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 Optical Society of America
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, 2]. 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 [3–5]. 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 , 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 . Use of FDML lasers has increased the OCT scanning rate from kilohertz to megahertz , 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, 10]. In Fourier domain OCT, the number of sampling points is determined by the number of illuminated pixels in the camera interfaced to the spectrometer . 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 . 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, 12] 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, 13] and Candes et al. [11, 14] 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) [15–18]. The prior work explored L1 minimization , homotopic L0 minimization , non-local total variation minimization  and regional differential sparsifying transformation of the original domain , to best account for the characteristics of different images.
Recently, the use of compressive sensing for reconstruction in high resolution OCT imagery has been reported by a number of different researchers such as Mohan et al. , Liu and Kang  and Young et al. . Those publications show promising reconstruction of OCT images from highly under-sampled k-space data, which has strong implications for data acquisition. However, those studies utilized either simulated OCT signals or actual images of onions, which have very different and much less complex morphology as compared to living biological tissues such as the human retina, cornea, skin, etc. In our previous work , a non-local strategy for sparse OCT reconstruction algorithm was evaluated on living tissues, and we have shown superior results compared with L1 minimization approach. However, the experimental results showed that for tissues with complex morphology, such as human retina, cornea and skin, at least 40%–50% of the originally sampled data is required to generate a reconstructed image of sufficiently good quality.
The research on compressive sensing (CS) is very active and its core research can be grouped into three major research areas,
- Image reconstruction: Efficient reconstruction methods have been used to reconstruct the original image. Common reconstruction methods, such as: greedy methods [28, 29], L1 minimization [11, 30, 31], non-local homotopic sparse reconstruction  and other have been proven efficient in finding the sparsest solution for CS . 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 . Those sensing distributions satisfy the restricted isometry property (RIP) with a controlled probability [13,14,34–36]. 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 . Therefore, it was proposed [15,37,38] 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  in the Fourier domain. It was also proposed  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  though a static sampling pattern. Those non-uniform sensing approaches are static, considering general representation domains properties and hardware constraints [40, 41] 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 [42,43] 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, 44–46] 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.
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:
According to the Nyquist criterion , the number of samples K in the k-space domain should obey
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 as47], and multi solutions exist if there is no regularization.
According to the emerging theory of compressive sensing [11, 12], 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 . To address this problem, pioneering work was done by Candes et al.  and Donoho , 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 .
Typically, a total variation (TV) penalty  is employed in the L1 minimization framework, which is known to have an edge-preservation effect . Both Mohan et al.  and Liu et al.  explored this L1 minimization framework, which represents the state-of-the-art in sparse OCT reconstruction, and their results show that OCT imagery can be reconstructed in a meaningful manner using sparsely sampled measurements in k-space. Our previous work used a homotopic non-local regularization reconstruction approach , and demonstrated that better SNR can be achieved under the same number of sample locations, but still needs large sample number to ensure reconstruction image quality.
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, 35]. In some cases  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:
Let denote a collection of M ≤ K discrete sampling functions. The linear measurements of F (Eq. (1)) can be generally written as:
In a more compact a matrix form:
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 :Eq. (11): Eq. (11) is pT (k) = 0 with probability (1 − πpS(k)).
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).
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)):51], can be used for sampling from the learned PDF. Since pS is an univariant distribution, Inverse Transform Sampling method can be used  which is define : 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)): 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
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.17], which has shown strong detail preservation in complex tissues.
3. Results and discussions
As with the proposed energy-guided approach (denoted as EGCS), the non-local sparse reconstruction framework  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 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):
The images were acquired with a research grade, high-speed, FD-OCT system , 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 . The testing methodology in this work, which employs pseudo-random masks , 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, 44–46].
To carry out a comprehensive and systematic assessment of the reconstruction performance of the different methods and for constancy with previous work , the peak signal to noise ratio (PSNR) was computed for a wide range of sampling rates.The PSNR metric was computed as follows:
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.
