{"id":"https://openalex.org/W2766477323","doi":"https://doi.org/10.1137/17m1153509","title":"Structural Variability from Noisy Tomographic Projections","display_name":"Structural Variability from Noisy Tomographic Projections","publication_year":2018,"publication_date":"2018-01-01","ids":{"openalex":"https://openalex.org/W2766477323","doi":"https://doi.org/10.1137/17m1153509","mag":"2766477323","pmid":"https://pubmed.ncbi.nlm.nih.gov/30555617","pmcid":"https://www.ncbi.nlm.nih.gov/pmc/articles/6294454"},"language":"en","primary_location":{"id":"doi:10.1137/17m1153509","doi":"https://doi.org/10.1137/17m1153509","is_oa":false,"landing_page_url":"https://doi.org/10.1137/17m1153509","pdf_url":null,"source":{"id":"https://openalex.org/S152600803","display_name":"SIAM Journal on Imaging Sciences","issn_l":"1936-4954","issn":["1936-4954"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["anvur-scientific","ccf-b","cwts-core","dongbi-b","jufo-2","norway-1","poland-140","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Imaging Sciences","raw_type":"journal-article"},"volume":null,"type":"article","indexed_in":["arxiv","crossref","datacite","pubmed"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/6294454","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5043290570","display_name":"Joakim And\u00e9n","orcid":"https://orcid.org/0000-0002-3377-813X","observed_orcids":["https://orcid.org/0000-0002-3377-813X"]},"institutions":[{"id":"https://openalex.org/I4387153999","display_name":"Flatiron Institute","ror":"https://ror.org/00sekdz59","country_code":"US","type":"nonprofit","lineage":["https://openalex.org/I4210107338","https://openalex.org/I4387153999"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Joakim And\u00e9n","raw_affiliation_strings":["Center for Computational Biology, Flatiron Institute, New York, NY 10100"],"raw_orcid":"https://orcid.org/0000-0002-3377-813X","affiliations":[{"raw_affiliation_string":"Center for Computational Biology, Flatiron Institute, New York, NY 10100","institution_ids":["https://openalex.org/I4387153999"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5062844820","display_name":"Amit Singer","orcid":"https://orcid.org/0000-0002-6975-7955","observed_orcids":["https://orcid.org/0000-0002-6975-7955"]},"institutions":[{"id":"https://openalex.org/I20089843","display_name":"Princeton University","ror":"https://ror.org/00hx57361","country_code":"US","type":"education","lineage":["https://openalex.org/I20089843"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Amit Singer","raw_affiliation_strings":["Department of Mathematics and Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Program in Applied and Computational Mathematics, Princeton University, Princeton, NJ 08544","institution_ids":["https://openalex.org/I20089843"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":3.0275,"has_fulltext":true,"cited_by_count":40,"citation_normalized_percentile":{"value":0.92926045,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":88,"max":99},"biblio":{"volume":"11","issue":"2","first_page":"1441","last_page":"1492"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10857","display_name":"Advanced Electron Microscopy Techniques and Applications","score":0.8403000235557556,"subfield":{"id":"https://openalex.org/subfields/1315","display_name":"Structural Biology"},"field":{"id":"https://openalex.org/fields/13","display_name":"Biochemistry, Genetics and Molecular Biology"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}},"topics":[{"id":"https://openalex.org/T10857","display_name":"Advanced Electron Microscopy Techniques and Applications","score":0.8403000235557556,"subfield":{"id":"https://openalex.org/subfields/1315","display_name":"Structural Biology"},"field":{"id":"https://openalex.org/fields/13","display_name":"Biochemistry, Genetics and Molecular Biology"},"domain":{"id":"https://openalex.org/domains/1","display_name":"Life Sciences"}},{"id":"https://openalex.org/T10688","display_name":"Image and Signal Denoising Methods","score":0.05649999901652336,"subfield":{"id":"https://openalex.org/subfields/1707","display_name":"Computer Vision and Pattern Recognition"},"field":{"id":"https://openalex.org/fields/17","display_name":"Computer Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T11205","display_name":"Numerical methods in inverse problems","score":0.0142000000923872,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/cryo-electron-microscopy","display_name":"cryo-electron microscopy","score":0.8399999737739563},{"id":"https://openalex.org/keywords/structural-variability","display_name":"structural