{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,26]],"date-time":"2026-05-26T08:06:08Z","timestamp":1779782768333,"version":"3.53.1"},"reference-count":44,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2025,4,10]],"date-time":"2025-04-10T00:00:00Z","timestamp":1744243200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,4,10]],"date-time":"2025-04-10T00:00:00Z","timestamp":1744243200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["AAECC"],"published-print":{"date-parts":[[2026,5]]},"DOI":"10.1007\/s00200-025-00682-w","type":"journal-article","created":{"date-parts":[[2025,4,10]],"date-time":"2025-04-10T03:08:43Z","timestamp":1744254523000},"page":"695-723","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Efficient computation of homology groups, betti numbers, and euler characteristics for 2D digital images"],"prefix":"10.1007","volume":"37","author":[{"given":"Ahmet","family":"\u00d6ztel","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Bayram","family":"Akg\u00fcl","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"\u0130smet","family":"Karaca","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"\u00d6zg\u00fcr","family":"Ege","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2025,4,10]]},"reference":[{"key":"682_CR1","doi-asserted-by":"crossref","unstructured":"Alsalibi, B., et al.: \u201cA bio-inspired software for homology groups of 2D digital images\u201d. In: Asian Conference on Membrane Computing ACMC (2014). IEEE. 2014, pp.1\u20134","DOI":"10.1109\/ACMC.2014.7065800"},{"key":"682_CR2","unstructured":"Arslan, H., Karaca, I., Oztel, A.: \u201cHomology groups of n-dimensional digital images\u201d. In: XXI Turkish National Mathematics Symposium. (2008), pp.1\u201313"},{"issue":"3","key":"682_CR3","first-page":"621","volume":"22","author":"AJ Blumberg","year":"2022","unstructured":"Blumberg, A.J., Lesnick, M.: Stability of 2-parameter persistent homology. Found. Comput. Math. 22(3), 621\u2013669 (2022)","journal-title":"Found. Comput. Math."},{"issue":"1","key":"682_CR4","doi-asserted-by":"publisher","first-page":"51","DOI":"10.1023\/A:1008370600456","volume":"10","author":"L Boxer","year":"1999","unstructured":"Boxer, L.: A classical construction for the digital fundamental group. J. Math. Imaging Vision 10(1), 51\u201362 (1999)","journal-title":"J. Math. Imaging Vision"},{"issue":"8","key":"682_CR5","doi-asserted-by":"publisher","first-page":"833","DOI":"10.1016\/0167-8655(94)90012-4","volume":"15","author":"L Boxer","year":"1994","unstructured":"Boxer, L.: Digitally continuous functions. Pattern Recognit. Lett. 15(8), 833\u2013839 (1994)","journal-title":"Pattern Recognit. Lett."},{"key":"682_CR6","unstructured":"Boxer, L.: \u201cVariants on covering maps\u201d. In: arXiv:2303 (2023), p.001150v2"},{"issue":"1","key":"682_CR7","doi-asserted-by":"publisher","first-page":"23","DOI":"10.1007\/s10851-008-0088-z","volume":"32","author":"L Boxer","year":"2008","unstructured":"Boxer, L., Karaca, I.: The classification of digital covering spaces. J. Math. Imaging Vision 32(1), 23\u201329 (2008)","journal-title":"J. Math. Imaging Vision"},{"issue":"2","key":"682_CR8","first-page":"109","volume":"11","author":"L Boxer","year":"2011","unstructured":"Boxer, L., Karaca, I., Oztel, A.: Topological invariants in digital images. J. Math. Sci.: Adv. Appl. 11(2), 109\u2013140 (2011)","journal-title":"J. Math. Sci.: Adv. Appl."},{"issue":"2","key":"682_CR9","doi-asserted-by":"publisher","first-page":"139","DOI":"10.4995\/agt.2016.4624","volume":"17","author":"L Boxer","year":"2016","unstructured":"Boxer, L., Staecker, P.C.: Fundamental groups and Euler characteristics of sphere-like digital images. Appl. General Topol. 17(2), 139\u2013158 (2016)","journal-title":"Appl. General Topol."