments

This project was funded in part by the Natural Sciences and Engineering Research Council of Canada, the Canadian Institutes for Health Research, and the University of Waterloo. The author would also like to thank the Chinese Council Scholarship and the Ontario Ministry of Economic Development and Innovation.

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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:
Medical Optics and Biotechnology

History
Original Manuscript: September 26, 2012
Revised Manuscript: November 16, 2012
Manuscript Accepted: December 15, 2012
Published: January 4, 2013

Virtual Issues
Vol. 8, Iss. 2 Virtual Journal for Biomedical Optics

Citation
Shimon Schwartz, Chenyi Liu, Alexander Wong, David A. Clausi, Paul Fieguth, and Kostadinka Bizheva, "Energy-guided learning approach to compressive FD-OCT," Opt. Express 21, 329-344 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-1-329


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References

  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]
  2. W. Drexler and J. G. Fujimoto, “Optical coherence tomography,” Springer BerlinHeidelberg (2008).
  3. 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]
  4. 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]
  5. 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]
  6. 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]
  7. 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]
  8. 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]
  9. 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]
  10. 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]
  11. 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]
  12. D. Donoho, “Compressive sensing,” IEEE Trans. Inf. Theory52, 1289–1306 (2006). [CrossRef]
  13. D. Donoho and J. Tanner, “Counting faces of randomly-projected polytopes when the projection radically lowers dimension,” J. AMS22, 1–5 (2009).
  14. E.J. Candes and T. Tao, “Near-optimal signal recovery from random projections: universal encoding strategies,” IEEE Trans. Inf. Theory52, 5406–5425 (2004). [CrossRef]
  15. 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]
  16. J. Trzasko and A. Manduca, “Highly undersampled magnetic resonance image reconstruction via homotopic l0-minimization,” IEEE Trans. Med. Image28, 106–121 (2009). [CrossRef]
  17. 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).
  18. D. Liang, H. Wang, and L. Ying, “SENSE reconstruction with nonlocal TV regularization,” Proc. IEEE Eng. Med. Biol. Soc.1032–1035 (2009).
  19. 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]
  20. 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]
  21. 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]
  22. 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]
  23. 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]
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