Topological reconstruction of sub-cellular motion with Ensemble Kalman velocimetry

Le Yin; Ioannis Sgouralis; Vasileios Maroulas

doi:10.3934/fods.2020007

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June 2020, 2(2): 101-121. doi: 10.3934/fods.2020007

Topological reconstruction of sub-cellular motion with Ensemble Kalman velocimetry

Le Yin ^1, , Ioannis Sgouralis ^2, and Vasileios Maroulas ^1,,

1.	Mathematics Dept., University of Tennessee, Knoxville, TN 37996, USA
2.	Center for Biological Physics, Arizona State University, Tempe, AZ 85281, USA

^* Corresponding author: vmaroula@utk.edu

Published June 2020

Microscopy imaging of plant cells allows the elaborate analysis of sub-cellular motions of organelles. The large video data set can be efficiently analyzed by automated algorithms. We develop a novel, data-oriented algorithm, which can track organelle movements and reconstruct their trajectories on stacks of image data. Our method proceeds with three steps: (ⅰ) identification, (ⅱ) localization, and (ⅲ) linking. This method combines topological data analysis and Ensemble Kalman Filtering, and does not assume a specific motion model. Application of this method on simulated data sets shows an agreement with ground truth. We also successfully test our method on real microscopy data.

Keywords: Multi-target tracking, velocimetry, topological data analysis, ensemble Kalman filter, fluorescence microscopy.

Mathematics Subject Classification: Primary: 62M20, 62R40; Secondary: 62H35.

Citation: Le Yin, Ioannis Sgouralis, Vasileios Maroulas. Topological reconstruction of sub-cellular motion with Ensemble Kalman velocimetry. Foundations of Data Science, 2020, 2 (2) : 101-121. doi: 10.3934/fods.2020007

References:

