{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T14:38:32Z","timestamp":1787236712122,"version":"build-2736575974"},"reference-count":54,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"1","funder":[{"DOI":"10.13039\/100000185","name":"Defense Advanced Research Projects Agency","doi-asserted-by":"publisher","award":["D15AP00109"],"award-info":[{"award-number":["D15AP00109"]}],"id":[{"id":"10.13039\/100000185","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000181","name":"Air Force Office of Scientific Research","doi-asserted-by":"publisher","award":["FA9550-13-1-0133"],"award-info":[{"award-number":["FA9550-13-1-0133"]}],"id":[{"id":"10.13039\/100000181","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000121","name":"Division of Mathematical Sciences","doi-asserted-by":"publisher","award":["1209136"],"award-info":[{"award-number":["1209136"]}],"id":[{"id":"10.13039\/100000121","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000121","name":"Division of Mathematical Sciences","doi-asserted-by":"publisher","award":["1057064"],"award-info":[{"award-number":["1057064"]}],"id":[{"id":"10.13039\/100000121","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["IIS 1546413"],"award-info":[{"award-number":["IIS 1546413"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Appl. Algebra Geometry"],"published-print":{"date-parts":[[2017,1]]},"abstract":"<jats:p>The goal of cryo-electron microscopy (EM) is to reconstruct the 3-dimensional structure of a molecule from a collection of its 2-dimensional projected images. In this paper, we show that the basic premise of cryo-EM---patching together 2-dimensional projections to reconstruct a 3-dimensional object---is naturally one of \u010cech cohomology with SO(2)-coefficients. We deduce that every cryo-EM reconstruction problem corresponds to an oriented circle bundle on a simplicial complex, allowing us to classify cryo-EM problems via principal bundles. In practice, the 2-dimensional images are noisy and a main task in cryo-EM is to denoise them. We will see how the aforementioned insights can be used towards this end.<\/jats:p>","DOI":"10.1137\/16m1070220","type":"journal-article","created":{"date-parts":[[2017,8,22]],"date-time":"2017-08-22T15:55:39Z","timestamp":1503417339000},"page":"507-535","source":"Crossref","is-referenced-by-count":7,"title":["Cohomology of Cryo-Electron Microscopy"],"prefix":"10.1137","volume":"1","author":[{"given":"Ke","family":"Ye","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Lek-Heng","family":"Lim","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2017,8,22]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1137\/050637492"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/21\/2\/004"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1088\/0266-5611\/15\/2\/022"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1088\/0031-9155\/42\/5\/008"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1006\/cviu.1999.0793"},{"key":"atypb6","unstructured":"A. S. Bandeira, Y. Chen, and A. Singer,\n                      Non-unique games over compact groups and orientation estimation in cryo-EM\n                      , preprint, 2015,http:\/\/arxiv.org\/abs\/1505.03840."},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1007\/s10107-016-0993-7"},{"key":"atypb8","doi-asserted-by":"crossref","unstructured":"C. Berge,\n                      Hypergraphs\n                      , North-Holland Math. Library 45, North-Holland, Amsterdam, 1989,https:\/\/doi.org\/10.1016\/S0924-6509(08)70093-X.","DOI":"10.1016\/S0924-6509(08)70093-X"},{"key":"atypb9","unstructured":"J.L. Brylinski,\n                      Loop Spaces, Characteristic Classes and Geometric Quantization\n                      , Birkha\u0308user Boston, Boston, 2008,htttps:\/\/doi.org\/10.1007\/978-0-8176-4731-5."