{"id":"https://openalex.org/W2139101145","doi":"https://doi.org/10.1109/tip.2009.2026678","title":"Color Image Segmentation by the Vector-Valued Allen\u2013Cahn Phase-Field Model: A Multigrid Solution","display_name":"Color Image Segmentation by the Vector-Valued Allen\u2013Cahn Phase-Field Model: A Multigrid Solution","publication_year":2009,"publication_date":"2009-07-07","ids":{"openalex":"https://openalex.org/W2139101145","doi":"https://doi.org/10.1109/tip.2009.2026678","mag":"2139101145","pmid":"https://pubmed.ncbi.nlm.nih.gov/19586818"},"language":"en","primary_location":{"id":"doi:10.1109/tip.2009.2026678","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tip.2009.2026678","pdf_url":null,"source":{"id":"https://openalex.org/S4210173141","display_name":"IEEE Transactions on Image Processing","issn_l":"1057-7149","issn":["1057-7149","1941-0042"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["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 Image Processing","raw_type":"journal-article"},"type":"article","indexed_in":["arxiv","crossref","pubmed"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/0710.0736","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5037754645","display_name":"David Kay","orcid":"https://orcid.org/0000-0003-4984-3669"},"institutions":[{"id":"https://openalex.org/I4210146410","display_name":"Science Oxford","ror":"https://ror.org/04j8yhy50","country_code":"GB","type":"nonprofit","lineage":["https://openalex.org/I4210146410"]}],"countries":["GB"],"is_corresponding":true,"raw_author_name":"D.A. Kay","raw_affiliation_strings":["Oxford Computing Laboratory, Wolfson Building, Oxford OX1 3QD, UK. david.kay@comlab.ox.ac.uk","Oxford Comput. Lab., Oxford, UK"],"affiliations":[{"raw_affiliation_string":"Oxford Computing Laboratory, Wolfson Building, Oxford OX1 3QD, UK. david.kay@comlab.ox.ac.uk","institution_ids":[]},{"raw_affiliation_string":"Oxford Comput. Lab., Oxford, UK","institution_ids":["https://openalex.org/I4210146410"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5072240392","display_name":"Alessandro Tomasi","orcid":"https://orcid.org/0000-0002-3518-9400"},"institutions":[{"id":"https://openalex.org/I162608824","display_name":"University of Sussex","ror":"https://ror.org/00ayhx656","country_code":"GB","type":"education","lineage":["https://openalex.org/I162608824"]}],"countries":["GB"],"is_corresponding":false,"raw_author_name":"A. Tomasi","raw_affiliation_strings":["Department of Mathematics, University of Sussex, Brighton, UK","Department of Mathematics, University of Sussex, Falmer, Brighton, U.K"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Sussex, Brighton, UK","institution_ids":["https://openalex.org/I162608824"]},{"raw_affiliation_string":"Department of Mathematics, University of Sussex, Falmer, Brighton, U.K","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5037754645"],"corresponding_institution_ids":["https://openalex.org/I4210146410"],"apc_list":null,"apc_paid":null,"fwci":1.4456,"has_fulltext":false,"cited_by_count":61,"citation_normalized_percentile":{"value":0.82228009,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":94,"max":99},"biblio":{"volume":"18","issue":"10","first_page":"2330","last_page":"2339"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11087","display_name":"Solidification and crystal growth phenomena","score":0.9975000023841858,"subfield":{"id":"https://openalex.org/subfields/2505","display_name":"Materials Chemistry"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T11087","display_name":"Solidification and crystal growth phenomena","score":0.9975000023841858,"subfield":{"id":"https://openalex.org/subfields/2505","display_name":"Materials Chemistry"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12100","display_name":"Advanced Mathematical Modeling in Engineering","score":0.9961000084877014,"subfield":{"id":"https://openalex.org/subfields/1703","display_name":"Computational Theory and Mathematics"},"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/T10670","display_name":"Aluminum Alloy Microstructure Properties","score":0.9240999817848206,"subfield":{"id":"https://openalex.org/subfields/2202","display_name":"Aerospace Engineering"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/multigrid-method","display_name":"Multigrid