[1]	P. Bendich, S. P. Chin, J. Clark, J. Desena, J. Harer, E. Munch, A. Newman, D. Porter, D. Rouse and N. Strawn et al., Topological and statistical behavior classifiers for tracking applications, IEEE Transactions on Aerospace and Electronic Systems, 52 (2016), 2644-2661.
[2]	G. Bishop, An Introduction to the Kalman Filter, Technical report, TR 95-041, Department of Computer Science, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3175, Monday, 2006.
[3]	S. S. Blackman, Multiple-target Tracking with Radar Applications, Dedham, MA, Artech House, Inc., 1986.
[4]	S. S. v. Braun and E. Schleiff, Movement of endosymbiotic organelles, Current Protein and Peptide Science, 8 (2007), 426-438.
[5]	G. Cai, L. Parrotta and M. Cresti, Organelle trafficking, the cytoskeleton, and pollen tube growth, Journal of Integrative Plant biology, 57 (2015), 63-78.
[6]	G. Carlsson, Topology and data, Bull. Amer. Math. Soc. (N.S.), 46 (2009), 255-308. doi: 10.1090/S0273-0979-09-01249-X.
[7]	G. Casella and R. L. Berger, Statistical Inference, The Wadsworth & Brooks/Cole Statistics/Probability Series. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1990.
[8]	N. Chenouard, I. Bloch and J.-C. Olivo-Marin, Multiple hypothesis tracking for cluttered biological image sequences, IEEE Transactions on Pattern Analysis and Machine Intelligence, 35 (2013), 2736-3750.
[9]	D. A. Collings, J. D. Harper, J. Marc, R. L. Overall and R. T. Mullen, Life in the fast lane: Actin-based motility of plant peroxisomes, Canadian Journal of Botany, 80 (2002), 430-441.
[10]	G. Danuser, Computer vision in cell biology, Cell, 147 (2011), 973-978.
[11]	J. Derksen, Pollen tubes: A Model system for plant cell growth, Botanica Acta, 109 (1996), 341-345.
[12]	H. Edelsbrunner and J. L. Harer, Computational Topology. An Introduction, American Mathematical Soc., Providence, RI, 2010.
[13]	G. B. Folland, Real Analysis: Modern Techniques and Their Applications, Pure and Applied Mathematics (New York). A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1999.
[14]	A. Gelman, J. B. Carlin, H. S. Stern, D. B. Dunson, A. Vehtari and D. B. Rubin, Bayesian Data Analysis, Texts in Statistical Science Series. CRC Press, Boca Raton, FL, 2014.
[15]	R. Gutierrez, J. J. Lindeboom, A. R. Paredez, A. M. C. Emons and D. W. Ehrhardt, Arabidopsis cortical microtubules position cellulose synthase delivery to the plasma membrane and interact with cellulose synthase trafficking compartments, Nature Cell Biology, 11 (2009), 797.
[16]	T. Hamada, M. Tominaga, T. Fukaya, M. Nakamura, A. Nakano, Y. Watanabe, T. Hashimoto and T. I. Baskin, Rna processing bodies, peroxisomes, golgi bodies, mitochondria, and endoplasmic reticulum tubule junctions frequently pause at cortical microtubules, Plant and Cell Physiology, 53 (2012), 699-708.
[17]	B. Herman and D. F. Albertini, A time-lapse video image intensification analysis of cytoplasmic organelle movements during endosome translocation, The Journal of Cell Biology, 98 (1984), 565-576.
[18]	M. Hirsch, R. J. Wareham, M. L. Martin-Fernandez, M. P. Hobson and D. J. Rolfe, A stochastic model for electron multiplication charge-coupled devices–from theory to practice, PloS One, 8 (2013), e53671.
[19]	B. Huang, M. Bates and X. Zhuang, Super-resolution fluorescence microscopy, Annual Review of Biochemistry, 78 (2009), 993-1016.
[20]	F. Huang, T. M. Hartwich, F. E. Rivera-Molina, Y. Lin, W. C. Duim, J. J. Long, P. D. Uchil, J. R. Myers, M. A. Baird, W. Mothes et al., Video-rate nanoscopy using sCMOS camera–specific single-molecule localization algorithms, Nature Methods, 10 (2013), 653.
[21]	A. Jaiswal, W. J. Godinez, M. J. Lehmann and K. Rohr, Direct combination of multi-scale detection and multi-frame association for tracking of virus particles in microscopy image data, in 2016 IEEE 13th International Symposium on Biomedical Imaging (ISBI), IEEE, 2016,976–979.
[22]	J. Janesick and T. Elliott, History and advancement of large array scientific ccd imagers, in Astronomical CCD Observing and Reduction Techniques, 23 (1992), 1.
[23]	S. Jazani, I. Sgouralis and S. Pressé, A method for single molecule tracking using a conventional single-focus confocal setup, The Journal of Chemical Physics, 150 (2019), 114108.
[24]	S. Jazani, I. Sgouralis, O. M. Shafraz, M. Levitus, S. Sivasankar and S. Pressé, An alternative framework for fluorescence correlation spectroscopy, Nature Communications, 10.
[25]	K. Kang, V. Maroulas, I. Schizas and F. Bao, Improved distributed particle filters for tracking in a wireless sensor network, Comput. Statist. Data Anal., 117 (2018), 90-108. doi: 10.1016/j.csda.2017.07.009.
[26]	K. Kang, V. Maroulas, I. D. Schizas and E. Blasch, A multilevel homotopy MCMC sequential Monte Carlo filter for multi-target tracking, in Information Fusion (FUSION), 2016 19th International Conference on, IEEE, (2016), 2015–2021.
[27]	A. Lee, K. Tsekouras, C. Calderon, C. Bustamante and S. Pressé, Unraveling the thousand word picture: An introduction to super-resolution data analysis, Chemical Reviews, 117 (2017), 7276-7330.
