Abstract

Sparse representation theory is an exciting area of research with recent applications in medical imaging and detection, segmentation, and quantitative analysis of biological processes. We present a variant on the robust-principal component analysis (RPCA) algorithm, called frequency constrained RPCA (FC-RPCA), for selectively segmenting dynamic phenomena that exhibit spectra within a user-defined range of frequencies. The algorithm lacks subjective parameter tuning and demonstrates robust segmentation in datasets containing multiple motion sources and high amplitude noise. When tested on 17 ex-vivo, time lapse optical coherence tomography (OCT) B-scans of human ciliated epithelium, segmentation accuracies ranged between 91–99% and consistently out-performed traditional RPCA.

© 2017 Optical Society of America

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2017 (6)

L. Fang, S. Li, D. Cunefare, and S. Farsiu, “Segmentation Based Sparse Reconstruction of Optical Coherence Tomography Images,” IEEE Transactions on Medical Imaging 36, 407–421 (2017).

Y. Sun, S. Li, and Z. Sun, “Fully automated macular pathology detection in retina optical coherence tomography images using sparse coding and dictionary learning,” J. Biomed. Opt. 22, 016012 (2017).
[Crossref]

K. E. Tipirneni, J. W. Grayson, S. Zhang, D.-Y. Cho, D. F. Skinner, D.-J. Lim, C. Mackey, G. J. Tearney, S. M. Rowe, and B. A. Woodworth, “Assessment of acquired mucociliary clearance defects using micro-optical coherence tomography,” Int. Forum Allergy Rhinol. 00, 1–6 (2017).

Y. Ling, X. Yao, U. A. Gamm, E. Arteaga-Solis, C. W. Emala, M. A. Choma, and C. P. Hendon, “Ex vivo visualization of human ciliated epithelium and quantitative analysis of induced flow dynamics by using optical coherence tomography,” Lasers Surg. Med. 49, 270–279 (2017).
[Crossref] [PubMed]

Y. Ling, X. Yao, and C. P. Hendon, “Highly phase-stable 200 kHz swept-source optical coherence tomography based on KTN electro-optic deflector,” Biomed. Opt. Express 8, 3687 (2017).
[Crossref]

N. C. Lin, C. P. Hendon, and E. S. Olson, “Signal competition in optical coherence tomography and its relevance for cochlear vibrometry,” J. Acoust. Soc. Am. 141, 395–405 (2017).
[Crossref] [PubMed]

2016 (2)

X. Yao, Y. Gan, C. C. Marboe, and C. P. Hendon, “Myocardial imaging using ultrahigh-resolution spectral domain optical coherence tomography,” J. Biomed. Opt. 21, 061006 (2016).
[Crossref]

R. Zareian, M. E. Susilo, J. A. Paten, J. P. McLean, J. Hollmann, D. Karamichos, C. S. Messer, D. T. Tambe, N. Saeidi, J. D. Zieske, and J. W. Ruberti, “Human Corneal Fibroblast Pattern Evolution and Matrix Synthesis on Mechanically Biased Substrates,” Tissue Eng. Part A 22, 1204–1217 (2016).
[Crossref] [PubMed]

2015 (5)

R. Otazo, E. Candès, and D. K. Sodickson, “Low-rank plus sparse matrix decomposition for accelerated dynamic MRI with separation of background and dynamic components,” Magn. Reson. Med. 73, 1125–1136 (2015).
[Crossref]

Y. Jia, S. T. Bailey, T. S. Hwang, S. M. McClintic, S. S. Gao, M. E. Pennesi, C. J. Flaxel, A. K. Lauer, D. J. Wilson, J. Hornegger, J. G. Fujimoto, and D. Huang, “Quantitative optical coherence tomography angiography of vascular abnormalities in the living human eye,” Proc. Natl. Acad. Sci. U. S. A. 112, E2395 (2015).
[Crossref] [PubMed]

S. Wang, J. C. Burton, R. R. Behringer, and I. V. Larina, “In vivo micro-scale tomography of ciliary behavior in the mammalian oviduct,” Sci. Rep. 5, 13216 (2015).
[Crossref] [PubMed]

B. K. Huang, U. A. Gamm, S. Jonas, M. K. Khokha, and M. A. Choma, “Quantitative optical coherence tomography imaging of intermediate flow defect phenotypes in ciliary physiology and pathophysiology,” J. Biomed. Opt. 20, 030502 (2015).
[Crossref] [PubMed]

H. Y. Lee, P. D. Raphael, J. Park, A. K. Ellerbee, B. E. Applegate, and J. S. Oghalai, “Noninvasive in vivo imaging reveals differences between tectorial membrane and basilar membrane traveling waves in the mouse cochlea,” Proc. Natl. Acad. Sci. 112, 3128–3133 (2015).
[Crossref] [PubMed]

2014 (1)

Y. Jia, S. T. Bailey, D. J. Wilson, O. Tan, M. L. Klein, C. J. Flaxel, B. Potsaid, J. J. Liu, C. D. Lu, M. F. Kraus, J. G. Fujimoto, and D. Huang, “Quantitative optical coherence tomography angiography of choroidal neovascularization in age-related macular degeneration,” Ophthalmology 121, 1435–1444 (2014).
[Crossref] [PubMed]

2012 (2)

A. L. Oldenburg, R. K. Chhetri, D. B. Hill, and B. Button, “Monitoring airway mucus flow and ciliary activity with optical coherence tomography,” Biomed. Opt. Express 3, 1978 (2012).
[Crossref] [PubMed]

T.-W. Su, L. Xue, and A. Ozcan, “High-throughput lensfree 3D tracking of human sperms reveals rare statistics of helical trajectories,” Supp. Mat. Proc. Natl. Acad. Sci. U. S. A. 109, 16018–22 (2012).
[Crossref]

2011 (2)

H. Gao, J.-F. Cai, Z. Shen, and H. Zhao, “Robust principal component analysis-based four-dimensional computed tomography,” Phys. Med. Biol. 56, 3181–3198 (2011).
[Crossref] [PubMed]

E. J. Candès, X. Li, Y. Ma, and J. Wright, “Robust principal component analysis?” J. ACM 58, 1–37 (2011).
[Crossref]

2010 (1)

S. Boyd, “Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers,” Found. Trends® Mach. Learn. 3, 1–122 (2010).
[Crossref]

2009 (1)

K. Baker and P. L. Beales, “Making sense of cilia in disease: The human ciliopathies,” Am. J. Med. Genet. Part C Semin. Med. Genet. 151, 281–295 (2009).
[Crossref]

2008 (2)

