{"id":"https://openalex.org/W4313226001","doi":"https://doi.org/10.3390/e25010048","title":"Why Shape Coding? Asymptotic Analysis of the Entropy Rate for Digital Images","display_name":"Why Shape Coding? Asymptotic Analysis of the Entropy Rate for Digital Images","publication_year":2022,"publication_date":"2022-12-27","ids":{"openalex":"https://openalex.org/W4313226001","doi":"https://doi.org/10.3390/e25010048","pmid":"https://pubmed.ncbi.nlm.nih.gov/36673189"},"language":"en","primary_location":{"id":"doi:10.3390/e25010048","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e25010048","pdf_url":"https://www.mdpi.com/1099-4300/25/1/48/pdf?version=1672137711","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},"type":"article","indexed_in":["crossref","doaj","pubmed"],"open_access":{"is_oa":true,"oa_status":"gold","oa_url":"https://www.mdpi.com/1099-4300/25/1/48/pdf?version=1672137711","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5066296680","display_name":"Gangtao Xin","orcid":"https://orcid.org/0000-0002-8168-6870"},"institutions":[{"id":"https://openalex.org/I99065089","display_name":"Tsinghua University","ror":"https://ror.org/03cve4549","country_code":"CN","type":"education","lineage":["https://openalex.org/I99065089"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Gangtao Xin","raw_affiliation_strings":["Beijing National Research Center for Information Science and Technology, Tsinghua University, Beijing 100084, China","Department of Electronic Engineering, Tsinghua University, Beijing 100084, China"],"raw_orcid":"https://orcid.org/0000-0002-8168-6870","affiliations":[{"raw_affiliation_string":"Beijing National Research Center for Information Science and Technology, Tsinghua University, Beijing 100084, China","institution_ids":["https://openalex.org/I99065089"]},{"raw_affiliation_string":"Department of Electronic Engineering, Tsinghua University, Beijing 100084, China","institution_ids":["https://openalex.org/I99065089"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5079233004","display_name":"Pingyi Fan","orcid":"https://orcid.org/0000-0002-0658-6079"},"institutions":[{"id":"https://openalex.org/I99065089","display_name":"Tsinghua University","ror":"https://ror.org/03cve4549","country_code":"CN","type":"education","lineage":["https://openalex.org/I99065089"]}],"countries":["CN"],"is_corresponding":true,"raw_author_name":"Pingyi Fan","raw_affiliation_strings":["Beijing National Research Center for Information Science and Technology, Tsinghua University, Beijing 100084, China","Department of Electronic Engineering, Tsinghua University, Beijing 100084, China"],"raw_orcid":"https://orcid.org/0000-0002-0658-6079","affiliations":[{"raw_affiliation_string":"Beijing National Research Center for Information Science and Technology, Tsinghua University, Beijing 100084, China","institution_ids":["https://openalex.org/I99065089"]},{"raw_affiliation_string":"Department of Electronic Engineering, Tsinghua University, Beijing 100084, China","institution_ids":["https://openalex.org/I99065089"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5079052203","display_name":"Khaled B. Letaief","orcid":"https://orcid.org/0000-0003-2519-6401"},"institutions":[{"id":"https://openalex.org/I200769079","display_name":"Hong Kong University of Science and Technology","ror":"https://ror.org/00q4vv597","country_code":"HK","type":"education","lineage":["https://openalex.org/I200769079"]}],"countries":["HK"],"is_corresponding":false,"raw_author_name":"Khaled B. Letaief","raw_affiliation_strings":["Department of Electrical and Computer Engineering, Hong Kong University of Science and Technology (HKUST), Hong Kong"],"raw_orcid":"https://orcid.org/0000-0003-2519-6401","affiliations":[{"raw_affiliation_string":"Department of Electrical and Computer Engineering, Hong Kong University of Science and Technology (HKUST), Hong Kong","institution_ids":["https://openalex.org/I200769079"]}]}],"institutions":[],"countries_distinct_count":2,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5079233004"],"corresponding_institution_ids":["https://openalex.org/I99065089"],"apc_list":{"value":2000,"currency":"CHF","value_usd":2165},"apc_paid":{"value":2000,"currency":"CHF","value_usd":2165},"fwci":1.5177,"has_fulltext":true,"cited_by_count":4,"citation_normalized_percentile":{"value":0.81146881,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":94,"max":96},"biblio":{"volume":"25","issue":"1","first_page":"48","last_page":"48"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.9983999729156494,"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"}},"topics":[{"id":"https://openalex.org/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.9983999729156494,"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"}},{"id":"https://openalex.org/T11269","display_name":"Algorithms