{"id":"https://openalex.org/W2154641379","doi":"https://doi.org/10.1109/cvprw.2008.4563000","title":"Dealing with uncertainty in the principal directions of tensors","display_name":"Dealing with uncertainty in the principal directions of tensors","publication_year":2008,"publication_date":"2008-06-01","ids":{"openalex":"https://openalex.org/W2154641379","doi":"https://doi.org/10.1109/cvprw.2008.4563000","mag":"2154641379"},"language":"en","primary_location":{"id":"doi:10.1109/cvprw.2008.4563000","is_oa":false,"landing_page_url":"https://doi.org/10.1109/cvprw.2008.4563000","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"2008 IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops","raw_type":"proceedings-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":false,"oa_status":"closed","oa_url":null,"any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5056449949","display_name":"Maxime Boucher","orcid":"https://orcid.org/0000-0002-0715-9909"},"institutions":[{"id":"https://openalex.org/I5023651","display_name":"McGill University","ror":"https://ror.org/01pxwe438","country_code":"CA","type":"education","lineage":["https://openalex.org/I5023651"]}],"countries":["CA"],"is_corresponding":true,"raw_author_name":"Maxime Boucher","raw_affiliation_strings":["School of Computer Science, McGill University, Montreal, Canada"],"affiliations":[{"raw_affiliation_string":"School of Computer Science, McGill University, Montreal, Canada","institution_ids":["https://openalex.org/I5023651"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5011305571","display_name":"Alan C. Evans","orcid":"https://orcid.org/0000-0003-3841-6098"},"institutions":[{"id":"https://openalex.org/I5023651","display_name":"McGill University","ror":"https://ror.org/01pxwe438","country_code":"CA","type":"education","lineage":["https://openalex.org/I5023651"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Alan Evans","raw_affiliation_strings":["McConnell Brain Imaging Center, McGill University, Montreal, Canada"],"affiliations":[{"raw_affiliation_string":"McConnell Brain Imaging Center, McGill University, Montreal, Canada","institution_ids":["https://openalex.org/I5023651"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5056449949"],"corresponding_institution_ids":["https://openalex.org/I5023651"],"apc_list":null,"apc_paid":null,"fwci":0.2488,"has_fulltext":false,"cited_by_count":1,"citation_normalized_percentile":{"value":0.67174619,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":"1","last_page":"8"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11304","display_name":"Advanced Neuroimaging Techniques and Applications","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2741","display_name":"Radiology, Nuclear Medicine and Imaging"},"field":{"id":"https://openalex.org/fields/27","display_name":"Medicine"},"domain":{"id":"https://openalex.org/domains/4","display_name":"Health Sciences"}},"topics":[{"id":"https://openalex.org/T11304","display_name":"Advanced Neuroimaging Techniques and Applications","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/2741","display_name":"Radiology, Nuclear Medicine and Imaging"},"field":{"id":"https://openalex.org/fields/27","display_name":"Medicine"},"domain":{"id":"https://openalex.org/domains/4","display_name":"Health Sciences"}},{"id":"https://openalex.org/T10378","display_name":"Advanced MRI Techniques and Applications","score":0.9901999831199646,"subfield":{"id":"https://openalex.org/subfields/2741","display_name":"Radiology, Nuclear Medicine and Imaging"},"field":{"id":"https://openalex.org/fields/27","display_name":"Medicine"},"domain":{"id":"https://openalex.org/domains/4","display_name":"Health Sciences"}},{"id":"https://openalex.org/T12603","display_name":"NMR spectroscopy and applications","score":0.9889000058174133,"subfield":{"id":"https://openalex.org/subfields/3106","display_name":"Nuclear and High Energy Physics"},"field":{"id":"https://openalex.org/fields/31","display_name":"Physics and Astronomy"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.8094197511672974},{"id":"https://openalex.org/keywords/tensor","display_name":"Tensor (intrinsic definition)","score":0.753061830997467},{"id":"https://openalex.org/keywords/isotropy","display_name":"Isotropy","score":0.6123037934303284},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.5853902697563171},{"id":"https://openalex.org/keywords/principal-component-analysis","display_name":"Principal component analysis","score":0.5780125260353088},{"id":"https://openalex.org/keywords/invariant","display_name":"Invariant (physics)","score":0.5399408936500549},{"id":"https://openalex.org/keywords/anisotropy","display_name":"Anisotropy","score":0.5255167484283447},{"id":"https://openalex.org/keywords/tensor-density","display_name":"Tensor density","score":0.4923931360244751},{"id":"https://openalex.org/keywords/monte-carlo-method","display_name":"Monte