},{"issue":"02","key":"682_CR10","doi-asserted-by":"publisher","first-page":"149","DOI":"10.1142\/S0218654305000761","volume":"11","author":"G Carlsson","year":"2005","unstructured":"Carlsson, G., et al.: Persistence barcodes for shapes. Int. J. Shape Model. 11(02), 149\u2013187 (2005)","journal-title":"Int. J. Shape Model."},{"issue":"2","key":"682_CR11","doi-asserted-by":"publisher","first-page":"376","DOI":"10.1007\/s00454-022-00476-8","volume":"70","author":"R Corbet","year":"2023","unstructured":"Corbet, R., et al.: Computing the multicover bifiltration. Discret. & Comput. Geom. 70(2), 376\u2013405 (2023)","journal-title":"Discret. & Comput. Geom."},{"key":"682_CR12","volume-title":"Computational topology: an introduction","author":"H Edelsbrunner","year":"2010","unstructured":"Edelsbrunner, H., Harer, J.: Computational topology: an introduction. American Mathematical Society, Rhode Island (2010)"},{"key":"682_CR13","doi-asserted-by":"publisher","first-page":"511","DOI":"10.1007\/s00454-002-2885-2","volume":"28","author":"H Edelsbrunner","year":"2002","unstructured":"Edelsbrunner, H., Letscher, D., Zomorodian, A.: Topological persistence and simplification. Discret. & Comput. Geom. 28, 511\u2013533 (2002)","journal-title":"Discret. & Comput. Geom."},{"key":"682_CR14","doi-asserted-by":"publisher","first-page":"1296","DOI":"10.1007\/s00454-021-00281-9","volume":"65","author":"H Edelsbrunner","year":"2021","unstructured":"Edelsbrunner, H., Osang, G.: The multi-cover persistence of Euclidean balls. Discret. & Comput. Geom. 65, 1296\u20131313 (2021)","journal-title":"Discret. & Comput. Geom."},{"issue":"3","key":"682_CR15","doi-asserted-by":"publisher","first-page":"237","DOI":"10.22436\/jnsa.008.03.08","volume":"8","author":"O Ege","year":"2015","unstructured":"Ege, O., Karaca, I.: Banach fixed point theorem for digital images. J. Nonlinear Sci. 8(3), 237\u2013245 (2015)","journal-title":"J. Nonlinear Sci."},{"key":"682_CR16","doi-asserted-by":"crossref","unstructured":"Gabrielsson, R.B., Carlsson, G.: \u201cExposition and interpretation of the topology of neural networks\u201d. In: 18th IEEE international conference on machine learning and applications (ICMLA). IEEE. 2019, 1069\u20131076 (2019)","DOI":"10.1109\/ICMLA.2019.00180"},{"issue":"16","key":"682_CR17","doi-asserted-by":"publisher","first-page":"3314","DOI":"10.1016\/j.ins.2006.12.013","volume":"177","author":"S-E Han","year":"2007","unstructured":"Han, S.-E.: Digital fundamental group and Euler characteristic of a connected sum of digital closed surfaces. Inf. Sci. 177(16), 3314\u20133326 (2007)","journal-title":"Inf. Sci."},{"key":"682_CR18","first-page":"1","volume":"20","author":"C Hofer","year":"2019","unstructured":"Hofer, C., et al.: Learning representations of persistence barcodes. J. Mach. Learn. Res. 20, 1\u201345 (2019)","journal-title":"J. Mach. Learn. Res."},{"key":"682_CR19","volume-title":"Computational homology","author":"T Kaczynski","year":"2006","unstructured":"Kaczynski, T., Mischaikow, K., Mrozek, M.: Computational homology. Springer Science & Business Media, Cham (2006)"},{"key":"682_CR20","first-page":"178","volume":"1","author":"I Karaca","year":"2012","unstructured":"Karaca, I., Ege, O.: Cubical homology in digital image. Int. J. Inf. Comput. Sci. 1, 178\u2013187 (2012)","journal-title":"Int. J. Inf. Comput. Sci."},{"issue":"8","key":"682_CR21","first-page":"198","volume":"1","author":"I Karaca","year":"2012","unstructured":"Karaca, I.: Some results on simplicial homology groups of 2D digital images. Int. J. Inf. Comput. Sci. 1(8), 198\u2013203 (2012)","journal-title":"Int. J. Inf. Comput. Sci."