The Theory of Multidimensional Persistence
Open accessIntegrity signals
Registry
Cites
Statistics
Claims the paper makes
Claims other papers make about this one
Each is checked against this paper’s own text: does it say what the citing paper says it does?
The isomorphism classification of n-persistence vector spaces with parameter set restricted to Z^n_+ is identical to the classification of n-graded modules over the n-graded ring k[t1,...,tn].
“it is shown in Carlsson and Zomorodian (2009) that the isomorphism classification of n -persistence vector spaces with the parameter set restricted to Z n + is identical to the classification of n -graded modules over the n -graded ring k [ t 1 , . . . , t n ].”
From Topological pattern recognition for point cloud data (2014) · cites this paper as Carlsson and Zomorodian (2009)
Cites
An Introduction to Homological Algebra
Weibel, C. A. (1994) · Cambridge University Press eBooks · doi:10.1017/cbo9781139644136 · Cited by 3,868 · see citing works
Mumford, D., Fogarty, J., & Kirwan, F. (1994) · Springer Grundlehren · doi:10.1007/978-3-642-57916-5 · Cited by 2,564 · see citing works
Gromov, M. (1987) · Mathematical Sciences Research Institute Publications · doi:10.1007/978-1-4613-9586-7_3 · Cited by 1,817 · see citing works
Topological Persistence and Simplification
Edelsbrunner, Letscher, & Zomorodian (2002) · Discrete & Computational Geometry · doi:10.1007/s00454-002-2885-2 · Cited by 1,779 · see citing works
Introduction to Commutative Algebra.
Lewis, D., McDonald, B. R., Atiyah, M. F., & MacDonald, I. G. (1970) · American Mathematical Monthly · doi:10.2307/2316241 · Cited by 1,709 · see citing works
Zomorodian, A., & Carlsson, G. (2005) · Discrete & Computational Geometry · doi:10.1007/s00454-004-1146-y · Cited by 1,591 · see citing works
Self-citationBarcodes: the persistent topology of data
Ghrist, R. (2007) · Bulletin of the American Mathematical Society · doi:10.1090/s0273-0979-07-01191-3 · Cited by 1,236 · see citing works
On the Local Behavior of Spaces of Natural Images
Carlsson, G., Ishkhanov, T., de Silva, V., & Zomorodian, A. (2008) · International Journal of Computer Vision · doi:10.1007/s11263-007-0056-x · Cited by 477 · see citing works
Self-citationPERSISTENCE BARCODES FOR SHAPES
CARLSSON, G., ZOMORODIAN, A., COLLINS, A., & GUIBAS, L. J. (2005) · International Journal of Shape Modeling · doi:10.1142/s0218654305000761 · Cited by 287 · see citing works
Self-citationThe Nonlinear Statistics of High-Contrast Patches in Natural Images
Lee, A. B., Pedersen, K. S., & Mumford, D. (2003) · International Journal of Computer Vision · doi:10.1023/a:1023705401078 · Cited by 234 · see citing works
A barcode shape descriptor for curve point cloud data
Collins, A., Zomorodian, A., Carlsson, G., & Guibas, L. J. (2004) · Computers & Graphics · doi:10.1016/j.cag.2004.08.015 · Cited by 132 · see citing works
Self-citationSerre, J.-P. (2000) · Springer monographs in mathematics · doi:10.1007/978-3-662-04203-8 · Cited by 103 · see citing works
Blind Swarms for Coverage in 2-D
de Silva, V., Ghrist, R., & Muhammad, A. (2005) · doi:10.15607/rss.2005.i.044 · Cited by 88 · see citing works
Size homotopy groups for computation of natural size distances
Frosini, P., & Mulazzani, M. (1999) · Bulletin of the Belgian Mathematical Society - Simon Stevin · doi:10.36045/bbms/1103065863 · Cited by 84 · see citing works
Chazal, F., & Lieutier, A. (2005) · Proceedings of the twenty-first annual symposium on Computational geometry · doi:10.1145/1064092.1064132 · Cited by 67 · see citing works
Zomorodian, A. J. (2009) · Chapman & Hall/CRC applied algorithms and data structures series · doi:10.1201/9781584888215-c3 · Cited by 46 · see citing works
Self-citationA refinement of multi-dimensional persistence
KNUDSON, K. P. (2008) · Homology Homotopy and Applications · doi:10.4310/hha.2008.v10.n1.a11 · Cited by 19 · see citing works