and Data Compression","score":0.9961000084877014,"subfield":{"id":"https://openalex.org/subfields/1702","display_name":"Artificial Intelligence"},"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/T10901","display_name":"Advanced Data Compression Techniques","score":0.9922000169754028,"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"}}],"keywords":[{"id":"https://openalex.org/keywords/lossless-compression","display_name":"Lossless compression","score":0.8305229544639587},{"id":"https://openalex.org/keywords/image-compression","display_name":"Image compression","score":0.5851560235023499},{"id":"https://openalex.org/keywords/encoder","display_name":"Encoder","score":0.5537004470825195},{"id":"https://openalex.org/keywords/entropy-encoding","display_name":"Entropy encoding","score":0.5511307120323181},{"id":"https://openalex.org/keywords/data-compression-ratio","display_name":"Data compression ratio","score":0.5288043022155762},{"id":"https://openalex.org/keywords/compression-ratio","display_name":"Compression ratio","score":0.5187766551971436},{"id":"https://openalex.org/keywords/ergodic-theory","display_name":"Ergodic theory","score":0.5056641101837158},{"id":"https://openalex.org/keywords/entropy","display_name":"Entropy (arrow of time)","score":0.5028981566429138},{"id":"https://openalex.org/keywords/mnist-database","display_name":"MNIST database","score":0.5024635791778564},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.49616536498069763},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.49405840039253235},{"id":"https://openalex.org/keywords/data-compression","display_name":"Data compression","score":0.4879259765148163},{"id":"https://openalex.org/keywords/pixel","display_name":"Pixel","score":0.468894898891449},{"id":"https://openalex.org/keywords/computer-vision","display_name":"Computer vision","score":0.44261273741722107},{"id":"https://openalex.org/keywords/artificial-intelligence","display_name":"Artificial intelligence","score":0.4367932975292206},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.434261679649353},{"id":"https://openalex.org/keywords/coding","display_name":"Coding (social sciences)","score":0.4108315408229828},{"id":"https://openalex.org/keywords/image","display_name":"Image (mathematics)","score":0.2882694602012634},{"id":"https://openalex.org/keywords/image-processing","display_name":"Image processing","score":0.27323204278945923},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.20791006088256836},{"id":"https://openalex.org/keywords/artificial-neural-network","display_name":"Artificial neural network","score":0.08366948366165161},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.08162331581115723},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.07764261960983276}],"concepts":[{"id":"https://openalex.org/C81081738","wikidata":"https://www.wikidata.org/wiki/Q55542","display_name":"Lossless compression","level":3,"score":0.8305229544639587},{"id":"https://openalex.org/C13481523","wikidata":"https://www.wikidata.org/wiki/Q412438","display_name":"Image compression","level":4,"score":0.5851560235023499},{"id":"https://openalex.org/C118505674","wikidata":"https://www.wikidata.org/wiki/Q42586063","display_name":"Encoder","level":2,"score":0.5537004470825195},{"id":"https://openalex.org/C1769480","wikidata":"https://www.wikidata.org/wiki/Q1345239","display_name":"Entropy encoding","level":3,"score":0.5511307120323181},{"id":"https://openalex.org/C94835093","wikidata":"https://www.wikidata.org/wiki/Q3113333","display_name":"Data compression ratio","level":5,"score":0.5288043022155762},{"id":"https://openalex.org/C25797200","wikidata":"https://www.wikidata.org/wiki/Q828137","display_name":"Compression ratio","level":3,"score":0.5187766551971436},{"id":"https://openalex.org/C122044880","wikidata":"https://www.wikidata.org/wiki/Q5498822","display_name":"Ergodic theory","level":2,"score":0.5056641101837158},{"id":"https://openalex.org/C106301342","wikidata":"https://www.wikidata.org/wiki/Q4117933","display_name":"Entropy (arrow of time)","level":2,"score":0.5028981566429138},{"id":"https://openalex.org/C190502265","wikidata":"https://www.wikidata.org/wiki/Q17069496","display_name":"MNIST database","level":3,"score":0.5024635791778564},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.49616536498069763},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.49405840039253235},{"id":"https://openalex.org/C78548338","wikidata":"https://www.wikidata.org/wiki/Q2493","display_name":"Data compression","level":2,"score":0.4879259765148163},{"id":"https://openalex.org/C160633673","wikidata":"https://www.wikidata.org/wiki/Q355198","display_name":"Pixel","level":2,"score":0.468894898891449},{"id":"https://openalex.org/C31972630","wikidata":"https://www.wikidata.org/wiki/Q844240","display_name":"Computer vision","level":1,"score":0.44261273741722107},{"id":"https://openalex.org/C154945302","wikidata":"https://www.wikidata.org/wiki/Q11660","display_name":"Artificial intelligence","level":1,"score":0.4367932975292206},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.434261679649353},{"id":"https://openalex.org/C179518139","wikidata":"https://www.wikidata.org/wiki/Q5140297","display_name":"Coding (social sciences)","level":2,"score":0.4108315408229828},{"id":"https://openalex.org/C115961682","wikidata":"https://www.wikidata.org/wiki/Q860623","display_name":"Image (mathematics)","level":2,"score":0.2882694602012634},{"id":"https://openalex.org/C9417928","wikidata":"https://www.wikidata.org/wiki/Q1070689","display_name":"Image processing","level":3,"score":0.27323204278945923},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.20791006088256836},{"id":"https://openalex.org/C50644808","wikidata":"https://www.wikidata.org/wiki/Q192776","display_name":"Artificial neural network","level":2,"score":0.08366948366165161},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.08162331581115723},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.07764261960983276},{"id":"https://openalex.org/C511840579","wikidata":"https://www.wikidata.org/wiki/Q12757","display_name":"Internal combustion engine","level":2,"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/C97355855","wikidata":"https://www.wikidata.org/wiki/Q11473","display_name":"Thermodynamics","level":1,"score":0.0}],"mesh":[],"locations_count":6,"locations":[{"id":"doi:10.3390/e25010048","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e25010048","pdf_url":"https://www.mdpi.com/1099-4300/25/1/48/pdf?version=1672137711","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},{"id":"pmid:36673189","is_oa":false,"landing_page_url":"https://pubmed.ncbi.nlm.nih.gov/36673189","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":"Entropy (Basel, Switzerland)","raw_type":null},{"id":"pmh:oai:repository.hkust.edu.hk:1783.1-124426","is_oa":false,"landing_page_url":"https://repository.hkust.edu.hk/ir/Record/1783.1-124426","pdf_url":null,"source":{"id":"https://openalex.org/S4306401796","display_name":"Rare & Special e-Zone (The Hong Kong University of Science and Technology)","issn_l":null,"issn":null,"is_oa":false,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I200769079","host_organization_name":"Hong Kong University of Science and Technology","host_organization_lineage":["https://openalex.org/I200769079"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"","raw_type":"Article"},{"id":"pmh:oai:doaj.org/article:264816375a874579ba7f811d7335550c","is_oa":false,"landing_page_url":"https://doaj.org/article/264816375a874579ba7f811d7335550c","pdf_url":null,"source":{"id":"https://openalex.org/S4306401280","display_name":"DOAJ (DOAJ: Directory of Open Access Journals)","issn_l":null,"issn":null,"is_oa":false,"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":"Entropy, Vol 25, Iss 1, p 48 (2022)","raw_type":"article"},{"id":"pmh:oai:mdpi.com:/1099-4300/25/1/48/","is_oa":true,"landing_page_url":"https://dx.doi.org/10.3390/e25010048","pdf_url":null,"source":{"id":"https://openalex.org/S4306400947","display_name":"MDPI (MDPI AG)","issn_l":null,"issn":null,"is_oa":true,"is_in_doaj":false,"is_core":false,"host_organization":"https://openalex.org/I4210097602","host_organization_name":"Multidisciplinary Digital Publishing Institute (Switzerland)","host_organization_lineage":["https://openalex.org/I4210097602"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy; Volume 25; Issue 1; Pages: 