},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1006\/cviu.2002.0984"},{"key":"atypb11","first-page":"114","author":"Chern S. S.","year":"1977","journal-title":"Berlin"},{"key":"atypb12","first-page":"10","volume":"54","author":"De Silva V.","year":"2007","journal-title":"Notices Amer. Math. Soc."},{"key":"atypb13","doi-asserted-by":"publisher","DOI":"10.1137\/S0036141002417814"},{"key":"atypb14","doi-asserted-by":"publisher","DOI":"10.1006\/cviu.1998.0721"},{"key":"atypb15","doi-asserted-by":"crossref","unstructured":"O. Forster,\n                      Lectures on Riemann surfaces\n                      , Grad. Texts Math. 81, Springer-Verlag, New York, 1991,https:\/\/doi.org\/10.1007\/978-1-4612-5961-9.","DOI":"10.1007\/978-1-4612-5961-9"},{"key":"atypb16","doi-asserted-by":"crossref","unstructured":"G. K. Francis,\n                      A Topological Picturebook\n                      , Springer, New York, 2007,https:\/\/doi.org\/10.1007\/978-0-387-68120-7.","DOI":"10.1007\/978-0-387-68120-7"},{"key":"atypb17","doi-asserted-by":"publisher","DOI":"10.1146\/annurev.biophys.31.082901.134202"},{"key":"atypb18","doi-asserted-by":"crossref","unstructured":"J. Frank,\n                      Three-Dimensional Electron Microscopy of Macromolecular Assemblies: Visualization of Biological Molecules in their Native State\n                      , Oxford University Press, New York, 2006,https:\/\/doi.org\/10.1093\/acprof:oso\/9780195182187.001.0001.","DOI":"10.1093\/acprof:oso\/9780195182187.003.0002"},{"key":"atypb19","doi-asserted-by":"crossref","unstructured":"P. Griffiths and J. Harris,\n                      Principles of Algebraic Geometry\n                      , Wiley, New York, 1994,https:\/\/doi.org\/10.1002\/9781118032527.","DOI":"10.1002\/9781118032527"},{"key":"atypb20","doi-asserted-by":"publisher","DOI":"10.4007\/annals.2011.174.2.11"},{"key":"atypb21","doi-asserted-by":"publisher","DOI":"10.1007\/s10208-011-9095-3"},{"key":"atypb22","doi-asserted-by":"crossref","unstructured":"R. Hartshorne,\n                      Algebraic Geometry\n                      , Grad. Texts Math. 52, Springer, New York, 1977,https:\/\/doi.org\/10.1007\/978-1-4757-3849-0.","DOI":"10.1007\/978-1-4757-3849-0"},{"key":"atypb23","unstructured":"A. Hatcher,\n                      Algebraic Topology\n                      , Cambridge University Press, Cambridge, UK, 2002,https:\/\/www.math.cornell.edu\/~hatcher\/AT\/AT.pdf."},{"key":"atypb24","doi-asserted-by":"crossref","unstructured":"T. W. Hungerford,\n                      Algebra\n                      , Grad. Texts Math. 73, Springer, New York, 1980,https:\/\/doi.org\/10.1007\/978-1-4612-6101-8.","DOI":"10.1007\/978-1-4612-6101-8"},{"key":"atypb25","doi-asserted-by":"crossref","unstructured":"D. Husemo\u0308ller,\n                      Fibre Bundles\n                      , 3rd ed., Grad. Texts Math. 20, Springer, New York, 1994,https:\/\/doi.org\/10.1007\/978-1-4757-2261-1.","DOI":"10.1007\/978-1-4757-2261-1_4"},{"key":"atypb26","doi-asserted-by":"crossref","unstructured":"D. Husemo\u0308ller, M. Joachim, B. Jurc\u030co, and M. Schottenloher,\n                      Basic Bundle Theory and $K$-Cohomology Invariants\n                      , Lecture Notes in Phys. 726, Springer, Berlin, 2008,https:\/\/doi.org\/10.1007\/978-3-540-74956-1.","DOI":"10.1007\/978-3-540-74956-1"},{"key":"atypb27","doi-asserted-by":"crossref","unstructured":"B. Iversen,\n                      Cohomology of Sheaves\n                      , Universitext, Springer-Verlag, Berlin, 1986,https:\/\/doi.org\/10.1007\/978-3-642-82783-9.","DOI":"10.1007\/978-3-642-82783-9"},{"key":"atypb28","doi-asserted-by":"crossref","unstructured":"F. John,\n                      Plane Waves and Spherical Means\n                      , Springer-Verlag, New York, 1981,https:\/\/doi.org\/10.1007\/978-1-4613-9453-2.","DOI":"10.1007\/978-1-4613-9453-2"},{"key":"atypb29","doi-asserted-by":"publisher","DOI":"10.2307\/2373408"},{"key":"atypb30","doi-asserted-by":"publisher","DOI":"10.1118\/1.597429"},{"key":"atypb31","unstructured":"L.H. Lim,\n                      Hodge Laplacians on Graphs\n                      ,\n                      Geometry and Topology in Statistical Inference\n                      , S. Mukherjee, ed., Proceedings of Symposia in Applied Math. 74, AMS, Providence, RI, 2017."