{"id":"https://openalex.org/W2963347434","doi":"https://doi.org/10.1090/s0025-5718-2012-02631-7","title":"\u2113\u2080 Minimization for wavelet frame based image restoration","display_name":"\u2113\u2080 Minimization for wavelet frame based image restoration","publication_year":2012,"publication_date":"2012-08-15","ids":{"openalex":"https://openalex.org/W2963347434","doi":"https://doi.org/10.1090/s0025-5718-2012-02631-7","mag":"2963347434"},"language":"en","primary_location":{"id":"doi:10.1090/s0025-5718-2012-02631-7","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-2012-02631-7","pdf_url":"https://www.ams.org/mcom/2013-82-282/S0025-5718-2012-02631-7/S0025-5718-2012-02631-7.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"bronze","oa_url":"https://www.ams.org/mcom/2013-82-282/S0025-5718-2012-02631-7/S0025-5718-2012-02631-7.pdf","any_repository_has_fulltext":false},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5039251328","display_name":"Yong Zhang","orcid":"https://orcid.org/0000-0003-1158-2248"},"institutions":[{"id":"https://openalex.org/I18014758","display_name":"Simon Fraser University","ror":"https://ror.org/0213rcc28","country_code":"CA","type":"education","lineage":["https://openalex.org/I18014758"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Yong Zhang","raw_affiliation_strings":["Department of Mathematics, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada","institution_ids":["https://openalex.org/I18014758"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5035683291","display_name":"Bin Dong","orcid":"https://orcid.org/0000-0003-1295-3362"},"institutions":[{"id":"https://openalex.org/I138006243","display_name":"University of Arizona","ror":"https://ror.org/03m2x1q45","country_code":"US","type":"education","lineage":["https://openalex.org/I138006243"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Bin Dong","raw_affiliation_strings":["Department of Mathematics, The University of Arizona, 617 N. Santa Rita Ave., Tucson, Arizona, 85721-0089"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, The University of Arizona, 617 N. Santa Rita Ave., Tucson, Arizona, 85721-0089","institution_ids":["https://openalex.org/I138006243"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5000609407","display_name":"Zhaosong Lu","orcid":"https://orcid.org/0000-0003-3277-7853"},"institutions":[{"id":"https://openalex.org/I18014758","display_name":"Simon Fraser University","ror":"https://ror.org/0213rcc28","country_code":"CA","type":"education","lineage":["https://openalex.org/I18014758"]}],"countries":["CA"],"is_corresponding":false,"raw_author_name":"Zhaosong Lu","raw_affiliation_strings":["Department of Mathematics, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada","institution_ids":["https://openalex.org/I18014758"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":6.5718,"has_fulltext":true,"cited_by_count":99,"citation_normalized_percentile":{"value":0.97089358,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":100},"biblio":{"volume":"82","issue":"282","first_page":"995","last_page":"1015"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10688","display_name":"Image and Signal Denoising Methods","score":0.9995999932289124,"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/T10688","display_name":"Image and Signal Denoising Methods","score":0.9995999932289124,"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/T12015","display_name":"Photoacoustic and Ultrasonic Imaging","score":0.9990000128746033,"subfield":{"id":"https://openalex.org/subfields/2204","display_name":"Biomedical Engineering"},"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/T10500","display_name":"Sparse and Compressive Sensing Techniques","score":0.9988999962806702,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.7683883309364319},{"id":"https://openalex.org/keywords/wavelet","display_name":"Wavelet","score":0.7563672661781311},{"id":"https://openalex.org/keywords/piecewise","display_name":"Piecewise","score":0.6614388823509216},{"id":"https://openalex.org/keywords/minification","display_name":"Minification","score":0.5800449848175049},{"id":"https://openalex.org/keywords/frame","display_name":"Frame (networking)","score":0.560236930847168},{"id":"https://openalex.org/keywords/image-restoration","display_name":"Image restoration","score":0.5490980744361877},{"id":"https://openalex.org/keywords/norm","display_name":"Norm (philosophy)","score":0.5121219158172607},{"id":"https://openalex.org/keywords/smoothness","display_name":"Smoothness","score":0.5063303709030151},{"id":"https://openalex.org/keywords/convergence","display_name":"Convergence (economics)","score":0.45576250553131104},{"id":"https://openalex.org/keywords/image","display_name":"Image (mathematics)","score":0.4522862136363983},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.43401777744293213},{"id":"https://openalex.org/keywords/multiresolution-analysis","display_name":"Multiresolution analysis","score":0.4240581691265106},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.41834506392478943},{"id":"https://openalex.org/keywords/image-processing","display_name":"Image