Carlo method","score":0.48664626479148865},{"id":"https://openalex.org/keywords/upper-and-lower-bounds","display_name":"Upper and lower bounds","score":0.464310884475708},{"id":"https://openalex.org/keywords/symmetric-tensor","display_name":"Symmetric tensor","score":0.4530458450317383},{"id":"https://openalex.org/keywords/principal-axis-theorem","display_name":"Principal axis theorem","score":0.44973224401474},{"id":"https://openalex.org/keywords/diffusion-mri","display_name":"Diffusion MRI","score":0.44791871309280396},{"id":"https://openalex.org/keywords/multivariate-statistics","display_name":"Multivariate statistics","score":0.4386640191078186},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.4367266893386841},{"id":"https://openalex.org/keywords/cartesian-tensor","display_name":"Cartesian tensor","score":0.436509907245636},{"id":"https://openalex.org/keywords/statistical-physics","display_name":"Statistical physics","score":0.41833046078681946},{"id":"https://openalex.org/keywords/tensor-field","display_name":"Tensor field","score":0.36482906341552734},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.3161865472793579},{"id":"https://openalex.org/keywords/exact-solutions-in-general-relativity","display_name":"Exact solutions in general relativity","score":0.2987350821495056},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.22636014223098755},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.16456955671310425},{"id":"https://openalex.org/keywords/quantum-mechanics","display_name":"Quantum mechanics","score":0.10747897624969482},{"id":"https://openalex.org/keywords/mathematical-physics","display_name":"Mathematical physics","score":0.082479327917099}],"concepts":[{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.8094197511672974},{"id":"https://openalex.org/C155281189","wikidata":"https://www.wikidata.org/wiki/Q3518150","display_name":"Tensor (intrinsic definition)","level":2,"score":0.753061830997467},{"id":"https://openalex.org/C184050105","wikidata":"https://www.wikidata.org/wiki/Q273163","display_name":"Isotropy","level":2,"score":0.6123037934303284},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.5853902697563171},{"id":"https://openalex.org/C27438332","wikidata":"https://www.wikidata.org/wiki/Q2873","display_name":"Principal component analysis","level":2,"score":0.5780125260353088},{"id":"https://openalex.org/C190470478","wikidata":"https://www.wikidata.org/wiki/Q2370229","display_name":"Invariant (physics)","level":2,"score":0.5399408936500549},{"id":"https://openalex.org/C85725439","wikidata":"https://www.wikidata.org/wiki/Q466686","display_name":"Anisotropy","level":2,"score":0.5255167484283447},{"id":"https://openalex.org/C148125525","wikidata":"https://www.wikidata.org/wiki/Q904927","display_name":"Tensor density","level":4,"score":0.4923931360244751},{"id":"https://openalex.org/C19499675","wikidata":"https://www.wikidata.org/wiki/Q232207","display_name":"Monte Carlo method","level":2,"score":0.48664626479148865},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.464310884475708},{"id":"https://openalex.org/C20178491","wikidata":"https://www.wikidata.org/wiki/Q2204117","display_name":"Symmetric tensor","level":3,"score":0.4530458450317383},{"id":"https://openalex.org/C161326058","wikidata":"https://www.wikidata.org/wiki/Q7245073","display_name":"Principal axis theorem","level":2,"score":0.44973224401474},{"id":"https://openalex.org/C149550507","wikidata":"https://www.wikidata.org/wiki/Q899360","display_name":"Diffusion MRI","level":3,"score":0.44791871309280396},{"id":"https://openalex.org/C161584116","wikidata":"https://www.wikidata.org/wiki/Q1952580","display_name":"Multivariate statistics","level":2,"score":0.4386640191078186},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.4367266893386841},{"id":"https://openalex.org/C64835786","wikidata":"https://www.wikidata.org/wiki/Q17004583","display_name":"Cartesian tensor","level":5,"score":0.436509907245636},{"id":"https://openalex.org/C121864883","wikidata":"https://www.wikidata.org/wiki/Q677916","display_name":"Statistical physics","level":1,"score":0.41833046078681946},{"id":"https://openalex.org/C166077713","wikidata":"https://www.wikidata.org/wiki/Q1758924","display_name":"Tensor field","level":3,"score":0.36482906341552734},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.3161865472793579},{"id":"https://openalex.org/C520416788","wikidata":"https://www.wikidata.org/wiki/Q5419229","display_name":"Exact solutions in general relativity","level":2,"score":0.2987350821495056},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.22636014223098755},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.16456955671310425},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","level":1,"score":0.10747897624969482},{"id":"https://openalex.org/C37914503","wikidata":"https://www.wikidata.org/wiki/Q156495","display_name":"Mathematical