{"id":"https://openalex.org/W2153791664","doi":"https://doi.org/10.1109/tip.2008.2008076","title":"Necessary and Sufficient Convergence Conditions for Algebraic Image Reconstruction Algorithms","display_name":"Necessary and Sufficient Convergence Conditions for Algebraic Image Reconstruction Algorithms","publication_year":2008,"publication_date":"2008-12-18","ids":{"openalex":"https://openalex.org/W2153791664","doi":"https://doi.org/10.1109/tip.2008.2008076","mag":"2153791664","pmid":"https://pubmed.ncbi.nlm.nih.gov/19095534"},"language":"en","primary_location":{"id":"doi:10.1109/tip.2008.2008076","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tip.2008.2008076","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":"letter","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/A5047966042","display_name":"Gangrong Qu","orcid":"https://orcid.org/0000-0002-2004-2457"},"institutions":[{"id":"https://openalex.org/I21193070","display_name":"Beijing Jiaotong University","ror":"https://ror.org/01yj56c84","country_code":"CN","type":"education","lineage":["https://openalex.org/I21193070"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Gangrong Qu","raw_affiliation_strings":["School of Science, Beijing Jiaotong University, Beijing, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"School of Science, Beijing Jiaotong University, Beijing, China","institution_ids":["https://openalex.org/I21193070"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5029516046","display_name":"Caifang Wang","orcid":null},"institutions":[{"id":"https://openalex.org/I20231570","display_name":"Peking University","ror":"https://ror.org/02v51f717","country_code":"CN","type":"education","lineage":["https://openalex.org/I20231570"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Caifang Wang","raw_affiliation_strings":["LMAM, School of Mathematical Sciences, Peking University, Beijing, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"LMAM, School of Mathematical Sciences, Peking University, Beijing, China","institution_ids":["https://openalex.org/I20231570"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5056308697","display_name":"Ming Jiang","orcid":"https://orcid.org/0000-0002-1661-0538"},"institutions":[{"id":"https://openalex.org/I20231570","display_name":"Peking University","ror":"https://ror.org/02v51f717","country_code":"CN","type":"education","lineage":["https://openalex.org/I20231570"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Ming Jiang","raw_affiliation_strings":["LMAM, School of Mathematical Sciences, Peking University, Beijing, China"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"LMAM, School of Mathematical Sciences, Peking University, Beijing, China","institution_ids":["https://openalex.org/I20231570"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":1.531,"has_fulltext":false,"cited_by_count":36,"citation_normalized_percentile":{"value":0.84202933,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":90,"max":98},"biblio":{"volume":"18","issue":"2","first_page":"435","last_page":"440"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9997000098228455,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9997000098228455,"subfield":{"id":"https://openalex.org/subfields/2206","display_name":"Computational Mechanics"},"field":{"id":"https://openalex.org/fields/22","display_name":"Engineering"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10522","display_name":"Medical Imaging Techniques and Applications","score":0.9994000196456909,"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/T11205","display_name":"Numerical methods in inverse problems","score":0.9991999864578247,"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/mathematics","display_name":"Mathematics","score":0.8136473894119263},{"id":"https://openalex.org/keywords/singular-value-decomposition","display_name":"Singular value decomposition","score":0.6880184412002563},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.5378619432449341},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.5220674872398376},{"id":"https://openalex.org/keywords/norm","display_name":"Norm (philosophy)","score":0.521547794342041},{"id":"https://openalex.org/keywords/algebraic-number","display_name":"Algebraic number","score":0.4847414195537567},{"id":"https://openalex.org/keywords/singular-value","display_name":"Singular