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.
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  provides lower PSNR when compared to the proposed method.
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.
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.
References and links
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. 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 Sci 52, 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. Ophthalmol 95(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 Imaging 42(4), 15–27 (2011). [CrossRef]
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]
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]
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. 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]
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. D. Donoho, “Compressive sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006). [CrossRef]
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. 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]
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]
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. SPIE 7570, 75700L–75700L-5 (2010). [CrossRef]
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. Express 2, 2690–2697 (2011) [CrossRef]
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]
24. D.L. Donoho and X. Huo, “Uncertainty principles and ideal atom decomposition,” IEEE Trans. Inf. Theory 47, 2845–2862 (2001). [CrossRef]
25. M. Elad and A.M. Bruckstein, “A generalized uncertainty principle and sparse representation in pairs of bases,” IEEE Trans. Inf. Theory 48, 2558–2567 (2002). [CrossRef]
26. R. Gribonval and M. Nielsen, “Sparse representations in unions of bases,” IEEE Trans. Inf. Theory 49, 3320–3325 (2003). [CrossRef]
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]
28. J. A. Tropp, “Greed is good: algorithmic results for sparse approximation,” IEEE Trans. Inf. Theory 50, 2231–2242 (2004). [CrossRef]
29. J. A. Tropp, “Just relax: convex programming methods for identifying sparse signals in noise,” IEEE Trans. Inf. Theory 51, 1030–1051 (2006). [CrossRef]
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. 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).
32. E. J. Candes and J. Romberg, “Sparsity and incoherence in compressive sampling,” Inverse Probl 23, 969–985 (2007). [CrossRef]
33. S. Mendelson, A. Pajor, and N. Tomczak-Jaegermann, “Uniform uncertainty principle for Bernoulli and subgaussian ensembles,” Constructive Approximation 28, 277–289 (2008). [CrossRef]
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. E. J. Candes, “Restricted isometry property and its implications for compressed sensing,” Comptes rendus - Mathematique 386, 589–592 (2008). [CrossRef]
36. M. Rudelson and R. Vershynin, “Sparse reconstruction by convex relaxation: Fourier and Gaussian measurements,” 40th An. Conf. Inf. Sc. Sys. 207–210 (2009).
37. Z. Wang and G. Arce, “Variable density compressed image sampling,”IEEE Tran. on Image Proc. 19, 264–270 (2010). [CrossRef]
38. G. Puy, P. Vandergheynst, and Y. Wiaux, “On variable density compressive sampling,” IEEE Sig. Proc. Lett 18, 595–598 (2011). [CrossRef]
39. X. Liu and J.U. Kang, “sparse OCT: optimizing compressed sensing in spectral domain optical coherence tomography,” Proc. SPIE 7904, 79041C (2011). [CrossRef]
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. 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]
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]
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).
45. 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. SPIE 4975, 115 (2003). [CrossRef]
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]
47. P. Fieguth, Statistical image processing and multidimensional modeling (SpringerNew York, 2010).
48. B. K. Natarajan, “Sparse approximate solutions to linear systems,” SIAM J. Comput. 24, 227–234 (1995). [CrossRef]
49. G. Gilboa and S. Osher, “Nonlocal operators with applications to image processing,” Tech. Rep. CAM Report 07–23, Univ. California, Los Angeles, 2007.
50. W. Guo and F. Huang, “Adaptive total variation based filtering for MRI images with spatially inhomogeneous noise and artifacts,” Int. Sym. Biomed Imag 101–104 (2009).
51. C. P. Robert and G. Casella, “Stable signal recovery from incomplete and inaccurate measurements,”Monte Carlo Statistical Methods, New York: Springer-Verlag (1999).
52. H. Chen, Tutorial on monte carlo sampling (The Ohio state university, department of chemcal and biomolecular engineering, technical report, 2005).
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]