Abstract

Optical coherence tomography (OCT) is a non-invasive optical imaging modality capable of high resolution imaging of internal tissue structures. It is widely believed that the high axial resolution in OCT systems requires a wide-bandwidth light source. As a result, often the potential advantages of narrow-bandwidth sources (in terms of cost and/or imaging speed) are understood to come at the cost of significant reduction in imaging resolution. In this paper, we argue that this trade-off between resolution and speed is a shortcoming imposed by the-state-of-the-art A-scan reconstruction algorithm, Fast Fourier Transform, and can be circumvented through use of alternative processing methods. In particular, we investigate the shortcomings of the FFT as well as previously proposed alternatives and demonstrate the first application of an iterative regularized re-weighted l2 norm method to improve the axial resolution of fast scan rate OCT systems in the narrow-bandwidth imaging conditions. We validate our claims via experimental results generated from a home-built OCT system used to image layered phantom and in vivo data. Our results rely on new, sophisticated signal processing algorithms to generate higher precision (i.e., higher resolution) OCT images at correspondingly fast scan rates. In other words, our work demonstrates the feasibility of simultaneously more reliable and more comfortable medical imaging systems for patients by reducing the overall scan time, without sacrificing image quality.

© 2016 Optical Society of America

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References

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2015 (1)

2014 (2)

C. S. Seelamantula and S. Mulleti, “Super-resolution reconstruction in frequency domain optical-coherence tomography using the finite-rate-of-innovation principle,” IEEE Trans. on Sig. Proc. 62(19), 5020–5029 (2014).
[Crossref]

K. L. Lurie, G. T. Smith, S. A. Khan, J. C. Liao, and A. K. Ellerbee, “Three-dimensional, distendable bladder phantom for optical coherence tomography and white light cystoscopy,” J. Biomed. Opts. 19(3), 036009 (2014).
[Crossref]

2013 (1)

2012 (2)

N. Zhang, T. Huo, C. Wang, T. Chen, J. Zheng, and P. Xue, “Compressed sensing with linear-in-wavenumber sampling in spectral-domain optical coherence tomography,” Opt. Lett. 37(15), 3075–3077 (2012).
[Crossref] [PubMed]

E. Bousi and C. Pitris, “Axial resolution improvement by modulated deconvolution in fourier domain optical coherence tomography,” J. Biomed. Opt. 17(7), 071307 (2012).
[Crossref] [PubMed]

2010 (2)

X. Liu and J. U. Kang, “Compressive SD-OCT: the application of compressed sensing in spectral domain optical coherence tomography,” Opt. Express 18(21), 22010–22019 (2010).
[Crossref] [PubMed]

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 (2010).
[Crossref]

2009 (1)

2007 (1)

2006 (4)

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE T. Inform. Theory 52(2), 489–509 (2006).
[Crossref]

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3(10), 793–796 (2006).
[Crossref] [PubMed]

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313(5793), 1642–1645 (2006).
[Crossref] [PubMed]

S. T. Hess, T. P. K. Giririjan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. 91(11), 4258–4272 (2006).
[Crossref] [PubMed]

2003 (1)

M. Stewart, “A Superfast toeplitz solver with improved numerical stability,” SIAM. J. Matrix Anal. A. 25(3), 669–693 (2003).
[Crossref]

2002 (1)

1999 (1)

J. M. Schmitt, “Optical coherence tomography (OCT): a review,” IEEE J. Sel. Top. Quant. 5(4), 1205–1215 (1999).
[Crossref]

1996 (1)

A.F. Ferchner, “Optical coherence tomography,” J. Biomed. Opt. 1(2), 157–173 (1996).
[Crossref]

1992 (1)

B. D. Rao and K. S. Arun, “Model based processing of signals: a state space approach,” Proc. IEEE 80(2), 283–309 (1992).
[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,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Arun, K. S.

B. D. Rao and K. S. Arun, “Model based processing of signals: a state space approach,” Proc. IEEE 80(2), 283–309 (1992).
[Crossref]

Bates, M.

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3(10), 793–796 (2006).
[Crossref] [PubMed]

Betzig, E.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313(5793), 1642–1645 (2006).
[Crossref] [PubMed]

Blu, T.