[28]	J. W. Lichtman and J.-A. Conchello, Fluorescence microscopy, Nature Methods, 2 (2005), 910.
[29]	I. Lichtscheidl and I. Foissner, Video microscopy of dynamic plant cell organelles: Principles of the technique and practical application, Journal of Microscopy, 181 (1996), 117-128.
[30]	S. Liu, M. J. Mlodzianoski, Z. Hu, Y. Ren, K. McElmurry, D. M. Suter and F. Huang, sCMOSnoise-correction algorithm for microscopy images, Nature Methods, 14 (2017), 760.
[31]	C. W. Lloyd, The plant cytoskeleton: The impact of fluorescence microscopy, Annual Review of Plant Physiology, 38 (1987), 119-137. doi: 10.1146/annurev.pp.38.060187.001003.
[32]	D. C. Logan and C. J. Leaver, Mitochondria-targeted GFP highlights the heterogeneity of mitochondrial shape, size and movement within living plant cells, Journal of Experimental Botany, 51 (2000), 865-871. doi: 10.1093/jexbot/51.346.865.
[33]	V. Maroulas and A. Nebenführ, Tracking rapid intracellular movements: A Bayesian random set approach, Ann. Appl. Stat., 9 (2015), 926-949. doi: 10.1214/15-AOAS819.
[34]	V. Maroulas and P. Stinis, Improved particle filters for multi-target tracking, J. Comput. Phys., 231 (2012), 602-611. doi: 10.1016/j.jcp.2011.09.023.
[35]	J. Munkres, Topology, Pearson Education, 2014.
[36]	A. Nebenführ, Identifying subcellular protein localization with fluorescent protein fusions after transient expression in onion epidermal cells, in Plant Cell Morphogenesis, Springer, (2014), 77–85.
[37]	A. Nebenführ and R. Dixit, Kinesins and myosins: Molecular motors that coordinate cellular functions in plants, Annual Review of Plant Biology, 69 (2018), 329-361.
[38]	A. Nebenführ, L. A. Gallagher, T. G. Dunahay, J. A. Frohlick, A. M. Mazurkiewicz, J. B. Meehl and L. A. Staehelin, Stop-and-go movements of plant golgi stacks are mediated by the acto-myosin system, Plant Physiology, 121 (1999), 1127-1141.
[39]	B. K. Nelson, X. Cai and A. Nebenführ, A multicolored set of in vivo organelle markers for co-localization studies in Arabidopsis and other plants, The Plant Journal, 51 (2007), 1126-1136. doi: 10.1111/j.1365-313X.2007.03212.x.
[40]	T. D. Pollard and J. A. Cooper, Actin, a central player in cell shape and movement, Science, 326 (2009), 1208-1212. doi: 10.1126/science.1175862.
[41]	G. Ren, V. Maroulas and I. Schizas, Distributed spatio-temporal association and tracking of multiple targets using multiple sensors, IEEE Transactions on Aerospace and Electronic Systems, 51 (2015), 2570-2589. doi: 10.1109/TAES.2015.140042.
[42]	G. Ren, V. Maroulas and I. D. Schizas, Exploiting sensor mobility and covariance sparsity for distributed tracking of multiple sparse targets, EURASIP Journal on Advances in Signal Processing, 2016 (2016), 53. doi: 10.1186/s13634-016-0354-y.
[43]	I. F. Sbalzarini and P. Koumoutsakos, Feature point tracking and trajectory analysis for video imaging in cell biology, Journal of Structural Biology, 151 (2005), 182-195. doi: 10.1016/j.jsb.2005.06.002.
[44]	I. Sgouralis, A. Nebenführ and V. Maroulas, A Bayesian topological framework for the identification and reconstruction of subcellular motion, SIAM J. Imaging Sci., 10 (2017), 871-899. doi: 10.1137/16M1095755.
[45]	D. M. Shotton, Video-enhanced light microscopy and its applications in cell biology, J. Cell Sci., 89 (1988), 129-150.
[46]	G. Singh, F. Mémoli and G. E. Carlsson, Topological methods for the analysis of high dimensional data sets and 3d object recognition, in SPBG, (2007), 91–100.
[47]	I. Smal, K. Draegestein, N. Galjart, W. Niessen and E. Meijering, Particle filtering for multiple object tracking in dynamic fluorescence microscopy images: Application to microtubule growth analysis, IEEE Transactions on Medical Imaging, 27 (2008), 789-804. doi: 10.1109/TMI.2008.916964.
[48]	I. Smal, W. Niessen and E. Meijering, Particle filtering for multiple object tracking in molecular cell biology, in Nonlinear Statistical Signal Processing Workshop, 2006 IEEE, IEEE, (2006), 129–132. doi: 10.1109/NSSPW.2006.4378836.
[49]	D. L. Snyder, A. M. Hammoud and R. L. White, Image recovery from data acquired with a charge-coupled-device camera, JOSA A, 10 (1993), 1014-1023. doi: 10.1364/JOSAA.10.001014.
[50]	E. H. Spanier, Algebraic Topology, vol. 55, Springer Science & Business Media, 1989.
[51]	I. A. Sparkes, Motoring around the plant cell: Insights from plant myosins, Biochem Soc. Trans., 38 (2010), 833–838. doi: 10.1042/BST0380833.
[52]	L. D. Stone, R. L. Streit, T. L. Corwin and K. L. Bell, Bayesian Multiple Target Tracking, Artech House, 2013.
[53]	M. Tavakoli, S. Jazani, I. Sgouralis, O. M. Shafraz, S. Sivasankar, B. Donaphon, M. Levitus and S. Pressé, Pitching single-focus confocal data analysis one photon at a time with bayesian nonparametrics, Phys. Rev. X, 10 (2020), 011021. doi: 10.1103/PhysRevX.10.011021.
[54]	J. K. Vick and A. Nebenführ, Putting on the breaks: Regulating organelle movements in plant cells, Journal of Integrative Plant Biology, 54 (2012), 868-874. doi: 10.1111/j.1744-7909.2012.01180.x.

