{"id":"https://openalex.org/W2603909116","doi":"https://doi.org/10.1137/15m1037457","title":"Image Restoration: Wavelet Frame Shrinkage, Nonlinear Evolution PDEs, and Beyond","display_name":"Image Restoration: Wavelet Frame Shrinkage, Nonlinear Evolution PDEs, and Beyond","publication_year":2017,"publication_date":"2017-01-01","ids":{"openalex":"https://openalex.org/W2603909116","doi":"https://doi.org/10.1137/15m1037457","mag":"2603909116"},"language":"en","primary_location":{"id":"doi:10.1137/15m1037457","is_oa":false,"landing_page_url":"https://doi.org/10.1137/15m1037457","pdf_url":null,"source":{"id":"https://openalex.org/S4357572","display_name":"Multiscale Modeling and Simulation","issn_l":"1540-3459","issn":["1540-3459","1540-3467"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Multiscale Modeling &amp; Simulation","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"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/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, University of Arizona, Tucson, AZ 85721, U.S.A"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, University of Arizona, Tucson, AZ 85721, U.S.A","institution_ids":["https://openalex.org/I138006243"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5027322245","display_name":"Qingtang Jiang","orcid":"https://orcid.org/0000-0002-0173-9988"},"institutions":[{"id":"https://openalex.org/I208333798","display_name":"University of Missouri\u2013St. Louis","ror":"https://ror.org/037cnag11","country_code":"US","type":"education","lineage":["https://openalex.org/I208333798"]}],"countries":["US"],"is_corresponding":false,"raw_author_name":"Qingtang Jiang","raw_affiliation_strings":["Department of Mathematics and Computer Science, University of Missouri-St. Louis, St. Louis, MO 63121, U.S.A"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics and Computer Science, University of Missouri-St. Louis, St. Louis, MO 63121, U.S.A","institution_ids":["https://openalex.org/I208333798"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5068391881","display_name":"Zuowei Shen","orcid":"https://orcid.org/0000-0002-1861-1884"},"institutions":[{"id":"https://openalex.org/I165932596","display_name":"National University of Singapore","ror":"https://ror.org/01tgyzw49","country_code":"SG","type":"education","lineage":["https://openalex.org/I165932596"]}],"countries":["SG"],"is_corresponding":false,"raw_author_name":"Zuowei Shen","raw_affiliation_strings":["Department of Mathematics, National University of Singapore 10 Lower Kent Ridge Road, Singapore, 119076"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Mathematics, National University of Singapore 10 Lower Kent Ridge Road, Singapore, 119076","institution_ids":["https://openalex.org/I165932596"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":3,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":3.3074,"has_fulltext":false,"cited_by_count":95,"citation_normalized_percentile":{"value":0.94883565,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":94,"max":99},"biblio":{"volume":"15","issue":"1","first_page":"606","last_page":"660"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10688","display_name":"Image and Signal Denoising Methods","score":0.9997000098228455,"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":0.9997000098228455,"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/T11105","display_name":"Advanced Image Processing Techniques","score":0.9976999759674072,"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/T10271","display_name":"Seismic Imaging and Inversion Techniques","score":0.9973999857902527,"subfield":{"id":"https://openalex.org/subfields/1908","display_name":"Geophysics"},"field":{"id":"https://openalex.org/fields/19","display_name":"Earth and Planetary Sciences"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/wavelet","display_name":"Wavelet","score":0.8313090801239014},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.6829895377159119},{"id":"https://openalex.org/keywords/partial-differential-equation","display_name":"Partial differential equation","score":0.6621492505073547},{"id":"https://openalex.org/keywords/nonlinear-system","display_name":"Nonlinear system","score":0.5657821297645569},{"id":"https://openalex.org/keywords/connection","display_name":"Connection (principal bundle)","score":0.5414244532585144},{"id":"https://openalex.org/keywords/frame","display_name":"Frame (networking)","score":0.5117576122283936},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.5047696828842163},{"id":"https://openalex.org/keywords/trace","display_name":"TRACE (psycholinguistics)","score":0.45617246627807617},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.4548455476760864},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.36169344186782837},{"id":"https://openalex.org/keywords/mathematical-optimization","display_name":"Mathematical optimization","score":0.33338218927383423},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.26996833086013794},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.18406051397323608},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.16525879502296448}],"concepts":[{"id":"https://openalex.org/C47432892","wikidata":"https://www.wikidata.org/wiki/Q831390","display_name":"Wavelet","level":2,"score":0.8313090801239014},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6829895377159119},{"id":"https://openalex.org/C93779851","wikidata":"https://www.wikidata.org/wiki/Q271977","display_name":"Partial differential equation","level":2,"score":0.6621492505073547},{"id":"https://openalex.org/C158622935","wikidata":"https://www.wikidata.org/wiki/Q660848","display_name":"Nonlinear system","level":2,"score":0.5657821297645569},{"id":"https://openalex.org/C13355873","wikidata":"https://www.wikidata.org/wiki/Q2920850","display_name":"Connection (principal bundle)","level":2,"score":0.5414244532585144},{"id":"https://openalex.org/C126042441","wikidata":"https://www.wikidata.org/wiki/Q1324888","display_name":"Frame (networking)","level":2,"score":0.5117576122283936},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.5047696828842163},{"id":"https://openalex.org/C75291252","wikidata":"https://www.wikidata.org/wiki/Q1315756","display_name":"TRACE (psycholinguistics)","level":2,"score":0.45617246627807617},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.4548455476760864},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.36169344186782837},{"id":"https://openalex.org/C126255220","wikidata":"https://www.wikidata.org/wiki/Q141495","display_name":"Mathematical optimization","level":1,"score":0.33338218927383423},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.26996833086013794},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.18406051397323608},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.16525879502296448},{"id":"https://openalex.org/C138885662","wikidata":"https://www.wikidata.org/wiki/Q5891","display_name":"Philosophy","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},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.0},{"id":"https://openalex.org/C76155785","wikidata":"https://www.wikidata.org/wiki/Q418","display_name":"Telecommunications","level":1,"score":0.0},{"id":"https://openalex.org/C41895202","wikidata":"https://www.wikidata.org/wiki/Q8162","display_name":"Linguistics","level":1,"score":0.0}],"mesh":[],"locations_count":2,"locations":[{"id":"doi:10.1137/15m1037457","is_oa":false,"landing_page_url":"https://doi.org/10.1137/15m1037457","pdf_url":null,"source":{"id":"https://openalex.org/S4357572","display_name":"Multiscale Modeling and Simulation","issn_l":"1540-3459","issn":["1540-3459","1540-3467"],"is_oa":false,"is_in_doaj":false,"is_core":true,"host_organization":"https://openalex.org/P4310320508","host_organization_name":"Society for Industrial and Applied Mathematics","host_organization_lineage":["https://openalex.org/P4310320508"],"host_organization_lineage_names":["Society for Industrial and Applied Mathematics"],"type":"journal"},"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Multiscale Modeling &amp; Simulation","raw_type":"journal-article"},{"id":"pmh:oai:irl.umsl.edu:mathstats-faculty-1005","is_oa":false,"landing_page_url":"https://irl.umsl.edu/mathstats-faculty/6","pdf_url":null,"source":{"id":"https://openalex.org/S4377196466","display_name":"IRL - University of Missouri, St. Louis (University of Missouri\u2013St. Louis)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I208333798","host_organization_name":"University of Missouri\u2013St. Louis","host_organization_lineage":["https://openalex.org/I208333798"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Mathematics and Statistics Faculty Works","raw_type":"text"}],"best_oa_location":null,"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/11","display_name":"Sustainable cities and