{"id":"https://openalex.org/W1975302208","doi":"https://doi.org/10.1109/tit.2014.2311903","title":"A Unified Formulation of Gaussian Versus Sparse Stochastic Processes\u2014Part II: Discrete-Domain Theory","display_name":"A Unified Formulation of Gaussian Versus Sparse Stochastic Processes\u2014Part II: Discrete-Domain Theory","publication_year":2014,"publication_date":"2014-03-14","ids":{"openalex":"https://openalex.org/W1975302208","doi":"https://doi.org/10.1109/tit.2014.2311903","mag":"1975302208"},"language":"en","primary_location":{"id":"doi:10.1109/tit.2014.2311903","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2014.2311903","pdf_url":null,"source":{"id":"https://openalex.org/S4502562","display_name":"IEEE Transactions on Information Theory","issn_l":"0018-9448","issn":["0018-9448","1557-9654"],"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 Information Theory","raw_type":"journal-article"},"type":"article","indexed_in":["crossref"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"http://infoscience.epfl.ch/record/199643","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5027830035","display_name":"Micha\u00ebl Unser","orcid":"https://orcid.org/0000-0003-1248-2513"},"institutions":[{"id":"https://openalex.org/I5124864","display_name":"\u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne","ror":"https://ror.org/02s376052","country_code":"CH","type":"education","lineage":["https://openalex.org/I2799323385","https://openalex.org/I5124864"]}],"countries":["CH"],"is_corresponding":true,"raw_author_name":"Michael Unser","raw_affiliation_strings":["Biomedical Imaging Group, \u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne, Lausanne, Switzerland","Biomed. Imaging Group, Ecole Polytech. Fed. de Lausanne, Lausanne, Switzerland"],"affiliations":[{"raw_affiliation_string":"Biomedical Imaging Group, \u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne, Lausanne, Switzerland","institution_ids":["https://openalex.org/I5124864"]},{"raw_affiliation_string":"Biomed. Imaging Group, Ecole Polytech. Fed. de Lausanne, Lausanne, Switzerland","institution_ids":["https://openalex.org/I5124864"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5113663395","display_name":"Pouya D. Tafti","orcid":null},"institutions":[{"id":"https://openalex.org/I5124864","display_name":"\u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne","ror":"https://ror.org/02s376052","country_code":"CH","type":"education","lineage":["https://openalex.org/I2799323385","https://openalex.org/I5124864"]}],"countries":["CH"],"is_corresponding":false,"raw_author_name":"Pouya D. Tafti","raw_affiliation_strings":["Biomedical Imaging Group, \u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne, Lausanne, Switzerland","Biomed. Imaging Group, Ecole Polytech. Fed. de Lausanne, Lausanne, Switzerland"],"affiliations":[{"raw_affiliation_string":"Biomedical Imaging Group, \u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne, Lausanne, Switzerland","institution_ids":["https://openalex.org/I5124864"]},{"raw_affiliation_string":"Biomed. Imaging Group, Ecole Polytech. Fed. de Lausanne, Lausanne, Switzerland","institution_ids":["https://openalex.org/I5124864"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5020881664","display_name":"Arash Amini","orcid":"https://orcid.org/0000-0002-7082-9581"},"institutions":[{"id":"https://openalex.org/I5124864","display_name":"\u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne","ror":"https://ror.org/02s376052","country_code":"CH","type":"education","lineage":["https://openalex.org/I2799323385","https://openalex.org/I5124864"]}],"countries":["CH"],"is_corresponding":false,"raw_author_name":"Arash Amini","raw_affiliation_strings":["Biomedical Imaging Group, \u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne, Lausanne, Switzerland","Biomed. Imaging Group, Ecole Polytech. Fed. de Lausanne, Lausanne, Switzerland"],"affiliations":[{"raw_affiliation_string":"Biomedical Imaging Group, \u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne, Lausanne, Switzerland","institution_ids":["https://openalex.org/I5124864"]},{"raw_affiliation_string":"Biomed. Imaging Group, Ecole Polytech. Fed. de Lausanne, Lausanne, Switzerland","institution_ids":["https://openalex.org/I5124864"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5060921980","display_name":"Hagai Kirshner","orcid":null},"institutions":[{"id":"https://openalex.org/I5124864","display_name":"\u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne","ror":"https://ror.org/02s376052","country_code":"CH","type":"education","lineage":["https://openalex.org/I2799323385","https://openalex.org/I5124864"]}],"countries":["CH"],"is_corresponding":false,"raw_author_name":"Hagai