processing","score":0.39934420585632324},{"id":"https://openalex.org/keywords/wavelet-transform","display_name":"Wavelet transform","score":0.34482598304748535},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.3331380784511566},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.22172224521636963},{"id":"https://openalex.org/keywords/discrete-wavelet-transform","display_name":"Discrete wavelet transform","score":0.21409469842910767},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.09593024849891663}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.7683883309364319},{"id":"https://openalex.org/C47432892","wikidata":"https://www.wikidata.org/wiki/Q831390","display_name":"Wavelet","level":2,"score":0.7563672661781311},{"id":"https://openalex.org/C164660894","wikidata":"https://www.wikidata.org/wiki/Q2037833","display_name":"Piecewise","level":2,"score":0.6614388823509216},{"id":"https://openalex.org/C147764199","wikidata":"https://www.wikidata.org/wiki/Q6865248","display_name":"Minification","level":2,"score":0.5800449848175049},{"id":"https://openalex.org/C126042441","wikidata":"https://www.wikidata.org/wiki/Q1324888","display_name":"Frame (networking)","level":2,"score":0.560236930847168},{"id":"https://openalex.org/C106430172","wikidata":"https://www.wikidata.org/wiki/Q6002272","display_name":"Image restoration","level":4,"score":0.5490980744361877},{"id":"https://openalex.org/C191795146","wikidata":"https://www.wikidata.org/wiki/Q3878446","display_name":"Norm (philosophy)","level":2,"score":0.5121219158172607},{"id":"https://openalex.org/C102634674","wikidata":"https://www.wikidata.org/wiki/Q868473","display_name":"Smoothness","level":2,"score":0.5063303709030151},{"id":"https://openalex.org/C2777303404","wikidata":"https://www.wikidata.org/wiki/Q759757","display_name":"Convergence (economics)","level":2,"score":0.45576250553131104},{"id":"https://openalex.org/C115961682","wikidata":"https://www.wikidata.org/wiki/Q860623","display_name":"Image (mathematics)","level":2,"score":0.4522862136363983},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.43401777744293213},{"id":"https://openalex.org/C121927907","wikidata":"https://www.wikidata.org/wiki/Q1952516","display_name":"Multiresolution analysis","level":5,"score":0.4240581691265106},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.41834506392478943},{"id":"https://openalex.org/C9417928","wikidata":"https://www.wikidata.org/wiki/Q1070689","display_name":"Image processing","level":3,"score":0.39934420585632324},{"id":"https://openalex.org/C196216189","wikidata":"https://www.wikidata.org/wiki/Q2867","display_name":"Wavelet transform","level":3,"score":0.34482598304748535},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.3331380784511566},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.22172224521636963},{"id":"https://openalex.org/C46286280","wikidata":"https://www.wikidata.org/wiki/Q2414958","display_name":"Discrete wavelet transform","level":4,"score":0.21409469842910767},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.09593024849891663},{"id":"https://openalex.org/C76155785","wikidata":"https://www.wikidata.org/wiki/Q418","display_name":"Telecommunications","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},{"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/C199539241","wikidata":"https://www.wikidata.org/wiki/Q7748","display_name":"Law","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1090/s0025-5718-2012-02631-7","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-2012-02631-7","pdf_url":"https://www.ams.org/mcom/2013-82-282/S0025-5718-2012-02631-7/S0025-5718-2012-02631-7.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"}],"best_oa_location":{"id":"doi:10.1090/s0025-5718-2012-02631-7","is_oa":true,"landing_page_url":"https://doi.org/10.1090/s0025-5718-2012-02631-7","pdf_url":"https://www.ams.org/mcom/2013-82-282/S0025-5718-2012-02631-7/S0025-5718-2012-02631-7.pdf","source":{"id":"https://openalex.org/S102439543","display_name":"Mathematics of Computation","issn_l":"0025-5718","issn":["0025-5718","1088-6842"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310315719","host_organization_name":"American Mathematical Society","host_organization_lineage":["https://openalex.org/P4310315719"],"host_organization_lineage_names":["American Mathematical Society"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Mathematics of Computation","raw_type":"journal-article"},"sustainable_development_goals":[{"score":0.5299999713897705,"display_name":"Sustainable cities and communities","id":"https://metadata.un.org/sdg/11"}],"awards":[],"funders":[{"id":"https://openalex.org/F4320334593","display_name":"Natural Sciences and Engineering Research Council of