S. C. Sekhar, H. Nazkani, T. Blu, and M. Unser, “A new technique for high-resolution frequency domain optical coherence tomography,” in Proceedings of IEEE conference on Acoustics, Speech and Signal Processing (IEEE, 2007), pp. I-425–I-428.

Bonifacino, J. S.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313(5793), 1642–1645 (2006).
[Crossref] [PubMed]

Bouma, B.

Bouma, B. E.

Bousi, E.

E. Bousi and C. Pitris, “Axial resolution improvement by modulated deconvolution in fourier domain optical coherence tomography,” J. Biomed. Opt. 17(7), 071307 (2012).
[Crossref] [PubMed]

Candes, E. J.

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE T. Inform. Theory 52(2), 489–509 (2006).
[Crossref]

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,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Chartrand, R.

R. Chartrand and W. Yin, “Iteratively reweighted algorithms for compressive sensing,” in Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing, (IEEE, 2008), pp. 3869–3872.

Chen, S.

Chen, T.

Cui, D.

Davidson, M. W.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313(5793), 1642–1645 (2006).
[Crossref] [PubMed]

Drexeler, W.

W. Drexeler and J. G. Fujimoto, Optical Coherence Tomography, Technology and Applications (Springer, 2008).
[Crossref]

Ellerbee, A. K.

K. L. Lurie, G. T. Smith, S. A. Khan, J. C. Liao, and A. K. Ellerbee, “Three-dimensional, distendable bladder phantom for optical coherence tomography and white light cystoscopy,” J. Biomed. Opts. 19(3), 036009 (2014).
[Crossref]

H. Y. Lee, H. Sudkamp, T. Marvdashti, and A. K. Ellerbee, “Interleaved optical coherence tomography,” Opt. Express 21(22), 26542–26556 (2013).
[Crossref] [PubMed]

Ferchner, A.F.

A.F. Ferchner, “Optical coherence tomography,” J. Biomed. Opt. 1(2), 157–173 (1996).
[Crossref]

Flotte, T.

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(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Fujimoto, J. G.

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(5035), 1178–1181 (1991).
[Crossref] [PubMed]

W. Drexeler and J. G. Fujimoto, Optical Coherence Tomography, Technology and Applications (Springer, 2008).
[Crossref]

Giririjan, T. P. K.

S. T. Hess, T. P. K. Giririjan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. 91(11), 4258–4272 (2006).
[Crossref] [PubMed]

Gregory, K.

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(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Hee, M. R.

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(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Hess, H. F.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313(5793), 1642–1645 (2006).
[Crossref] [PubMed]

Hess, S. T.

S. T. Hess, T. P. K. Giririjan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. 91(11), 4258–4272 (2006).
[Crossref] [PubMed]

Huang, D.

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(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Huo, T.

Kang, J. U.

Karl, W. C.

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 (2010).
[Crossref]

Khan, S. A.

K. L. Lurie, G. T. Smith, S. A. Khan, J. C. Liao, and A. K. Ellerbee, “Three-dimensional, distendable bladder phantom for optical coherence tomography and white light cystoscopy,” J. Biomed. Opts. 19(3), 036009 (2014).
[Crossref]

Lee, H. Y.

Liang, Y.

Liao, J. C.

K. L. Lurie, G. T. Smith, S. A. Khan, J. C. Liao, and A. K. Ellerbee, “Three-dimensional, distendable bladder phantom for optical coherence tomography and white light cystoscopy,” J. Biomed. Opts. 19(3), 036009 (2014).
[Crossref]

Lin, C. P.

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(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Lindwasser, O. W.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313(5793), 1642–1645 (2006).
[Crossref] [PubMed]

Lippincott-Schwartz, J.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313(5793), 1642–1645 (2006).
[Crossref] [PubMed]

Liu, L.

Liu, X.

Liu, Y.

Lurie, K. L.

K. L. Lurie, G. T. Smith, S. A. Khan, J. C. Liao, and A. K. Ellerbee, “Three-dimensional, distendable bladder phantom for optical coherence tomography and white light cystoscopy,” J. Biomed. Opts. 19(3), 036009 (2014).
[Crossref]

Marple, S. L.

S. L. Marple, Digital Spectral Analysis: With Applications, (Prentice Hall, 1986).

Marvdashti, T.