Terao, H. (2002) · Topology and its Applications · doi:10.1016/s0166-8641(01)00054-2 · Cited by 16 · see citing works
Gauss–Man i n Conne ction s for Arrangements, I E igenvalues
Cohen, D. C., & Orlik, P. (2003) · Compositio Mathematica · doi:10.1023/a:1023262022279 · Cited by 7 · see citing works
Flagged first, then by citations
Cited by
Sorted by citations
Carlsson, G. (2009) · Bulletin of the American Mathematical Society · doi:10.1090/s0273-0979-09-01249-x · Cited by 2,302 · see citing works
Self-citationNetworks beyond pairwise interactions: Structure and dynamics
Battiston, F., Cencetti, G., Iacopini, I., Latora, V., Lucas, M., Patania, A., Young, J.-G., & Petri, G. (2020) · Physics Reports · doi:10.1016/j.physrep.2020.05.004 · Cited by 1,739 · see citing works
Networks beyond pairwise interactions: Structure and dynamics
Battiston, F., Cencetti, G., Iacopini, I., Latora, V., Lucas, M., Patania, A., Young, J., & Petri, G. (2020) · UCL Discovery (University College London) · Cited by 1,179 · see citing works
A roadmap for the computation of persistent homology
Otter, N., Porter, M. A., Tillmann, U., Grindrod, P., & Harrington, H. A. (2017) · EPJ Data Science · doi:10.1140/epjds/s13688-017-0109-5 · Cited by 769 · see citing works
What are higher-order networks?
Bick, C., Gross, E., Harrington, H. A., & Schaub, M. T. (2023) · SIAM Review · doi:10.1137/21m1414024 · Cited by 420 · see citing works
Cang, Z., & Wei, G.-W. (2017) · PLOS Computational Biology · doi:10.1371/journal.pcbi.1005690 · Cited by 330 · see citing works
Cang, Z., Mu, L., & Wei, G.-W. (2018) · PLOS Computational Biology · doi:10.1371/journal.pcbi.1005929 · Cited by 257 · see citing works
Confidence sets for persistence diagrams
Fasy, B. T., Lecci, F., Rinaldo, A., Wasserman, L., Balakrishnan, S., & Singh, A. (2014) · The Annals of Statistics · doi:10.1214/14-aos1252 · Cited by 251 · see citing works
Topological pattern recognition for point cloud data
Carlsson, G. (2014) · Acta Numerica · doi:10.1017/s0962492914000051 · Cited by 230 · see citing works
Self-citationTopological Strata of Weighted Complex Networks
Petri, G., Scolamiero, M., Donato, I., & Vaccarino, F. (2013) · PLoS ONE · doi:10.1371/journal.pone.0066506 · Cited by 214 · see citing works
The Theory of the Interleaving Distance on Multidimensional Persistence Modules
Lesnick, M. (2015) · Foundations of Computational Mathematics · doi:10.1007/s10208-015-9255-y · Cited by 207 · see citing works
Armstrong, R. T., McClure, J. E., Robins, V., Liu, Z., Arns, C. H., Schlüter, S., & Berg, S. (2018) · Transport in Porous Media · doi:10.1007/s11242-018-1201-4 · Cited by 204 · see citing works
Topological Data Analysis of Biological Aggregation Models
Topaz, C. M., Ziegelmeier, L., & Halverson, T. (2015) · PLoS ONE · doi:10.1371/journal.pone.0126383 · Cited by 187 · see citing works
A Topological Representation of Branching Neuronal Morphologies
Kanari, L., Dłotko, P., Scolamiero, M., Levi, R., Shillcock, J. C., Hess, K. P., & Markram, H. (2017) · Neuroinformatics · doi:10.1007/s12021-017-9341-1 · Cited by 179 · see citing works
Cang, Z., & Wei, G. (2018) · International Journal for Numerical Methods in Biomedical Engineering · doi:10.1002/cnm.2914 · Cited by 172 · see citing works
Persistent Homology Analysis for Materials Research and Persistent Homology Software: HomCloud
Obayashi, I., Nakamura, T., & Hiraoka, Y. (2022) · Journal of the Physical Society of Japan · doi:10.7566/jpsj.91.091013 · Cited by 149 · see citing works
Betti numbers in multidimensional persistent homology are stable functions
Cerri, A., Fabio, B. D., Ferri, M., Frosini, P., & Landi, C. (2013) · Mathematical Methods in the Applied Sciences · doi:10.1002/mma.2704 · Cited by 138 · see citing works
Chatter detection in turning using persistent homology