communities","score":0.6000000238418579}],"awards":[],"funders":[],"has_content":{"grobid_xml":false,"pdf":false},"content_urls":null,"referenced_works_count":75,"referenced_works":["https://openalex.org/W1480546554","https://openalex.org/W1492459669","https://openalex.org/W1496099826","https://openalex.org/W1497523050","https://openalex.org/W1502486124","https://openalex.org/W1570378235","https://openalex.org/W1582617215","https://openalex.org/W1608138259","https://openalex.org/W1966083533","https://openalex.org/W1968773088","https://openalex.org/W1973700755","https://openalex.org/W1983772857","https://openalex.org/W1989619792","https://openalex.org/W1989893875","https://openalex.org/W1993253771","https://openalex.org/W1996092669","https://openalex.org/W1997431226","https://openalex.org/W1999788007","https://openalex.org/W1999829977","https://openalex.org/W1999905919","https://openalex.org/W2002917479","https://openalex.org/W2006262045","https://openalex.org/W2015080174","https://openalex.org/W2019043293","https://openalex.org/W2021535288","https://openalex.org/W2024178729","https://openalex.org/W2025263014","https://openalex.org/W2027191314","https://openalex.org/W2032240925","https://openalex.org/W2035541560","https://openalex.org/W2040573322","https://openalex.org/W2042961821","https://openalex.org/W2045280866","https://openalex.org/W2057220024","https://openalex.org/W2062024414","https://openalex.org/W2065401661","https://openalex.org/W2069947795","https://openalex.org/W2086670019","https://openalex.org/W2089202208","https://openalex.org/W2090390875","https://openalex.org/W2090525223","https://openalex.org/W2100556411","https://openalex.org/W2103559027","https://openalex.org/W2108454251","https://openalex.org/W2110505738","https://openalex.org/W2122287830","https://openalex.org/W2125555749","https://openalex.org/W2127436071","https://openalex.org/W2135036909","https://openalex.org/W2142058898","https://openalex.org/W2145003640","https://openalex.org/W2146052399","https://openalex.org/W2146842127","https://openalex.org/W2147850450","https://openalex.org/W2150134853","https://openalex.org/W2154996879","https://openalex.org/W2156575092","https://openalex.org/W2157434051","https://openalex.org/W2161214672","https://openalex.org/W2165939338","https://openalex.org/W2167400582","https://openalex.org/W2295936755","https://openalex.org/W2322609174","https://openalex.org/W2340039847","https://openalex.org/W2349369506","https://openalex.org/W2407415358","https://openalex.org/W2608329453","https://openalex.org/W2623831436","https://openalex.org/W2963347434","https://openalex.org/W3011612142","https://openalex.org/W3021722416","https://openalex.org/W3106045672","https://openalex.org/W3161057735","https://openalex.org/W4248590282","https://openalex.org/W4285719527"],"related_works":["https://openalex.org/W587735977","https://openalex.org/W1978142926","https://openalex.org/W3157395178","https://openalex.org/W2131958170","https://openalex.org/W2382174632","https://openalex.org/W2061122711","https://openalex.org/W2273754158","https://openalex.org/W2129959498","https://openalex.org/W184455116","https://openalex.org/W2152739222"],"abstract_inverted_index":{"In":[0],"the":[1,17,25,99,109,115,153,166,175,184,230,240,283,292,300,303,310,315,347,373,378,401,411,417,453,529,561],"past":[2],"few":[3],"decades,":[4],"mathematics":[5],"based":[6,22,78,112,117,144,147,169,177,197,233,247,350,371,404,414],"approaches":[7,85,135,148,473],"have":[8,136,149,395],"been":[9,151],"widely":[10],"adopted":[11],"in":[12,104,138,209,291,313,321,380,434,550],"various":[13,441],"image":[14,304,369,489,532],"restoration":[15,370],"problems;":[16],"partial":[18],"differential":[19],"equation":[20],"(PDE)":[21],"approach":[23,79,113,118,170,178,198,234,248,351,405,415],"(e.g.,":[24,416],"total":[26,201,284],"variation":[27,202,285],"model":[28,203,265,281,286],"[L.":[29],"Rudin,":[30],"S.":[31,214],"Osher,":[32,215],"and":[33,42,49,75,91,114,145,204,216,330,352,358,364,424,431,439,443,462,468,482,515,536,564],"E.":[34],"Fatemi,":[35],"Phys.":[36],"D,":[37],"60":[38],"(1992),":[39,71],"pp.":[40,60,72,225,338,391],"259--268]":[41],"its":[43,205],"generalizations,":[44],"nonlinear":[45,353,421,463,494],"diffusions":[46],"[P.":[47],"Perona":[48],"J.":[50,67,219,385],"Malik,":[51],"IEEE":[52],"Trans.":[53],"Pattern":[54],"Anal.":[55],"Mach.":[56],"Intel.,":[57],"12":[58],"(1990),":[59],"629--639;":[61],"F.":[62],"Catt\u00e9":[63],"et":[64,255,323,328,383],"al.,":[65,384],"SIAM":[66],"Numer.":[68,334],"Anal.,":[69],"29":[70],"182--193],":[73],"etc.)":[74],"wavelet":[76,110,