variability","score":0.7419999837875366},{"id":"https://openalex.org/keywords/low-rank-covariance","display_name":"low-rank covariance","score":0.7020000219345093},{"id":"https://openalex.org/keywords/linear-inverse-problems","display_name":"linear inverse problems","score":0.7480000257492065},{"id":"https://openalex.org/keywords/least-squares-estimation","display_name":"least squares estimation","score":0.7229999899864197},{"id":"https://openalex.org/keywords/deconvolution","display_name":"deconvolution","score":0.8640000224113464},{"id":"https://openalex.org/keywords/conjugate-gradient-method","display_name":"conjugate gradient method","score":0.8040000200271606},{"id":"https://openalex.org/keywords/circulant-preconditioners","display_name":"circulant preconditioners","score":0.9639999866485596},{"id":"https://openalex.org/keywords/tomographic-projections","display_name":"tomographic projections","score":0.5989999771118164}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6262819766998291},{"id":"https://openalex.org/C178650346","wikidata":"https://www.wikidata.org/wiki/Q201984","display_name":"Covariance","level":2,"score":0.604127049446106},{"id":"https://openalex.org/C185429906","wikidata":"https://www.wikidata.org/wiki/Q1130160","display_name":"Estimator","level":2,"score":0.5741205811500549},{"id":"https://openalex.org/C185142706","wikidata":"https://www.wikidata.org/wiki/Q1134404","display_name":"Covariance matrix","level":2,"score":0.5593871474266052},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.5580407381057739},{"id":"https://openalex.org/C81184566","wikidata":"https://www.wikidata.org/wiki/Q1191895","display_name":"Conjugate gradient method","level":2,"score":0.523657500743866},{"id":"https://openalex.org/C174576160","wikidata":"https://www.wikidata.org/wiki/Q1183700","display_name":"Deconvolution","level":2,"score":0.49814486503601074},{"id":"https://openalex.org/C180877172","wikidata":"https://www.wikidata.org/wiki/Q5401390","display_name":"Estimation of covariance matrices","level":3,"score":0.49363625049591064},{"id":"https://openalex.org/C118006245","wikidata":"https://www.wikidata.org/wiki/Q6792079","display_name":"Mat\u00e9rn covariance function","level":5,"score":0.4397442936897278},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.42980048060417175},{"id":"https://openalex.org/C164226766","wikidata":"https://www.wikidata.org/wiki/Q7293202","display_name":"Rank (graph theory)","level":2,"score":0.4256754219532013},{"id":"https://openalex.org/C83042196","wikidata":"https://www.wikidata.org/wiki/Q5178898","display_name":"Covariance intersection","level":4,"score":0.30724209547042847},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.19689041376113892},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.18598371744155884},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.10409331321716309},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.0}],"mesh":[],"locations_count":6,"locations":[{"id":"doi:10.1137/17m1153509","doi":"https://doi.org/10.1137/17m1153509","is_oa":false,"landing_page_url":"https://doi.org/10.1137/17m1153509","pdf_url":null,"source":{"id":"https://openalex.org/S152600803","display_name":"SIAM Journal on Imaging Sciences","issn_l":"1936-4954","issn":["1936-4954"],"is_oa":false,"is_in_doaj":false,"is_core":true,"listed_in":["anvur-scientific","ccf-b","cwts-core","dongbi-b","jufo-2","norway-1","poland-140","russia-white-list-1","vabb-shw"],"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM Journal on Imaging Sciences","raw_type":"journal-article"},{"id":"pmid:30555617","doi":null,"is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/30555617","pdf_url":null,"source":{"id":"https://openalex.org/S4306525036","display_name":"PubMed","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"SIAM journal on imaging sciences","raw_type":"Journal Article"},{"id":"pmh:oai:europepmc.org:5262910","doi":null,"is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/6294454","pdf_url":null,"source":{"id":"https://openalex.org/S4306400806","display_name":"Europe PMC (PubMed Central)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I1303153112","host_organization_name":"European Bioinformatics Institute","host_organization_lineage":["https://openalex.org/I1303153112"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"acceptedVersion","is_accepted":true,"is_published":false,"raw_source_name":null,"raw_type":"Text"},{"id":"pmh:oai:pubmedcentral.nih.gov:6294454","doi":null,"is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/6294454","pdf_url":null,"source":{"id":"https://openalex.org/S2764455111","display_name":"PubMed