Canada","ror":"https://ror.org/01h531d29"}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W2963347434.pdf","grobid_xml":"https://content.openalex.org/works/W2963347434.grobid-xml"},"referenced_works_count":64,"referenced_works":["https://openalex.org/W40037269","https://openalex.org/W136272581","https://openalex.org/W207373900","https://openalex.org/W1496099826","https://openalex.org/W1540764732","https://openalex.org/W1579559187","https://openalex.org/W1897767665","https://openalex.org/W1972163814","https://openalex.org/W1973700755","https://openalex.org/W1973734200","https://openalex.org/W1978333359","https://openalex.org/W1980454827","https://openalex.org/W1983772857","https://openalex.org/W1992609597","https://openalex.org/W1996726072","https://openalex.org/W1999905919","https://openalex.org/W2019043293","https://openalex.org/W2021535288","https://openalex.org/W2025263014","https://openalex.org/W2027191314","https://openalex.org/W2035541560","https://openalex.org/W2039939700","https://openalex.org/W2055483062","https://openalex.org/W2062024414","https://openalex.org/W2069033451","https://openalex.org/W2086670019","https://openalex.org/W2089202208","https://openalex.org/W2090390875","https://openalex.org/W2093834886","https://openalex.org/W2097924771","https://openalex.org/W2100556411","https://openalex.org/W2103559027","https://openalex.org/W2103955025","https://openalex.org/W2108454251","https://openalex.org/W2108982210","https://openalex.org/W2110505738","https://openalex.org/W2125555749","https://openalex.org/W2127436071","https://openalex.org/W2129131372","https://openalex.org/W2129638195","https://openalex.org/W2142058898","https://openalex.org/W2145003640","https://openalex.org/W2145096794","https://openalex.org/W2147850450","https://openalex.org/W2154996879","https://openalex.org/W2156575092","https://openalex.org/W2157434051","https://openalex.org/W2169472615","https://openalex.org/W2296616510","https://openalex.org/W2322609174","https://openalex.org/W2608329453","https://openalex.org/W2623831436","https://openalex.org/W2795407492","https://openalex.org/W2798817241","https://openalex.org/W2952230283","https://openalex.org/W2963322354","https://openalex.org/W3106045672","https://openalex.org/W4230684562","https://openalex.org/W4237182695","https://openalex.org/W4253960796","https://openalex.org/W4285719527","https://openalex.org/W6643536926","https://openalex.org/W6659198754","https://openalex.org/W7019313871"],"related_works":["https://openalex.org/W2393022482","https://openalex.org/W2377346130","https://openalex.org/W2361092061","https://openalex.org/W34817178","https://openalex.org/W2319775965","https://openalex.org/W2357314690","https://openalex.org/W2795976185","https://openalex.org/W2191886813","https://openalex.org/W4302598424","https://openalex.org/W3097078770"],"abstract_inverted_index":{"The":[0,31],"theory":[1],"of":[2,33,50,119,132,145,185,194],"(tight)":[3],"wavelet":[4,34,71],"frames":[5],"has":[6],"been":[7],"extensively":[8],"studied":[9],"in":[10,183],"the":[11,60,97,120,127,138,146,158,163,195],"past":[12],"twenty":[13],"years":[14],"and":[15,24,28,41,63,134],"they":[16],"are":[17],"currently":[18],"widely":[19],"used":[20],"for":[21],"image":[22,26],"restoration":[23],"other":[25],"processing":[27],"analysis":[29,42,64,144,177],"problems.":[30],"success":[32],"frame":[35,72,121],"based":[36,43,65,73,178],"models,":[37],"including":[38],"balanced":[39,61,181],"approach":[40,62,179,182],"approach,":[44,66],"is":[45,123],"due":[46],"to":[47,136],"their":[48],"capability":[49],"sparsely":[51],"approximating":[52],"piecewise":[53],"smooth":[54],"functions":[55],"like":[56],"images.":[57,197],"Motivated":[58],"by":[59,162,175],"we":[67],"shall":[68],"propose":[69],"a":[70],"<inline-formula":[74,98],"content-type=\"math/mathml\">":[75,99],"<mml:math":[76,100],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[77,101],"alttext=\"script":[78,102],"l":[79,103],"0\">":[80,104],"<mml:semantics>":[81,105],"<mml:msub>":[82,106],"<mml:mi>":[83,107],"\u2113":[84,108],"</mml:mi>":[85,109],"<mml:mn>0</mml:mn>":[86,110],"</mml:msub>":[87,111],"<mml:annotation":[88,112],"encoding=\"application/x-tex\">\\ell":[89,113],"_0</mml:annotation>":[90,114],"</mml:semantics>":[91,115],"</mml:math>":[92,116],"</inline-formula>":[93,117],"minimization":[94],"model,":[95],"where":[96],"\u201cnorm\u201d":[118],"coefficients":[122],"penalized.":[124],"We":[125],"adapt":[126],"penalty":[128],"decomposition":[129],"(PD)":[130],"method":[131,149,165],"Lu":[133],"Zhang":[135],"solve":[137],"proposed":[139,159],"optimization":[140],"problem.":[141],"Some":[142],"convergence":[143],"adapted":[147],"PD":[148,164],"will":[150],"also":[151],"be":[152],"provided.":[153],"Numerical":[154],"results":[155],"showed":[156],"that":[157],"model":[160],"solved":[161],"can":[166],"generate":[167],"images":[168],"with":[169],"better":[170],"quality":[171],"than":[172],"those":[173],"obtained":[174],"either":[176],"or":[180],"terms":[184],"restoring":[186],"sharp":[187],"features":[188],"as":[189,191],"well":[190],"maintaining":[192],"smoothness":[193],"recovered":[196]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":3},{"year":2024,"cited_by_count":1},{"year":2023,"cited_by_count":6},{"year":2022,"cited_by_count":3},{"year":2021,"cited_by_count":8},{"year":2020,"cited_by_count":5},{"year":2019,"cited_by_count":8},{"year":2018,"cited_by_count":14},{"year":2017,"cited_by_count":17},{"year":2016,"cited_by_count":9},{"year":2015,"cited_by_count":8},{"year":2014,"cited_by_count":7},{"year":2013,"cited_by_count":6},{"year":2012,"cited_by_count":2}],"updated_date":"2026-07-25T15:57:00.446498","created_date":"2025-10-10T00:00:00"}
