{"id":"https://openalex.org/W2003945700","doi":"https://doi.org/10.1109/tpami.1986.4767767","title":"Image Analysis Using Multigrid Relaxation Methods","display_name":"Image Analysis Using Multigrid Relaxation Methods","publication_year":1986,"publication_date":"1986-03-01","ids":{"openalex":"https://openalex.org/W2003945700","doi":"https://doi.org/10.1109/tpami.1986.4767767","mag":"2003945700","pmid":"https://pubmed.ncbi.nlm.nih.gov/21869332"},"language":"en","primary_location":{"id":"doi:10.1109/tpami.1986.4767767","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tpami.1986.4767767","pdf_url":null,"source":{"id":"https://openalex.org/S199944782","display_name":"IEEE Transactions on Pattern Analysis and Machine Intelligence","issn_l":"0162-8828","issn":["0162-8828","1939-3539","2160-9292"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320439","host_organization_name":"IEEE Computer Society","host_organization_lineage":["https://openalex.org/P4310320439","https://openalex.org/P4310319808"],"host_organization_lineage_names":["IEEE Computer Society","Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Pattern Analysis and Machine Intelligence","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","pubmed"],"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/A5034397232","display_name":"Demetri Terzopoulos","orcid":null},"institutions":[{"id":"https://openalex.org/I63966007","display_name":"Massachusetts Institute of Technology","ror":"https://ror.org/042nb2s44","country_code":"US","type":"education","lineage":["https://openalex.org/I63966007"]}],"countries":["US"],"is_corresponding":true,"raw_author_name":"Demetri Terzopoulos","raw_affiliation_strings":["M.I.T Artificial Intelligence Laboratory, 545 Technology Square, Cambridge, MA 02139","[M.I.T Artificial Intelligence Laboratory, 545 Technology Square, Cambridge, MA 02139.]"],"affiliations":[{"raw_affiliation_string":"M.I.T Artificial Intelligence Laboratory, 545 Technology Square, Cambridge, MA 02139","institution_ids":[]},{"raw_affiliation_string":"[M.I.T Artificial Intelligence Laboratory, 545 Technology Square, Cambridge, MA 02139.]","institution_ids":["https://openalex.org/I63966007"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":["https://openalex.org/A5034397232"],"corresponding_institution_ids":["https://openalex.org/I63966007"],"apc_list":null,"apc_paid":null,"fwci":18.1106,"has_fulltext":false,"cited_by_count":372,"citation_normalized_percentile":{"value":0.99281371,"is_in_top_1_percent":true,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"PAMI-8","issue":"2","first_page":"129","last_page":"139"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10531","display_name":"Advanced Vision and Imaging","score":0.9998999834060669,"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"}},"topics":[{"id":"https://openalex.org/T10531","display_name":"Advanced Vision and Imaging","score":0.9998999834060669,"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/T10481","display_name":"Computer Graphics and Visualization Techniques","score":0.9988999962806702,"subfield":{"id":"https://openalex.org/subfields/1704","display_name":"Computer Graphics and Computer-Aided Design"},"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/T11019","display_name":"Image Enhancement Techniques","score":0.9986000061035156,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/multigrid-method","display_name":"Multigrid method","score":0.8984491229057312},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.69171541929245},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.5688719749450684},{"id":"https://openalex.org/keywords/image-processing","display_name":"Image processing","score":0.49624162912368774},{"id":"https://openalex.org/keywords/iterative-reconstruction","display_name":"Iterative reconstruction","score":0.4955400228500366},{"id":"https://openalex.org/keywords/relaxation","display_name":"Relaxation (psychology)","score":0.49082764983177185},{"id":"https://openalex.org/keywords/iterative-method","display_name":"Iterative method","score":0.47407737374305725},{"id":"https://openalex.org/keywords/optical-flow","display_name":"Optical