show all references

References:

[1]	P. Bendich, S. P. Chin, J. Clark, J. Desena, J. Harer, E. Munch, A. Newman, D. Porter, D. Rouse and N. Strawn et al., Topological and statistical behavior classifiers for tracking applications, IEEE Transactions on Aerospace and Electronic Systems, 52 (2016), 2644-2661.
[2]	G. Bishop, An Introduction to the Kalman Filter, Technical report, TR 95-041, Department of Computer Science, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3175, Monday, 2006.
[3]	S. S. Blackman, Multiple-target Tracking with Radar Applications, Dedham, MA, Artech House, Inc., 1986.
[4]	S. S. v. Braun and E. Schleiff, Movement of endosymbiotic organelles, Current Protein and Peptide Science, 8 (2007), 426-438.
[5]	G. Cai, L. Parrotta and M. Cresti, Organelle trafficking, the cytoskeleton, and pollen tube growth, Journal of Integrative Plant biology, 57 (2015), 63-78.
[6]	G. Carlsson, Topology and data, Bull. Amer. Math. Soc. (N.S.), 46 (2009), 255-308. doi: 10.1090/S0273-0979-09-01249-X.
[7]	G. Casella and R. L. Berger, Statistical Inference, The Wadsworth & Brooks/Cole Statistics/Probability Series. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1990.
[8]	N. Chenouard, I. Bloch and J.-C. Olivo-Marin, Multiple hypothesis tracking for cluttered biological image sequences, IEEE Transactions on Pattern Analysis and Machine Intelligence, 35 (2013), 2736-3750.
[9]	D. A. Collings, J. D. Harper, J. Marc, R. L. Overall and R. T. Mullen, Life in the fast lane: Actin-based motility of plant peroxisomes, Canadian Journal of Botany, 80 (2002), 430-441.
[10]	G. Danuser, Computer vision in cell biology, Cell, 147 (2011), 973-978.
[11]	J. Derksen, Pollen tubes: A Model system for plant cell growth, Botanica Acta, 109 (1996), 341-345.
[12]	H. Edelsbrunner and J. L. Harer, Computational Topology. An Introduction, American Mathematical Soc., Providence, RI, 2010.
[13]	G. B. Folland, Real Analysis: Modern Techniques and Their Applications, Pure and Applied Mathematics (New York). A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1999.
[14]	A. Gelman, J. B. Carlin, H. S. Stern, D. B. Dunson, A. Vehtari and D. B. Rubin, Bayesian Data Analysis, Texts in Statistical Science Series. CRC Press, Boca Raton, FL, 2014.
[15]	R. Gutierrez, J. J. Lindeboom, A. R. Paredez, A. M. C. Emons and D. W. Ehrhardt, Arabidopsis cortical microtubules position cellulose synthase delivery to the plasma membrane and interact with cellulose synthase trafficking compartments, Nature Cell Biology, 11 (2009), 797.
[16]	T. Hamada, M. Tominaga, T. Fukaya, M. Nakamura, A. Nakano, Y. Watanabe, T. Hashimoto and T. I. Baskin, Rna processing bodies, peroxisomes, golgi bodies, mitochondria, and endoplasmic reticulum tubule junctions frequently pause at cortical microtubules, Plant and Cell Physiology, 53 (2012), 699-708.
[17]	B. Herman and D. F. Albertini, A time-lapse video image intensification analysis of cytoplasmic organelle movements during endosome translocation, The Journal of Cell Biology, 98 (1984), 565-576.
[18]	M. Hirsch, R. J. Wareham, M. L. Martin-Fernandez, M. P. Hobson and D. J. Rolfe, A stochastic model for electron multiplication charge-coupled devices–from theory to practice, PloS One, 8 (2013), e53671.
[19]	B. Huang, M. Bates and X. Zhuang, Super-resolution fluorescence microscopy, Annual Review of Biochemistry, 78 (2009), 993-1016.