Central","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I1299303238","host_organization_name":"National Institutes of Health","host_organization_lineage":["https://openalex.org/I1299303238"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"acceptedVersion","is_accepted":true,"is_published":false,"raw_source_name":"SIAM J Imaging Sci","raw_type":"Text"},{"id":"pmh:oai:arXiv.org:1710.09791","doi":null,"is_oa":true,"landing_page_url":"http://arxiv.org/abs/1710.09791","pdf_url":"https://arxiv.org/pdf/1710.09791","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.1710.09791","doi":"https://doi.org/10.48550/arxiv.1710.09791","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.1710.09791","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:europepmc.org:5262910","doi":null,"is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/6294454","pdf_url":null,"source":{"id":"https://openalex.org/S4306400806","display_name":"Europe PMC (PubMed Central)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I1303153112","host_organization_name":"European Bioinformatics Institute","host_organization_lineage":["https://openalex.org/I1303153112"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"acceptedVersion","is_accepted":true,"is_published":false,"raw_source_name":null,"raw_type":"Text"},"sustainable_development_goals":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G3276902854","display_name":"Improved Algorithms for Macromolecular Structure Determination by cyro-EM","funder_award_id":"R01GM090200","funder_id":"https://openalex.org/F4320337354","funder_display_name":"National Institute of General Medical Sciences"}],"funders":[{"id":"https://openalex.org/F4320306164","display_name":"Simons Foundation","ror":"https://ror.org/01cmst727"},{"id":"https://openalex.org/F4320306202","display_name":"Gordon and Betty Moore Foundation","ror":"https://ror.org/006wxqw41"},{"id":"https://openalex.org/F4320337354","display_name":"National Institute of General Medical Sciences","ror":"https://ror.org/04q48ey07"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2766477323.pdf","grobid_xml":"https://content.openalex.org/works/W2766477323.grobid-xml"},"referenced_works_count":87,"referenced_works":["https://openalex.org/W36187607","https://openalex.org/W93530381","https://openalex.org/W575681570","https://openalex.org/W1483313280","https://openalex.org/W1483804921","https://openalex.org/W1492556495","https://openalex.org/W1506342804","https://openalex.org/W1520752838","https://openalex.org/W1533471646","https://openalex.org/W1548938959","https://openalex.org/W1568854817","https://openalex.org/W1576783654","https://openalex.org/W1623890683","https://openalex.org/W1941957828","https://openalex.org/W1967374108","https://openalex.org/W1970377488","https://openalex.org/W1975696503","https://openalex.org/W1978885539","https://openalex.org/W1982584811","https://openalex.org/W2001923477","https://openalex.org/W2005041367","https://openalex.org/W2005240547","https://openalex.org/W2006025109","https://openalex.org/W2010122118","https://openalex.org/W2011665065","https://openalex.org/W2012300893","https://openalex.org/W2016982898","https://openalex.org/W2017508876","https://openalex.org/W2021703585","https://openalex.org/W2036823849","https://openalex.org/W2040957249","https://openalex.org/W2044098317","https://openalex.org/W2047902460","https://openalex.org/W2048826637","https://openalex.org/W2049633694","https://openalex.org/W2051925784","https://openalex.org/W2053860607","https://openalex.org/W2057602562","https://openalex.org/W2058583833","https://openalex.org/W2060581589","https://openalex.org/W2061171222","https://openalex.org/W2072834874","https://openalex.org/W2073949924","https://openalex.org/W2080097309","https://openalex.org/W2092709807","https://openalex.org/W2104234755","https://openalex.org/W2105367101","https://openalex.org/W2106005123","https://openalex.org/W2115755118","https://openalex.org/W2117948104","https://openalex.org/W2121752514","https://openalex.org/W2121947440","https://openalex.org/W2128801186","https://openalex.org/W2133307020","https://openalex.org/W2134189568","https://openalex.org/W2139916860","https://openalex.org/W2142158597","https://openalex.org/W2143962756","https://openalex.org/W2147001810","https://openalex.org/W2148575499","https://openalex.org/W2150593711","https://openalex.org/W2150790075","https://openalex.org/W2150938882","https://openalex.org/W2158044763","https://openalex.org/W2160880244","https://openalex.org