method","score":0.7136322259902954},{"id":"https://openalex.org/keywords/image-segmentation","display_name":"Image segmentation","score":0.6315188407897949},{"id":"https://openalex.org/keywords/finite-element-method","display_name":"Finite element method","score":0.5408594608306885},{"id":"https://openalex.org/keywords/vector-field","display_name":"Vector field","score":0.5103771090507507},{"id":"https://openalex.org/keywords/segmentation","display_name":"Segmentation","score":0.4970095455646515},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.45913392305374146},{"id":"https://openalex.org/keywords/scale-space-segmentation","display_name":"Scale-space segmentation","score":0.43027758598327637},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.41830235719680786},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.41297444701194763},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.37844744324684143},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.36685997247695923},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.3467028737068176},{"id":"https://openalex.org/keywords/computer-vision","display_name":"Computer vision","score":0.3265515863895416},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.30386272072792053},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.26775842905044556},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.2274036407470703},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.10323584079742432}],"concepts":[{"id":"https://openalex.org/C137119250","wikidata":"https://www.wikidata.org/wiki/Q1413101","display_name":"Multigrid method","level":3,"score":0.7136322259902954},{"id":"https://openalex.org/C124504099","wikidata":"https://www.wikidata.org/wiki/Q56933","display_name":"Image segmentation","level":3,"score":0.6315188407897949},{"id":"https://openalex.org/C135628077","wikidata":"https://www.wikidata.org/wiki/Q220184","display_name":"Finite element method","level":2,"score":0.5408594608306885},{"id":"https://openalex.org/C91188154","wikidata":"https://www.wikidata.org/wiki/Q186247","display_name":"Vector field","level":2,"score":0.5103771090507507},{"id":"https://openalex.org/C89600930","wikidata":"https://www.wikidata.org/wiki/Q1423946","display_name":"Segmentation","level":2,"score":0.4970095455646515},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.45913392305374146},{"id":"https://openalex.org/C65885262","wikidata":"https://www.wikidata.org/wiki/Q7429708","display_name":"Scale-space segmentation","level":4,"score":0.43027758598327637},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.41830235719680786},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.41297444701194763},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.37844744324684143},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.36685997247695923},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.3467028737068176},{"id":"https://openalex.org/C31972630","wikidata":"https://www.wikidata.org/wiki/Q844240","display_name":"Computer vision","level":1,"score":0.3265515863895416},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.30386272072792053},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.26775842905044556},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.2274036407470703},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.10323584079742432},{"id":"https://openalex.org/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.0}],"mesh":[{"descriptor_ui":"D000465","descriptor_name":"Algorithms","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true},{"descriptor_ui":"D000465","descriptor_name":"Algorithms","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true},{"descriptor_ui":"D000465","descriptor_name":"Algorithms","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true},{"descriptor_ui":"D001185","descriptor_name":"Artificial Intelligence","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D001185","descriptor_name":"Artificial Intelligence","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D001185","descriptor_name":"Artificial