142,167,195,231,263,268,348,402,459,484,506,512,540,574],"frame":[77,111,143,168,196,232,264,269,349,403,460,485,507,513,541],"are":[80],"some":[81],"successful":[82],"examples.":[83],"These":[84],"were":[86],"developed":[87],"through":[88],"different":[89,96,134,139],"paths":[90],"generally":[92],"provided":[93],"understanding":[94],"from":[95],"angles":[97],"of":[98,108,156,193,262,277,302,361,399,471,524,531,547,566,573],"same":[100,154,241],"problem.":[101],"As":[102],"shown":[103],"numerical":[105,126,130,435,567],"simulations,":[106],"implementations":[107],"PDE":[116,146,176,246,413,444,480,496],"quite":[119],"often":[120],"end":[121],"up":[122],"solving":[123,548],"a":[124,194,200,235,245,250,259,278,288,343,396,429,544,555],"similar":[125,129],"problem":[127],"with":[128,158,174,199,249,377,509],"behaviors,":[131],"even":[132],"though":[133],"advantages":[137],"applications.":[140],"Since":[141],"all":[150,183],"modeling":[152],"types":[155],"problems":[157],"success,":[159],"it":[160,243],"is":[161,171,309,318,527,543],"natural":[162],"to":[163,187,306,409,437,477],"ask":[164],"whether":[165],"fundamentally":[172],"connected":[173],"when":[179],"we":[180,393,450],"trace":[181],"them":[182],"way":[185,546],"back":[186],"their":[188],"roots.":[189],"A":[190,295,492],"fundamental":[191,344],"connection":[192,228,345],"generalizations":[206],"was":[207],"established":[208],"[J.":[210,381],"Cai,":[211],"B.":[212],"Dong,":[213],"Z.":[217],"Shen,":[218],"Amer.":[220,386],"Math.":[221,387],"Soc.,":[222,388],"25":[223,389],"(2012),":[224,337,390],"1033--1089].":[226],"This":[227],"gives":[229],"geometric":[236],"explanation":[237],"and,":[238],"at":[239],"time,":[242],"equips":[244],"time":[251],"frequency":[252],"analysis.":[253],"Cai":[254,322,327,382],"al.":[256,324,329],"showed":[257],"that":[258,474,539,558],"special":[260,289],"type":[261],"using":[266],"generic":[267,279,493],"systems":[270,514],"can":[271,406,425,502],"be":[272,407,426,447,503],"regarded":[273],"as":[274,287,299,428,569,571],"an":[275],"approximation":[276],"variational":[280,418,442],"(with":[282],"case)":[290],"discrete":[293],"setting.":[294],"systematic":[296],"convergence":[297],"analysis,":[298],"resolution":[301],"goes":[305],"infinity,":[307],"which":[308,552],"key":[311],"step":[312],"linking":[314],"two":[316],"approaches,":[317],"also":[319],"given":[320],"Motivated":[325],"by":[326,505],"[Q.":[331],"Jiang,":[332],"Appl.":[333],"Math.,":[335],"62":[336],"51--66],":[339],"this":[340,525],"paper":[341],"establishes":[342],"between":[346,458],"evolution":[354,422,464,495],"PDEs,":[355,568],"provides":[356],"interpretations":[357,470],"analytical":[359],"studies":[360],"such":[362],"connections,":[363],"proposes":[365],"new":[366,374,430,467,479,545,556],"algorithms":[367,487],"for":[368,488],"on":[372],"understandings.":[375],"Together":[376],"results":[379],"1033--1089],":[392],"now":[394],"better":[397],"picture":[398],"how":[400],"used":[408,427],"interpret":[410],"general":[412],"models":[419,481],"or":[420,499],"PDEs)":[423],"useful":[432],"tool":[433],"analysis":[436,535],"discretize":[438],"solve":[440],"models.":[445],"To":[446],"more":[448],"precise,":[449],"shall":[451],"establish":[452],"following:":[454],"(1)":[455],"The":[456,521],"connections":[457],"shrinkage":[461,486,508,518,542],"PDEs":[465,549],"provide":[466,554],"inspiring":[469],"both":[472],"enable":[475],"us":[476],"derive":[478],"(better)":[483],"restoration.":[490,533],"(2)":[491],"(of":[497],"parabolic":[498],"hyperbolic":[500],"type)":[501],"approximated":[504],"properly":[510],"chosen":[511],"carefully":[516],"designed":[517],"functions.":[519],"(3)":[520],"main":[522],"idea":[523],"work":[526],"beyond":[528],"scope":[530],"Our":[534],"discussions":[537],"indicate":[538],"general,":[551],"will":[553,559],"insight":[557],"enrich":[560],"existing":[562],"theory":[563],"applications":[565],"well":[570],"those":[572],"frames.":[575]},"counts_by_year":[{"year":2026,"cited_by_count":2},{"year":2025,"cited_by_count":9},{"year":2024,"cited_by_count":10},{"year":2023,"cited_by_count":6},{"year":2022,"cited_by_count":8},{"year":2021,"cited_by_count":11},{"year":2020,"cited_by_count":14},{"year":2019,"cited_by_count":9},{"year":2018,"cited_by_count":5},{"year":2017,"cited_by_count":3},{"year":2016,"cited_by_count":7},{"year":2015,"cited_by_count":9},{"year":2014,"cited_by_count":2}],"updated_date":"2026-08-01T09:00:35.917206","created_date":"2025-10-10T00:00:00"}