K. Jaqaman, D. Loerke, M. Mettlen, H. Kuwata, S. Grinstein, S. L. Schmid, and G. Danuser, “Robust single-particle tracking in live-cell time-lapse sequences,” Nat. Methods 5, 695–702 (2008).
[Crossref] [PubMed]

M. Guizar-Sicairos, S. T. Thurman, and J. R. Fienup, “Efficient subpixel image registration algorithms,” Opt. Lett. 33, 156 (2008).
[Crossref] [PubMed]

2005 (1)

2003 (1)

J. Bang, T. Dahl, A. Bruinsma, J. H. Kaspersen, T. A. Nagelhus Hernes, and H. O. Myhre, “A new method for analysis of motion of carotid plaques from RF ultrasound images,” Ultrasound Med. Biol. 29, 967–976 (2003).
[Crossref] [PubMed]

2000 (1)

1999 (1)

W. Tvaruskó, M. Bentele, T. Misteli, R. Rudolf, C. Kaether, D. L. Spector, H. H. Gerdes, and R. Eils, “Time-resolved analysis and visualization of dynamic processes in living cells,” Proc. Natl. Acad. Sci. U. S. A. 96, 7950–7955 (1999).
[Crossref] [PubMed]

1991 (1)

J. Ophir, I. Céspedes, H. Ponnekanti, Y. Yazdi, and X. Li, “Elastography: A quantitative method for imaging the elasticity of biological tissues,” Ultrason. Imaging 13, 111–134 (1991).
[Crossref] [PubMed]

Applegate, B. E.

H. Y. Lee, P. D. Raphael, J. Park, A. K. Ellerbee, B. E. Applegate, and J. S. Oghalai, “Noninvasive in vivo imaging reveals differences between tectorial membrane and basilar membrane traveling waves in the mouse cochlea,” Proc. Natl. Acad. Sci. 112, 3128–3133 (2015).
[Crossref] [PubMed]

Arteaga-Solis, E.

Y. Ling, X. Yao, U. A. Gamm, E. Arteaga-Solis, C. W. Emala, M. A. Choma, and C. P. Hendon, “Ex vivo visualization of human ciliated epithelium and quantitative analysis of induced flow dynamics by using optical coherence tomography,” Lasers Surg. Med. 49, 270–279 (2017).
[Crossref] [PubMed]

Baghaie, A.

A. Baghaie, R. M. D’Souza, and Z. Yu, “Sparse and low rank decomposition based batch image alignment for speckle reduction of retinal OCT images,” Proc. - Int. Symp. Biomed. Imaging2015-July, 226–230 (2015).

Bailey, S. T.

Y. Jia, S. T. Bailey, T. S. Hwang, S. M. McClintic, S. S. Gao, M. E. Pennesi, C. J. Flaxel, A. K. Lauer, D. J. Wilson, J. Hornegger, J. G. Fujimoto, and D. Huang, “Quantitative optical coherence tomography angiography of vascular abnormalities in the living human eye,” Proc. Natl. Acad. Sci. U. S. A. 112, E2395 (2015).
[Crossref] [PubMed]

Y. Jia, S. T. Bailey, D. J. Wilson, O. Tan, M. L. Klein, C. J. Flaxel, B. Potsaid, J. J. Liu, C. D. Lu, M. F. Kraus, J. G. Fujimoto, and D. Huang, “Quantitative optical coherence tomography angiography of choroidal neovascularization in age-related macular degeneration,” Ophthalmology 121, 1435–1444 (2014).
[Crossref] [PubMed]

Baker, K.

K. Baker and P. L. Beales, “Making sense of cilia in disease: The human ciliopathies,” Am. J. Med. Genet. Part C Semin. Med. Genet. 151, 281–295 (2009).
[Crossref]

Bang, J.

J. Bang, T. Dahl, A. Bruinsma, J. H. Kaspersen, T. A. Nagelhus Hernes, and H. O. Myhre, “A new method for analysis of motion of carotid plaques from RF ultrasound images,” Ultrasound Med. Biol. 29, 967–976 (2003).
[Crossref] [PubMed]

Barton, J.

Beales, P. L.

K. Baker and P. L. Beales, “Making sense of cilia in disease: The human ciliopathies,” Am. J. Med. Genet. Part C Semin. Med. Genet. 151, 281–295 (2009).
[Crossref]

Behringer, R. R.

S. Wang, J. C. Burton, R. R. Behringer, and I. V. Larina, “In vivo micro-scale tomography of ciliary behavior in the mammalian oviduct,” Sci. Rep. 5, 13216 (2015).
[Crossref] [PubMed]

Bentele, M.

W. Tvaruskó, M. Bentele, T. Misteli, R. Rudolf, C. Kaether, D. L. Spector, H. H. Gerdes, and R. Eils, “Time-resolved analysis and visualization of dynamic processes in living cells,” Proc. Natl. Acad. Sci. U. S. A. 96, 7950–7955 (1999).
[Crossref] [PubMed]

Boppart, S. a.

D. L. Marks, T. S. Ralston, and S. a. Boppart, “Data Analysis and Signal Postprocessing for Optical Coherence Tomography,” Optical Coherence Tomagraphy (Springer,2008).
[Crossref]

Boyd, S.

S. Boyd, “Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers,” Found. Trends® Mach. Learn. 3, 1–122 (2010).
[Crossref]

Bruinsma, A.

J. Bang, T. Dahl, A. Bruinsma, J. H. Kaspersen, T. A. Nagelhus Hernes, and H. O. Myhre, “A new method for analysis of motion of carotid plaques from RF ultrasound images,” Ultrasound Med. Biol. 29, 967–976 (2003).
[Crossref] [PubMed]

Burton, J. C.

S. Wang, J. C. Burton, R. R. Behringer, and I. V. Larina, “In vivo micro-scale tomography of ciliary behavior in the mammalian oviduct,” Sci. Rep. 5, 13216 (2015).
[Crossref] [PubMed]

Button, B.

Cai, J.-F.

H. Gao, J.-F. Cai, Z. Shen, and H. Zhao, “Robust principal component analysis-based four-dimensional computed tomography,” Phys. Med. Biol. 56, 3181–3198 (2011).
[Crossref] [PubMed]

Candès, E.

R. Otazo, E. Candès, and D. K. Sodickson, “Low-rank plus sparse matrix decomposition for accelerated dynamic MRI with separation of background and dynamic components,” Magn. Reson. Med. 73, 1125–1136 (2015).
[Crossref]

Candès, E. J.

E. J. Candès, X. Li, Y. Ma, and J. Wright, “Robust principal component analysis?” J. ACM 58, 1–37 (2011).
[Crossref]

Céspedes, I.