flow","score":0.46220895648002625},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.4588850140571594},{"id":"https://openalex.org/keywords/regularization","display_name":"Regularization (linguistics)","score":0.4568506181240082},{"id":"https://openalex.org/keywords/inverse-problem","display_name":"Inverse problem","score":0.4215902090072632},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.39547407627105713},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.33007311820983887},{"id":"https://openalex.org/keywords/image","display_name":"Image (mathematics)","score":0.3278311491012573},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.2561836242675781}],"concepts":[{"id":"https://openalex.org/C137119250","wikidata":"https://www.wikidata.org/wiki/Q1413101","display_name":"Multigrid method","level":3,"score":0.8984491229057312},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.69171541929245},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.5688719749450684},{"id":"https://openalex.org/C9417928","wikidata":"https://www.wikidata.org/wiki/Q1070689","display_name":"Image processing","level":3,"score":0.49624162912368774},{"id":"https://openalex.org/C141379421","wikidata":"https://www.wikidata.org/wiki/Q6094427","display_name":"Iterative reconstruction","level":2,"score":0.4955400228500366},{"id":"https://openalex.org/C2776029896","wikidata":"https://www.wikidata.org/wiki/Q3935810","display_name":"Relaxation (psychology)","level":2,"score":0.49082764983177185},{"id":"https://openalex.org/C159694833","wikidata":"https://www.wikidata.org/wiki/Q2321565","display_name":"Iterative method","level":2,"score":0.47407737374305725},{"id":"https://openalex.org/C155542232","wikidata":"https://www.wikidata.org/wiki/Q736111","display_name":"Optical flow","level":3,"score":0.46220895648002625},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.4588850140571594},{"id":"https://openalex.org/C2776135515","wikidata":"https://www.wikidata.org/wiki/Q17143721","display_name":"Regularization (linguistics)","level":2,"score":0.4568506181240082},{"id":"https://openalex.org/C135252773","wikidata":"https://www.wikidata.org/wiki/Q1567213","display_name":"Inverse problem","level":2,"score":0.4215902090072632},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.39547407627105713},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.33007311820983887},{"id":"https://openalex.org/C115961682","wikidata":"https://www.wikidata.org/wiki/Q860623","display_name":"Image (mathematics)","level":2,"score":0.3278311491012573},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.2561836242675781},{"id":"https://openalex.org/C77805123","wikidata":"https://www.wikidata.org/wiki/Q161272","display_name":"Social psychology","level":1,"score":0.0},{"id":"https://openalex.org/C15744967","wikidata":"https://www.wikidata.org/wiki/Q9418","display_name":"Psychology","level":0,"score":0.0},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1109/tpami.1986.4767767","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tpami.1986.4767767","pdf_url":null,"source":{"id":"https://openalex.org/S199944782","display_name":"IEEE Transactions on Pattern Analysis and Machine Intelligence","issn_l":"0162-8828","issn":["0162-8828","1939-3539","2160-9292"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320439","host_organization_name":"IEEE Computer Society","host_organization_lineage":["https://openalex.org/P4310320439","https://openalex.org/P4310319808"],"host_organization_lineage_names":["IEEE Computer Society","Institute of Electrical and Electronics Engineers"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"IEEE Transactions on Pattern Analysis and Machine Intelligence","raw_type":"journal-article"},{"id":"pmid:21869332","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/21869332","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,"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":"IEEE transactions on pattern analysis and machine