Khasawneh, F. A., & Munch, E. (2015) · Mechanical Systems and Signal Processing · doi:10.1016/j.ymssp.2015.09.046 · Cited by 137 · see citing works
Persistent-homology-based machine learning: a survey and a comparative study
Pun, C. S., Lee, S. X., & Xia, K. (2022) · Artificial Intelligence Review · doi:10.1007/s10462-022-10146-z · Cited by 223 · see citing works
Metrics for Generalized Persistence Modules
Bubenik, P., de Silva, V., & Scott, J. (2014) · Foundations of Computational Mathematics · doi:10.1007/s10208-014-9229-5 · Cited by 120 · see citing works
The topology of the cosmic web in terms of persistent Betti numbers
Pranav, P., Edelsbrunner, H., van de Weygaert, R., Vegter, G., Kerber, M., Jones, B. J. T., & Wintraecken, M. (2016) · Monthly Notices of the Royal Astronomical Society · doi:10.1093/mnras/stw2862 · Cited by 118 · see citing works
Objective Morphological Classification of Neocortical Pyramidal Cells
Kanari, L., Ramaswamy, S., Shi, Y., Morand, S., Meystre, J., Perin, R., Abdellah, M., Wang, Y., Hess, K., & Markram, H. (2019) · Cerebral Cortex · doi:10.1093/cercor/bhy339 · Cited by 94 · see citing works
Classification of hepatic lesions using the matching metric
Adcock, A., Rubin, D. L., & Carlsson, G. (2014) · Computer Vision and Image Understanding · doi:10.1016/j.cviu.2013.10.014 · Cited by 98 · see citing works
Self-citationPersistent homology for random fields and complexes
Adler, R. J., Bobrowski, O., Borman, M. S., Subag, E., & Weinberger, S. (2010) · Institute of Mathematical Statistics Collections · doi:10.1214/10-imscoll609 · Cited by 91 · see citing works
Persistent-Homology-Based Machine Learning and Its Applications -- A SurveyPreprint
Pun, C. S., Xia, K., & Lee, S. X. (2018) · SSRN Electronic Journal · doi:10.2139/ssrn.3275996 · Cited by 89 · see citing works
Yan, L., Masood, T. B., Sridharamurthy, R., Rasheed, F., Natarajan, V., Hotz, I., & Wang, B. (2021) · Computer Graphics Forum · doi:10.1111/cgf.14331 · Cited by 81 · see citing works
Topological methods for data modelling
Carlsson, G. (2020) · Nature Reviews Physics · doi:10.1038/s42254-020-00249-3 · Cited by 79 · see citing works
Self-citationMultiscale topology classifies cells in subcellular spatial transcriptomics
Benjamin, K., Bhandari, A., Kepple, J. D., Qi, R., Shang, Z., Xing, Y., An, Y., Zhang, N., Hou, Y., Crockford, T. L., et al. (2024) · Nature · doi:10.1038/s41586-024-07563-1 · Cited by 81 · see citing works
Multiparameter persistent homology landscapes identify immune cell spatial patterns in tumors
Vipond, O., Bull, J. A., Macklin, P. S., Tillmann, U., Pugh, C. W., Byrne, H. M., & Harrington, H. A. (2021) · Proceedings of the National Academy of Sciences · doi:10.1073/pnas.2102166118 · Cited by 76 · see citing works
Autonomous Hamiltonian flows, Hofer’s geometry and persistence modules
Polterovich, L., & Shelukhin, E. (2015) · Selecta Mathematica · doi:10.1007/s00029-015-0201-2 · Cited by 74 · see citing works
Multidimensional persistence in biomolecular data
Xia, K., & Wei, G. (2015) · Journal of Computational Chemistry · doi:10.1002/jcc.23953 · Cited by 74 · see citing works
Algebraic stability of zigzag persistence modules
Botnan, M. B., & Lesnick, M. (2018) · Algebraic & Geometric Topology · doi:10.2140/agt.2018.18.3133 · Cited by 73 · see citing works
Robins, V., & Turner, K. (2016) · Physica D Nonlinear Phenomena · doi:10.1016/j.physd.2016.03.007 · Cited by 68 · see citing works
Topological deep learning: a review of an emerging paradigm
Zia, A., Khamis, A., Nichols, J. D., Tayab, U. B., Hayder, Z., Rolland, V., Stone, E. A., & Petersson, L. (2024) · Artificial Intelligence Review · doi:10.1007/s10462-024-10710-9 · Cited by 74 · see citing works
Computing Multidimensional Persistence
Carlsson, G., Singh, G., & Zomorodian, A. J. (2009) · Lecture notes in computer science · doi:10.1007/978-3-642-10631-6_74 · Cited by 65 · see citing works