J. Ophir, I. Céspedes, H. Ponnekanti, Y. Yazdi, and X. Li, “Elastography: A quantitative method for imaging the elasticity of biological tissues,” Ultrason. Imaging 13, 111–134 (1991).
[Crossref] [PubMed]

Chen, Z.

Chhetri, R. K.

Cho, D.-Y.

K. E. Tipirneni, J. W. Grayson, S. Zhang, D.-Y. Cho, D. F. Skinner, D.-J. Lim, C. Mackey, G. J. Tearney, S. M. Rowe, and B. A. Woodworth, “Assessment of acquired mucociliary clearance defects using micro-optical coherence tomography,” Int. Forum Allergy Rhinol. 00, 1–6 (2017).

Choma, M. A.

Y. Ling, X. Yao, U. A. Gamm, E. Arteaga-Solis, C. W. Emala, M. A. Choma, and C. P. Hendon, “Ex vivo visualization of human ciliated epithelium and quantitative analysis of induced flow dynamics by using optical coherence tomography,” Lasers Surg. Med. 49, 270–279 (2017).
[Crossref] [PubMed]

B. K. Huang, U. A. Gamm, S. Jonas, M. K. Khokha, and M. A. Choma, “Quantitative optical coherence tomography imaging of intermediate flow defect phenotypes in ciliary physiology and pathophysiology,” J. Biomed. Opt. 20, 030502 (2015).
[Crossref] [PubMed]

Cunefare, D.

L. Fang, S. Li, D. Cunefare, and S. Farsiu, “Segmentation Based Sparse Reconstruction of Optical Coherence Tomography Images,” IEEE Transactions on Medical Imaging 36, 407–421 (2017).

D’Souza, R. M.

A. Baghaie, R. M. D’Souza, and Z. Yu, “Sparse and low rank decomposition based batch image alignment for speckle reduction of retinal OCT images,” Proc. - Int. Symp. Biomed. Imaging2015-July, 226–230 (2015).

Dabov, K.

K. Dabov, R. Foi, V. Katkovnik, and K. Egiazarian, “BM3D image denoising with shape-adaptive principal component analysis,” Proc. Work. Signal Process. with Adapt. Sparse Struct. Represent. p. 6 (2009).

Dahl, T.

J. Bang, T. Dahl, A. Bruinsma, J. H. Kaspersen, T. A. Nagelhus Hernes, and H. O. Myhre, “A new method for analysis of motion of carotid plaques from RF ultrasound images,” Ultrasound Med. Biol. 29, 967–976 (2003).
[Crossref] [PubMed]

Danuser, G.

K. Jaqaman, D. Loerke, M. Mettlen, H. Kuwata, S. Grinstein, S. L. Schmid, and G. Danuser, “Robust single-particle tracking in live-cell time-lapse sequences,” Nat. Methods 5, 695–702 (2008).
[Crossref] [PubMed]

de Boer, J. F.

Egiazarian, K.

K. Dabov, R. Foi, V. Katkovnik, and K. Egiazarian, “BM3D image denoising with shape-adaptive principal component analysis,” Proc. Work. Signal Process. with Adapt. Sparse Struct. Represent. p. 6 (2009).

Eils, R.

W. Tvaruskó, M. Bentele, T. Misteli, R. Rudolf, C. Kaether, D. L. Spector, H. H. Gerdes, and R. Eils, “Time-resolved analysis and visualization of dynamic processes in living cells,” Proc. Natl. Acad. Sci. U. S. A. 96, 7950–7955 (1999).
[Crossref] [PubMed]

Ellerbee, A. K.

H. Y. Lee, P. D. Raphael, J. Park, A. K. Ellerbee, B. E. Applegate, and J. S. Oghalai, “Noninvasive in vivo imaging reveals differences between tectorial membrane and basilar membrane traveling waves in the mouse cochlea,” Proc. Natl. Acad. Sci. 112, 3128–3133 (2015).
[Crossref] [PubMed]

Emala, C. W.

Y. Ling, X. Yao, U. A. Gamm, E. Arteaga-Solis, C. W. Emala, M. A. Choma, and C. P. Hendon, “Ex vivo visualization of human ciliated epithelium and quantitative analysis of induced flow dynamics by using optical coherence tomography,” Lasers Surg. Med. 49, 270–279 (2017).
[Crossref] [PubMed]

Fang, L.

L. Fang, S. Li, D. Cunefare, and S. Farsiu, “Segmentation Based Sparse Reconstruction of Optical Coherence Tomography Images,” IEEE Transactions on Medical Imaging 36, 407–421 (2017).

Farsiu, S.

L. Fang, S. Li, D. Cunefare, and S. Farsiu, “Segmentation Based Sparse Reconstruction of Optical Coherence Tomography Images,” IEEE Transactions on Medical Imaging 36, 407–421 (2017).

Fienup, J. R.

Flaxel, C. J.

Y. Jia, S. T. Bailey, T. S. Hwang, S. M. McClintic, S. S. Gao, M. E. Pennesi, C. J. Flaxel, A. K. Lauer, D. J. Wilson, J. Hornegger, J. G. Fujimoto, and D. Huang, “Quantitative optical coherence tomography angiography of vascular abnormalities in the living human eye,” Proc. Natl. Acad. Sci. U. S. A. 112, E2395 (2015).
[Crossref] [PubMed]

Y. Jia, S. T. Bailey, D. J. Wilson, O. Tan, M. L. Klein, C. J. Flaxel, B. Potsaid, J. J. Liu, C. D. Lu, M. F. Kraus, J. G. Fujimoto, and D. Huang, “Quantitative optical coherence tomography angiography of choroidal neovascularization in age-related macular degeneration,” Ophthalmology 121, 1435–1444 (2014).
[Crossref] [PubMed]

Foi, R.

K. Dabov, R. Foi, V. Katkovnik, and K. Egiazarian, “BM3D image denoising with shape-adaptive principal component analysis,” Proc. Work. Signal Process. with Adapt. Sparse Struct. Represent. p. 6 (2009).

Fujimoto, J. G.

Y. Jia, S. T. Bailey, T. S. Hwang, S. M. McClintic, S. S. Gao, M. E. Pennesi, C. J. Flaxel, A. K. Lauer, D. J. Wilson, J. Hornegger, J. G. Fujimoto, and D. Huang, “Quantitative optical coherence tomography angiography of vascular abnormalities in the living human eye,” Proc. Natl. Acad. Sci. U. S. A. 112, E2395 (2015).
[Crossref] [PubMed]

Y. Jia, S. T. Bailey, D. J. Wilson, O. Tan, M. L. Klein, C. J. Flaxel, B. Potsaid, J. J. Liu, C. D. Lu, M. F. Kraus, J. G. Fujimoto, and D. Huang, “Quantitative optical coherence tomography angiography of choroidal neovascularization in age-related macular degeneration,” Ophthalmology 121, 1435–1444 (2014).
[Crossref] [PubMed]

Gamm, U. A.