physics","level":1,"score":0.082479327917099},{"id":"https://openalex.org/C143409427","wikidata":"https://www.wikidata.org/wiki/Q161238","display_name":"Magnetic resonance imaging","level":2,"score":0.0},{"id":"https://openalex.org/C126838900","wikidata":"https://www.wikidata.org/wiki/Q77604","display_name":"Radiology","level":1,"score":0.0},{"id":"https://openalex.org/C71924100","wikidata":"https://www.wikidata.org/wiki/Q11190","display_name":"Medicine","level":0,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1109/cvprw.2008.4563000","is_oa":false,"landing_page_url":"https://doi.org/10.1109/cvprw.2008.4563000","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"2008 IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops","raw_type":"proceedings-article"},{"id":"pmh:oai:CiteSeerX.psu:10.1.1.419.8206","is_oa":false,"landing_page_url":"http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.419.8206","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"http://mplab.ucsd.edu/wp-content/uploads/CVPR2008/WorkShops/data/papers/051.pdf","raw_type":"text"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":18,"referenced_works":["https://openalex.org/W1528905581","https://openalex.org/W1541153235","https://openalex.org/W1544006236","https://openalex.org/W1983496390","https://openalex.org/W1989592929","https://openalex.org/W2011842664","https://openalex.org/W2013959399","https://openalex.org/W2100177310","https://openalex.org/W2123738261","https://openalex.org/W2131339155","https://openalex.org/W2143017382","https://openalex.org/W2144929975","https://openalex.org/W2161913873","https://openalex.org/W2796910762","https://openalex.org/W3015237271","https://openalex.org/W3022774728","https://openalex.org/W4234362502","https://openalex.org/W4301861531"],"related_works":["https://openalex.org/W1965338607","https://openalex.org/W2981910625","https://openalex.org/W2765690628","https://openalex.org/W2607194831","https://openalex.org/W2058505222","https://openalex.org/W4360921185","https://openalex.org/W2403348579","https://openalex.org/W2010106354","https://openalex.org/W4236556086","https://openalex.org/W2066112700"],"abstract_inverted_index":{"The":[0,52],"variability":[1],"of":[2,17,36,45,77,99,129,156,161,165,183,209],"tensor":[3,47,57,81,131],"fields":[4],"is":[5,58,64,109,117,148,171],"usually":[6],"analyzed":[7,189],"with":[8],"multivariate":[9],"statistical":[10,40,201],"distributions.":[11],"Multivariate":[12],"distributions":[13],"model":[14],"every":[15],"component":[16],"the":[18,33,37,43,46,75,103,106,115,127,145,154,162,168,184,192,196,213,221],"tensor,":[19],"which":[20],"are":[21,82],"not":[22],"invariant":[23],"under":[24],"rotation.":[25],"They":[26],"therefore":[27],"tell":[28],"very":[29],"little":[30],"information":[31],"about":[32],"true":[34],"shape":[35],"tensor.":[38],"A":[39],"analysis":[41,164],"on":[42,167,204],"eigenvalues":[44,53,76,128,193,205],"would":[48],"be":[49,120,133,188,224],"more":[50],"revealing.":[51],"determine":[54],"if":[55],"a":[56,78,111,122,130,137],"uniaxial,":[59],"i.e.:":[60],"only":[61,218],"one":[62,219],"eigenvalue":[63],"different":[65],"from":[66],"zero,":[67],"isotropic,":[68],"volume":[69],"preserving":[70],"or":[71,231],"purely":[72],"anisotropic.":[73],"However,":[74],"normally":[79,86,134],"distributed":[80,135],"not,":[83],"in":[84,173,206,217],"general,":[85],"distributed.":[87],"In":[88],"this":[89,93,174,177],"paper,":[90],"we":[91,125,179],"solve":[92],"problem":[94],"directly":[95,203],"for":[96],"small":[97],"sizes":[98],"samples":[100],"by":[101,190],"determining":[102],"probability":[104],"that":[105,144,181],"maximum":[107],"error":[108,116],"within":[110,121],"reasonable":[112,123],"bound.":[113],"When":[114],"likely":[118],"to":[119,132,226],"bound,":[124],"consider":[126],"along":[136],"mean":[138],"eigendirection.":[139],"Monte":[140],"Carlo":[141],"simulation":[142],"shows":[143],"computed":[146],"bound":[147],"tight":[149],"and":[150],"becomes":[151],"tighter":[152],"when":[153],"number":[155],"sample":[157],"increases.":[158],"An":[159],"application":[160,227],"method,":[163],"deformations":[166,186],"cortical":[169],"surface,":[170],"presented":[172],"paper.":[175],"On":[176],"data,":[178],"found":[180],"80%":[182],"anisotropic":[185],"could":[187,223],"modeling":[191],"directly.":[194],"Thus,":[195],"proposed":[197],"method":[198,214,222],"allows":[199],"formulating":[200],"hypothesis":[202],"many":[207],"cases":[208],"measured":[210],"deformations.":[211],"Although":[212],"was":[215],"used":[216],"application,":[220],"extended":[225],"involving":[228,235],"diffusion":[229],"MRI":[230],"other":[232],"imaging":[233],"technique":[234],"tensors.":[236]},"counts_by_year":[],"updated_date":"2026-04-05T17:49:38.594831","created_date":"2025-10-10T00:00:00"}