{"id":"https://openalex.org/W2891037709","doi":"https://doi.org/10.1109/tmi.2018.2870968","title":"A Convergence Proof of MLEM and MLEM-3 With Fixed Background","display_name":"A Convergence Proof of MLEM and MLEM-3 With Fixed Background","publication_year":2018,"publication_date":"2018-09-18","ids":{"openalex":"https://openalex.org/W2891037709","doi":"https://doi.org/10.1109/tmi.2018.2870968","mag":"2891037709","pmid":"https://pubmed.ncbi.nlm.nih.gov/30235122"},"language":"en","primary_location":{"id":"doi:10.1109/tmi.2018.2870968","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tmi.2018.2870968","pdf_url":null,"source":{"id":"https://openalex.org/S58069681","display_name":"IEEE Transactions on Medical Imaging","issn_l":"0278-0062","issn":["0278-0062","1558-254X"],"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 Medical Imaging","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/A5073630730","display_name":"Koen Salvo","orcid":"https://orcid.org/0000-0001-7962-1069"},"institutions":[{"id":"https://openalex.org/I13469542","display_name":"Vrije Universiteit Brussel","ror":"https://ror.org/006e5kg04","country_code":"BE","type":"education","lineage":["https://openalex.org/I13469542"]}],"countries":["BE"],"is_corresponding":false,"raw_author_name":"Koen Salvo","raw_affiliation_strings":["Department of Nuclear Medicine, Vrije Universiteit Brussel, Brussels, Belgium"],"raw_orcid":"https://orcid.org/0000-0001-7962-1069","affiliations":[{"raw_affiliation_string":"Department of Nuclear Medicine, Vrije Universiteit Brussel, Brussels, Belgium","institution_ids":["https://openalex.org/I13469542"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5036773393","display_name":"Michel Defrise","orcid":"https://orcid.org/0000-0001-8999-2920"},"institutions":[{"id":"https://openalex.org/I13469542","display_name":"Vrije Universiteit Brussel","ror":"https://ror.org/006e5kg04","country_code":"BE","type":"education","lineage":["https://openalex.org/I13469542"]}],"countries":["BE"],"is_corresponding":false,"raw_author_name":"Michel Defrise","raw_affiliation_strings":["Department of Nuclear Medicine, Vrije Universiteit Brussel, Brussels, Belgium"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Nuclear Medicine, Vrije Universiteit Brussel, Brussels, Belgium","institution_ids":["https://openalex.org/I13469542"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.3702,"has_fulltext":false,"cited_by_count":6,"citation_normalized_percentile":{"value":0.65455695,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"38","issue":"3","first_page":"721","last_page":"729"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10522","display_name":"Medical Imaging Techniques and Applications","score":1.0,"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/T10522","display_name":"Medical Imaging Techniques and Applications","score":1.0,"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":0.9944000244140625,"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/T10136","display_name":"Statistical Methods and Inference","score":0.9926000237464905,"subfield":{"id":"https://openalex.org/subfields/2613","display_name":"Statistics and Probability"},"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/convergence","display_name":"Convergence (economics)","score":0.6264855265617371},{"id":"https://openalex.org/keywords/iterative-reconstruction","display_name":"Iterative reconstruction","score":0.5168838500976562},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.4703039526939392},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4008263349533081},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.37348344922065735},{"id":"https://openalex.org/keywords/calculus","display_name":"Calculus (dental)","score":0.3465248942375183},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.3088592290878296},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3057214319705963},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.20404699444770813},{"id":"https://openalex.org/keywords/medicine","display_name":"Medicine","score":0.12374752759933472}],"concepts":[{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.6264855265617371},{"id":"https://openalex.org/C141379421","wikidata":"https://www.wikidata.org/wiki/Q6094427","display_name":"Iterative reconstruction","level":2,"score":0.5168838500976562},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.4703039526939392},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4008263349533081},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.37348344922065735},{"id":"https://openalex.org/C2777686260","wikidata":"https://www.wikidata.org/wiki/Q144037","display_name":"Calculus (dental)","level":2,"score":0.3465248942375183},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.3088592290878296},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3057214319705963},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.20404699444770813},{"id":"https://openalex.org/C71924100","wikidata":"https://www.wikidata.org/wiki/Q11190","display_name":"Medicine","level":0,"score":0.12374752759933472},{"id":"https://openalex.org/C199343813","wikidata":"https://www.wikidata.org/wiki/Q12128","display_name":"Dentistry","level":1,"score":0.0},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"id":"https://openalex.org/C50522688","wikidata":"https://www.wikidata.org/wiki/Q189833","display_name":"Economic growth","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":"D006801","descriptor_name":"Humans","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D006801","descriptor_name":"Humans","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D006801","descriptor_name":"Humans","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D007091","descriptor_name":"Image Processing, Computer-Assisted","qualifier_ui":"Q000379","qualifier_name":"methods","is_major_topic":false},{"descriptor_ui":"D007091","descriptor_name":"Image Processing, Computer-Assisted","qualifier_ui":"Q000379","qualifier_name":"methods","is_major_topic":false},{"descriptor_ui":"D007091","descriptor_name":"Image