Y. Ling, X. Yao, U. A. Gamm, E. Arteaga-Solis, C. W. Emala, M. A. Choma, and C. P. Hendon, “Ex vivo visualization of human ciliated epithelium and quantitative analysis of induced flow dynamics by using optical coherence tomography,” Lasers Surg. Med. 49, 270–279 (2017).
[Crossref] [PubMed]

B. K. Huang, U. A. Gamm, S. Jonas, M. K. Khokha, and M. A. Choma, “Quantitative optical coherence tomography imaging of intermediate flow defect phenotypes in ciliary physiology and pathophysiology,” J. Biomed. Opt. 20, 030502 (2015).
[Crossref] [PubMed]

Gan, Y.

X. Yao, Y. Gan, C. C. Marboe, and C. P. Hendon, “Myocardial imaging using ultrahigh-resolution spectral domain optical coherence tomography,” J. Biomed. Opt. 21, 061006 (2016).
[Crossref]

Gao, H.

H. Gao, J.-F. Cai, Z. Shen, and H. Zhao, “Robust principal component analysis-based four-dimensional computed tomography,” Phys. Med. Biol. 56, 3181–3198 (2011).
[Crossref] [PubMed]

Gao, S. S.

Y. Jia, S. T. Bailey, T. S. Hwang, S. M. McClintic, S. S. Gao, M. E. Pennesi, C. J. Flaxel, A. K. Lauer, D. J. Wilson, J. Hornegger, J. G. Fujimoto, and D. Huang, “Quantitative optical coherence tomography angiography of vascular abnormalities in the living human eye,” Proc. Natl. Acad. Sci. U. S. A. 112, E2395 (2015).
[Crossref] [PubMed]

Gerdes, H. H.

W. Tvaruskó, M. Bentele, T. Misteli, R. Rudolf, C. Kaether, D. L. Spector, H. H. Gerdes, and R. Eils, “Time-resolved analysis and visualization of dynamic processes in living cells,” Proc. Natl. Acad. Sci. U. S. A. 96, 7950–7955 (1999).
[Crossref] [PubMed]

Grayson, J. W.

K. E. Tipirneni, J. W. Grayson, S. Zhang, D.-Y. Cho, D. F. Skinner, D.-J. Lim, C. Mackey, G. J. Tearney, S. M. Rowe, and B. A. Woodworth, “Assessment of acquired mucociliary clearance defects using micro-optical coherence tomography,” Int. Forum Allergy Rhinol. 00, 1–6 (2017).

Grinstein, S.

K. Jaqaman, D. Loerke, M. Mettlen, H. Kuwata, S. Grinstein, S. L. Schmid, and G. Danuser, “Robust single-particle tracking in live-cell time-lapse sequences,” Nat. Methods 5, 695–702 (2008).
[Crossref] [PubMed]

Guizar-Sicairos, M.

Haldar, J. P.

J. P. Haldar and Z. P. Liang, “Spatiotemporal imaging with partially separable functions: A matrix recovery approach,” 2010 7th IEEE Int. Symp. Biomed. Imaging From Nano to Macro, ISBI 2010 - Proc. pp. 716–719 (2010).

Hendon, C. P.

Y. Ling, X. Yao, U. A. Gamm, E. Arteaga-Solis, C. W. Emala, M. A. Choma, and C. P. Hendon, “Ex vivo visualization of human ciliated epithelium and quantitative analysis of induced flow dynamics by using optical coherence tomography,” Lasers Surg. Med. 49, 270–279 (2017).
[Crossref] [PubMed]

N. C. Lin, C. P. Hendon, and E. S. Olson, “Signal competition in optical coherence tomography and its relevance for cochlear vibrometry,” J. Acoust. Soc. Am. 141, 395–405 (2017).
[Crossref] [PubMed]

Y. Ling, X. Yao, and C. P. Hendon, “Highly phase-stable 200 kHz swept-source optical coherence tomography based on KTN electro-optic deflector,” Biomed. Opt. Express 8, 3687 (2017).
[Crossref]

X. Yao, Y. Gan, C. C. Marboe, and C. P. Hendon, “Myocardial imaging using ultrahigh-resolution spectral domain optical coherence tomography,” J. Biomed. Opt. 21, 061006 (2016).
[Crossref]

Hill, D. B.

Hollmann, J.

R. Zareian, M. E. Susilo, J. A. Paten, J. P. McLean, J. Hollmann, D. Karamichos, C. S. Messer, D. T. Tambe, N. Saeidi, J. D. Zieske, and J. W. Ruberti, “Human Corneal Fibroblast Pattern Evolution and Matrix Synthesis on Mechanically Biased Substrates,” Tissue Eng. Part A 22, 1204–1217 (2016).
[Crossref] [PubMed]

Hornegger, J.

Y. Jia, S. T. Bailey, T. S. Hwang, S. M. McClintic, S. S. Gao, M. E. Pennesi, C. J. Flaxel, A. K. Lauer, D. J. Wilson, J. Hornegger, J. G. Fujimoto, and D. Huang, “Quantitative optical coherence tomography angiography of vascular abnormalities in the living human eye,” Proc. Natl. Acad. Sci. U. S. A. 112, E2395 (2015).
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Huang, B. K.

B. K. Huang, U. A. Gamm, S. Jonas, M. K. Khokha, and M. A. Choma, “Quantitative optical coherence tomography imaging of intermediate flow defect phenotypes in ciliary physiology and pathophysiology,” J. Biomed. Opt. 20, 030502 (2015).
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Huang, D.

Y. Jia, S. T. Bailey, T. S. Hwang, S. M. McClintic, S. S. Gao, M. E. Pennesi, C. J. Flaxel, A. K. Lauer, D. J. Wilson, J. Hornegger, J. G. Fujimoto, and D. Huang, “Quantitative optical coherence tomography angiography of vascular abnormalities in the living human eye,” Proc. Natl. Acad. Sci. U. S. A. 112, E2395 (2015).
[Crossref] [PubMed]

Y. Jia, S. T. Bailey, D. J. Wilson, O. Tan, M. L. Klein, C. J. Flaxel, B. Potsaid, J. J. Liu, C. D. Lu, M. F. Kraus, J. G. Fujimoto, and D. Huang, “Quantitative optical coherence tomography angiography of choroidal neovascularization in age-related macular degeneration,” Ophthalmology 121, 1435–1444 (2014).
[Crossref] [PubMed]

Hwang, T. S.