Mason, M. D.

S. T. Hess, T. P. K. Giririjan, and M. D. Mason, “Ultra-high resolution imaging by fluorescence photoactivation localization microscopy,” Biophys. J. 91(11), 4258–4272 (2006).
[Crossref] [PubMed]

Mohan, N.

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 (2010).
[Crossref]

Moses, R.

P. Stoica and R. Moses, Introduction to Spectral Analysis (Prentice Hall, 2000).

Motz, J.

Mu, G.

Mulleti, S.

C. S. Seelamantula and S. Mulleti, “Super-resolution reconstruction in frequency domain optical-coherence tomography using the finite-rate-of-innovation principle,” IEEE Trans. on Sig. Proc. 62(19), 5020–5029 (2014).
[Crossref]

Nazkani, H.

S. C. Sekhar, H. Nazkani, T. Blu, and M. Unser, “A new technique for high-resolution frequency domain optical coherence tomography,” in Proceedings of IEEE conference on Acoustics, Speech and Signal Processing (IEEE, 2007), pp. I-425–I-428.

Olenych, S.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313(5793), 1642–1645 (2006).
[Crossref] [PubMed]

Patterson, G. H.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313(5793), 1642–1645 (2006).
[Crossref] [PubMed]

Pitris, C.

E. Bousi and C. Pitris, “Axial resolution improvement by modulated deconvolution in fourier domain optical coherence tomography,” J. Biomed. Opt. 17(7), 071307 (2012).
[Crossref] [PubMed]

Puliafito, C. A.

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(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Rao, B. D.

B. D. Rao and K. S. Arun, “Model based processing of signals: a state space approach,” Proc. IEEE 80(2), 283–309 (1992).
[Crossref]

Romberg, J.

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE T. Inform. Theory 52(2), 489–509 (2006).
[Crossref]

Rust, M. J.

M. J. Rust, M. Bates, and X. Zhuang, “Sub-diffraction-limit imaging by stochastic optical reconstruction microscopy (STORM),” Nat. Methods 3(10), 793–796 (2006).
[Crossref] [PubMed]

Saleh, B. E. A.

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 (2010).
[Crossref]

Schmitt, J. M.

J. M. Schmitt, “Optical coherence tomography (OCT): a review,” IEEE J. Sel. Top. Quant. 5(4), 1205–1215 (1999).
[Crossref]

Schuman, J. S.

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(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Seelamantula, C. S.

C. S. Seelamantula and S. Mulleti, “Super-resolution reconstruction in frequency domain optical-coherence tomography using the finite-rate-of-innovation principle,” IEEE Trans. on Sig. Proc. 62(19), 5020–5029 (2014).
[Crossref]

Sekhar, S. C.

S. C. Sekhar, H. Nazkani, T. Blu, and M. Unser, “A new technique for high-resolution frequency domain optical coherence tomography,” in Proceedings of IEEE conference on Acoustics, Speech and Signal Processing (IEEE, 2007), pp. I-425–I-428.

Shishkov, M.

Smith, G. T.

K. L. Lurie, G. T. Smith, S. A. Khan, J. C. Liao, and A. K. Ellerbee, “Three-dimensional, distendable bladder phantom for optical coherence tomography and white light cystoscopy,” J. Biomed. Opts. 19(3), 036009 (2014).
[Crossref]

Sougrat, R.

E. Betzig, G. H. Patterson, R. Sougrat, O. W. Lindwasser, S. Olenych, J. S. Bonifacino, M. W. Davidson, J. Lippincott-Schwartz, and H. F. Hess, “Imaging intracellular fluorescent proteins at nanometer resolution,” Science 313(5793), 1642–1645 (2006).
[Crossref] [PubMed]

Stewart, M.

M. Stewart, “A Superfast toeplitz solver with improved numerical stability,” SIAM. J. Matrix Anal. A. 25(3), 669–693 (2003).
[Crossref]

Stinson, W. G.

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(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Stoica, P.

P. Stoica and R. Moses, Introduction to Spectral Analysis (Prentice Hall, 2000).

Stojanovic, I.

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 (2010).
[Crossref]

Sudkamp, H.

Swanson, E. A.

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(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Tao, T.

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE T. Inform. Theory 52(2), 489–509 (2006).
[Crossref]

Tearney, G.