48","raw_type":"Text"},{"id":"pmh:oai:pubmedcentral.nih.gov:9857653","is_oa":true,"landing_page_url":"https://www.ncbi.nlm.nih.gov/pmc/articles/9857653","pdf_url":null,"source":{"id":"https://openalex.org/S2764455111","display_name":"PubMed Central","issn_l":null,"issn":null,"is_oa":true,"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":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":"Entropy (Basel)","raw_type":"Text"}],"best_oa_location":{"id":"doi:10.3390/e25010048","is_oa":true,"landing_page_url":"https://doi.org/10.3390/e25010048","pdf_url":"https://www.mdpi.com/1099-4300/25/1/48/pdf?version=1672137711","source":{"id":"https://openalex.org/S195231649","display_name":"Entropy","issn_l":"1099-4300","issn":["1099-4300"],"is_oa":true,"is_in_doaj":true,"is_core":true,"host_organization":"https://openalex.org/P4310310987","host_organization_name":"Multidisciplinary Digital Publishing Institute","host_organization_lineage":["https://openalex.org/P4310310987"],"host_organization_lineage_names":["Multidisciplinary Digital Publishing Institute"],"type":"journal"},"license":"cc-by","license_id":"https://openalex.org/licenses/cc-by","version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"Entropy","raw_type":"journal-article"},"sustainable_development_goals":[],"awards":[{"id":"https://openalex.org/G3957290171","display_name":null,"funder_award_id":"YFA1000500","funder_id":"https://openalex.org/F4320335777","funder_display_name":"National Key Research and Development Program of China"},{"id":"https://openalex.org/G6358204205","display_name":null,"funder_award_id":"2021.YFA1000500(4)","funder_id":"https://openalex.org/F4320335777","funder_display_name":"National Key Research and Development Program of China"}],"funders":[{"id":"https://openalex.org/F4320335777","display_name":"National Key Research and Development Program of China","ror":null}],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W4313226001.pdf","grobid_xml":"https://content.openalex.org/works/W4313226001.grobid-xml"},"referenced_works_count":21,"referenced_works":["https://openalex.org/W2027176594","https://openalex.org/W2103756638","https://openalex.org/W2122962290","https://openalex.org/W2137349251","https://openalex.org/W2149961130","https://openalex.org/W2777013526","https://openalex.org/W2920965512","https://openalex.org/W2936788968","https://openalex.org/W2963451384","https://openalex.org/W3088505943","https://openalex.org/W3096470384","https://openalex.org/W3098033783","https://openalex.org/W3125548804","https://openalex.org/W3134214575","https://openalex.org/W3134748538","https://openalex.org/W3167058553","https://openalex.org/W4200348960","https://openalex.org/W4307252178","https://openalex.org/W4312141206","https://openalex.org/W4317418669","https://openalex.org/W6773727879"],"related_works":["https://openalex.org/W2128890367","https://openalex.org/W4381744720","https://openalex.org/W2993349805","https://openalex.org/W2167897635","https://openalex.org/W4287817374","https://openalex.org/W2155738200","https://openalex.org/W3080614128","https://openalex.org/W4200061735","https://openalex.org/W1901494946","https://openalex.org/W2164077395"],"abstract_inverted_index":{"This":[0],"paper":[1],"focuses":[2],"on":[3,45],"the":[4,28,36,40,46,69,76,82,88],"ultimate":[5],"limit":[6],"theory":[7],"of":[8,72,101],"image":[9,16,50,83],"compression.":[10],"It":[11],"proves":[12],"that":[13,25],"for":[14,63,98],"an":[15,49],"source,":[17],"there":[18],"exists":[19],"a":[20,32,95],"coding":[21,51,93],"method":[22],"with":[23,53,91],"shapes":[24,54],"can":[26],"achieve":[27],"entropy":[29],"rate":[30],"under":[31],"certain":[33],"condition":[34,70],"where":[35],"shape-pixel":[37,73],"ratio":[38,74],"in":[39,75,81],"encoder/decoder":[41,77],"is":[42,55,94],"O(1/logt).":[43],"Based":[44],"new":[47],"finding,":[48],"framework":[52],"proposed":[56],"and":[57,65],"proved":[58],"to":[59],"be":[60],"asymptotically":[61],"optimal":[62],"stationary":[64],"ergodic":[66],"processes.":[67],"Moreover,":[68],"O(1/logt)":[71],"has":[78],"been":[79],"confirmed":[80],"database":[84],"MNIST,":[85],"which":[86],"illustrates":[87],"soft":[89],"compression":[90,100],"shape":[92],"near-optimal":[96],"scheme":[97],"lossless":[99],"images.":[102]},"counts_by_year":[{"year":2024,"cited_by_count":2},{"year":2023,"cited_by_count":2}],"updated_date":"2026-05-21T06:26:12.895304","created_date":"2025-10-10T00:00:00"}