},{"key":"atypb32","unstructured":"J. P. May,\n                      A Concise Course in Algebraic Topology\n                      , Chicago Lectures in Mathematics, University of Chicago Press, Chicago, 1999,https:\/\/www.math.uchicago.edu\/ may\/CONCISE\/ConciseRevised.pdf."},{"key":"atypb33","doi-asserted-by":"crossref","unstructured":"J. W. Milnor and J. D. Stasheff,\n                      Characteristic Classes\n                      , Princeton University Press, Princeton, NJ, 1974.","DOI":"10.1515\/9781400881826"},{"key":"atypb34","doi-asserted-by":"crossref","unstructured":"S. Morita,\n                      Geometry of Characteristic Classes\n                      , Transl. Math. Monogr. 199, AMS, Providence, RI, 2001.","DOI":"10.1090\/mmono\/199"},{"key":"atypb35","doi-asserted-by":"crossref","unstructured":"F. Natterer,\n                      The Mathematics of Computerized Tomography\n                      , Classics Appl. Math. 32, SIAM, Philadelphia, 2001,https:\/\/doi.org\/10.1137\/1.9780898719284.","DOI":"10.1137\/1.9780898719284"},{"key":"atypb36","doi-asserted-by":"publisher","DOI":"10.1109\/TIP.2009.2013081"},{"key":"atypb37","doi-asserted-by":"publisher","DOI":"10.1155\/IJMMS.2005.3487"},{"key":"atypb38","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3991(96)00037-X"},{"key":"atypb39","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3991(92)90233-A"},{"key":"atypb40","first-page":"11","volume":"17","author":"Penrose R.","year":"1991","journal-title":"Structural Topology"},{"key":"atypb41","doi-asserted-by":"crossref","unstructured":"V. V. Prasolov,\n                      Elements of Homology Theory\n                      , Grad. Stud. Math., 81, AMS, Providence, RI, 2007.","DOI":"10.1090\/gsm\/081"},{"key":"atypb42","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3991(94)90003-5"},{"key":"atypb43","doi-asserted-by":"publisher","DOI":"10.1088\/0031-9155\/44\/11\/302"},{"key":"atypb44","doi-asserted-by":"publisher","DOI":"10.1016\/j.acha.2010.02.001"},{"key":"atypb45","doi-asserted-by":"publisher","DOI":"10.1137\/090767777"},{"key":"atypb46","doi-asserted-by":"publisher","DOI":"10.1137\/090764657"},{"key":"atypb47","doi-asserted-by":"publisher","DOI":"10.1002\/cpa.21395"},{"key":"atypb48","doi-asserted-by":"publisher","DOI":"10.1137\/090778390"},{"key":"atypb49","doi-asserted-by":"crossref","unstructured":"E. H. Spanier,\n                      Algebraic Topology\n                      , Springer, New York, 1981,https:\/\/doi.org\/10.1007\/978-1-4684-9322-1.","DOI":"10.1007\/978-1-4684-9322-1"},{"key":"atypb50","doi-asserted-by":"publisher","DOI":"10.2307\/1969267"},{"key":"atypb51","first-page":"278","volume":"31","author":"Vainshtein B.","year":"1986","journal-title":"Sov. Phys. Dok."},{"key":"atypb52","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3991(87)90078-7"},{"key":"atypb53","doi-asserted-by":"publisher","DOI":"10.1007\/BF01447877"},{"key":"atypb54","doi-asserted-by":"publisher","DOI":"10.1109\/TMI.2002.801176"}],"container-title":["SIAM Journal on Applied Algebra and Geometry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/16M1070220","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,20]],"date-time":"2026-08-20T14:04:08Z","timestamp":1787234648000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/16M1070220"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,1]]},"references-count":54,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2017,1]]}},"alternative-id":["10.1137\/16M1070220"],"URL":"https:\/\/doi.org\/10.1137\/16m1070220","relation":{},"ISSN":["2470-6566"],"issn-type":[{"value":"2470-6566","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,1]]}}}