Kirshner","raw_affiliation_strings":["Biomedical Imaging Group, \u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne, Lausanne, Switzerland","Biomed. Imaging Group, Ecole Polytech. Fed. de Lausanne, Lausanne, Switzerland"],"affiliations":[{"raw_affiliation_string":"Biomedical Imaging Group, \u00c9cole Polytechnique F\u00e9d\u00e9rale de Lausanne, Lausanne, Switzerland","institution_ids":["https://openalex.org/I5124864"]},{"raw_affiliation_string":"Biomed. Imaging Group, Ecole Polytech. Fed. de Lausanne, Lausanne, Switzerland","institution_ids":["https://openalex.org/I5124864"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":["https://openalex.org/A5027830035"],"corresponding_institution_ids":["https://openalex.org/I5124864"],"apc_list":null,"apc_paid":null,"fwci":7.7934,"has_fulltext":false,"cited_by_count":37,"citation_normalized_percentile":{"value":0.97262171,"is_in_top_1_percent":false,"is_in_top_10_percent":true},"cited_by_percentile_year":{"min":90,"max":99},"biblio":{"volume":"60","issue":"5","first_page":"3036","last_page":"3051"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10067","display_name":"Stochastic processes and financial applications","score":0.9925000071525574,"subfield":{"id":"https://openalex.org/subfields/2003","display_name":"Finance"},"field":{"id":"https://openalex.org/fields/20","display_name":"Economics, Econometrics and Finance"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},"topics":[{"id":"https://openalex.org/T10067","display_name":"Stochastic processes and financial applications","score":0.9925000071525574,"subfield":{"id":"https://openalex.org/subfields/2003","display_name":"Finance"},"field":{"id":"https://openalex.org/fields/20","display_name":"Economics, Econometrics and Finance"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T10928","display_name":"Probabilistic and Robust Engineering Design","score":0.9912999868392944,"subfield":{"id":"https://openalex.org/subfields/1804","display_name":"Statistics, Probability and Uncertainty"},"field":{"id":"https://openalex.org/fields/18","display_name":"Decision Sciences"},"domain":{"id":"https://openalex.org/domains/2","display_name":"Social Sciences"}},{"id":"https://openalex.org/T10968","display_name":"Statistical Distribution Estimation and Applications","score":0.9779999852180481,"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/mathematics","display_name":"Mathematics","score":0.6372056007385254},{"id":"https://openalex.org/keywords/gaussian","display_name":"Gaussian","score":0.5273498296737671},{"id":"https://openalex.org/keywords/gaussian-process","display_name":"Gaussian process","score":0.5041114091873169},{"id":"https://openalex.org/keywords/applied-mathematics","display_name":"Applied mathematics","score":0.483864426612854},{"id":"https://openalex.org/keywords/stochastic-process","display_name":"Stochastic process","score":0.47467041015625},{"id":"https://openalex.org/keywords/autoregressive-model","display_name":"Autoregressive model","score":0.44875016808509827},{"id":"https://openalex.org/keywords/discrete-time-and-continuous-time","display_name":"Discrete time and continuous time","score":0.41516950726509094}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.6372056007385254},{"id":"https://openalex.org/C163716315","wikidata":"https://www.wikidata.org/wiki/Q901177","display_name":"Gaussian","level":2,"score":0.5273498296737671},{"id":"https://openalex.org/C61326573","wikidata":"https://www.wikidata.org/wiki/Q1496376","display_name":"Gaussian process","level":3,"score":0.5041114091873169},{"id":"https://openalex.org/C28826006","wikidata":"https://www.wikidata.org/wiki/Q33521","display_name":"Applied mathematics","level":1,"score":0.483864426612854},{"id":"https://openalex.org/C8272713","wikidata":"https://www.wikidata.org/wiki/Q176737","display_name":"Stochastic process","level":2,"score":0.47467041015625},{"id":"https://openalex.org/C159877910","wikidata":"https://www.wikidata.org/wiki/Q2202883","display_name":"Autoregressive model","level":2,"score":0.44875016808509827},{"id":"https://openalex.org/C55689738","wikidata":"https://www.wikidata.org/wiki/Q15963867","display_name":"Discrete time and continuous