Self-citationBergomi, M. G., Frosini, P., Giorgi, D., & Quercioli, N. (2019) · Nature Machine Intelligence · doi:10.1038/s42256-019-0087-3 · Cited by 60 · see citing works
Decomposition of persistence modules
Botnan, M. B., & Crawley-Boevey, W. (2019) · Proceedings of the American Mathematical Society · doi:10.1090/proc/14790 · Cited by 60 · see citing works
Topological data analysis for the string landscape
Cole, A., & Shiu, G. (2019) · Journal of High Energy Physics · doi:10.1007/jhep03(2019)054 · Cited by 60 · see citing works
Topological methods for genomics: Present and future directions
Cámara, P. G. (2016) · Current Opinion in Systems Biology · doi:10.1016/j.coisb.2016.12.007 · Cited by 56 · see citing works
Persistent Betti numbers for a noise tolerant shape-based approach to image retrieval
Frosini, P., & Landi, C. (2012) · Pattern Recognition Letters · doi:10.1016/j.patrec.2012.10.015 · Cited by 54 · see citing works
Stratifying Multiparameter Persistent Homology
Harrington, H. A., Otter, N., Schenck, H., & Tillmann, U. (2019) · SIAM Journal on Applied Algebra and Geometry · doi:10.1137/18m1224350 · Cited by 52 · see citing works
One-dimensional reduction of multidimensional persistent homology
Cagliari, F., Fabio, B. D., & Ferri, M. (2010) · Proceedings of the American Mathematical Society · doi:10.1090/s0002-9939-10-10312-8 · Cited by 52 · see citing works
Generalized persistence diagrams for persistence modules over posets
Kim, W., & Mémoli, F. (2021) · Journal of Applied and Computational Topology · doi:10.1007/s41468-021-00075-1 · Cited by 49 · see citing works
Persistence Modules on Commutative Ladders of Finite Type
Escolar, E. G., & Hiraoka, Y. (2015) · Discrete & Computational Geometry · doi:10.1007/s00454-015-9746-2 · Cited by 48 · see citing works
Multidimensional Persistence and Noise
Scolamiero, M., Chachólski, W., Lundman, A., Ramanujam, R., & Öberg, S. (2016) · Foundations of Computational Mathematics · doi:10.1007/s10208-016-9323-y · Cited by 48 · see citing works
Integrated multimodal network approach to PET and MRI based on multidimensional persistent homology
Lee, H., Kang, H., Chung, M. K., Lim, S., Kim, B., & Lee, D. S. (2016) · Human Brain Mapping · doi:10.1002/hbm.23461 · Cited by 47 · see citing works
An introduction to multiparameter persistence
Botnan, M. B., & Lesnick, M. (2023) · EMS series of congress reports · doi:10.4171/ecr/19/4 · Cited by 45 · see citing works
Topological data analysis and machine learning
Leykam, D., & Angelakis, D. G. (2023) · Advances in Physics X · doi:10.1080/23746149.2023.2202331 · Cited by 47 · see citing works
Multiscale Persistent Functions for Biomolecular Structure Characterization
Xia, K., Li, Z., & Mu, L. (2017) · Bulletin of Mathematical Biology · doi:10.1007/s11538-017-0362-6 · Cited by 40 · see citing works
Computing Minimal Presentations and Bigraded Betti Numbers of 2-Parameter Persistent Homology
Lesnick, M., & Wright, M. (2022) · SIAM Journal on Applied Algebra and Geometry · doi:10.1137/20m1388425 · Cited by 39 · see citing works
An output-sensitive algorithm for persistent homology
Chen, C., & Kerber, M. (2012) · Computational Geometry · doi:10.1016/j.comgeo.2012.02.010 · Cited by 38 · see citing works
Ren, S., Wu, C., & Wu, J. (2018) · Rocky Mountain Journal of Mathematics · doi:10.1216/rmj-2018-48-8-2661 · Cited by 38 · see citing works
A topological approach to selecting models of biological experiments
Ulmer, M. P., Ziegelmeier, L., & Topaz, C. M. (2019) · PLoS ONE · doi:10.1371/journal.pone.0213679 · Cited by 36 · see citing works
The Rank Invariant Stability via Interleavings
Landi, C. (2018) · Association for Women in Mathematics series · doi:10.1007/978-3-319-89593-2_1 · Cited by 32 · see citing works
Persistent Topology and Metastable State in Conformational Dynamics