Intelligence","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D003116","descriptor_name":"Color","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true},{"descriptor_ui":"D003116","descriptor_name":"Color","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true},{"descriptor_ui":"D003116","descriptor_name":"Color","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true},{"descriptor_ui":"D003124","descriptor_name":"Colorimetry","qualifier_ui":"Q000379","qualifier_name":"methods","is_major_topic":false},{"descriptor_ui":"D003124","descriptor_name":"Colorimetry","qualifier_ui":"Q000379","qualifier_name":"methods","is_major_topic":false},{"descriptor_ui":"D003124","descriptor_name":"Colorimetry","qualifier_ui":"Q000379","qualifier_name":"methods","is_major_topic":false},{"descriptor_ui":"D007089","descriptor_name":"Image Enhancement","qualifier_ui":"Q000379","qualifier_name":"methods","is_major_topic":false},{"descriptor_ui":"D007089","descriptor_name":"Image Enhancement","qualifier_ui":"Q000379","qualifier_name":"methods","is_major_topic":false},{"descriptor_ui":"D007089","descriptor_name":"Image Enhancement","qualifier_ui":"Q000379","qualifier_name":"methods","is_major_topic":false},{"descriptor_ui":"D007090","descriptor_name":"Image Interpretation, Computer-Assisted","qualifier_ui":"Q000379","qualifier_name":"methods","is_major_topic":false},{"descriptor_ui":"D007090","descriptor_name":"Image Interpretation, Computer-Assisted","qualifier_ui":"Q000379","qualifier_name":"methods","is_major_topic":false},{"descriptor_ui":"D007090","descriptor_name":"Image Interpretation, Computer-Assisted","qualifier_ui":"Q000379","qualifier_name":"methods","is_major_topic":false},{"descriptor_ui":"D010363","descriptor_name":"Pattern Recognition, Automated","qualifier_ui":"Q000379","qualifier_name":"methods","is_major_topic":false},{"descriptor_ui":"D010363","descriptor_name":"Pattern Recognition, Automated","qualifier_ui":"Q000379","qualifier_name":"methods","is_major_topic":false},{"descriptor_ui":"D010363","descriptor_name":"Pattern Recognition, Automated","qualifier_ui":"Q000379","qualifier_name":"methods","is_major_topic":false},{"descriptor_ui":"D012680","descriptor_name":"Sensitivity and Specificity","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D012680","descriptor_name":"Sensitivity and Specificity","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D012680","descriptor_name":"Sensitivity and Specificity","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D013382","descriptor_name":"Subtraction Technique","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true},{"descriptor_ui":"D013382","descriptor_name":"Subtraction Technique","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true},{"descriptor_ui":"D013382","descriptor_name":"Subtraction Technique","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true},{"descriptor_ui":"D015203","descriptor_name":"Reproducibility of Results","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D015203","descriptor_name":"Reproducibility of Results","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D015203","descriptor_name":"Reproducibility of Results","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false}],"locations_count":5,"locations":[{"id":"doi:10.1109/tip.2009.2026678","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tip.2009.2026678","pdf_url":null,"source":{"id":"https://openalex.org/S4210173141","display_name":"IEEE Transactions on Image Processing","issn_l":"1057-7149","issn":["1057-7149","1941-0042"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310319808","host_organization_name":"Institute of Electrical and Electronics Engineers","host_organization_lineage":["https://openalex.org/P4310319808"],"host_organization_lineage_names":["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 Image Processing","raw_type":"journal-article"},{"id":"pmid:19586818","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/19586818","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 image processing : a publication of the IEEE Signal Processing Society","raw_type":null},{"id":"pmh:oai:arXiv.org:0710.0736","is_oa":true,"landing_page_url":"http://arxiv.org/abs/0710.0736","pdf_url":"https://arxiv.org/pdf/0710.0736","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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":"pmh:oai:ora.ox.ac.uk:uuid:3040412b-4fff-4cb1-a385-2ecc7055da58","is_oa":false,"landing_page_url":"https://ora.ox.ac.uk/objects/uuid:3040412b-4fff-4cb1-a385-2ecc7055da58","pdf_url":null,"source":{"id":"https://openalex.org/S4306402636","display_name":"Oxford University