Tearney, G. J.

Teich, M. C.

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 (2010).
[Crossref]

Unser, M.

S. C. Sekhar, H. Nazkani, T. Blu, and M. Unser, “A new technique for high-resolution frequency domain optical coherence tomography,” in Proceedings of IEEE conference on Acoustics, Speech and Signal Processing (IEEE, 2007), pp. I-425–I-428.

Wang, C.

White, W.

Xue, P.

Yelin, D.

Yin, W.

R. Chartrand and W. Yin, “Iteratively reweighted algorithms for compressive sensing,” in Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing, (IEEE, 2008), pp. 3869–3872.

Yu, X.

Yun, S.

Zhang, N.

Zheng, J.

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Figures (4)

Fig. 1
Fig. 1 The Gaussian shape shows the source S(k) versus the wavenumber, k. Arrows with O and X (in red and blue respectively) show samples of the source, collected for two interferograms (i.e., two lateral positions). As can be noted from the figures, in the case of random sampling, samples are collected from the whole bandwidth, whereas in the case of consecutive sampling, the collected samples occupy a sub-band of the source. Therefore, in the latter scheme, data from multiple lateral positions can be collected in only one source scan.
Fig. 2
Fig. 2 Qualitative comparison of algorithms with various source bandwidths and phantom data. Note that the ground truth is reconstructed with the maximum available source bandwidth of 100 nm and is presented in the first row. The red box portion is zoomed in to the bottom left corner of each image for easier visual comparison. Note the smearing effect in FFT images where LR deconvolution is not able to compensate for it well enough when the source bandwidth is as narrow as 5–10 nm. Our proposed method performs significantly better with narrow bandwidth sources and has comparable performance to the other algorithms as the source bandwidth increases to 20–35 nm.
Fig. 3
Fig. 3 Qualitative comparison of algorithms with various source bandwidths and finger data. Note that the ground truth (on top) is reconstructed with the maximum available source bandwidth of 100 nm. The region inside the red box appears in zoomed form at the top right corner of each image for easier visual comparison. At 1 nm, feature a is well reconstructed for both re-weighted l2 and Prony; however, the Prony result is less informative for features b and c. With the 10-nm source, the re-weighted l2 solution results in an image very similar to the ground truth. Beyond that, all algorithms show the features of interest very well, except for FFT, which still shows a smearing effect that is particularly strong at feature a.
Fig. 4
Fig. 4 A-scan comparison of mirror data for various source bandwidths for FFT and the proposed algorithm. Ground truth data correspond to the FFT-generated result with a 100-nm source bandwidth. The improvement in resolution obtained with the current algorithm (i.e., the super-resolution factor) is greatest for narrow bandwidths.

Tables (5)

Tables Icon

Algorithm 1: Regularized re-weighted l2 algorithm.

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Table 1 Quantitative comparison of layered phantom data for various source bandwidths, and different algorithms (FFT, Prony, LR deconvolution, and re-weighted l2). In each row, the number in bold indicates the largest PSNR and smallest MSE, which indicate the best performing algorithm for each source bandwidth.

Tables Icon

Table 2 Quantitative comparison of finger data for various source bandwidths, and different algorithms (FFT, Prony, LR deconvolution, and re-weighted l2). In each row, the number in bold indicates the largest PSNR and smallest MSE, which indicate the best performing algorithm for each source bandwidth.

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Table 3 Comparison of resolution between FFT and the proposed method for various bandwidths. As can be noted, resolution improves as the source bandwidth increases for re-weighted l2 and FFT algorithm. However, the proposed method has significantly improved resolution in the narrow bandwidth cases. The theoretical axial resolution obtained with FFT processing is included for reference.

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Table 4 Computation time comparison of the proposed algorithm with FFT, Prony and LR.

Equations (4)

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I ( n ) = S ( k ) × | r R e j 2 k z R + 0 z max r S ( z S ) e j 2 k z S d z S | 2 + ε ( k ) | k = k 0 K / 2 + n K / N
r S ( z S ) m = 0 M 1 r ( m ) δ ( z S 2 π N m K M ) .
I ( n ) = m = 0 M 1 r ( m ) e j 2 π m n M , n = 1 , , N .
Ar = I .

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