{"id":"https://openalex.org/W4403469729","doi":"https://doi.org/10.3934/mfc.2024040","title":"A mathematical explanation of UNet","display_name":"A mathematical explanation of UNet","publication_year":2024,"publication_date":"2024-10-16","ids":{"openalex":"https://openalex.org/W4403469729","doi":"https://doi.org/10.3934/mfc.2024040"},"language":"en","primary_location":{"id":"doi:10.3934/mfc.2024040","is_oa":true,"landing_page_url":"https://doi.org/10.3934/mfc.2024040","pdf_url":"https://www.aimsciences.org/data/article/export-pdf?id=670ce03f2d4d1a4052cb7bb3","source":{"id":"https://openalex.org/S4210238810","display_name":"Mathematical Foundations of Computing","issn_l":"2577-8838","issn":["2577-8838"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315844","host_organization_name":"American Institute of Mathematical Sciences","host_organization_lineage":["https://openalex.org/P4310315844"],"host_organization_lineage_names":["American Institute of Mathematical Sciences"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematical Foundations of Computing","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"diamond","oa_url":"https://www.aimsciences.org/data/article/export-pdf?id=670ce03f2d4d1a4052cb7bb3","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5101951979","display_name":"Xue\u2010Cheng Tai","orcid":"https://orcid.org/0000-0003-3359-9104"},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Xue-Cheng Tai","raw_affiliation_strings":[],"affiliations":[]},{"author_position":"middle","author":{"id":"https://openalex.org/A5100458873","display_name":"Hao Liu","orcid":"https://orcid.org/0000-0003-0954-5405"},"institutions":[{"id":"https://openalex.org/I141568987","display_name":"Hong Kong Baptist University","ror":"https://ror.org/0145fw131","country_code":"HK","type":"education","lineage":["https://openalex.org/I141568987"]}],"countries":["HK"],"is_corresponding":false,"raw_author_name":"Hao Liu","raw_affiliation_strings":["Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong SAR"],"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Hong Kong Baptist University, Kowloon Tong, Hong Kong SAR","institution_ids":["https://openalex.org/I141568987"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5004126693","display_name":"Raymond H. Chan","orcid":"https://orcid.org/0000-0003-0910-4685"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Raymond H. Chan","raw_affiliation_strings":["Lingnan University, Tuen Mun, Hong Kong SAR"],"affiliations":[{"raw_affiliation_string":"Lingnan University, Tuen Mun, Hong Kong SAR","institution_ids":[]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5100774102","display_name":"Lingfeng Li","orcid":"https://orcid.org/0000-0003-1351-5911"},"institutions":[],"countries":[],"is_corresponding":false,"raw_author_name":"Lingfeng Li","raw_affiliation_strings":["Hong Kong Center for Cerebro-Cardiovascular Health Engineering, Hong Kong SAR"],"affiliations":[{"raw_affiliation_string":"Hong Kong Center for Cerebro-Cardiovascular Health Engineering, Hong Kong SAR","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5101951979"],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":2.5713,"has_fulltext":true,"cited_by_count":7,"citation_normalized_percentile":{"value":0.90838814,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":97,"max":99},"biblio":{"volume":"8","issue":"5","first_page":"874","last_page":"889"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T11697","display_name":"Numerical Methods and Algorithms","score":0.9854999780654907,"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"}},"topics":[{"id":"https://openalex.org/T11697","display_name":"Numerical Methods and Algorithms","score":0.9854999780654907,"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/T12002","display_name":"Computability, Logic, AI Algorithms","score":0.9611999988555908,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.4050242006778717}],"concepts":[{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.4050242006778717}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.3934/mfc.2024040","is_oa":true,"landing_page_url":"https://doi.org/10.3934/mfc.2024040","pdf_url":"https://www.aimsciences.org/data/article/export-pdf?id=670ce03f2d4d1a4052cb7bb3","source":{"id":"https://openalex.org/S4210238810","display_name":"Mathematical Foundations of Computing","issn_l":"2577-8838","issn":["2577-8838"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315844","host_organization_name":"American Institute of Mathematical Sciences","host_organization_lineage":["https://openalex.org/P4310315844"],"host_organization_lineage_names":["American Institute of Mathematical Sciences"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematical Foundations of Computing","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.3934/mfc.2024040","is_oa":true,"landing_page_url":"https://doi.org/10.3934/mfc.2024040","pdf_url":"https://www.aimsciences.org/data/article/export-pdf?id=670ce03f2d4d1a4052cb7bb3","source":{"id":"https://openalex.org/S4210238810","display_name":"Mathematical Foundations of Computing","issn_l":"2577-8838","issn":["2577-8838"],"is_oa":true,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315844","host_organization_name":"American Institute of Mathematical Sciences","host_organization_lineage":["https://openalex.org/P4310315844"],"host_organization_lineage_names":["American