Abstract

Endoscopic optical coherence tomography (OCT) devices are capable of generating high-resolution images of esophageal structures at high speed. To make the obtained data easy to interpret and reveal the clinical significance, an automatic segmentation algorithm is needed. This work proposes a fast algorithm combining sparse Bayesian learning and graph search (termed as SBGS) to automatically identify six layer boundaries on esophageal OCT images. The SBGS first extracts features, including multi-scale gradients, averages and Gabor wavelet coefficients, to train the sparse Bayesian classifier, which is used to generate probability maps indicating boundary positions. Given these probability maps, the graph search method is employed to create the final continuous smooth boundaries. The segmentation performance of the proposed SBGS algorithm was verified by esophageal OCT images from healthy guinea pigs and the eosinophilic esophagitis (EoE) models. Experiments confirmed that the SBGS method is able to implement robust esophageal segmentation for all the tested cases. In addition, benefiting from the sparse model of SBGS, the segmentation efficiency is significantly improved compared to other widely used techniques.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2018 (6)

K. Yu, F. Shi, E. T. Gao, W. F. Zhu, H. Y. Chen, and X. J. Chen, “Shared-hole graph search with adaptive constraints for 3d optic nerve head optical coherence tomography image segmentation,” Biomed. Opt. Express 9, 962–983 (2018).
[Crossref] [PubMed]

B. T. Soetikno, L. Beckmann, X. Zhang, A. A. Fawzi, and H. F. Zhang, “Visible-light optical coherence tomography oximetry based on circumpapillary scan and graph-search segmentation,” Biomed. Opt. Express 9, 3640 (2018).
[Crossref] [PubMed]

F. G. Venhuizen, B. van Ginneken, B. Liefers, F. van Asten, V. Schreur, S. Fauser, C. Hoyng, T. Theelen, and C. I. Sanchez, “Deep learning approach for the detection and quantification of intraretinal cystoid fluid in multivendor optical coherence tomography,” Biomed. Opt. Express 9, 1545–1569 (2018).
[Crossref] [PubMed]

H. Z. Fu, J. Cheng, Y. W. Xu, D. W. K. Wong, J. Liu, and X. C. Cao, “Joint optic disc and cup segmentation based on multi-label deep network and polar transformation,” IEEE Transactions on Med. Imaging 37, 1597–1605 (2018).
[Crossref]

S. K. Devalla, P. K. Renukanand, B. K. Sreedhar, G. Subramanian, L. Zhang, S. Perera, J. M. Mari, K. S. Chin, T. A. Tun, N. G. Strouthidis, T. Aung, A. H. Thiery, and M. J. A. Girard, “Drunet: a dilated-residual u-net deep learning network to segment optic nerve head tissues in optical coherence tomography images,” Biomed. Opt. Express 9, 3244–3265 (2018).
[Crossref] [PubMed]

M. Gan, C. Wang, T. Yang, N. Yang, M. Zhang, W. Yuan, X. D. Li, and L. R. Wang, “Robust layer segmentation of esophageal OCT images based on graph search using edge-enhanced weights,” Biomed. Opt. Express 9, 4481–4495 (2018).
[Crossref]

2017 (6)

J. L. Zhang, W. Yuan, W. X. Liang, S. Y. Yu, Y. M. Liang, Z. Y. Xu, Y. X. Wei, and X. D. Li, “Automatic and robust segmentation of endoscopic OCT images and optical staining,” Biomed. Opt. Express 8, 2697–2708 (2017).
[Crossref] [PubMed]

L. Y. Fang, D. Cunefare, C. Wang, R. H. Guymer, S. T. Li, and S. Farsiu, “Automatic segmentation of nine retinal layer boundaries in OCT images of non-exudative amd patients using deep learning and graph search,” Biomed. Opt. Express 8, 2732–2744 (2017).
[Crossref] [PubMed]

J. P. McLean, Y. Ling, and C. P. Hendon, “Frequency-constrained robust principal component analysis: a sparse representations approach to segmentation of dynamic features in optical coherence tomography imaging,” Opt. Express 25, 25819–25830 (2017).
[Crossref] [PubMed]

J. K. Zhang, B. M. Williams, S. Lawman, D. Atkinson, Z. J. Zhang, Y. C. Shen, and Y. L. Zheng, “Non-destructive analysis of flake properties in automotive paints with full-field optical coherence tomography and 3d segmentation,” Opt. Express 25, 18614–18628 (2017).
[Crossref] [PubMed]