/W2164659462","https://openalex.org/W2167034439","https://openalex.org/W2316564661","https://openalex.org/W2432427150","https://openalex.org/W2469490828","https://openalex.org/W2498179566","https://openalex.org/W2528518535","https://openalex.org/W2569750427","https://openalex.org/W2574455081","https://openalex.org/W2587625522","https://openalex.org/W2588224366","https://openalex.org/W2592692890","https://openalex.org/W2606074844","https://openalex.org/W2609504638","https://openalex.org/W2754811825","https://openalex.org/W2783817513","https://openalex.org/W2950967305","https://openalex.org/W2952274186","https://openalex.org/W2963669613","https://openalex.org/W2963927955","https://openalex.org/W4213367101","https://openalex.org/W4233363368"],"related_works":["https://openalex.org/W2963669613","https://openalex.org/W1941957828","https://openalex.org/W1568854817","https://openalex.org/W7130762989","https://openalex.org/W2010134232","https://openalex.org/W2072951627","https://openalex.org/W4366817303","https://openalex.org/W2204858460","https://openalex.org/W4306662177","https://openalex.org/W2039827057"],"abstract_inverted_index":{"In":[0],"cryo-electron":[1],"microscopy,":[2],"the":[3,60,64,68,73,112,119,126,129,146,158,167,188,243,246],"three-dimensional":[4],"(3D)":[5],"electric":[6],"potentials":[7,29],"of":[8,11,35,67,75,92,166,191,245,252],"an":[9,98,151,239],"ensemble":[10],"molecules":[12],"are":[13,132],"projected":[14],"along":[15],"arbitrary":[16],"viewing":[17],"directions":[18],"to":[19,58,134,181,224],"yield":[20],"noisy":[21,171],"two-dimensional":[22],"images.":[23],"The":[24,155],"volume":[25],"maps":[26],"representing":[27],"these":[28],"typically":[30],"exhibit":[31],"a":[32,80,85,135,202],"great":[33],"deal":[34],"structural":[36,253],"variability,":[37],"which":[38],"is":[39,50,116,142,157],"described":[40],"by":[41,145,187],"their":[42],"3D":[43,168],"covariance":[44,48,77,103,169,197],"matrix.":[45],"Typically,":[46],"this":[47,76,102],"matrix":[49,78,198],"approximately":[51],"low":[52],"rank":[53],"and":[54,219],"can":[55],"be":[56],"used":[57],"cluster":[59],"volumes":[61,237],"or":[62],"estimate":[63],"intrinsic":[65],"geometry":[66],"conformation":[69],"space.":[70],"We":[71,212,231],"formulate":[72],"estimation":[74,165],"as":[79],"linear":[81],"inverse":[82],"problem,":[83],"yielding":[84],"consistent":[86,164],"least-squares":[87],"estimator.":[88],"For":[89],"$n$":[90],"images":[91],"size":[93],"$N$-by-$N$":[94],"pixels,":[95],"we":[96,200],"propose":[97],"algorithm":[99,162,248],"for":[100,163,195,249],"calculating":[101],"estimator":[104],"with":[105,150,179],"computational":[106],"complexity":[107],"$\\mathcal{O}(nN^4+\\sqrt{\\kappa}N^6":[108],"\\log":[109],"N)$,":[110],"where":[111],"condition":[113],"number":[114],"$\\kappa$":[115],"empirically":[117],"in":[118,138,177,227,238],"range":[120],"10--200.":[121],"Its":[122],"efficiency":[123],"relies":[124],"on":[125,216,235],"observation":[127],"that":[128,205],"normal":[130],"equations":[131],"equivalent":[133],"deconvolution":[136],"problem":[137],"six":[139],"dimensions.":[140],"This":[141],"then":[143],"solved":[144],"conjugate":[147],"gradient":[148],"method":[149],"appropriate":[152],"circulant":[153],"preconditioner.":[154],"result":[156],"first":[159],"computationally":[160],"efficient":[161],"from":[170],"projections.":[172],"It":[173],"also":[174,232],"compares":[175],"favorably":[176],"runtime":[178],"respect":[180],"previously":[182],"proposed":[183,247],"nonconsistent":[184],"estimators.":[185],"Motivated":[186],"recent":[189],"success":[190],"eigenvalue":[192],"shrinkage":[193,203],"procedures":[194],"high-dimensional":[196],"estimation,":[199],"incorporate":[201],"procedure":[204],"improves":[206],"accuracy":[207],"at":[208],"lower":[209],"signal-to-noise":[210],"ratios.":[211],"evaluate":[213],"our":[214],"methods":[215,226],"simulated":[217],"datasets":[218],"achieve":[220],"classification":[221],"results":[222,234],"comparable":[223],"state-of-the-art":[225],"shorter":[228],"running":[229],"time.":[230],"present":[233],"clustering":[236],"experimental":[240],"dataset,":[241],"illustrating":[242],"power":[244],"practical":[250],"determination":[251],"variability.":[254]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":5},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":5},{"year":2022,"cited_by_count":1},{"year":2021,"cited_by_count":11},{"year":2020,"cited_by_count":6},{"year":2019,"cited_by_count":8},{"year":2018,"cited_by_count":1}],"updated_date":"2026-11-05T12:25:35.769452","created_date":"2017-11-10T00:00:00"}