value","score":0.48428481817245483},{"id":"https://openalex.org/keywords/relaxation","display_name":"Relaxation (psychology)","score":0.48177167773246765},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.4755632281303406},{"id":"https://openalex.org/keywords/projection","display_name":"Projection (relational algebra)","score":0.4493900537490845},{"id":"https://openalex.org/keywords/iterative-reconstruction","display_name":"Iterative reconstruction","score":0.4217141568660736},{"id":"https://openalex.org/keywords/iterative-method","display_name":"Iterative method","score":0.4188714623451233},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.2021351158618927},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.11540365219116211},{"id":"https://openalex.org/keywords/eigenvalues-and-eigenvectors","display_name":"Eigenvalues and eigenvectors","score":0.07921433448791504}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8136473894119263},{"id":"https://openalex.org/C22789450","wikidata":"https://www.wikidata.org/wiki/Q420904","display_name":"Singular value decomposition","level":2,"score":0.6880184412002563},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.5378619432449341},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.5220674872398376},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.521547794342041},{"id":"https://openalex.org/C9376300","wikidata":"https://www.wikidata.org/wiki/Q168817","display_name":"Algebraic number","level":2,"score":0.4847414195537567},{"id":"https://openalex.org/C109282560","wikidata":"https://www.wikidata.org/wiki/Q4166054","display_name":"Singular value","level":3,"score":0.48428481817245483},{"id":"https://openalex.org/C2776029896","wikidata":"https://www.wikidata.org/wiki/Q3935810","display_name":"Relaxation (psychology)","level":2,"score":0.48177167773246765},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.4755632281303406},{"id":"https://openalex.org/C57493831","wikidata":"https://www.wikidata.org/wiki/Q3134666","display_name":"Projection (relational algebra)","level":2,"score":0.4493900537490845},{"id":"https://openalex.org/C141379421","wikidata":"https://www.wikidata.org/wiki/Q6094427","display_name":"Iterative reconstruction","level":2,"score":0.4217141568660736},{"id":"https://openalex.org/C159694833","wikidata":"https://www.wikidata.org/wiki/Q2321565","display_name":"Iterative method","level":2,"score":0.4188714623451233},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.2021351158618927},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.11540365219116211},{"id":"https://openalex.org/C158693339","wikidata":"https://www.wikidata.org/wiki/Q190524","display_name":"Eigenvalues and eigenvectors","level":2,"score":0.07921433448791504},{"id":"https://openalex.org/C162324750","wikidata":"https://www.wikidata.org/wiki/Q8134","display_name":"Economics","level":0,"score":0.0},{"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/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.0},{"id":"https://openalex.org/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.0},{"id":"https://openalex.org/C17744445","wikidata":"https://www.wikidata.org/wiki/Q36442","display_name":"Political science","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},{"id":"https://openalex.org/C15744967","wikidata":"https://www.wikidata.org/wiki/Q9418","display_name":"Psychology","level":0,"score":0.0},{"id":"https://openalex.org/C62520636","wikidata":"https://www.wikidata.org/wiki/Q944","display_name":"Quantum mechanics","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":"D003198","descriptor_name":"Computer Simulation","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D003198","descriptor_name":"Computer Simulation","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D003198","descriptor_name":"Computer Simulation","qualifier_ui":null,"qualifier_name":null,"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":"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":"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},{"descriptor_ui":"D015233","descriptor_name":"Models, Statistical","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D015233","descriptor_name":"Models, Statistical","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false},{"descriptor_ui":"D015233","descriptor_name":"Models, Statistical","qualifier_ui":null,"qualifier_name":null,"is_major_topic":false}],"locations_count":2,"locations":[{"id":"doi:10.1109/tip.2008.2008076","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tip.2008.2008076","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:19095534","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/19095534","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}],"best_oa_location":null,"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/15","score":0.4000000059604645,"display_name":"Life