L. Y. Fang, S. T. Li, D. Cunefare, and S. Farsiu, “Segmentation based sparse reconstruction of optical coherence tomography images,” IEEE Transactions on Med. Imaging 36, 407–421 (2017).
[Crossref]

W. Yuan, R. Brown, W. Mitzner, L. Yarmus, and X. D. Li, “Super-achromatic monolithic microprobe for ultrahigh-resolution endoscopic optical coherence tomography at 800 nm,” Nat. Commun. 8, 1531 (2017).
[Crossref]

2016 (3)

2015 (2)

L. Y. Fang, S. T. Li, X. D. Kang, J. A. Izatt, and S. Farsiu, “3-d adaptive sparsity based image compression with applications to optical coherence tomography,” IEEE Transactions on Med. Imaging 34, 1306–1320 (2015).
[Crossref]

Z. Y. Liu, Y. T. Hu, X. Y. Yu, J. F. Xi, X. M. Fan, C. M. Tse, A. C. Myers, P. J. Pasricha, X. D. Li, and S. Y. Yu, “Allergen challenge sensitizes trpa1 in vagal sensory neurons and afferent c-fiber subtypes in guinea pig esophagus,” Am. J. Physiol. Liver Physiol. 308, G482–G488 (2015).

2014 (3)

Z. Y. Liu, J. F. Xi, M. Tse, A. C. Myers, X. D. Li, P. J. Pasricha, and S. Y. Yu, “Allergic inflammation-induced structural and functional changes in esophageal epithelium in a guinea pig model of eosinophilic esophagitis,” Gastroenterology 146, S92 (2014).
[Crossref]

M. J. Suter, M. J. Gora, G. Y. Lauwers, T. Arnason, J. Sauk, K. A. Gallagher, L. Kava, K. M. Tan, A. R. Soomro, T. P. Gallagher, J. A. Gardecki, B. E. Bouma, M. Rosenberg, N. S. Nishioka, and G. J. Tearney, “Esophageal-guided biopsy with volumetric laser endomicroscopy and laser cautery marking: a pilot clinical study,” Gastrointest Endosc. 79, 886–896 (2014).
[Crossref] [PubMed]

J. F. Xi, A. Q. Zhang, Z. Y. Liu, W. X. Liang, L. Y. Lin, S. Y. Yu, and X. D. Li, “Diffractive catheter for ultrahigh-resolution spectral-domain volumetric OCT imaging,” Opt. Lett. 39, 2016–2019 (2014).
[Crossref] [PubMed]

2013 (1)

2012 (1)

C. Dong and L. F. Tian, “Accelerating relevance-vector-machine-based classification of hyperspectral image with parallel computing,” Math. Probl. Eng. 2018, 252979 (2012).
[Crossref]

2011 (2)

2010 (3)

2009 (1)

A. Yazdanpanah, G. Hamarneh, B. Smith, and M. Sarunic, “Intra-retinal layer segmentation in optical coherence tomography using an active contour approach,” Med. Image Comput. Comput. Interv. - Miccai 2009, Pt Ii, Proc. 5762, 649 (2009).

2008 (2)

M. K. Garvin, M. D. Abramoff, R. Kardon, S. R. Russell, X. D. Wu, and M. Sonka, “Intraretinal layer segmentation of macular optical coherence tomography images using optimal 3-d graph search,” IEEE Transactions on Med. Imaging 27, 1495–1505 (2008).
[Crossref]

V. J. Srinivasan, B. K. Monson, M. Wojtkowski, R. A. Bilonick, I. Gorczynska, R. Chen, J. S. Duker, J. S. Schuman, and J. G. Fujimoto, “Characterization of outer retinal morphology with high-speed, ultrahigh-resolution optical coherence tomography,” Investig. Ophthalmol. & Vis. Sci. 49, 1571–1579 (2008).
[Crossref]

2006 (2)

P. A. Yushkevich, J. Piven, H. C. Hazlett, R. G. Smith, S. Ho, J. C. Gee, and G. Gerig, “User-guided 3D active contour segmentation of anatomical structures: Significantly improved efficiency and reliability,” Neuroimage 31, 1116–1128 (2006).
[Crossref] [PubMed]

S. H. Yun, G. J. Tearney, B. J. Vakoc, M. Shishkov, W. Y. Oh, A. E. Desjardins, M. J. Suter, R. C. Chan, J. A. Evans, I. K. Jang, N. S. Nishioka, J. F. de Boer, and B. E. Bouma, “Comprehensive volumetric optical microscopy in vivo,” Nat. Medicine 12, 1429–1433 (2006).
[Crossref]

2005 (3)

2004 (1)

M. E. Tipping, “Bayesian inference: An introduction to principles and practice in machine learning,” Adv. Lect. on Mach. Learn. 3176, 41–62 (2004).