intelligence","raw_type":null}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":47,"referenced_works":["https://openalex.org/W2751752","https://openalex.org/W112287601","https://openalex.org/W1481069933","https://openalex.org/W1539768692","https://openalex.org/W1578285471","https://openalex.org/W1623955465","https://openalex.org/W1660110636","https://openalex.org/W1965774677","https://openalex.org/W1979622972","https://openalex.org/W1979966786","https://openalex.org/W1981025738","https://openalex.org/W1990779621","https://openalex.org/W1994794884","https://openalex.org/W2003368332","https://openalex.org/W2020999234","https://openalex.org/W2022140147","https://openalex.org/W2024060531","https://openalex.org/W2031555400","https://openalex.org/W2046064079","https://openalex.org/W2048330959","https://openalex.org/W2049935762","https://openalex.org/W2066780173","https://openalex.org/W2084943008","https://openalex.org/W2107792892","https://openalex.org/W2114429593","https://openalex.org/W2119241866","https://openalex.org/W2139762693","https://openalex.org/W2150629612","https://openalex.org/W2151317657","https://openalex.org/W2154661161","https://openalex.org/W2164847484","https://openalex.org/W2321310861","https://openalex.org/W2498633332","https://openalex.org/W2570617923","https://openalex.org/W2751023760","https://openalex.org/W2795512983","https://openalex.org/W3022071797","https://openalex.org/W3142915818","https://openalex.org/W4250530934","https://openalex.org/W4254380507","https://openalex.org/W4255090195","https://openalex.org/W4285719527","https://openalex.org/W4300938688","https://openalex.org/W6650990417","https://openalex.org/W6662777261","https://openalex.org/W6671747336","https://openalex.org/W6677234374"],"related_works":["https://openalex.org/W2077078424","https://openalex.org/W2107386309","https://openalex.org/W1928301487","https://openalex.org/W2151220638","https://openalex.org/W2725829804","https://openalex.org/W2545869789","https://openalex.org/W2105149350","https://openalex.org/W2058165188","https://openalex.org/W1964555484","https://openalex.org/W2141090006"],"abstract_inverted_index":{"Image":[0],"analysis":[1,145,157],"problems,":[2],"posed":[3],"mathematically":[4],"as":[5,9,84],"variational":[6],"principles":[7],"or":[8],"partial":[10],"differential":[11],"equations,":[12],"are":[13,22,30,46],"amenable":[14],"to":[15,139],"numerical":[16],"solution":[17,149],"by":[18,50,69],"relaxation":[19],"algorithms":[20,45,101,116],"that":[21,124],"local,":[23],"iterative,":[24],"and":[25,106,109],"often":[26],"parallel.":[27],"Although":[28],"they":[29,63],"well":[31],"suited":[32],"structurally":[33],"for":[34,102,146],"implementation":[35],"on":[36,90],"massively":[37],"parallel,":[38],"locally":[39],"interconnected":[40],"computational":[41],"architectures,":[42],"such":[43],"distributed":[44],"seriously":[47],"handi":[48],"capped":[49],"an":[51,136],"inherent":[52],"inefficiency":[53],"at":[54],"propagating":[55],"constraints":[56],"between":[57],"widely":[58],"separated":[59],"processing":[60],"elements.":[61],"Hence,":[62],"converge":[64],"extremely":[65],"slowly":[66],"when":[67],"confronted":[68],"the":[70,112,147],"large":[71],"representations":[72],"of":[73,77,114,150,154],"early":[74,131],"vision.":[75,132],"Application":[76],"multigrid":[78,122],"methods":[79],"can":[80],"overcome":[81],"this":[82,95],"drawback,":[83],"we":[85,97,110,125],"showed":[86],"in":[87,130,141],"previous":[88],"work":[89],"3-D":[91],"surface":[92],"reconstruction.":[93],"In":[94],"paper,":[96],"develop":[98],"multiresolution":[99],"iterative":[100],"computing":[103],"lightness,":[104],"shape-from-shading,":[105],"optical":[107],"flow,":[108],"examine":[111],"efficiency":[113],"these":[115],"using":[117],"synthetic":[118],"image":[119,156],"inputs.":[120],"The":[121],"methodology":[123],"describe":[126],"is":[127,135],"broadly":[128],"applicable":[129],"Notably,":[133],"it":[134],"appealing":[137],"strategy":[138],"use":[140],"conjunction":[142],"with":[143],"regularization":[144],"efficient":[148],"a":[151],"wide":[152],"range":[153],"ill-posed":[155],"problems.":[158]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":1},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":1},{"year":2022,"cited_by_count":5},{"year":2021,"cited_by_count":3},{"year":2020,"cited_by_count":4},{"year":2019,"cited_by_count":6},{"year":2018,"cited_by_count":3},{"year":2017,"cited_by_count":3},{"year":2015,"cited_by_count":4},{"year":2014,"cited_by_count":7},{"year":2013,"cited_by_count":4},{"year":2012,"cited_by_count":6}],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}