Research Archive (ORA) (University of Oxford)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I40120149","host_organization_name":"University of Oxford","host_organization_lineage":["https://openalex.org/I40120149"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"PubMed (http://www.ncbi.nlm.nih.gov/pubmed/)","raw_type":"Journal article"},{"id":"pmh:oai:ora.ox.ac.uk:uuid:741e81ee-f59d-4dd2-95ef-261e19f07609","is_oa":false,"landing_page_url":"https://ora.ox.ac.uk/objects/uuid:741e81ee-f59d-4dd2-95ef-261e19f07609","pdf_url":null,"source":{"id":"https://openalex.org/S4306402636","display_name":"Oxford University Research Archive (ORA) (University of Oxford)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I40120149","host_organization_name":"University of Oxford","host_organization_lineage":["https://openalex.org/I40120149"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Department of Computer Science","raw_type":"Journal article"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:0710.0736","is_oa":true,"landing_page_url":"http://arxiv.org/abs/0710.0736","pdf_url":"https://arxiv.org/pdf/0710.0736","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"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"},"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":35,"referenced_works":["https://openalex.org/W45888090","https://openalex.org/W207373900","https://openalex.org/W1580269359","https://openalex.org/W1596247731","https://openalex.org/W1979167448","https://openalex.org/W1984648213","https://openalex.org/W2000379836","https://openalex.org/W2017085690","https://openalex.org/W2026912579","https://openalex.org/W2041001120","https://openalex.org/W2042226984","https://openalex.org/W2044243502","https://openalex.org/W2044619502","https://openalex.org/W2055487369","https://openalex.org/W2066036475","https://openalex.org/W2080474884","https://openalex.org/W2084683687","https://openalex.org/W2096201928","https://openalex.org/W2114487471","https://openalex.org/W2116040950","https://openalex.org/W2128931829","https://openalex.org/W2132218491","https://openalex.org/W2132363464","https://openalex.org/W2133542147","https://openalex.org/W2144193479","https://openalex.org/W2150134853","https://openalex.org/W2172029033","https://openalex.org/W2258361296","https://openalex.org/W2340382982","https://openalex.org/W4240520845","https://openalex.org/W4285719527","https://openalex.org/W4388292704","https://openalex.org/W6608410706","https://openalex.org/W6634563179","https://openalex.org/W6635464864"],"related_works":["https://openalex.org/W1986655823","https://openalex.org/W2185902295","https://openalex.org/W2103507220","https://openalex.org/W3144569342","https://openalex.org/W3011384228","https://openalex.org/W2945274617","https://openalex.org/W3199300986","https://openalex.org/W4313052709","https://openalex.org/W4298131179","https://openalex.org/W2375430703"],"abstract_inverted_index":{"We":[0,69],"present":[1,71],"an":[2,58],"efficient":[3,59],"numerical":[4,16,43],"solution":[5,44],"of":[6,18,51,66,74],"a":[7,48,52],"PDE-driven":[8],"model":[9],"for":[10,63],"color":[11],"image":[12],"segmentation":[13,65,83],"and":[14,39,60],"give":[15],"examples":[17],"the":[19,24,64,72,82],"results.":[20],"The":[21],"method":[22,62],"combines":[23],"vector-valued":[25],"Allen-Cahn":[26],"phase":[27],"field":[28],"equation":[29],"with":[30,35],"initial":[31],"data":[32],"fitting":[33],"terms":[34],"prescribed":[36],"interface":[37],"width":[38],"fidelity":[40],"constants.":[41],"Efficient":[42],"is":[45],"achieved":[46],"using":[47],"multigrid":[49],"splitting":[50],"finite":[53],"element":[54],"space,":[55],"thereby":[56],"producing":[57],"robust":[61],"large":[67],"images.":[68],"also":[70],"use":[73],"adaptive":[75],"mesh":[76],"refinement":[77],"to":[78],"further":[79],"speed":[80],"up":[81],"process.":[84]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":5},{"year":2024,"cited_by_count":5},{"year":2023,"cited_by_count":9},{"year":2022,"cited_by_count":7},{"year":2021,"cited_by_count":7},{"year":2020,"cited_by_count":3},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":3},{"year":2015,"cited_by_count":3},{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":2},{"year":2012,"cited_by_count":6}],"updated_date":"2026-04-04T16:13:02.066488","created_date":"2025-10-10T00:00:00"}