time","level":2,"score":0.41516950726509094},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","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/C149782125","wikidata":"https://www.wikidata.org/wiki/Q160039","display_name":"Econometrics","level":1,"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":[],"locations_count":3,"locations":[{"id":"doi:10.1109/tit.2014.2311903","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tit.2014.2311903","pdf_url":null,"source":{"id":"https://openalex.org/S4502562","display_name":"IEEE Transactions on Information Theory","issn_l":"0018-9448","issn":["0018-9448","1557-9654"],"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 Information Theory","raw_type":"journal-article"},{"id":"pmh:oai:infoscience.epfl.ch:199643","is_oa":true,"landing_page_url":"http://infoscience.epfl.ch/record/199643","pdf_url":null,"source":{"id":"https://openalex.org/S4306400487","display_name":"Infoscience (Ecole Polytechnique F\u00e9d\u00e9rale de Lausanne)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Text"},{"id":"pmh:oai:infoscience.tind.io:199643","is_oa":true,"landing_page_url":"http://bigwww.epfl.ch/publications/unser1402.html","pdf_url":null,"source":{"id":"https://openalex.org/S4306400487","display_name":"Infoscience (Ecole Polytechnique F\u00e9d\u00e9rale de Lausanne)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"research article"}],"best_oa_location":{"id":"pmh:oai:infoscience.epfl.ch:199643","is_oa":true,"landing_page_url":"http://infoscience.epfl.ch/record/199643","pdf_url":null,"source":{"id":"https://openalex.org/S4306400487","display_name":"Infoscience (Ecole Polytechnique F\u00e9d\u00e9rale de Lausanne)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":null,"host_organization_name":null,"host_organization_lineage":[],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Text"},"sustainable_development_goals":[{"id":"https://metadata.un.org/sdg/9","score":0.6299999952316284,"display_name":"Industry, innovation and infrastructure"}],"awards":[{"id":"https://openalex.org/G1250623024","display_name":null,"funder_award_id":"200020-144355","funder_id":"https://openalex.org/F4320320924","funder_display_name":"Schweizerischer Nationalfonds zur F\u00f6rderung der Wissenschaftlichen Forschung"},{"id":"https://openalex.org/G1441312571","display_name":null,"funder_award_id":"00020","funder_id":"https://openalex.org/F4320320924","funder_display_name":"Schweizerischer Nationalfonds zur F\u00f6rderung der Wissenschaftlichen Forschung"},{"id":"https://openalex.org/G1826283034","display_name":"Erweiterung der entscheidungstheoretischen Ordnungstheorie um dieSystemwissenschaft.","funder_award_id":"26743","funder_id":"https://openalex.org/F4320320924","funder_display_name":"Schweizerischer Nationalfonds zur F\u00f6rderung der Wissenschaftlichen Forschung"},{"id":"https://openalex.org/G3046754738","display_name":"Spline methods for the continuous/discrete processing of images","funder_award_id":"144355","funder_id":"https://openalex.org/F4320320924","funder_display_name":"Schweizerischer Nationalfonds zur F\u00f6rderung der Wissenschaftlichen Forschung"},{"id":"https://openalex.org/G4247904154","display_name":null,"funder_award_id":"European Research Council","funder_id":"https://openalex.org/F4320320924","funder_display_name":"Schweizerischer Nationalfonds zur F\u00f6rderung der Wissenschaftlichen Forschung"},{"id":"https://openalex.org/G4836148144","display_name":null,"funder_award_id":"20002","funder_id":"https://openalex.org/F4320320924","funder_display_name":"Schweizerischer Nationalfonds zur F\u00f6rderung der Wissenschaftlichen Forschung"},{"id":"https://openalex.org/G5599649393","display_name":null,"funder_award_id":"200020","funder_id":"https://openalex.org/F4320306076","funder_display_name":"National Science Foundation"},{"id":"https://openalex.org/G628552264","display_name":null,"funder_award_id":"200020","funder_id":"https://openalex.org/F4320320924","funder_display_name":"Schweizerischer Nationalfonds zur F\u00f6rderung der Wissenschaftlichen Forschung"}],"funders":[{"id":"https://openalex.org/F4320306076","display_name":"National Science Foundation","ror":"https://ror.org/021nxhr62"},{"id":"https://openalex.org/F4320320924","display_name":"Schweizerischer Nationalfonds zur F\u00f6rderung der Wissenschaftlichen