Chang, H.-W., Bacallado, S., Pande, V., & Carlsson, G. (2013) · PLoS ONE · doi:10.1371/journal.pone.0058699 · Cited by 31 · see citing works
Self-citationTopological spaces of persistence modules and their properties
Bubenik, P., & Vergili, T. (2018) · Journal of Applied and Computational Topology · doi:10.1007/s41468-018-0022-4 · Cited by 32 · see citing works
PHOG: Photometric and geometric functions for textured shape retrieval
Biasotti, S., Cerri, A., Giorgi, D., & Spagnuolo, M. (2013) · Computer Graphics Forum · doi:10.1111/cgf.12168 · Cited by 31 · see citing works
A Scale Space Based Persistence Measure for Critical Points in 2D Scalar Fields
Reininghaus, J., Kotava, N., Günther, D., Kasten, J., Hagen, H., & Hotz, I. (2011) · IEEE Transactions on Visualization and Computer Graphics · doi:10.1109/tvcg.2011.159 · Cited by 31 · see citing works
The observable structure of persistence modulesPreprint
Crawley-Boevey, W., Vin, & de Silva, V. (2014) · Cited by 31 · see citing works
A fast algorithm for constructing topological structure in large data
Liu, X., Xie, Z. Y., & Yi, D. (2012) · Homology Homotopy and Applications · doi:10.4310/hha.2012.v14.n1.a11 · Cited by 30 · see citing works
A kernel for multi-parameter persistent homology
Corbet, R., Fugacci, U., Kerber, M., Landi, C., & Wang, B. (2019) · Computers & Graphics X · doi:10.1016/j.cagx.2019.100005 · Cited by 30 · see citing works
Dey, T. K., Shi, D., & Wang, Y. (2019) · ACM Journal of Experimental Algorithmics · doi:10.1145/3284360 · Cited by 29 · see citing works
Spatiotemporal Persistent Homology for Dynamic Metric Spaces
Kim, W., & Mémoli, F. (2020) · Discrete & Computational Geometry · doi:10.1007/s00454-019-00168-w · Cited by 28 · see citing works
The Normalized Graph Cut and Cheeger Constant: From Discrete to Continuous
Ery Arias-castro, A., Bruno Pelletier, B., & Pudlo, P. (2012) · Advances in Applied Probability · doi:10.1239/aap/1354716583 · Cited by 26 · see citing works
On the Stability of Interval Decomposable Persistence Modules
Bjerkevik, H. B. (2021) · Discrete & Computational Geometry · doi:10.1007/s00454-021-00298-0 · Cited by 26 · see citing works
Towards Integrative Machine Learning and Knowledge Extraction
Holzinger, A., Goebel, R. G., Palade, V., & Ferri, M. (2017) · Lecture notes in computer science · doi:10.1007/978-3-319-69775-8_1 · Cited by 26 · see citing works
Stability of 2-Parameter Persistent Homology
Blumberg, A. J., & Lesnick, M. (2022) · Foundations of Computational Mathematics · doi:10.1007/s10208-022-09576-6 · Cited by 25 · see citing works
The Persistence Space in Multidimensional Persistent Homology
Cerri, A., & Landi, C. (2013) · Lecture notes in computer science · doi:10.1007/978-3-642-37067-0_16 · Cited by 24 · see citing works
Generalized persistence algorithm for decomposing multiparameter persistence modules
Dey, T. K., & Xin, C. (2022) · Journal of Applied and Computational Topology · doi:10.1007/s41468-022-00087-5 · Cited by 24 · see citing works
Aspects of topological approaches for data science
Grbić, J., Wu, J., Xia, K., & Wei, G. (2022) · Foundations of Data Science · doi:10.3934/fods.2022002 · Cited by 24 · see citing works
Combining Persistent Homology and Invariance Groups for Shape Comparison
Frosini, P., & Jabłoński, G. (2016) · Discrete & Computational Geometry · doi:10.1007/s00454-016-9761-y · Cited by 24 · see citing works
Ephemeral persistence modules and distance comparison
Berkouk, N., & Petit, F. (2021) · Algebraic & Geometric Topology · doi:10.2140/agt.2021.21.247 · Cited by 23 · see citing works
Combinatorial presentation of multidimensional persistent homology
Chachólski, W., Scolamiero, M., & Vaccarino, F. (2016) · Journal of Pure and Applied Algebra · doi:10.1016/j.jpaa.2016.09.001 · Cited by 23 · see citing works
Why Topology for Machine Learning and Knowledge Extraction?
