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A density-based approach to feature detection in persistence diagrams for firn data
1. | Program of Informatics and Analytics, University of North Carolina at Greensboro, Greensboro, NC 27402, USA |
2. | Department of Mathematics, University of Maryland, College Park, MD 20742, USA |
3. | Department of Mathematics and Statistics, University of North Carolina at Greensboro, Greensboro, NC 27402, USA |
4. | Department of Geological Sciences and Engineering, University of Nevada, Reno, Reno, NV 89557, USA |
5. | Department of Mathematics, William & Mary, Williamsburg, VA 23185, USA |
Topological data analysis, and in particular persistence diagrams, are gaining popularity as tools for extracting topological information from noisy point cloud and digital data. Persistence diagrams track topological features in the form of $ k $-dimensional holes in the data. Here, we construct a new, automated approach for identifying persistence diagram points that represent robust long-life features. These features may be used to provide a more accurate estimate of Betti numbers for the underlying space. This approach extends the established practice of using a lifespan cutoff on the features in order to take advantage of the observation that noisy features typically appear in clusters in the persistence diagram. We show that this approach offers more flexibility in partitioning features in the persistence diagram, resulting in greater accuracy in computed Betti numbers, especially in the case of high noise levels and varying image illumination. This work is motivated by 3-dimensional Micro-CT imaging of ice core samples, and is applicable for separating noise from robust signals in persistence diagrams from noisy data.
References:
[1] |
R. J. Adler, O. Bobrowski, M. S. Borman, E. Subag and S. Weinberger, Persistent homology for random fields and complexes, in Borrowing Strength: Theory Powering Applications–A Festschrift for Lawrence D. Brown, Inst. Math. Stat. (IMS) Collect., 6, Inst. Math. Statist., Beachwood, OH, 2010,124–143.
doi: 10.1214/10-IMSCOLL609. |
[2] |
N. Atienza, R. Gonzalez-Diaz and M. Rucco,
Persistent entropy for separating topological features from noise in vietoris-rips complexes, J. Intelligent Information Systems, 52 (2019), 637-655.
doi: 10.1007/s10844-017-0473-4. |
[3] |
N. Atienza, R. Gonzalez-Diaz and M. Rucco, Separating topological noise from features using persistent entropy, in Software Technologies: Applications and Foundations, Lecture Notes in Computer Science, 9946, Springer, Cham, 2016, 3–12.
doi: 10.1007/978-3-319-50230-4_1. |
[4] |
G. Carlsson,
Topology and data, Bull. Amer. Math. Soc.(N.S.), 46 (2009), 255-308.
doi: 10.1090/S0273-0979-09-01249-X. |
[5] |
F. Chazal, V. de Silva and S. Oudot,
Persistence stability for geometric complexes, Geom. Dedicata, 173 (2014), 193-214.
doi: 10.1007/s10711-013-9937-z. |
[6] |
F. Chazal and B. Michel, An introduction to Topological Data Analysis: Fundamental and practical aspects for data scientists, preprint, arXiv: 1710.04019. |
[7] |
Y.-M. Chung and S. Day,
Topological fidelity and image thresholding: A persistent homology approach, J. Math. Imaging Vision, 60 (2018), 1167-1179.
doi: 10.1007/s10851-018-0802-4. |
[8] |
D. Cohen-Steiner, H. Edelsbrunner and J. Harer,
Stability of persistence diagrams, Discrete Comput. Geom., 37 (2007), 103-120.
doi: 10.1007/s00454-006-1276-5. |
[9] |
P. Dlotko, Cubical complex, in GUDHI User and Reference Manual, GUDHI Editorial Board, 3.3.0 edition, 2020. Available from: https://gudhi.inria.fr/. |
[10] |
H. Edelsbrunner and J. L. Harer, Computational Topology. An Introduction, American Mathematical Society, Providence, RI, 2010.
doi: 10.1090/mbk/069. |
[11] |
M. Ester, H.-P. Kriegel, J. Sander and X. Xu, A density-based algorithm for discovering clusters in large spatial databases with noise, in KDD-96 Proceedings, 1996,226–231. |
[12] |
B. T. Fasy, F. Lecci, A. Rinaldo, L. Wasserman, S. Balakrishnan and A. Singh,
Confidence sets for persistence diagrams, Ann. Statist., 42 (2014), 2301-2339.
doi: 10.1214/14-AOS1252. |
[13] |
A. Garin and G. Tauzin, A topological "reading" lesson: Classification of MNIST using TDA, 2019 18th IEEE International Conference on Machine Learning and Applications (ICMLA), Boca Raton, FL, 2019, 1551–1556.