Y. Jia, S. T. Bailey, T. S. Hwang, S. M. McClintic, S. S. Gao, M. E. Pennesi, C. J. Flaxel, A. K. Lauer, D. J. Wilson, J. Hornegger, J. G. Fujimoto, and D. Huang, “Quantitative optical coherence tomography angiography of vascular abnormalities in the living human eye,” Proc. Natl. Acad. Sci. U. S. A. 112, E2395 (2015).
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Jaqaman, K.

K. Jaqaman, D. Loerke, M. Mettlen, H. Kuwata, S. Grinstein, S. L. Schmid, and G. Danuser, “Robust single-particle tracking in live-cell time-lapse sequences,” Nat. Methods 5, 695–702 (2008).
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Jia, Y.

Y. Jia, S. T. Bailey, T. S. Hwang, S. M. McClintic, S. S. Gao, M. E. Pennesi, C. J. Flaxel, A. K. Lauer, D. J. Wilson, J. Hornegger, J. G. Fujimoto, and D. Huang, “Quantitative optical coherence tomography angiography of vascular abnormalities in the living human eye,” Proc. Natl. Acad. Sci. U. S. A. 112, E2395 (2015).
[Crossref] [PubMed]

Y. Jia, S. T. Bailey, D. J. Wilson, O. Tan, M. L. Klein, C. J. Flaxel, B. Potsaid, J. J. Liu, C. D. Lu, M. F. Kraus, J. G. Fujimoto, and D. Huang, “Quantitative optical coherence tomography angiography of choroidal neovascularization in age-related macular degeneration,” Ophthalmology 121, 1435–1444 (2014).
[Crossref] [PubMed]

Jonas, S.

B. K. Huang, U. A. Gamm, S. Jonas, M. K. Khokha, and M. A. Choma, “Quantitative optical coherence tomography imaging of intermediate flow defect phenotypes in ciliary physiology and pathophysiology,” J. Biomed. Opt. 20, 030502 (2015).
[Crossref] [PubMed]

Kaether, C.

W. Tvaruskó, M. Bentele, T. Misteli, R. Rudolf, C. Kaether, D. L. Spector, H. H. Gerdes, and R. Eils, “Time-resolved analysis and visualization of dynamic processes in living cells,” Proc. Natl. Acad. Sci. U. S. A. 96, 7950–7955 (1999).
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Karamichos, D.

R. Zareian, M. E. Susilo, J. A. Paten, J. P. McLean, J. Hollmann, D. Karamichos, C. S. Messer, D. T. Tambe, N. Saeidi, J. D. Zieske, and J. W. Ruberti, “Human Corneal Fibroblast Pattern Evolution and Matrix Synthesis on Mechanically Biased Substrates,” Tissue Eng. Part A 22, 1204–1217 (2016).
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Kaspersen, J. H.

J. Bang, T. Dahl, A. Bruinsma, J. H. Kaspersen, T. A. Nagelhus Hernes, and H. O. Myhre, “A new method for analysis of motion of carotid plaques from RF ultrasound images,” Ultrasound Med. Biol. 29, 967–976 (2003).
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Khokha, M. K.

B. K. Huang, U. A. Gamm, S. Jonas, M. K. Khokha, and M. A. Choma, “Quantitative optical coherence tomography imaging of intermediate flow defect phenotypes in ciliary physiology and pathophysiology,” J. Biomed. Opt. 20, 030502 (2015).
[Crossref] [PubMed]

Klein, M. L.

Y. Jia, S. T. Bailey, D. J. Wilson, O. Tan, M. L. Klein, C. J. Flaxel, B. Potsaid, J. J. Liu, C. D. Lu, M. F. Kraus, J. G. Fujimoto, and D. Huang, “Quantitative optical coherence tomography angiography of choroidal neovascularization in age-related macular degeneration,” Ophthalmology 121, 1435–1444 (2014).
[Crossref] [PubMed]

Kraus, M. F.

Y. Jia, S. T. Bailey, D. J. Wilson, O. Tan, M. L. Klein, C. J. Flaxel, B. Potsaid, J. J. Liu, C. D. Lu, M. F. Kraus, J. G. Fujimoto, and D. Huang, “Quantitative optical coherence tomography angiography of choroidal neovascularization in age-related macular degeneration,” Ophthalmology 121, 1435–1444 (2014).
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Kuwata, H.

K. Jaqaman, D. Loerke, M. Mettlen, H. Kuwata, S. Grinstein, S. L. Schmid, and G. Danuser, “Robust single-particle tracking in live-cell time-lapse sequences,” Nat. Methods 5, 695–702 (2008).
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S. Wang, J. C. Burton, R. R. Behringer, and I. V. Larina, “In vivo micro-scale tomography of ciliary behavior in the mammalian oviduct,” Sci. Rep. 5, 13216 (2015).
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Lauer, A. K.

Y. Jia, S. T. Bailey, T. S. Hwang, S. M. McClintic, S. S. Gao, M. E. Pennesi, C. J. Flaxel, A. K. Lauer, D. J. Wilson, J. Hornegger, J. G. Fujimoto, and D. Huang, “Quantitative optical coherence tomography angiography of vascular abnormalities in the living human eye,” Proc. Natl. Acad. Sci. U. S. A. 112, E2395 (2015).
[Crossref] [PubMed]

Lee, H. Y.

H. Y. Lee, P. D. Raphael, J. Park, A. K. Ellerbee, B. E. Applegate, and J. S. Oghalai, “Noninvasive in vivo imaging reveals differences between tectorial membrane and basilar membrane traveling waves in the mouse cochlea,” Proc. Natl. Acad. Sci. 112, 3128–3133 (2015).
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Li, S.

Y. Sun, S. Li, and Z. Sun, “Fully automated macular pathology detection in retina optical coherence tomography images using sparse coding and dictionary learning,” J. Biomed. Opt. 22, 016012 (2017).
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L. Fang, S. Li, D. Cunefare, and S. Farsiu, “Segmentation Based Sparse Reconstruction of Optical Coherence Tomography Images,” IEEE Transactions on Medical Imaging 36, 407–421 (2017).

Li, X.

E. J. Candès, X. Li, Y. Ma, and J. Wright, “Robust principal component analysis?” J. ACM 58, 1–37 (2011).
[Crossref]

J. Ophir, I. Céspedes, H. Ponnekanti, Y. Yazdi, and X. Li, “Elastography: A quantitative method for imaging the elasticity of biological tissues,” Ultrason. Imaging 13, 111–134 (1991).
[Crossref] [PubMed]

Liang, Z. P.