2001 (3)

D. Koozekanani, K. Boyer, and C. Roberts, “Retinal thickness measurements from optical coherence tomography using a markov boundary model,” IEEE Transactions on Med. Imaging 20, 900–916 (2001).
[Crossref]

J. M. Poneros, S. Brand, B. E. Bouma, G. J. Tearney, C. C. Compton, and N. S. Nishioka, “Diagnosis of specialized intestinal metaplasia by optical coherence tomography,” Gastroenterology 120, 7–12 (2001).
[Crossref] [PubMed]

M. E. Tipping, “Sparse bayesian learning and the relevance vector machine,” J. Mach. Learn. Res. 1, 211–244 (2001).

1998 (1)

Y. Lecun, L. Bottou, Y. Bengio, and P. Haffner, “Gradient-based learning applied to document recognition,” Proc. IEEE 86, 2278–2324 (1998).
[Crossref]

1997 (1)

G. J. Tearney, M. E. Brezinski, B. E. Bouma, S. A. Boppart, C. Pitris, J. F. Southern, and J. G. Fujimoto, “In vivo endoscopic optical biopsy with optical coherence tomography,” Science 276, 2037–2039 (1997).
[Crossref] [PubMed]

1996 (1)

T. S. Lee, “Image representation using 2d gabor wavelets,” IEEE Transactions on Pattern Analysis Mach. Intell. 18, 959–971 (1996).
[Crossref]

1995 (1)

M. R. Hee, J. A. Izatt, E. A. Swanson, D. Huang, J. S. Schuman, C. P. Lin, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography of the human retina,” Arch. Ophthalmol. 113, 325–332 (1995).
[Crossref] [PubMed]

1992 (1)

D. J. C. Mackay, “The evidence framework applied to classification networks,” Neural Comput. 4, 720–736 (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, 1178–1181 (1991).
[Crossref] [PubMed]

Abramoff, M. D.

M. K. Garvin, M. D. Abramoff, R. Kardon, S. R. Russell, X. D. Wu, and M. Sonka, “Intraretinal layer segmentation of macular optical coherence tomography images using optimal 3-d graph search,” IEEE Transactions on Med. Imaging 27, 1495–1505 (2008).
[Crossref]

Akkin, T.

Araie, M.

Arnason, T.

M. J. Suter, M. J. Gora, G. Y. Lauwers, T. Arnason, J. Sauk, K. A. Gallagher, L. Kava, K. M. Tan, A. R. Soomro, T. P. Gallagher, J. A. Gardecki, B. E. Bouma, M. Rosenberg, N. S. Nishioka, and G. J. Tearney, “Esophageal-guided biopsy with volumetric laser endomicroscopy and laser cautery marking: a pilot clinical study,” Gastrointest Endosc. 79, 886–896 (2014).
[Crossref] [PubMed]

Atkinson, D.

Aung, T.

Beaton, S.

H. Ishikawa, D. M. Stein, G. Wollstein, S. Beaton, J. G. Fujimoto, and J. S. Schuman, “Macular segmentation with optical coherence tomography,” Investig. Ophthalmol. & Vis. Sci. 46, 2012–2017 (2005).
[Crossref]

Beckmann, L.

Bengio, Y.

Y. Lecun, L. Bottou, Y. Bengio, and P. Haffner, “Gradient-based learning applied to document recognition,” Proc. IEEE 86, 2278–2324 (1998).
[Crossref]

Bilonick, R. A.

V. J. Srinivasan, B. K. Monson, M. Wojtkowski, R. A. Bilonick, I. Gorczynska, R. Chen, J. S. Duker, J. S. Schuman, and J. G. Fujimoto, “Characterization of outer retinal morphology with high-speed, ultrahigh-resolution optical coherence tomography,” Investig. Ophthalmol. & Vis. Sci. 49, 1571–1579 (2008).
[Crossref]

Boppart, S. A.