Ferri, M. (2018) · Machine Learning and Knowledge Extraction · doi:10.3390/make1010006 · Cited by 23 · see citing works
Homological Algebra for Persistence Modules
Bubenik, P., & Milićević, N. (2021) · Foundations of Computational Mathematics · doi:10.1007/s10208-020-09482-9 · Cited by 23 · see citing works
Decomposition of Exact pfd Persistence Bimodules
Cochoy, J., & Oudot, S. (2019) · Discrete & Computational Geometry · doi:10.1007/s00454-019-00165-z · Cited by 6 · see citing works
Persistent Cohomology for Data With Multicomponent Heterogeneous Information
Cang, Z., & Wei, G. (2020) · SIAM Journal on Mathematics of Data Science · doi:10.1137/19m1272226 · Cited by 24 · see citing works
Capturing dynamics of time-varying data via topology
Xian, L., Adams, H., Topaz, C. M., & Ziegelmeier, L. (2021) · Foundations of Data Science · doi:10.3934/fods.2021033 · Cited by 21 · see citing works
Persistent Topology for Natural Data Analysis — A Survey
Ferri, M. (2017) · Lecture notes in computer science · doi:10.1007/978-3-319-69775-8_6 · Cited by 21 · see citing works
Topological data analysis and cosheaves
Curry, J. M. (2015) · Japan Journal of Industrial and Applied Mathematics · doi:10.1007/s13160-015-0173-9 · Cited by 21 · see citing works
A Topological Machine Learning Pipeline for Classification
Conti, F., Moroni, D., & Pascali, M. A. (2022) · Mathematics · doi:10.3390/math10173086 · Cited by 20 · see citing works
Nudged elastic band in topological data analysis
Atanasov, A., Carlsson, G., & Adams, H. (2015) · Topological Methods in Nonlinear Analysis · doi:10.12775/tmna.2015.013 · Cited by 20 · see citing works
Self-citationPersistent homology index as a robust quantitative measure of immunohistochemical scoring
Takiyama, A., Teramoto, T., Suzuki, H., Yamashiro, K., & Tanaka, S. (2017) · Scientific Reports · doi:10.1038/s41598-017-14392-y · Cited by 19 · see citing works
Evolutionary homology on coupled dynamical systems with applications to protein flexibility analysis
Cang, Z., Munch, E., & Wei, G. (2020) · Journal of Applied and Computational Topology · doi:10.1007/s41468-020-00057-9 · Cited by 19 · see citing works
Bottleneck stability for generalized persistence diagrams
McCleary, A., & Patel, A. (2019) · Proceedings of the American Mathematical Society · doi:10.1090/proc/14929 · Cited by 19 · see citing works
On interval decomposability of 2D persistence modules
Asashiba, H., Buchet, M., Escolar, E. G., Nakashima, K., & Yoshiwaki, M. (2022) · Computational Geometry · doi:10.1016/j.comgeo.2022.101879 · Cited by 19 · see citing works
A refinement of multi-dimensional persistence
KNUDSON, K. P. (2008) · Homology Homotopy and Applications · doi:10.4310/hha.2008.v10.n1.a11 · Cited by 19 · see citing works
Baas, N. A., Carlsson, G., Quick, G., Szymik, M., & Thaule, M. (2020) · Abel symposia · doi:10.1007/978-3-030-43408-3 · Cited by 18 · see citing works
Self-citationOn approximation of 2D persistence modules by interval-decomposables
Asashiba, H., Escolar, E. G., Nakashima, K., & Yoshiwaki, M. (2023) · Journal of Computational Algebra · doi:10.1016/j.jaca.2023.100007 · Cited by 22 · see citing works
The Weighted Euler Curve Transform for Shape and Image Analysis
Jiang, Q., Kurtek, S., & Needham, T. (2020) · IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops (CVPRW) · doi:10.1109/cvprw50498.2020.00430 · Cited by 17 · see citing works