},{"issue":"1","key":"682_CR22","first-page":"1","volume":"1","author":"E Khalimsky","year":"1987","unstructured":"Khalimsky, E.: Topological structures in computer science. Int. J. Stoch. Anal. 1(1), 1\u201316 (1987)","journal-title":"Int. J. Stoch. Anal."},{"issue":"1","key":"682_CR23","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/0166-8641(90)90031-V","volume":"36","author":"E Khalimsky","year":"1990","unstructured":"Khalimsky, E., Kopperman, R., Meyer, P.R.: Computer graphics and connected topologies on finite ordered sets. Topol. Appl. 36(1), 1\u201317 (1990)","journal-title":"Topol. Appl."},{"key":"682_CR24","unstructured":"Tat, Y.K.: \u201cDigital topology with applications to image-processing\u201d. PhD thesis. University of Oxford, (1985)"},{"issue":"3","key":"682_CR25","doi-asserted-by":"publisher","first-page":"357","DOI":"10.1016\/0734-189X(89)90147-3","volume":"48","author":"YK Tat","year":"1989","unstructured":"Tat, Y.K., Rosenfeld, A.: Digital topology: introduction and survey. Comput. Vision, Gr., Image Process. 48(3), 357\u2013393 (1989)","journal-title":"Comput. Vision, Gr., Image Process."},{"issue":"2","key":"682_CR26","doi-asserted-by":"publisher","first-page":"149","DOI":"10.1007\/s00200-019-00398-8","volume":"31","author":"D-W Lee","year":"2020","unstructured":"Lee, D.-W.: Digital H-spaces and actions in the pointed digital homotopy category. Appl. Algebra Eng. Commun. Comput. 31(2), 149\u2013169 (2020)","journal-title":"Appl. Algebra Eng. Commun. Comput."},{"issue":"1","key":"682_CR27","first-page":"39","volume":"88","author":"D-W Lee","year":"2014","unstructured":"Lee, D.-W.: Digital singular homology groups of digital images. Far East J. Math. Sci. 88(1), 39 (2014)","journal-title":"Far East J. Math. Sci."},{"issue":"3","key":"682_CR28","doi-asserted-by":"publisher","first-page":"261","DOI":"10.1007\/s00200-017-0341-z","volume":"29","author":"D-W Lee","year":"2018","unstructured":"Lee, D.-W.: Near-rings on digital Hopf groups. Appl. Algebra Eng. Commun. Comput. 29(3), 261\u2013282 (2018)","journal-title":"Appl. Algebra Eng. Commun. Comput."},{"key":"682_CR29","doi-asserted-by":"publisher","first-page":"20","DOI":"10.1016\/j.topol.2023.108644","volume":"338","author":"D-W Lee","year":"2023","unstructured":"Lee, D.-W., Staecker, P.C.: Digital topological groups. Topol. Appl. 338, 20\u2013108644 (2023)","journal-title":"Topol. Appl."},{"issue":"5","key":"682_CR30","doi-asserted-by":"publisher","first-page":"2417","DOI":"10.12785\/amis\/080537","volume":"8","author":"ET Meric","year":"2014","unstructured":"Meric, E.T., Karaca, I.: Computing higher dimensional digital homotopy groups. Appl. Math. Inf. Sci. 8(5), 2417\u20132425 (2014)","journal-title":"Appl. Math. Inf. Sci."},{"key":"682_CR31","unstructured":"Moor, M., et al.: \u201cTopological autoencoders\u201d. In: International Conference on Machine Learning (2020), pp.7045\u20137054"},{"issue":"1","key":"682_CR32","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1140\/epjds\/s13688-017-0109-5","volume":"6","author":"Nina Otter","year":"2017","unstructured":"Otter, Nina, et al.: A roadmap for the computation of persistent homology. EPJ Data Sci. 6(1), 17 (2017)","journal-title":"EPJ Data Sci."},{"key":"682_CR33","unstructured":"\u00d6ztel, A.: \u201cHomology Groups of Digital Images\u201d. Master\u2019s Thesis. Bornova, \u0130zmir: Ege University, Graduate School of Natural and Applied Sciences, Feb. (2010). url: https:\/\/tez.yok.gov.tr\/UlusalTezMerkezi\/TezGoster?key=NtBAevXNhYaNqJFoAcdBdi7lnZ35iccrHMiTEvmQBgU_EDPwyukln1Uy4T61-7Ei%3E"},{"key":"682_CR34","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.media.2019.03.014","volume":"55","author":"T Qaiser","year":"2019","unstructured":"Qaiser, T., et al.: Fast and accurate tumor segmentation of histology images using persistent homology and deep convolutional features. Med. Image Anal. 55, 1\u201314 (2019)","journal-title":"Med. Image Anal."