Processing, Computer-Assisted","qualifier_ui":"Q000379","qualifier_name":"methods","is_major_topic":false},{"descriptor_ui":"D016013","descriptor_name":"Likelihood Functions","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true},{"descriptor_ui":"D016013","descriptor_name":"Likelihood Functions","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true},{"descriptor_ui":"D016013","descriptor_name":"Likelihood Functions","qualifier_ui":null,"qualifier_name":null,"is_major_topic":true},{"descriptor_ui":"D049268","descriptor_name":"Positron-Emission Tomography","qualifier_ui":"Q000379","qualifier_name":"methods","is_major_topic":false},{"descriptor_ui":"D049268","descriptor_name":"Positron-Emission Tomography","qualifier_ui":"Q000379","qualifier_name":"methods","is_major_topic":false},{"descriptor_ui":"D049268","descriptor_name":"Positron-Emission Tomography","qualifier_ui":"Q000379","qualifier_name":"methods","is_major_topic":false}],"locations_count":5,"locations":[{"id":"doi:10.1109/tmi.2018.2870968","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tmi.2018.2870968","pdf_url":null,"source":{"id":"https://openalex.org/S58069681","display_name":"IEEE Transactions on Medical Imaging","issn_l":"0278-0062","issn":["0278-0062","1558-254X"],"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 Medical Imaging","raw_type":"journal-article"},{"id":"pmid:30235122","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/30235122","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 medical imaging","raw_type":null},{"id":"pmh:oai:vubissmart:VUBISSMART:2000:117798","is_oa":false,"landing_page_url":"https://biblio.vub.ac.be/vubir/a-convergence-proof-of-mlem-and-mlem3-with-fixed-background(f4763320-af46-42bc-868c-cfaf3f76f26c).html","pdf_url":null,"source":{"id":"https://openalex.org/S4306402573","display_name":"VUBIR (Vrije Universiteit Brussel)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I13469542","host_organization_name":"Vrije Universiteit Brussel","host_organization_lineage":["https://openalex.org/I13469542"],"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":"article"},{"id":"pmh:oai:vubissmart:VUBISSMART:2000:117798","is_oa":false,"landing_page_url":"https://doi.org/10.1109/TMI.2018.2870968","pdf_url":null,"source":{"id":"https://openalex.org/S4306402573","display_name":"VUBIR (Vrije Universiteit Brussel)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I13469542","host_organization_name":"Vrije Universiteit Brussel","host_organization_lineage":["https://openalex.org/I13469542"],"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":"article"},{"id":"pmh:oai:vubissmart:VUBISSMART:2000:212417","is_oa":false,"landing_page_url":null,"pdf_url":null,"source":{"id":"https://openalex.org/S4306402573","display_name":"VUBIR (Vrije Universiteit Brussel)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I13469542","host_organization_name":"Vrije Universiteit Brussel","host_organization_lineage":["https://openalex.org/I13469542"],"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":"article"}],"best_oa_location":null,"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/10","score":0.47999998927116394,"display_name":"Reduced inequalities"}],"awards":[{"id":"https://openalex.org/G6987426773","display_name":null,"funder_award_id":"G027514N","funder_id":"https://openalex.org/F4320321730","funder_display_name":"Fonds Wetenschappelijk Onderzoek"}],"funders":[{"id":"https://openalex.org/F4320321730","display_name":"Fonds Wetenschappelijk Onderzoek","ror":"https://ror.org/03qtxy027"},{"id":"https://openalex.org/F4320322634","display_name":"Vrije Universiteit Brussel","ror":"https://ror.org/006e5kg04"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":22,"referenced_works":["https://openalex.org/W1647554959","https://openalex.org/W1965555277","https://openalex.org/W1975198675","https://openalex.org/W2014116058","https://openalex.org/W2039284087","https://openalex.org/W2049633694","https://openalex.org/W2065390928","https://openalex.org/W2069629287","https://openalex.org/W2088909704","https://openalex.org/W2117853077","https://openalex.org/W2136334809","https://openalex.org/W2149251622","https://openalex.org/W2151517435","https://openalex.org/W2152693305","https://openalex.org/W2154679219","https://openalex.org/W2170608748","https://openalex.org/W2800027815","https://openalex.org/W3193262203","https://openalex.org/W4234848677","https://openalex.org/W4247706771","https://openalex.org/W4396738453","https://openalex.org/W6636952455"],"related_works":["https://openalex.org/W1506216622","https://openalex.org/W1987033902","https://openalex.org/W2207380610","https://openalex.org/W2298446516","https://openalex.org/W3126152140","https://openalex.org/W1985163180","https://openalex.org/W1970820012","https://openalex.org/W2785418248","https://openalex.org/W1563906813","https://openalex.org/W3129082682"],"abstract_inverted_index":{"Maximum":[0],"likelihood":[1],"expectation-maximization":[2],"(MLEM)":[3],"is":[4],"a":[5,20,52],"popular":[6],"algorithm":[7],"to":[8,85],"reconstruct":[9],"the":[10,24,47,72,83],"activity":[11],"image":[12],"in":[13,41],"positron":[14],"emission":[15],"tomography.":[16],"This":[17],"paper":[18],"introduces":[19],"\"fundamental":[21],"equality\"":[22],"for":[23,51,71,82],"MLEM":[25,50],"complete":[26],"data":[27],"from":[28],"which":[29],"two":[30],"key":[31],"properties":[32],"easily":[33],"follow":[34],"that":[35],"allows":[36],"us":[37],"to:":[38],"1)":[39],"prove":[40],"an":[42],"elegant":[43],"and":[44,63,66,79],"compact":[45],"way":[46],"convergence":[48],"of":[49],"forward":[53],"model":[54],"with":[55],"fixed":[56],"background":[57],"(i.e.,":[58],"counts":[59],"such":[60],"as":[61],"random":[62],"scatter":[64],"coincidences)":[65],"2)":[67],"generalize":[68],"this":[69],"proof":[70],"MLEM-3":[73],"algorithm.":[74],"Moreover,":[75],"we":[76],"give":[77],"necessary":[78],"sufficient":[80],"conditions":[81],"solution":[84],"be":[86],"unique.":[87]},"counts_by_year":[{"year":2025,"cited_by_count":2},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":1},{"year":2019,"cited_by_count":1}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}