J. P. Haldar and Z. P. Liang, “Spatiotemporal imaging with partially separable functions: A matrix recovery approach,” 2010 7th IEEE Int. Symp. Biomed. Imaging From Nano to Macro, ISBI 2010 - Proc. pp. 716–719 (2010).

Lim, D.-J.

K. E. Tipirneni, J. W. Grayson, S. Zhang, D.-Y. Cho, D. F. Skinner, D.-J. Lim, C. Mackey, G. J. Tearney, S. M. Rowe, and B. A. Woodworth, “Assessment of acquired mucociliary clearance defects using micro-optical coherence tomography,” Int. Forum Allergy Rhinol. 00, 1–6 (2017).

Lin, N. C.

N. C. Lin, C. P. Hendon, and E. S. Olson, “Signal competition in optical coherence tomography and its relevance for cochlear vibrometry,” J. Acoust. Soc. Am. 141, 395–405 (2017).
[Crossref] [PubMed]

Ling, Y.

Y. Ling, X. Yao, and C. P. Hendon, “Highly phase-stable 200 kHz swept-source optical coherence tomography based on KTN electro-optic deflector,” Biomed. Opt. Express 8, 3687 (2017).
[Crossref]

Y. Ling, X. Yao, U. A. Gamm, E. Arteaga-Solis, C. W. Emala, M. A. Choma, and C. P. Hendon, “Ex vivo visualization of human ciliated epithelium and quantitative analysis of induced flow dynamics by using optical coherence tomography,” Lasers Surg. Med. 49, 270–279 (2017).
[Crossref] [PubMed]

Liu, J. J.

Y. Jia, S. T. Bailey, D. J. Wilson, O. Tan, M. L. Klein, C. J. Flaxel, B. Potsaid, J. J. Liu, C. D. Lu, M. F. Kraus, J. G. Fujimoto, and D. Huang, “Quantitative optical coherence tomography angiography of choroidal neovascularization in age-related macular degeneration,” Ophthalmology 121, 1435–1444 (2014).
[Crossref] [PubMed]

Loerke, D.

K. Jaqaman, D. Loerke, M. Mettlen, H. Kuwata, S. Grinstein, S. L. Schmid, and G. Danuser, “Robust single-particle tracking in live-cell time-lapse sequences,” Nat. Methods 5, 695–702 (2008).
[Crossref] [PubMed]

Lu, C. D.

Y. Jia, S. T. Bailey, D. J. Wilson, O. Tan, M. L. Klein, C. J. Flaxel, B. Potsaid, J. J. Liu, C. D. Lu, M. F. Kraus, J. G. Fujimoto, and D. Huang, “Quantitative optical coherence tomography angiography of choroidal neovascularization in age-related macular degeneration,” Ophthalmology 121, 1435–1444 (2014).
[Crossref] [PubMed]

Ma, Y.

E. J. Candès, X. Li, Y. Ma, and J. Wright, “Robust principal component analysis?” J. ACM 58, 1–37 (2011).
[Crossref]

Mackey, C.

K. E. Tipirneni, J. W. Grayson, S. Zhang, D.-Y. Cho, D. F. Skinner, D.-J. Lim, C. Mackey, G. J. Tearney, S. M. Rowe, and B. A. Woodworth, “Assessment of acquired mucociliary clearance defects using micro-optical coherence tomography,” Int. Forum Allergy Rhinol. 00, 1–6 (2017).

Marboe, C. C.

X. Yao, Y. Gan, C. C. Marboe, and C. P. Hendon, “Myocardial imaging using ultrahigh-resolution spectral domain optical coherence tomography,” J. Biomed. Opt. 21, 061006 (2016).
[Crossref]

Marks, D. L.

D. L. Marks, T. S. Ralston, and S. a. Boppart, “Data Analysis and Signal Postprocessing for Optical Coherence Tomography,” Optical Coherence Tomagraphy (Springer,2008).
[Crossref]

McClintic, S. M.

Y. Jia, S. T. Bailey, T. S. Hwang, S. M. McClintic, S. S. Gao, M. E. Pennesi, C. J. Flaxel, A. K. Lauer, D. J. Wilson, J. Hornegger, J. G. Fujimoto, and D. Huang, “Quantitative optical coherence tomography angiography of vascular abnormalities in the living human eye,” Proc. Natl. Acad. Sci. U. S. A. 112, E2395 (2015).
[Crossref] [PubMed]

McLean, J. P.

R. Zareian, M. E. Susilo, J. A. Paten, J. P. McLean, J. Hollmann, D. Karamichos, C. S. Messer, D. T. Tambe, N. Saeidi, J. D. Zieske, and J. W. Ruberti, “Human Corneal Fibroblast Pattern Evolution and Matrix Synthesis on Mechanically Biased Substrates,” Tissue Eng. Part A 22, 1204–1217 (2016).
[Crossref] [PubMed]

Messer, C. S.

R. Zareian, M. E. Susilo, J. A. Paten, J. P. McLean, J. Hollmann, D. Karamichos, C. S. Messer, D. T. Tambe, N. Saeidi, J. D. Zieske, and J. W. Ruberti, “Human Corneal Fibroblast Pattern Evolution and Matrix Synthesis on Mechanically Biased Substrates,” Tissue Eng. Part A 22, 1204–1217 (2016).
[Crossref] [PubMed]

Mettlen, M.

K. Jaqaman, D. Loerke, M. Mettlen, H. Kuwata, S. Grinstein, S. L. Schmid, and G. Danuser, “Robust single-particle tracking in live-cell time-lapse sequences,” Nat. Methods 5, 695–702 (2008).
[Crossref] [PubMed]

Misteli, T.

W. Tvaruskó, M. Bentele, T. Misteli, R. Rudolf, C. Kaether, D. L. Spector, H. H. Gerdes, and R. Eils, “Time-resolved analysis and visualization of dynamic processes in living cells,” Proc. Natl. Acad. Sci. U. S. A. 96, 7950–7955 (1999).
[Crossref] [PubMed]

Myhre, H. O.

J. Bang, T. Dahl, A. Bruinsma, J. H. Kaspersen, T. A. Nagelhus Hernes, and H. O. Myhre, “A new method for analysis of motion of carotid plaques from RF ultrasound images,” Ultrasound Med. Biol. 29, 967–976 (2003).
[Crossref] [PubMed]

Nagelhus Hernes, T. A.

J. Bang, T. Dahl, A. Bruinsma, J. H. Kaspersen, T. A. Nagelhus Hernes, and H. O. Myhre, “A new method for analysis of motion of carotid plaques from RF ultrasound images,” Ultrasound Med. Biol. 29, 967–976 (2003).
[Crossref] [PubMed]

Nelson, J. S.

Oghalai, J. S.