},{"key":"682_CR35","doi-asserted-by":"crossref","unstructured":"Rabad\u00e1n, R., Blumberg, A.J.: Topological data analysis for genomics and evolution: topology in biology. Cambridge University Press, (2019)","DOI":"10.1017\/9781316671665"},{"key":"682_CR36","unstructured":"Rieck, B., et al.: \u201cNeural persistence: A complexity measure for deep neural networks using algebraic topology\u201d. In: arXiv preprint arXiv:1812.09764 (2019)"},{"issue":"3","key":"682_CR37","doi-asserted-by":"publisher","first-page":"177","DOI":"10.1016\/0167-8655(86)90017-6","volume":"4","author":"A Rosenfeld","year":"1986","unstructured":"Rosenfeld, A.: Continuous functions on digital pictures. Pattern Recogn. Lett. 4(3), 177\u2013184 (1986)","journal-title":"Pattern Recogn. Lett."},{"key":"682_CR38","doi-asserted-by":"crossref","unstructured":"Rosenfeld, A.: \u201cDigital topology\u201d. In: American Mathematical Monthly (1979), pp.621\u2013630","DOI":"10.1080\/00029890.1979.11994873"},{"issue":"2","key":"682_CR39","doi-asserted-by":"publisher","first-page":"119","DOI":"10.1016\/S0019-9958(81)90177-7","volume":"50","author":"A Rosenfeld","year":"1981","unstructured":"Rosenfeld, A.: Three-dimensional digital topology. Inf. Control 50(2), 119\u2013127 (1981)","journal-title":"Inf. Control"},{"key":"682_CR40","volume-title":"An introduction to algebraic topology","author":"J Rotman","year":"2013","unstructured":"Rotman, J.: An introduction to algebraic topology. Springer Science & Business Media, Cham (2013)"},{"issue":"12","key":"682_CR41","doi-asserted-by":"publisher","first-page":"1955","DOI":"10.1016\/0031-3203(95)00058-5","volume":"28","author":"P Saha","year":"1995","unstructured":"Saha, P., Chaudhuri, B.B.: A new approach to computing the Euler characteristic. Pattern Recogn. 28(12), 1955\u20131963 (1995)","journal-title":"Pattern Recogn."},{"key":"682_CR42","unstructured":"Sato, H.: \u201cAlgebraic topology: an intuitive approach\u201d. In: Mathematical Monographs 183 (1996)"},{"key":"682_CR43","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511546945","volume-title":"Topology for computing","author":"A Zomorodian","year":"2005","unstructured":"Zomorodian, A.: Topology for computing. Cambridge University Press, Cambridge (2005)"},{"issue":"2","key":"682_CR44","doi-asserted-by":"publisher","first-page":"249","DOI":"10.1007\/s00454-004-1146-y","volume":"33","author":"A Zomorodian","year":"2005","unstructured":"Zomorodian, A., Carlsson, G.: Computing persistent homology. Discret. & Comput. Geom. 33(2), 249\u2013274 (2005)","journal-title":"Discret. & Comput. Geom."}],"container-title":["Applicable Algebra in Engineering, Communication and Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00200-025-00682-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00200-025-00682-w","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00200-025-00682-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,5,26]],"date-time":"2026-05-26T07:46:50Z","timestamp":1779781610000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00200-025-00682-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,10]]},"references-count":44,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2026,5]]}},"alternative-id":["682"],"URL":"https:\/\/doi.org\/10.1007\/s00200-025-00682-w","relation":{},"ISSN":["0938-1279","1432-0622"],"issn-type":[{"value":"0938-1279","type":"print"},{"value":"1432-0622","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,4,10]]},"assertion":[{"value":"11 December 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 January 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 April 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}