doi: 10.1109/ICMLA.2019.00256. |
[14] |
R. Ghrist,
Barcodes: The persistent topology of data, Bull. Amer. Math. Soc., 45 (2008), 61-75.
doi: 10.1090/S0273-0979-07-01191-3. |
[15] |
J. F. Jardine, Data and homotopy types, preprint, arXiv: 1908.06323. |
[16] |
T. Kaczynski, K. Mischaikow and M. Mrozek, Computational Homology, Applied Mathematical Sciences, 157, Springer-Verlag, New York, 2004.
doi: 10.1007/b97315. |
[17] |
K. Keegan, M. R. Albert and I. Baker,
The impact of ice layers on gas transport through firn at the North Greenland Eemian Ice Drilling (NEEM) site, Greenland, The Cryosphere, 8 (2014), 1801-1806.
doi: 10.5194/tc-8-1801-2014. |
[18] |
A. Landais, J. M. Barnola, K. Kawamura, N. Caillon and M. Delmotte,
Firn-air $\delta$15n in modern polar sites and glacial–interglacial ice: A model-data mismatch during glacial periods in Antarctica?, Quaternary Science Reviews, 25 (2006), 49-62.
doi: 10.1016/j.quascirev.2005.06.007. |
[19] |
M. Lesnick and M. Wright, Interactive visualization of 2-D persistence modules, preprint, arXiv: 1512.00180. |
[20] |
J. M. D. Lundin, C. M. Stevens, R. Arthern, C. Buizert and A. Orsi,
Firn model intercomparison experiment (FirnMICE), J. Glaciology, 63 (2017), 401-422.
doi: 10.1017/jog.2016.114. |
[21] |
I. Obayashi, Y. Hiraoka and M. Kimura,
Persistence diagrams with linear machine learning models, J. Appl. Comput. Topol., 1 (2018), 421-449.
doi: 10.1007/s41468-018-0013-5. |
[22] |
N. Otter, M. A. Porter, U. Tillmann, P. Grindrod and H. A. Harrington, A roadmap for the computation of persistent homology, EPJ Data Science, 6 (2017).
doi: 10.1140/epjds/s13688-017-0109-5. |
[23] |
A. Patania, F. Vaccarino and G. Petri, Topological analysis of data, EPJ Data Science, 6 (2017).
doi: 10.1140/epjds/s13688-017-0104-x. |
[24] |
J. Schwander and B. Stauffer,
Age difference between polar ice and the air trapped in its bubbles, Nature, 311 (1984), 45-47.
doi: 10.1038/311045a0. |
[25] |
L. Wasserman,
Topological data analysis, Annu. Rev. Stat. Appl., 5 (2018), 501-535.
doi: 10.1146/annurev-statistics-031017-100045. |
[26] |
A. Zomorodian, Topological data analysis, in Advances in Applied and Computational Topology, Proc. Sympos. Appl. Math., 70, Amer. Math. Soc., Providence, RI, 2012, 1–39.
doi: 10.1090/psapm/070/587. |
show all references
References:
[1] |
R. J. Adler, O. Bobrowski, M. S. Borman, E. Subag and S. Weinberger, Persistent homology for random fields and complexes, in Borrowing Strength: Theory Powering Applications–A Festschrift for Lawrence D. Brown, Inst. Math. Stat. (IMS) Collect., 6, Inst. Math. Statist., Beachwood, OH, 2010,124–143.
doi: 10.1214/10-IMSCOLL609. |
[2] |
N. Atienza, R. Gonzalez-Diaz and M. Rucco,
Persistent entropy for separating topological features from noise in vietoris-rips complexes, J. Intelligent Information Systems, 52 (2019), 637-655.
doi: 10.1007/s10844-017-0473-4. |
[3] |
N. Atienza, R. Gonzalez-Diaz and M. Rucco, Separating topological noise from features using persistent entropy, in Software Technologies: Applications and Foundations, Lecture Notes in Computer Science, 9946, Springer, Cham, 2016, 3–12.
doi: 10.1007/978-3-319-50230-4_1. |
[4] |
G. Carlsson,
Topology and data, Bull. Amer. Math. Soc.(N.S.), 46 (2009), 255-308.
doi: 10.1090/S0273-0979-09-01249-X. |
[5] |
F. Chazal, V. de Silva and S. Oudot,
Persistence stability for geometric complexes, Geom. Dedicata, 173 (2014), 193-214.
doi: 10.1007/s10711-013-9937-z. |
[6] |
F. Chazal and B. Michel, An introduction to Topological Data Analysis: Fundamental and practical aspects for data scientists, preprint, arXiv: 1710.04019. |
[7] |
Y.-M. Chung and S. Day,
Topological fidelity and image thresholding: A persistent homology approach, J. Math. Imaging Vision, 60 (2018), 1167-1179.