in Land"}],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":39,"referenced_works":["https://openalex.org/W16591383","https://openalex.org/W1210424432","https://openalex.org/W1514248469","https://openalex.org/W1965018134","https://openalex.org/W1969075481","https://openalex.org/W1975826938","https://openalex.org/W1988539966","https://openalex.org/W1989633043","https://openalex.org/W1990003751","https://openalex.org/W2006262045","https://openalex.org/W2019027232","https://openalex.org/W2019065055","https://openalex.org/W2021059663","https://openalex.org/W2024978582","https://openalex.org/W2040100614","https://openalex.org/W2040793430","https://openalex.org/W2048584772","https://openalex.org/W2049772201","https://openalex.org/W2058550915","https://openalex.org/W2058583833","https://openalex.org/W2066034539","https://openalex.org/W2084928869","https://openalex.org/W2090824426","https://openalex.org/W2100503136","https://openalex.org/W2111152057","https://openalex.org/W2131100802","https://openalex.org/W2133560833","https://openalex.org/W2138324694","https://openalex.org/W2146583368","https://openalex.org/W2150195104","https://openalex.org/W2158065297","https://openalex.org/W2159909156","https://openalex.org/W2795407492","https://openalex.org/W3035235874","https://openalex.org/W4229540245","https://openalex.org/W4240518936","https://openalex.org/W4249918849","https://openalex.org/W4292414053","https://openalex.org/W4298264214"],"related_works":["https://openalex.org/W4319586039","https://openalex.org/W4382583540","https://openalex.org/W1607100495","https://openalex.org/W2147072251","https://openalex.org/W1990844505","https://openalex.org/W2148568324","https://openalex.org/W4386721910","https://openalex.org/W4378770618","https://openalex.org/W2164147222","https://openalex.org/W2038393145"],"abstract_inverted_index":{"The":[0,17],"Landweber":[1,21,44,91,120],"scheme":[2,22,45,92,121],"is":[3,23,87],"an":[4,38,78,143],"algebraic":[5],"reconstruction":[6],"method":[7],"and":[8,27,46,51,63,142],"includes":[9],"several":[10],"important":[11],"algorithms":[12],"as":[13],"its":[14,55],"special":[15],"cases.":[16],"convergence":[18,72,124],"of":[19,24,65,108,110,118,132,137,146,154],"the":[20,31,43,49,61,90,99,106,111,116,119,130,133,147,151,155],"both":[25],"theoretical":[26],"practical":[28],"importance.":[29],"Using":[30],"singular":[32,113],"value":[33],"decomposition":[34],"(SVD),":[35],"we":[36,69],"derive":[37],"iterative":[39],"representation":[40],"formula":[41],"for":[42,54],"consequently":[47],"establish":[48],"necessary":[50],"sufficient":[52],"conditions":[53,73],"convergence.":[56],"In":[57],"addition":[58],"to":[59,104,128],"verifying":[60],"necessity":[62],"sufficiency":[64],"known":[66,83],"convergent":[67],"conditions,":[68],"find":[70],"new":[71],"allowing":[74],"relaxation":[75,100],"coefficients":[76,101],"in":[77,122],"interval":[79],"not":[80],"covered":[81],"by":[82],"results.":[84],"Moreover,":[85],"it":[86],"found":[88],"that":[89],"can":[93],"converge":[94],"within":[95],"finite":[96],"iterations":[97],"when":[98],"are":[102,126],"chosen":[103],"be":[105,129],"inverses":[107],"squares":[109],"nonzero":[112],"values.":[114],"Furthermore,":[115],"limits":[117],"all":[123],"cases":[125],"shown":[127],"sum":[131],"minimum":[134],"norm":[135],"solution":[136],"a":[138],"weighted":[139],"least-squares":[140],"problem":[141],"oblique":[144],"projection":[145],"initial":[148],"image":[149],"onto":[150],"null":[152],"space":[153],"system":[156],"matrix.":[157]},"counts_by_year":[{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":3},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":4},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":1},{"year":2018,"cited_by_count":2},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":4},{"year":2015,"cited_by_count":3},{"year":2014,"cited_by_count":2},{"year":2013,"cited_by_count":1},{"year":2012,"cited_by_count":2}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}