[20]	F. Huang, T. M. Hartwich, F. E. Rivera-Molina, Y. Lin, W. C. Duim, J. J. Long, P. D. Uchil, J. R. Myers, M. A. Baird, W. Mothes et al., Video-rate nanoscopy using sCMOS camera–specific single-molecule localization algorithms, Nature Methods, 10 (2013), 653.
[21]	A. Jaiswal, W. J. Godinez, M. J. Lehmann and K. Rohr, Direct combination of multi-scale detection and multi-frame association for tracking of virus particles in microscopy image data, in 2016 IEEE 13th International Symposium on Biomedical Imaging (ISBI), IEEE, 2016,976–979.
[22]	J. Janesick and T. Elliott, History and advancement of large array scientific ccd imagers, in Astronomical CCD Observing and Reduction Techniques, 23 (1992), 1.
[23]	S. Jazani, I. Sgouralis and S. Pressé, A method for single molecule tracking using a conventional single-focus confocal setup, The Journal of Chemical Physics, 150 (2019), 114108.
[24]	S. Jazani, I. Sgouralis, O. M. Shafraz, M. Levitus, S. Sivasankar and S. Pressé, An alternative framework for fluorescence correlation spectroscopy, Nature Communications, 10.
[25]	K. Kang, V. Maroulas, I. Schizas and F. Bao, Improved distributed particle filters for tracking in a wireless sensor network, Comput. Statist. Data Anal., 117 (2018), 90-108. doi: 10.1016/j.csda.2017.07.009.
[26]	K. Kang, V. Maroulas, I. D. Schizas and E. Blasch, A multilevel homotopy MCMC sequential Monte Carlo filter for multi-target tracking, in Information Fusion (FUSION), 2016 19th International Conference on, IEEE, (2016), 2015–2021.
[27]	A. Lee, K. Tsekouras, C. Calderon, C. Bustamante and S. Pressé, Unraveling the thousand word picture: An introduction to super-resolution data analysis, Chemical Reviews, 117 (2017), 7276-7330.
[28]	J. W. Lichtman and J.-A. Conchello, Fluorescence microscopy, Nature Methods, 2 (2005), 910.
[29]	I. Lichtscheidl and I. Foissner, Video microscopy of dynamic plant cell organelles: Principles of the technique and practical application, Journal of Microscopy, 181 (1996), 117-128.
[30]	S. Liu, M. J. Mlodzianoski, Z. Hu, Y. Ren, K. McElmurry, D. M. Suter and F. Huang, sCMOSnoise-correction algorithm for microscopy images, Nature Methods, 14 (2017), 760.
[31]	C. W. Lloyd, The plant cytoskeleton: The impact of fluorescence microscopy, Annual Review of Plant Physiology, 38 (1987), 119-137. doi: 10.1146/annurev.pp.38.060187.001003.
[32]	D. C. Logan and C. J. Leaver, Mitochondria-targeted GFP highlights the heterogeneity of mitochondrial shape, size and movement within living plant cells, Journal of Experimental Botany, 51 (2000), 865-871. doi: 10.1093/jexbot/51.346.865.
[33]	V. Maroulas and A. Nebenführ, Tracking rapid intracellular movements: A Bayesian random set approach, Ann. Appl. Stat., 9 (2015), 926-949. doi: 10.1214/15-AOAS819.
[34]	V. Maroulas and P. Stinis, Improved particle filters for multi-target tracking, J. Comput. Phys., 231 (2012), 602-611. doi: 10.1016/j.jcp.2011.09.023.
[35]	J. Munkres, Topology, Pearson Education, 2014.
[36]	A. Nebenführ, Identifying subcellular protein localization with fluorescent protein fusions after transient expression in onion epidermal cells, in Plant Cell Morphogenesis, Springer, (2014), 77–85.
[37]	A. Nebenführ and R. Dixit, Kinesins and myosins: Molecular motors that coordinate cellular functions in plants, Annual Review of Plant Biology, 69 (2018), 329-361.