doi: 10.1007/s10851-018-0802-4. |
[8] |
D. Cohen-Steiner, H. Edelsbrunner and J. Harer,
Stability of persistence diagrams, Discrete Comput. Geom., 37 (2007), 103-120.
doi: 10.1007/s00454-006-1276-5. |
[9] |
P. Dlotko, Cubical complex, in GUDHI User and Reference Manual, GUDHI Editorial Board, 3.3.0 edition, 2020. Available from: https://gudhi.inria.fr/. |
[10] |
H. Edelsbrunner and J. L. Harer, Computational Topology. An Introduction, American Mathematical Society, Providence, RI, 2010.
doi: 10.1090/mbk/069. |
[11] |
M. Ester, H.-P. Kriegel, J. Sander and X. Xu, A density-based algorithm for discovering clusters in large spatial databases with noise, in KDD-96 Proceedings, 1996,226–231. |
[12] |
B. T. Fasy, F. Lecci, A. Rinaldo, L. Wasserman, S. Balakrishnan and A. Singh,
Confidence sets for persistence diagrams, Ann. Statist., 42 (2014), 2301-2339.
doi: 10.1214/14-AOS1252. |
[13] |
A. Garin and G. Tauzin, A topological "reading" lesson: Classification of MNIST using TDA, 2019 18th IEEE International Conference on Machine Learning and Applications (ICMLA), Boca Raton, FL, 2019, 1551–1556.
doi: 10.1109/ICMLA.2019.00256. |
[14] |
R. Ghrist,
Barcodes: The persistent topology of data, Bull. Amer. Math. Soc., 45 (2008), 61-75.
doi: 10.1090/S0273-0979-07-01191-3. |
[15] |
J. F. Jardine, Data and homotopy types, preprint, arXiv: 1908.06323. |
[16] |
T. Kaczynski, K. Mischaikow and M. Mrozek, Computational Homology, Applied Mathematical Sciences, 157, Springer-Verlag, New York, 2004.
doi: 10.1007/b97315. |
[17] |
K. Keegan, M. R. Albert and I. Baker,
The impact of ice layers on gas transport through firn at the North Greenland Eemian Ice Drilling (NEEM) site, Greenland, The Cryosphere, 8 (2014), 1801-1806.
doi: 10.5194/tc-8-1801-2014. |
[18] |
A. Landais, J. M. Barnola, K. Kawamura, N. Caillon and M. Delmotte,
Firn-air $\delta$15n in modern polar sites and glacial–interglacial ice: A model-data mismatch during glacial periods in Antarctica?, Quaternary Science Reviews, 25 (2006), 49-62.
doi: 10.1016/j.quascirev.2005.06.007. |
[19] |
M. Lesnick and M. Wright, Interactive visualization of 2-D persistence modules, preprint, arXiv: 1512.00180. |
[20] |
J. M. D. Lundin, C. M. Stevens, R. Arthern, C. Buizert and A. Orsi,
Firn model intercomparison experiment (FirnMICE), J. Glaciology, 63 (2017), 401-422.
doi: 10.1017/jog.2016.114. |
[21] |
I. Obayashi, Y. Hiraoka and M. Kimura,
Persistence diagrams with linear machine learning models, J. Appl. Comput. Topol., 1 (2018), 421-449.
doi: 10.1007/s41468-018-0013-5. |
[22] |
N. Otter, M. A. Porter, U. Tillmann, P. Grindrod and H. A. Harrington, A roadmap for the computation of persistent homology, EPJ Data Science, 6 (2017).
doi: 10.1140/epjds/s13688-017-0109-5. |
[23] |
A. Patania, F. Vaccarino and G. Petri, Topological analysis of data, EPJ Data Science, 6 (2017).
doi: 10.1140/epjds/s13688-017-0104-x. |
[24] |
J. Schwander and B. Stauffer,
Age difference between polar ice and the air trapped in its bubbles, Nature, 311 (1984), 45-47.
doi: 10.1038/311045a0. |
[25] |
L. Wasserman,
Topological data analysis, Annu. Rev. Stat. Appl., 5 (2018), 501-535.
doi: 10.1146/annurev-statistics-031017-100045. |
[26] |
A. Zomorodian, Topological data analysis, in Advances in Applied and Computational Topology, Proc. Sympos. Appl. Math., 70, Amer. Math. Soc., Providence, RI, 2012, 1–39.
doi: 10.1090/psapm/070/587. |










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