H. Y. Lee, P. D. Raphael, J. Park, A. K. Ellerbee, B. E. Applegate, and J. S. Oghalai, “Noninvasive in vivo imaging reveals differences between tectorial membrane and basilar membrane traveling waves in the mouse cochlea,” Proc. Natl. Acad. Sci. 112, 3128–3133 (2015).
[Crossref] [PubMed]

Oldenburg, A. L.

Olson, E. S.

N. C. Lin, C. P. Hendon, and E. S. Olson, “Signal competition in optical coherence tomography and its relevance for cochlear vibrometry,” J. Acoust. Soc. Am. 141, 395–405 (2017).
[Crossref] [PubMed]

Ophir, J.

J. Ophir, I. Céspedes, H. Ponnekanti, Y. Yazdi, and X. Li, “Elastography: A quantitative method for imaging the elasticity of biological tissues,” Ultrason. Imaging 13, 111–134 (1991).
[Crossref] [PubMed]

Otazo, R.

R. Otazo, E. Candès, and D. K. Sodickson, “Low-rank plus sparse matrix decomposition for accelerated dynamic MRI with separation of background and dynamic components,” Magn. Reson. Med. 73, 1125–1136 (2015).
[Crossref]

Ozcan, A.

T.-W. Su, L. Xue, and A. Ozcan, “High-throughput lensfree 3D tracking of human sperms reveals rare statistics of helical trajectories,” Supp. Mat. Proc. Natl. Acad. Sci. U. S. A. 109, 16018–22 (2012).
[Crossref]

Park, J.

H. Y. Lee, P. D. Raphael, J. Park, A. K. Ellerbee, B. E. Applegate, and J. S. Oghalai, “Noninvasive in vivo imaging reveals differences between tectorial membrane and basilar membrane traveling waves in the mouse cochlea,” Proc. Natl. Acad. Sci. 112, 3128–3133 (2015).
[Crossref] [PubMed]

Paten, J. A.

R. Zareian, M. E. Susilo, J. A. Paten, J. P. McLean, J. Hollmann, D. Karamichos, C. S. Messer, D. T. Tambe, N. Saeidi, J. D. Zieske, and J. W. Ruberti, “Human Corneal Fibroblast Pattern Evolution and Matrix Synthesis on Mechanically Biased Substrates,” Tissue Eng. Part A 22, 1204–1217 (2016).
[Crossref] [PubMed]

Pennesi, M. E.

Y. Jia, S. T. Bailey, T. S. Hwang, S. M. McClintic, S. S. Gao, M. E. Pennesi, C. J. Flaxel, A. K. Lauer, D. J. Wilson, J. Hornegger, J. G. Fujimoto, and D. Huang, “Quantitative optical coherence tomography angiography of vascular abnormalities in the living human eye,” Proc. Natl. Acad. Sci. U. S. A. 112, E2395 (2015).
[Crossref] [PubMed]

Ponnekanti, H.

J. Ophir, I. Céspedes, H. Ponnekanti, Y. Yazdi, and X. Li, “Elastography: A quantitative method for imaging the elasticity of biological tissues,” Ultrason. Imaging 13, 111–134 (1991).
[Crossref] [PubMed]

Potsaid, B.

Y. Jia, S. T. Bailey, D. J. Wilson, O. Tan, M. L. Klein, C. J. Flaxel, B. Potsaid, J. J. Liu, C. D. Lu, M. F. Kraus, J. G. Fujimoto, and D. Huang, “Quantitative optical coherence tomography angiography of choroidal neovascularization in age-related macular degeneration,” Ophthalmology 121, 1435–1444 (2014).
[Crossref] [PubMed]

Ralston, T. S.

D. L. Marks, T. S. Ralston, and S. a. Boppart, “Data Analysis and Signal Postprocessing for Optical Coherence Tomography,” Optical Coherence Tomagraphy (Springer,2008).
[Crossref]

Raphael, P. D.

H. Y. Lee, P. D. Raphael, J. Park, A. K. Ellerbee, B. E. Applegate, and J. S. Oghalai, “Noninvasive in vivo imaging reveals differences between tectorial membrane and basilar membrane traveling waves in the mouse cochlea,” Proc. Natl. Acad. Sci. 112, 3128–3133 (2015).
[Crossref] [PubMed]

Rowe, S. M.

K. E. Tipirneni, J. W. Grayson, S. Zhang, D.-Y. Cho, D. F. Skinner, D.-J. Lim, C. Mackey, G. J. Tearney, S. M. Rowe, and B. A. Woodworth, “Assessment of acquired mucociliary clearance defects using micro-optical coherence tomography,” Int. Forum Allergy Rhinol. 00, 1–6 (2017).

Ruberti, J. W.

R. Zareian, M. E. Susilo, J. A. Paten, J. P. McLean, J. Hollmann, D. Karamichos, C. S. Messer, D. T. Tambe, N. Saeidi, J. D. Zieske, and J. W. Ruberti, “Human Corneal Fibroblast Pattern Evolution and Matrix Synthesis on Mechanically Biased Substrates,” Tissue Eng. Part A 22, 1204–1217 (2016).
[Crossref] [PubMed]

Rudolf, R.

W. Tvaruskó, M. Bentele, T. Misteli, R. Rudolf, C. Kaether, D. L. Spector, H. H. Gerdes, and R. Eils, “Time-resolved analysis and visualization of dynamic processes in living cells,” Proc. Natl. Acad. Sci. U. S. A. 96, 7950–7955 (1999).
[Crossref] [PubMed]

Saeidi, N.

R. Zareian, M. E. Susilo, J. A. Paten, J. P. McLean, J. Hollmann, D. Karamichos, C. S. Messer, D. T. Tambe, N. Saeidi, J. D. Zieske, and J. W. Ruberti, “Human Corneal Fibroblast Pattern Evolution and Matrix Synthesis on Mechanically Biased Substrates,” Tissue Eng. Part A 22, 1204–1217 (2016).
[Crossref] [PubMed]

Saxer, C.

Schmid, S. L.

K. Jaqaman, D. Loerke, M. Mettlen, H. Kuwata, S. Grinstein, S. L. Schmid, and G. Danuser, “Robust single-particle tracking in live-cell time-lapse sequences,” Nat. Methods 5, 695–702 (2008).
[Crossref] [PubMed]

Shen, Q.

Shen, Z.

H. Gao, J.-F. Cai, Z. Shen, and H. Zhao, “Robust principal component analysis-based four-dimensional computed tomography,” Phys. Med. Biol. 56, 3181–3198 (2011).
[Crossref] [PubMed]

Skinner, D. F.