[38]	A. Nebenführ, L. A. Gallagher, T. G. Dunahay, J. A. Frohlick, A. M. Mazurkiewicz, J. B. Meehl and L. A. Staehelin, Stop-and-go movements of plant golgi stacks are mediated by the acto-myosin system, Plant Physiology, 121 (1999), 1127-1141.
[39]	B. K. Nelson, X. Cai and A. Nebenführ, A multicolored set of in vivo organelle markers for co-localization studies in Arabidopsis and other plants, The Plant Journal, 51 (2007), 1126-1136. doi: 10.1111/j.1365-313X.2007.03212.x.
[40]	T. D. Pollard and J. A. Cooper, Actin, a central player in cell shape and movement, Science, 326 (2009), 1208-1212. doi: 10.1126/science.1175862.
[41]	G. Ren, V. Maroulas and I. Schizas, Distributed spatio-temporal association and tracking of multiple targets using multiple sensors, IEEE Transactions on Aerospace and Electronic Systems, 51 (2015), 2570-2589. doi: 10.1109/TAES.2015.140042.
[42]	G. Ren, V. Maroulas and I. D. Schizas, Exploiting sensor mobility and covariance sparsity for distributed tracking of multiple sparse targets, EURASIP Journal on Advances in Signal Processing, 2016 (2016), 53. doi: 10.1186/s13634-016-0354-y.
[43]	I. F. Sbalzarini and P. Koumoutsakos, Feature point tracking and trajectory analysis for video imaging in cell biology, Journal of Structural Biology, 151 (2005), 182-195. doi: 10.1016/j.jsb.2005.06.002.
[44]	I. Sgouralis, A. Nebenführ and V. Maroulas, A Bayesian topological framework for the identification and reconstruction of subcellular motion, SIAM J. Imaging Sci., 10 (2017), 871-899. doi: 10.1137/16M1095755.
[45]	D. M. Shotton, Video-enhanced light microscopy and its applications in cell biology, J. Cell Sci., 89 (1988), 129-150.
[46]	G. Singh, F. Mémoli and G. E. Carlsson, Topological methods for the analysis of high dimensional data sets and 3d object recognition, in SPBG, (2007), 91–100.
[47]	I. Smal, K. Draegestein, N. Galjart, W. Niessen and E. Meijering, Particle filtering for multiple object tracking in dynamic fluorescence microscopy images: Application to microtubule growth analysis, IEEE Transactions on Medical Imaging, 27 (2008), 789-804. doi: 10.1109/TMI.2008.916964.
[48]	I. Smal, W. Niessen and E. Meijering, Particle filtering for multiple object tracking in molecular cell biology, in Nonlinear Statistical Signal Processing Workshop, 2006 IEEE, IEEE, (2006), 129–132. doi: 10.1109/NSSPW.2006.4378836.
[49]	D. L. Snyder, A. M. Hammoud and R. L. White, Image recovery from data acquired with a charge-coupled-device camera, JOSA A, 10 (1993), 1014-1023. doi: 10.1364/JOSAA.10.001014.
[50]	E. H. Spanier, Algebraic Topology, vol. 55, Springer Science & Business Media, 1989.
[51]	I. A. Sparkes, Motoring around the plant cell: Insights from plant myosins, Biochem Soc. Trans., 38 (2010), 833–838. doi: 10.1042/BST0380833.
[52]	L. D. Stone, R. L. Streit, T. L. Corwin and K. L. Bell, Bayesian Multiple Target Tracking, Artech House, 2013.
[53]	M. Tavakoli, S. Jazani, I. Sgouralis, O. M. Shafraz, S. Sivasankar, B. Donaphon, M. Levitus and S. Pressé, Pitching single-focus confocal data analysis one photon at a time with bayesian nonparametrics, Phys. Rev. X, 10 (2020), 011021. doi: 10.1103/PhysRevX.10.011021.
[54]	J. K. Vick and A. Nebenführ, Putting on the breaks: Regulating organelle movements in plant cells, Journal of Integrative Plant Biology, 54 (2012), 868-874. doi: 10.1111/j.1744-7909.2012.01180.x.