Institute of Mathematical Sciences"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematical Foundations of Computing","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G1486628861","display_name":null,"funder_award_id":"C1013-21GF","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G2087396116","display_name":null,"funder_award_id":"China","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G2651498393","display_name":null,"funder_award_id":"HKRGC","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G3236912370","display_name":null,"funder_award_id":"HKRGC ECS 22302123","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G3317480652","display_name":null,"funder_award_id":"Science","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G3693056921","display_name":null,"funder_award_id":"22302123","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G391238517","display_name":null,"funder_award_id":", and","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G5994120800","display_name":null,"funder_award_id":"Natural","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G6557754292","display_name":null,"funder_award_id":"CityU1101120","funder_id":"https://openalex.org/F4320309893","funder_display_name":"City University of Hong Kong"},{"id":"https://openalex.org/G6736763787","display_name":null,"funder_award_id":"12201530","funder_id":"https://openalex.org/F4320321001","funder_display_name":"National Natural Science Foundation of China"},{"id":"https://openalex.org/G8543516176","display_name":null,"funder_award_id":"CityU11309922","funder_id":"https://openalex.org/F4320309893","funder_display_name":"City University of Hong Kong"}],"funders":[{"id":"https://openalex.org/F4320309893","display_name":"City University of Hong Kong","ror":"https://ror.org/03q8dnn23"},{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4403469729.pdf","grobid_xml":"https://content.openalex.org/works/W4403469729.grobid-xml"},"referenced_works_count":47,"referenced_works":["https://openalex.org/W599684198","https://openalex.org/W1901129140","https://openalex.org/W1903029394","https://openalex.org/W1984648213","https://openalex.org/W2003413173","https://openalex.org/W2017085690","https://openalex.org/W2104126767","https://openalex.org/W2116040950","https://openalex.org/W2143612262","https://openalex.org/W2166116275","https://openalex.org/W2412782625","https://openalex.org/W2508457857","https://openalex.org/W2528305538","https://openalex.org/W2569134578","https://openalex.org/W2570940516","https://openalex.org/W2600297185","https://openalex.org/W2612252966","https://openalex.org/W2788421913","https://openalex.org/W2893817145","https://openalex.org/W2900512984","https://openalex.org/W2912389156","https://openalex.org/W2963626582","https://openalex.org/W2963881378","https://openalex.org/W2964110066","https://openalex.org/W2974916071","https://openalex.org/W2981253580","https://openalex.org/W2983315964","https://openalex.org/W2990121169","https://openalex.org/W2996290406","https://openalex.org/W2997997872","https://openalex.org/W3098011980","https://openalex.org/W3102569296","https://openalex.org/W3104544580","https://openalex.org/W3129492101","https://openalex.org/W3168159543","https://openalex.org/W3189175238","https://openalex.org/W4224024937","https://openalex.org/W4285806732","https://openalex.org/W4288038760","https://openalex.org/W4313134080","https://openalex.org/W4383815414","https://openalex.org/W4388398069","https://openalex.org/W4392550756","https://openalex.org/W4392881389","https://openalex.org/W4396881213","https://openalex.org/W4399888444","https://openalex.org/W4406363697"],"related_works":["https://openalex.org/W4391375266","https://openalex.org/W2748952813","https://openalex.org/W2390279801","https://openalex.org/W4391913857","https://openalex.org/W2358668433","https://openalex.org/W4396701345","https://openalex.org/W2376932109","https://openalex.org/W2001405890","https://openalex.org/W4396696052","https://openalex.org/W4402327032"],"abstract_inverted_index":{"The":[0],"UNet":[1,57,85,91],"architecture":[2,81],"has":[3],"transformed":[4],"image":[5],"segmentation.":[6],"UNet's":[7],"versatility":[8],"and":[9,33,45],"accuracy":[10],"have":[11],"driven":[12],"its":[13],"widespread":[14],"adoption,":[15],"significantly":[16],"advancing":[17],"fields":[18],"reliant":[19],"on":[20],"machine":[21],"learning":[22],"problems":[23],"with":[24],"images.":[25],"In":[26],"this":[27],"work,":[28],"we":[29],"gave":[30],"a":[31,60,93],"clear":[32],"concise":[34],"mathematical":[35],"explanation":[36],"of":[37,47,49,52],"UNet.":[38,53],"We":[39,54,63],"explained":[40],"what":[41],"is":[42],"the":[43,50,65,78,84,98],"meaning":[44],"function":[46],"each":[48],"components":[51],"showed":[55,89],"that":[56,90],"was":[58,92],"solving":[59],"control":[61,66,99],"problem.":[62,100],"decomposed":[64],"variables":[67],"using":[68],"multigrid":[69],"methods.":[70],"Then,":[71],"operator-splitting":[72,95],"techniques":[73],"were":[74],"used":[75],"to":[76],"solve":[77],"problem,":[79],"whose":[80],"exactly":[82],"recovered":[83],"architecture.":[86],"Our":[87],"result":[88],"one-step":[94],"algorithm":[96],"for":[97]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":6}],"updated_date":"2026-03-22T08:09:32.410652","created_date":"2025-10-10T00:00:00"}