K. E. Tipirneni, J. W. Grayson, S. Zhang, D.-Y. Cho, D. F. Skinner, D.-J. Lim, C. Mackey, G. J. Tearney, S. M. Rowe, and B. A. Woodworth, “Assessment of acquired mucociliary clearance defects using micro-optical coherence tomography,” Int. Forum Allergy Rhinol. 00, 1–6 (2017).

Sodickson, D. K.

R. Otazo, E. Candès, and D. K. Sodickson, “Low-rank plus sparse matrix decomposition for accelerated dynamic MRI with separation of background and dynamic components,” Magn. Reson. Med. 73, 1125–1136 (2015).
[Crossref]

Spector, D. L.

W. Tvaruskó, M. Bentele, T. Misteli, R. Rudolf, C. Kaether, D. L. Spector, H. H. Gerdes, and R. Eils, “Time-resolved analysis and visualization of dynamic processes in living cells,” Proc. Natl. Acad. Sci. U. S. A. 96, 7950–7955 (1999).
[Crossref] [PubMed]

Stromski, S.

Su, T.-W.

T.-W. Su, L. Xue, and A. Ozcan, “High-throughput lensfree 3D tracking of human sperms reveals rare statistics of helical trajectories,” Supp. Mat. Proc. Natl. Acad. Sci. U. S. A. 109, 16018–22 (2012).
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Supplementary Material (3)

NameDescription
» Visualization 1       FC-RPCA sparse component output (color) overlayed on time-varying OCT B-scan (gray). FC-RPCA image processed using 3 - 14 Hz frequency constraint.
» Visualization 2       FC-RPCA sparse component output (color) overlayed on time-varying OCT B-scan (gray). FC-RPCA image processed using frequency constraint that only rejects DC component. Noise and other unwanted features in FC-RPCA result are clearly visible.
» Visualization 3       Differences between FC-RPCA and Speckle Variance. Left image is the FC-RPCA overlay (the same image shown in manuscript Fig. 2(c)) and the right is the Speckle Variance overlay for the same dataset.

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Figures (3)

Fig. 1
Fig. 1

Visualization of the input and output variables for the proposed Frequency Constrained RPCA algorithm. Inputs (blue box) include an image stack of time-lapse B-scans (Y), two scalar weights (λ, µ) which provide tuning of the 1 and 2 regularizers, respectively, and the indicator set (Γ) which allows the user to select frequencies to reject from the sparse component during the decomposition. The algorithm outputs (orange box) are two image stacks (L, S) of the same size as input Y that correspond to the low-rank background and sparse foreground, respectively.

Fig. 2
Fig. 2

(a) Ex-vivo OCT B-scan image of ciliated epithelium from human trachea (Sample 5) and (b) corresponding histology. (c) The Maximum Intensity Projection of the FC-RPCA sparse output painted over the image in (a) using MATLAB’s jet colormap. (d) A closer look at a dense area of cilia located directly under a mucus cloud from (c). In contrast, (f) examines a sparsely populated area of cilia where the cells may be damaged or non-functioning. (e) and (g) are closer views of the histology corresponding to (d) and (f), respectively. The locations of these two regions are marked with the corresponding figure letter in the full view histology (b) and OCT B-scan (c) images. Colorbar indicates normalized intensity of the FC-RPCA sparse component. The arrows and corresponding labels mark key physiological regions (CE = ciliated epithelium, BM = basement membrane, M = mucus).

Fig. 3
Fig. 3

Comparison between FC-RPCA sparse components created using two different sets of frequency constraints on data from Sample 1. (a) Single OCT B-scan before FC-RPCA processing. (b) Cilia region segmented using frequencies constrained between 3 and 14 Hz (see Visualization 1). (c) Cilia region segmented with only rejection of the DC component (see Visualization 2). Arrows in (a) and (c) point to a bright mucus cloud that appears in the unconstrained output (c), but was successfully rejected from the frequency constrained output (b). Additionally, many bright pixels in (c) associated with undesirable noise from static tissue were rejected in (b) due to the frequency constraint. The arrows and corresponding labels mark key physiological features (CE = ciliated epithelium, M = mucus).

Tables (2)

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Table 1 Frequency-Constrained RPCA Algorithm

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Table 2 Comparison of average cilia segmentation accuracy across datasets for multiple frequency constraint conditions

Equations (15)

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minimize L * + λ S 1 subject to L + S = Y
minimize L * + λ S F T 1 subject to L + S = Y , S F T Γ = 0
minimize L * + λ S ¯ F T 1 subject to L + S = Y , S F T Γ = 0 , S ¯ S = 0
g ( ν , ρ ) = inf L , S μ ( L , S , ν , ρ )
μ ( L , S , S ¯ , ν , γ , ρ ) = L * + λ S ¯ F T 1 + L + S Y , ν + S ¯ S , γ + S F T Γ , ρ + μ 2 ( L + S Y F 2 + S ¯ S F 2 + S F T Γ F 2 )
L ( k + 1 ) arg min L μ ( L , S ( k ) , S ¯ ( k ) , ν ( k ) , γ ( k ) , ρ ( k ) )
S ¯ ( k + 1 ) arg min S ¯ μ ( L ( k + 1 ) , S ( k ) , S ¯ , ν ( k ) , γ ( k ) , ρ ( k ) )
S ( k + 1 ) arg min S μ ( L ( k + 1 ) , S , S ¯ ( k + 1 ) , ν ( k ) , γ ( k ) , ρ ( k ) )
ν ( k + 1 ) = ν ( k ) + μ ( L ( k + 1 ) + S ( k + 1 ) Y )
γ ( k + 1 ) = γ ( k ) + μ ( S ¯ ( k + 1 ) S ( k + 1 ) )
ρ ( k + 1 ) = ρ ( k ) + μ ( S ( k + 1 ) F T Γ )
prox h / μ = arg min x { h ( x ) + μ 2 x w F 2 }
L ( k + 1 ) = arg min L L * + L + S Y , ν + μ 2 L + S Y F 2 = prox * ( Y S ( k ) μ 1 ν ( k ) )
S ¯ ( k + 1 ) = arg min S ¯ λ S ¯ F T 1 + S ¯ S , γ + μ 2 S ¯ S F 2 = arg min S ¯ λ S ¯ F T 1 + μ 2 n S ¯ F T ( S μ 1 ν ) F T F 2 = prox n λ 1 ( [ S ( k ) μ 1 γ ( k ) ] F T ) ( F T ) 1
S ( k + 1 ) = μ 1 [ ( γ ( k ) ν ( k ) ρ ( k ) Γ * ( F T ) * ) + ( Y L ( k + 1 ) ) + S ¯ ( k + 1 ) ] × [ 2 I + ( F T Γ Γ * ( F T ) * ) ] 1