Noise robustness of persistent homology on greyscale images, across filtrations and signatures
Turkeš, R., Nys, J., Verdonck, T., & Latré, S. (2021) · PLoS ONE · doi:10.1371/journal.pone.0257215 · Cited by 17 · see citing works
A Primer on Topological Data Analysis to Support Image Analysis Tasks in Environmental Science
Hoef, L. V., Adams, H., King, E. J., & Ebert‐Uphoff, I. (2022) · Artificial Intelligence for the Earth Systems · doi:10.1175/aies-d-22-0039.1 · Cited by 17 · see citing works
Topological Fidelity and Image Thresholding: A Persistent Homology Approach
Chung, Y.-M., & Day, S. (2018) · Journal of Mathematical Imaging and Vision · doi:10.1007/s10851-018-0802-4 · Cited by 17 · see citing works
An output-sensitive algorithm for persistent homology
Chen, C., & Kerber, M. (2011) · Annual Symposium on Computational Geometry (SoCG) · doi:10.1145/1998196.1998228 · Cited by 15 · see citing works
Homological approximations in persistence theory
Blanchette, B., Brüstle, T., & Hanson, E. J. (2022) · Canadian Journal of Mathematics · doi:10.4153/s0008414x22000657 · Cited by 16 · see citing works
Numeric invariants from multidimensional persistence
Skryzalin, J., & Carlsson, G. (2017) · Journal of Applied and Computational Topology · doi:10.1007/s41468-017-0003-z · Cited by 16 · see citing works
Self-citationSigned Barcodes for Multi-parameter Persistence via Rank Decompositions and Rank-Exact Resolutions
Botnan, M. B., Oppermann, S., & Oudot, S. (2024) · Foundations of Computational Mathematics · doi:10.1007/s10208-024-09672-9 · Cited by 15 · see citing works
Homological Algebra of Modules over Posets
Miller, E. (2025) · SIAM Journal on Applied Algebra and Geometry · doi:10.1137/22m1516361 · Cited by 15 · see citing works
On the Stability of Multigraded Betti Numbers and Hilbert Functions
Oudot, S., & Scoccola, L. (2024) · SIAM Journal on Applied Algebra and Geometry · doi:10.1137/22m1489150 · Cited by 15 · see citing works
Govc, D., & Hepworth, R. (2020) · Journal of Pure and Applied Algebra · doi:10.1016/j.jpaa.2020.106517 · Cited by 14 · see citing works
- Authors
- Carlsson, G., & Zomorodian, A.
- Affiliations
- Stanford University (US)
- Dartmouth College (US)
- Corresponding author
- Zomorodian, Afra
- ORCID
- Carlsson, Gunnar
- Venue
- Discrete & Computational Geometry
- Published
- 2009-04-24
- Volume
- 42
- Issue
- 1
- Pages
- 71-93
- ISSN
- 0179-5376 (ISSN-L)
- 1432-0444
- OpenAlex
- W4251980118
- Cited by
- 270
- References
- 19
- Citations by year
- 2026: 11 · 2025: 29 · 2024: 21 · 2023: 25 · 2022: 31 · 2021: 23 · 2020: 22 · 2019: 13 · 2018: 16 · 2017: 13 · 2016: 12 · 2015: 12 · 2014: 11 · 2013: 8 · 2012: 9
- FWCI
- 7.16
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Licence
- www.springer.com/tdm (tdm)
- Locations
- Discrete & Computational Geometry (publishedVersion)
- Funders
- National Science Foundation (0354543)
- Defense Advanced Research Projects Agency
- Office of Naval Research
- Topics
- Physical Sciences › Computer Science › Computational Theory and Mathematics › Topological and Geometric Data Analysis
- Physical Sciences › Mathematics › Mathematical Physics › Homotopy and Cohomology in Algebraic Topology
- Physical Sciences › Mathematics › Geometry and Topology › Geometric and Algebraic Topology
- Keywords
- persistent homology, Betti numbers, discrete invariants, topological data analysis
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review