G. J. Tearney, M. E. Brezinski, B. E. Bouma, S. A. Boppart, C. Pitris, J. F. Southern, and J. G. Fujimoto, “In vivo endoscopic optical biopsy with optical coherence tomography,” Science 276, 2037–2039 (1997).
[Crossref] [PubMed]

Bottou, L.

Y. Lecun, L. Bottou, Y. Bengio, and P. Haffner, “Gradient-based learning applied to document recognition,” Proc. IEEE 86, 2278–2324 (1998).
[Crossref]

Bouma, B. E.

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G. J. Tearney, M. E. Brezinski, B. E. Bouma, S. A. Boppart, C. Pitris, J. F. Southern, and J. G. Fujimoto, “In vivo endoscopic optical biopsy with optical coherence tomography,” Science 276, 2037–2039 (1997).
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W. Yuan, R. Brown, W. Mitzner, L. Yarmus, and X. D. Li, “Super-achromatic monolithic microprobe for ultrahigh-resolution endoscopic optical coherence tomography at 800 nm,” Nat. Commun. 8, 1531 (2017).
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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).
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V. J. Srinivasan, B. K. Monson, M. Wojtkowski, R. A. Bilonick, I. Gorczynska, R. Chen, J. S. Duker, J. S. Schuman, and J. G. Fujimoto, “Characterization of outer retinal morphology with high-speed, ultrahigh-resolution optical coherence tomography,” Investig. Ophthalmol. & Vis. Sci. 49, 1571–1579 (2008).
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H. Z. Fu, J. Cheng, Y. W. Xu, D. W. K. Wong, J. Liu, and X. C. Cao, “Joint optic disc and cup segmentation based on multi-label deep network and polar transformation,” IEEE Transactions on Med. Imaging 37, 1597–1605 (2018).
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J. M. Poneros, S. Brand, B. E. Bouma, G. J. Tearney, C. C. Compton, and N. S. Nishioka, “Diagnosis of specialized intestinal metaplasia by optical coherence tomography,” Gastroenterology 120, 7–12 (2001).
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L. Y. Fang, S. T. Li, D. Cunefare, and S. Farsiu, “Segmentation based sparse reconstruction of optical coherence tomography images,” IEEE Transactions on Med. Imaging 36, 407–421 (2017).
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L. Y. Fang, D. Cunefare, C. Wang, R. H. Guymer, S. T. Li, and S. Farsiu, “Automatic segmentation of nine retinal layer boundaries in OCT images of non-exudative amd patients using deep learning and graph search,” Biomed. Opt. Express 8, 2732–2744 (2017).
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L. Y. Fang, S. T. Li, D. Cunefare, and S. Farsiu, “Segmentation based sparse reconstruction of optical coherence tomography images,” IEEE Transactions on Med. Imaging 36, 407–421 (2017).
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L. Y. Fang, D. Cunefare, C. Wang, R. H. Guymer, S. T. Li, and S. Farsiu, “Automatic segmentation of nine retinal layer boundaries in OCT images of non-exudative amd patients using deep learning and graph search,” Biomed. Opt. Express 8, 2732–2744 (2017).
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L. Y. Fang, S. T. Li, D. Cunefare, and S. Farsiu, “Segmentation based sparse reconstruction of optical coherence tomography images,” IEEE Transactions on Med. Imaging 36, 407–421 (2017).
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L. Y. Fang, D. Cunefare, C. Wang, R. H. Guymer, S. T. Li, and S. Farsiu, “Automatic segmentation of nine retinal layer boundaries in OCT images of non-exudative amd patients using deep learning and graph search,” Biomed. Opt. Express 8, 2732–2744 (2017).
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L. Y. Fang, S. T. Li, X. D. Kang, J. A. Izatt, and S. Farsiu, “3-d adaptive sparsity based image compression with applications to optical coherence tomography,” IEEE Transactions on Med. Imaging 34, 1306–1320 (2015).
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H. Z. Fu, J. Cheng, Y. W. Xu, D. W. K. Wong, J. Liu, and X. C. Cao, “Joint optic disc and cup segmentation based on multi-label deep network and polar transformation,” IEEE Transactions on Med. Imaging 37, 1597–1605 (2018).
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V. J. Srinivasan, B. K. Monson, M. Wojtkowski, R. A. Bilonick, I. Gorczynska, R. Chen, J. S. Duker, J. S. Schuman, and J. G. Fujimoto, “Characterization of outer retinal morphology with high-speed, ultrahigh-resolution optical coherence tomography,” Investig. Ophthalmol. & Vis. Sci. 49, 1571–1579 (2008).
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V. J. Srinivasan, B. K. Monson, M. Wojtkowski, R. A. Bilonick, I. Gorczynska, R. Chen, J. S. Duker, J. S. Schuman, and J. G. Fujimoto, “Characterization of outer retinal morphology with high-speed, ultrahigh-resolution optical coherence tomography,” Investig. Ophthalmol. & Vis. Sci. 49, 1571–1579 (2008).
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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).
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P. A. Yushkevich, J. Piven, H. C. Hazlett, R. G. Smith, S. Ho, J. C. Gee, and G. Gerig, “User-guided 3D active contour segmentation of anatomical structures: Significantly improved efficiency and reliability,” Neuroimage 31, 1116–1128 (2006).
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M. R. Hee, J. A. Izatt, E. A. Swanson, D. Huang, J. S. Schuman, C. P. Lin, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography of the human retina,” Arch. Ophthalmol. 113, 325–332 (1995).
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P. A. Yushkevich, J. Piven, H. C. Hazlett, R. G. Smith, S. Ho, J. C. Gee, and G. Gerig, “User-guided 3D active contour segmentation of anatomical structures: Significantly improved efficiency and reliability,” Neuroimage 31, 1116–1128 (2006).
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Huang, D.

