{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T14:38:13Z","timestamp":1787323093338,"version":"build-2736575974"},"reference-count":32,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Imaging Sci."],"published-print":{"date-parts":[[2011,1]]},"abstract":"<jats:p>The cryo-electron microscopy (cryo-EM) reconstruction problem is to find the three-dimensional structure of a macromolecule given noisy versions of its two-dimensional projection images at unknown random directions. We introduce a new algorithm for identifying noisy cryo-EM images of nearby viewing angles. This identification is an important first step in three-dimensional structure determination of macromolecules from cryo-EM, because once identified, these images can be rotationally aligned and averaged to produce \u201cclass averages\u201d of better quality. The main advantage of our algorithm is its extreme robustness to noise. The algorithm is also very efficient in terms of running time and memory requirements, because it is based on the computation of the top few eigenvectors of a specially designed sparse Hermitian matrix. These advantages are demonstrated in numerous numerical experiments.<\/jats:p>","DOI":"10.1137\/090778390","type":"journal-article","created":{"date-parts":[[2011,6,24]],"date-time":"2011-06-24T12:42:51Z","timestamp":1308919371000},"page":"723-759","source":"Crossref","is-referenced-by-count":52,"title":["Viewing Angle Classification of Cryo-Electron Microscopy Images Using Eigenvectors"],"prefix":"10.1137","volume":"4","author":[{"given":"A.","family":"Singer","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Z.","family":"Zhao","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Y.","family":"Shkolnisky","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"R.","family":"Hadani","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2011,6,23]]},"reference":[{"key":"R1","doi-asserted-by":"publisher","DOI":"10.1109\/83.846252"},{"key":"R2","doi-asserted-by":"publisher","DOI":"10.1109\/83.846251"},{"key":"R3","doi-asserted-by":"publisher","DOI":"10.1016\/j.acha.2006.04.006"},{"key":"R4","doi-asserted-by":"publisher","DOI":"10.1016\/j.acha.2009.11.003"},{"key":"R5","doi-asserted-by":"publisher","DOI":"10.1109\/TIP.2008.2002305"},{"key":"R6","doi-asserted-by":"publisher","DOI":"10.1016\/j.jsb.2005.08.006"},{"key":"R7","unstructured":"M. P. Do Carmo,\n                      Differential Geometry of Curves and Surfaces\n                      , Prentice\u2013Hall, Englewood Cliffs, NJ, 1976."},{"key":"R8","doi-asserted-by":"publisher","DOI":"10.1126\/science.280.5360.69"},{"key":"R9","unstructured":"C. L. Epstein,\n                      Introduction to the Mathematics of Medical Imaging\n                      , Pearson Education, Upper Saddle River, NJ, 2003."},{"key":"R10","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."},{"key":"R11","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.0735871100"},{"key":"R12","unstructured":"R. Hadani and A. Singer,\n                      Representation theoretic patterns in cryo electron microscopy\n                      I:\n                      The intrinsic reconstitution algorithm\n                      , Ann. of Math. (2), to appear."},{"key":"R13","unstructured":"R. Hadani and A. Singer,\n                      Representation theoretic patterns in cryo electron microscopy\n                      II:\n                      The class averaging problem\n                      , Found. Comput. Math., to appear."},{"key":"R14","doi-asserted-by":"publisher","DOI":"10.1016\/S0304-3991(01)00154-1"},{"key":"R15","doi-asserted-by":"crossref","unstructured":"O. Khorunzhiy,\n                      Rooted trees and moments of large sparse random matrices\n                      , in Discrete Random Walks, Discrete Math. Theor. Comput. Sci. Proc., AC, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, France, 2003, pp. 145\u2013154.","DOI":"10.46298\/dmtcs.3346"},{"key":"R16","doi-asserted-by":"publisher","DOI":"10.1239\/aap\/999187900"},{"key":"R17","doi-asserted-by":"publisher","DOI":"10.1007\/s10540-004-7190-2"},{"key":"R18","doi-asserted-by":"publisher","DOI":"10.2307\/2320860"},{"key":"R19","doi-asserted-by":"crossref","unstructured":"F. Natterer,\n                      The Mathematics of Computerized Tomography\n                      , Classics Appl. Math. 32, SIAM, Philadelphia, 2001.","DOI":"10.1137\/1.9780898719284"},{"key":"R20","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3991(96)00037-X"},{"key":"R21","unstructured":"C. Ponce and A. Singer,\n                      Computing steerable principal components of a large set of images and their rotations\n                      , IEEE Trans. Image Process., to appear."},{"key":"R22","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3991(94)90003-5"},{"key":"R23","doi-asserted-by":"publisher","DOI":"10.1016\/j.acha.2010.02.001"},{"key":"R24","doi-asserted-by":"publisher","DOI":"10.1137\/090767777"},{"key":"R25","doi-asserted-by":"crossref","unstructured":"H. Solomon,\n                      Geometric Probability\n                      , SIAM, Philadelphia, 1978.","DOI":"10.1137\/1.9781611970418"},{"key":"R26","unstructured":"B. Vainshtein and A. Goncharov,\n                      Determination of the spatial orientation of arbitrarily arranged identical particles of an unknown structure from their projections\n                      , in Proceedings of the 11th International Congress on Electron Microscopy, T. Imura, S. Maruse, and T. Suzuki, eds., Japan Soc. Electron Microscopy, Tokyo, Japan, 1986, pp. 459\u2013460."},{"key":"R27","doi-asserted-by":"publisher","DOI":"10.1016\/0304-3991(87)90078-7"},{"key":"R28","doi-asserted-by":"publisher","DOI":"10.1017\/S0033583500003644"},{"key":"R29","doi-asserted-by":"publisher","DOI":"10.1038\/30918"},{"key":"R30","doi-asserted-by":"publisher","DOI":"10.2307\/1970079"},{"key":"R31","doi-asserted-by":"publisher","DOI":"10.2307\/1970008"},{"key":"R32","unstructured":"Z. Zhao and A. Singer,\n                      Fast Rotational Alignment of Cryo-EM Images Using Steerable Principal Components\n                      , preprint."}],"container-title":["SIAM Journal on Imaging Sciences"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/090778390","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T13:41:33Z","timestamp":1787319693000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/090778390"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2011,1]]},"references-count":32,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2011,1]]}},"alternative-id":["10.1137\/090778390"],"URL":"https:\/\/doi.org\/10.1137\/090778390","relation":{},"ISSN":["1936-4954"],"issn-type":[{"value":"1936-4954","type":"electronic"}],"subject":[],"published":{"date-parts":[[2011,1]]}}}