Forschung","ror":"https://ror.org/00yjd3n13"}],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":38,"referenced_works":["https://openalex.org/W1499969990","https://openalex.org/W1519038128","https://openalex.org/W1536997931","https://openalex.org/W1542026384","https://openalex.org/W1570180454","https://openalex.org/W1591798773","https://openalex.org/W1667165204","https://openalex.org/W1996124770","https://openalex.org/W2002530172","https://openalex.org/W2022894179","https://openalex.org/W2031817855","https://openalex.org/W2036488048","https://openalex.org/W2037489965","https://openalex.org/W2046245205","https://openalex.org/W2048209332","https://openalex.org/W2062114965","https://openalex.org/W2077791698","https://openalex.org/W2099205567","https://openalex.org/W2140749977","https://openalex.org/W2141436719","https://openalex.org/W2153058280","https://openalex.org/W2156621264","https://openalex.org/W2158537680","https://openalex.org/W2160014433","https://openalex.org/W2171440547","https://openalex.org/W2567032824","https://openalex.org/W2579190672","https://openalex.org/W2751862591","https://openalex.org/W2790825170","https://openalex.org/W2797641211","https://openalex.org/W2799137445","https://openalex.org/W2983836991","https://openalex.org/W3022439851","https://openalex.org/W3023184745","https://openalex.org/W4210426676","https://openalex.org/W4246756383","https://openalex.org/W4247340585","https://openalex.org/W6634360839"],"related_works":["https://openalex.org/W2900278465","https://openalex.org/W1991387996","https://openalex.org/W2160749004","https://openalex.org/W2145556071","https://openalex.org/W2135954451","https://openalex.org/W2538359481","https://openalex.org/W4288093948","https://openalex.org/W2979489168","https://openalex.org/W2158798129","https://openalex.org/W3123852743"],"abstract_inverted_index":{"This":[0],"paper":[1],"is":[2,175,192],"devoted":[3],"to":[4,66,165,194,221],"the":[5,62,67,82,85,88,101,104,109,118,132,135,143,153,176,185,205,208,211,233],"characterization":[6],"of":[7,11,21,38,59,84,112,134,142,146,155,178,200,207,217,226],"an":[8,167],"extended":[9],"family":[10],"continuous-time":[12],"autoregressive":[13],"moving":[14],"average":[15],"(CARMA)":[16],"processes":[17,74],"that":[18,42,55,129,229],"are":[19,31,75,103,108,230],"solutions":[20],"stochastic":[22],"differential":[23],"equations":[24],"driven":[25],"by":[26],"white":[27],"L\u00e9vy":[28,105],"innovations.":[29],"These":[30],"completely":[32],"specified":[33],"by:":[34],"1)":[35],"a":[36,49,97,125,179],"set":[37],"poles":[39,86],"and":[40,47,123,138,210],"zeros":[41],"fixes":[43],"their":[44,57],"correlation":[45,136,209],"structure":[46],"2)":[48],"canonical":[50],"infinitely":[51],"divisible":[52],"probability":[53,140],"distribution":[54],"controls":[56],"degree":[58],"sparsity":[60],"(with":[61,96],"Gaussian":[63],"model":[64],"corresponding":[65],"least":[68],"sparse":[69,227],"scenario).":[70],"The":[71,91],"generalized":[72,156],"CARMA":[73],"either":[76],"stationary":[77],"or":[78],"nonstationary,":[79],"depending":[80],"on":[81,117,152],"location":[83],"in":[87,198],"complex":[89],"plane.":[90],"most":[92,199],"basic":[93],"nonstationary":[94],"representatives":[95],"single":[98],"pole":[99],"at":[100,204],"origin)":[102],"processes,":[106],"which":[107,159,191],"non-Gaussian":[110],"counterparts":[111],"Brownian":[113],"motion.":[114],"We":[115,149,214],"focus":[116],"general":[119],"analog-to-discrete":[120],"conversion":[121],"problem":[122],"introduce":[124],"novel":[126],"spline-based":[127],"formalism":[128],"greatly":[130],"simplifies":[131],"derivation":[133],"properties":[137],"joint":[139],"distributions":[141],"discrete":[144],"versions":[145],"these":[147,218],"processes.":[148],"also":[150],"rely":[151],"concept":[154],"increment":[157],"process,":[158],"suppresses":[160],"all":[161],"long":[162],"range":[163],"dependencies,":[164],"specify":[166],"equivalent":[168],"discrete-domain":[169,219],"innovation":[170],"model.":[171,235],"A":[172],"crucial":[173],"ingredient":[174],"existence":[177],"minimally":[180],"supported":[181],"function":[182],"associated":[183],"with":[184,232],"whitening":[186],"operator":[187],"L;":[188],"this":[189],"B-spline,":[190],"fundamental":[193],"our":[195,201],"formulation,":[196],"appears":[197],"formulas,":[202],"both":[203],"level":[206],"characteristic":[212],"function.":[213],"make":[215],"use":[216],"results":[220],"numerically":[222],"generate":[223],"illustrative":[224],"examples":[225],"signals":[228],"consistent":[231],"continuous-domain":[234]},"counts_by_year":[{"year":2026,"cited_by_count":1},{"year":2025,"cited_by_count":1},{"year":2023,"cited_by_count":1},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":2},{"year":2019,"cited_by_count":2},{"year":2018,"cited_by_count":3},{"year":2017,"cited_by_count":2},{"year":2016,"cited_by_count":4},{"year":2015,"cited_by_count":7},{"year":2014,"cited_by_count":4},{"year":2013,"cited_by_count":3},{"year":2012,"cited_by_count":5}],"updated_date":"2026-04-21T08:09:41.155169","created_date":"2025-10-10T00:00:00"}