M. R. Hee, J. A. Izatt, E. A. Swanson, D. Huang, J. S. Schuman, C. P. Lin, C. A. Puliafito, and J. G. Fujimoto, “Optical coherence tomography of the human retina,” Arch. Ophthalmol. 113, 325–332 (1995).
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H. Ishikawa, D. M. Stein, G. Wollstein, S. Beaton, J. G. Fujimoto, and J. S. Schuman, “Macular segmentation with optical coherence tomography,” Investig. Ophthalmol. & Vis. Sci. 46, 2012–2017 (2005).
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L. Y. Fang, S. T. Li, X. D. Kang, J. A. Izatt, and S. Farsiu, “3-d adaptive sparsity based image compression with applications to optical coherence tomography,” IEEE Transactions on Med. Imaging 34, 1306–1320 (2015).
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S. J. Chiu, X. T. Li, P. Nicholas, C. A. Toth, J. A. Izatt, and S. Farsiu, “Automatic segmentation of seven retinal layers in SD-OCT images congruent with expert manual segmentation,” Opt. Express 18, 19413–19428 (2010).
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L. Y. Fang, S. T. Li, X. D. Kang, J. A. Izatt, and S. Farsiu, “3-d adaptive sparsity based image compression with applications to optical coherence tomography,” IEEE Transactions on Med. Imaging 34, 1306–1320 (2015).
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M. K. Garvin, M. D. Abramoff, R. Kardon, S. R. Russell, X. D. Wu, and M. Sonka, “Intraretinal layer segmentation of macular optical coherence tomography images using optimal 3-d graph search,” IEEE Transactions on Med. Imaging 27, 1495–1505 (2008).
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M. J. Suter, M. J. Gora, G. Y. Lauwers, T. Arnason, J. Sauk, K. A. Gallagher, L. Kava, K. M. Tan, A. R. Soomro, T. P. Gallagher, J. A. Gardecki, B. E. Bouma, M. Rosenberg, N. S. Nishioka, and G. J. Tearney, “Esophageal-guided biopsy with volumetric laser endomicroscopy and laser cautery marking: a pilot clinical study,” Gastrointest Endosc. 79, 886–896 (2014).
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D. Koozekanani, K. Boyer, and C. Roberts, “Retinal thickness measurements from optical coherence tomography using a markov boundary model,” IEEE Transactions on Med. Imaging 20, 900–916 (2001).
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M. J. Suter, M. J. Gora, G. Y. Lauwers, T. Arnason, J. Sauk, K. A. Gallagher, L. Kava, K. M. Tan, A. R. Soomro, T. P. Gallagher, J. A. Gardecki, B. E. Bouma, M. Rosenberg, N. S. Nishioka, and G. J. Tearney, “Esophageal-guided biopsy with volumetric laser endomicroscopy and laser cautery marking: a pilot clinical study,” Gastrointest Endosc. 79, 886–896 (2014).
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L. Y. Fang, S. T. Li, X. D. Kang, J. A. Izatt, and S. Farsiu, “3-d adaptive sparsity based image compression with applications to optical coherence tomography,” IEEE Transactions on Med. Imaging 34, 1306–1320 (2015).
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Z. Y. Liu, J. F. Xi, M. Tse, A. C. Myers, X. D. Li, P. J. Pasricha, and S. Y. Yu, “Allergic inflammation-induced structural and functional changes in esophageal epithelium in a guinea pig model of eosinophilic esophagitis,” Gastroenterology 146, S92 (2014).
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Figures (12)