Figure 1. The motion of organelles, during an experiment starting at $ t_1 = 0 $ ending at $ t_N = T $, is identified at discrete times $ t_n $ (dots). For simplicity, space is represented with one dimension, although real datasets are two dimensional. The black dots represent the locations of organelles at different time levels. $ \tilde {\mathcal{R}} $ is the set contains the locations of all black dots

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Figure 2. Here, $ \bar x^j $ is the position of an organelle producing the images shown (gray), $ \bar f_{n, +} $ and $ \bar f_{n, -} $ illustrate 1-level forward displacement and backward displacement of $ \bar x^j $, respectively. For clarity, the image produced by the organelle are shown as multi-peaked and space as 1D

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Figure 3. The relations of forward fields and backward fields are indicated here. (a) shows the approach depiction of forward displacement fields, (b) shows the approach depiction of backward displacement fields. For clarity, time marches forward in (a) and backward in (b)

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Figure 4. (a) shows $ \tilde {\mathcal{R}} $ as black dots and $ {\mathcal{R}} $ as gray lines; (b) shows $ {\mathcal{R}} $, $ {\mathcal{T}}_n $ in Eq. (12) and $ P_ {\mathcal{R}}^{-1}( {\mathcal{T}}_n) $ as blue segments; (c) shows $ {\mathcal{R}} $, $ {\mathcal{T}}_n $, $ P_ {\mathcal{R}}^{-1}( {\mathcal{T}}_n) $, $ \tilde R $ and reconstructed discrete trajectories. For visualization purpose, space is shown in 1D

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Figure 5. Case I: The frame size is 320 by 320 pixels. Trajectories of 20 organelles are in red spanning from time $ t = 0\; $s to $ t = 99\; $s. Their motion is described by a diffusion process containing both a diffusion and a drift term. The starting distance of any two adjacent organelles at $ t = 0\; $s is 10 pixels

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Figure 6. Case I: Four histograms of mean error of each frame. Each one compares estimated forward and backward displacement with ground truth along $ x $-axis and $ y $-axis, respectively

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Figure 7. Case I: Linking result of all trajectories in red. The accuracy rate is 100%

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Figure 8. Case I: Positions of organelles over time after adding perturbation $ U(-\epsilon, \epsilon) $ when $ \epsilon = 0, \ 1, \ 1.5, \ 2, \ 2.5, \ 3, \ 3.5, \ 4\; $pixels, respectively. If $ \epsilon $ increases, it is more difficult to detect trajectories, especially, when $ \epsilon = 4\; $ pixels, there are no clear patterns for all trajectories to be reconstructed

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Figure 9. Case II: The left shows the rough detection result, the right show the locations after correction. The red dots in the left penal and blue pentagons in the right penal are the original locations before Bayesian identification. The red pentagons in the right panel are the fitted location after Bayesian identification

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Figure 10. Case II: Estimated displacement fields of 17th frame using EnKF. Panel (a) shows the estimated displacement fields for the entire focal plane. Panel (b) shows the enlarged area of $ [140,230]\times[260,350] $ in Penal (a)

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Figure 11. Case II: Trajectories reconstruction result

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Figure 12. Case II: Four specific sets of trajectory reconstructions vs ground truth. Each panel shows reconstructions versus one true trajectory. The upper left is amplified from the area $ [290,380]\times[40,130] $ in Fig. 11; the upper right is amplified from the area $ [40,190]\times[190,340] $ in Fig. 11; the bottom left is amplified from the area $ [80,210]\times[200,330] $ in Fig. 11; the bottom right is amplified from the area $ [230,380]\times[200,350] $ in Fig. 11;

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Figure 13. Case III: Panel (a) is the first frame of the video. Panel (b) exhibits all estimated trajectories in red. Panel (c) further shows each estimated trajectory in different colors

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Table 1. Case I: Table of detection result

$ \epsilon $ (in pixels)	total	$> 10\; s $	$ =100\% $	$ \geq 90\% $	$ \geq 50\% $
$ \epsilon=1 $	20	20	20	20	20
$ \epsilon=1.5 $	20	20	20	20	20
$ \epsilon=2 $	20	20	20	20	20
$ \epsilon=2.5 $	24	24	14	17	20
$ \epsilon=3 $	32	27	9	14	17
$ \epsilon=3.5 $	33	30	5	10	15
$ \epsilon=4 $	53	40	1	4	14

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$ \epsilon $ (in pixels)	total	$> 10\; s $	$ =100\% $	$ \geq 90\% $	$ \geq 50\% $
$ \epsilon=1 $	20	20	20	20	20
$ \epsilon=1.5 $	20	20	20	20	20
$ \epsilon=2 $	20	20	20	20	20
$ \epsilon=2.5 $	24	24	14	17	20
$ \epsilon=3 $	32	27	9	14	17
$ \epsilon=3.5 $	33	30	5	10	15
$ \epsilon=4 $	53	40	1	4	14

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