Fig. 1
Fig. 1 A manual segmented esophageal OCT image from the guinea pig.
Fig. 2
Fig. 2 Flowchart of the SBGS algorithm.
Fig. 3
Fig. 3 Plastic sheath removal: (a) position of P 1 and P 2; (b) image with the plastic sheath removed.
Fig. 4
Fig. 4 Demonstration of (a) position of B 1 and (b) the flattened OCT image with marked selected region.
Fig. 5
Fig. 5 Demonstration of intensity averages and gradients at different scales.
Fig. 6
Fig. 6 Demonstration of the Gabor wavelet coefficients.
Fig. 7
Fig. 7 Demonstration of Gabor wavelet features for esophageal OCT images.
Fig. 8
Fig. 8 Flowchart of the boundary identification process.
Fig. 9
Fig. 9 Presentation of the boundary position indicated by sparse Bayesian classification. The probability map has a border painted with the same color as the corresponding boundary shown by the image on top left.
Fig. 10
Fig. 10 Demonstration of (a) SBGS segmentation result for OCT image with regular tissues; (b) comparisons of SBGS (red line) and GTDP (blue line) results; (c) SBGS segmentation result for OCT images with tissue irregularities and (d) comparisons of SBGS(red line) and GTDP (blue line) results.
Fig. 11
Fig. 11 Representative segmentation result by SBGS of (a) Case 3 and (b) Case 4 from the testing healthy guinea pig (Subject 2); (c) Case 7 and (d) Case 8 from the testing EoE model (Subject 4).
Fig. 12
Fig. 12 Statistical results of layer thicknesses from different guinea pigs.

Tables (5)

Tables Icon

Table 1 Information of the dataset used in this study.

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Table 2 Unsigned border position differences of different esophageal OCT image segmentation algorithms.

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Table 3 Computation time for OCT image segmentation.

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Table 4 Comparisons of Nr and the classification time for SVM and sparse Bayesian classifier (denoted by SBC).

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Table 5 The CNN architecture used in this experiment.

Equations (12)

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h d ( z ) = 1 2 d Δ y = 1 2 d 1 1 + 2 d 1 I ( y + Δ y )
g d ( y ) = h d ( y + 2 d 1 ) h d ( y 2 d 1 )
f g a = [ h 0 ( y ) , g 0 ( y ) , h 1 ( y ) , g 1 ( y ) , , h d ( y ) , g d ( y ) ]
W I ( a , θ , x 0 , y 0 ) = a 1 I ( x , y ) ψ θ ( x x 0 a , y y 0 a ) d x d y
G ( x , y ) = f 2 π γ η exp   ( x r 2 + γ 2 y r 2 2 σ 2 ) exp   ( j 2 π f x r + ϕ ) x r = x cos   θ + y sin   θ y r = x sin   θ + y cos   θ
{ G f u , θ v ( x , y ) } f u = f m a x   2 u , u = 0 , 1 , U 1 θ v = v V π , v = 0 , , V 1
  P ( t | w ) = n = 1 N σ { y ( x n ; w ) } t n [ 1 σ { y ( x n ; w ) } ] 1 t n
y ( x ; w ) = i = 1 N w i K ( x , x i ) + w 0
K ( x 1 , x 2 ) = exp   { γ x 1 x 2 2 2 }
p ( w | α ) = i = 0 N N ( w i | 0 , α i 1 )
p ( t ˜ = 1 | w ) = σ { y ( x ˜ ; w ) } = σ [ i = 1 N r w i K ( x , x i ) + w 0 ]
W a b i = 2 ( P a i + P b i ) + w min

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