{"id":"https://openalex.org/W2097539475","doi":"https://doi.org/10.1109/tim.2009.2015540","title":"Modified Factorial-Free Direct Methods for Zernike and Pseudo-Zernike Moment Computation","display_name":"Modified Factorial-Free Direct Methods for Zernike and Pseudo-Zernike Moment Computation","publication_year":2009,"publication_date":"2009-06-02","ids":{"openalex":"https://openalex.org/W2097539475","doi":"https://doi.org/10.1109/tim.2009.2015540","mag":"2097539475"},"language":"en","primary_location":{"id":"doi:10.1109/tim.2009.2015540","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tim.2009.2015540","pdf_url":null,"source":{"id":"https://openalex.org/S10892749","display_name":"IEEE Transactions on Instrumentation and Measurement","issn_l":"0018-9456","issn":["0018-9456","1557-9662"],"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 Instrumentation and Measurement","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/A5032725664","display_name":"George A. Papakostas","orcid":"https://orcid.org/0000-0001-5545-1499"},"institutions":[{"id":"https://openalex.org/I147962203","display_name":"Democritus University of Thrace","ror":"https://ror.org/03bfqnx40","country_code":"GR","type":"education","lineage":["https://openalex.org/I147962203"]}],"countries":["GR"],"is_corresponding":false,"raw_author_name":"G.A. Papakostas","raw_affiliation_strings":["Department of Electrical and Computer Engineering, Democritus University of Thrace, Xanthi, Greece"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Electrical and Computer Engineering, Democritus University of Thrace, Xanthi, Greece","institution_ids":["https://openalex.org/I147962203"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5011949101","display_name":"Yiannis S. Boutalis","orcid":null},"institutions":[{"id":"https://openalex.org/I147962203","display_name":"Democritus University of Thrace","ror":"https://ror.org/03bfqnx40","country_code":"GR","type":"education","lineage":["https://openalex.org/I147962203"]},{"id":"https://openalex.org/I231025917","display_name":"Hellenic Open University","ror":"https://ror.org/02kq26x23","country_code":"GR","type":"education","lineage":["https://openalex.org/I231025917"]}],"countries":["GR"],"is_corresponding":false,"raw_author_name":"Y.S. Boutalis","raw_affiliation_strings":["Department of Electrical and Computer Engineering, Democritus University of Thrace, Xanthi, Greece","Department of Informatics, Hellenic Open University, Patras, Greece"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Electrical and Computer Engineering, Democritus University of Thrace, Xanthi, Greece","institution_ids":["https://openalex.org/I147962203"]},{"raw_affiliation_string":"Department of Informatics, Hellenic Open University, Patras, Greece","institution_ids":["https://openalex.org/I231025917"]}]},{"author_position":"middle","author":{"id":"https://openalex.org/A5002850928","display_name":"D.A. Karras","orcid":"https://orcid.org/0000-0002-2759-8482"},"institutions":[{"id":"https://openalex.org/I231025917","display_name":"Hellenic Open University","ror":"https://ror.org/02kq26x23","country_code":"GR","type":"education","lineage":["https://openalex.org/I231025917"]}],"countries":["GR"],"is_corresponding":false,"raw_author_name":"D.A. Karras","raw_affiliation_strings":["Department of Informatics, Hellenic Open University, Patras, Greece"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Informatics, Hellenic Open University, Patras, Greece","institution_ids":["https://openalex.org/I231025917"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5068105234","display_name":"B.G. Mertzios","orcid":null},"institutions":[{"id":"https://openalex.org/I77990126","display_name":"Alexander Technological Educational Institute of Thessaloniki","ror":"https://ror.org/04h36ea57","country_code":"GR","type":"education","lineage":["https://openalex.org/I77990126"]}],"countries":["GR"],"is_corresponding":false,"raw_author_name":"B.G. Mertzios","raw_affiliation_strings":["Department of Automation, Technological Educational Institute of Thessaloniki, Thessaloniki, Greece"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Department of Automation, Technological Educational Institute of Thessaloniki, Thessaloniki, Greece","institution_ids":["https://openalex.org/I77990126"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":4,"corresponding_author_ids":[],"corresponding_institution_ids":[],"apc_list":null,"apc_paid":null,"fwci":0.9855,"has_fulltext":false,"cited_by_count":15,"citation_normalized_percentile":{"value":0.78088944,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":97},"biblio":{"volume":"58","issue":"7","first_page":"2121","last_page":"2131"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10627","display_name":"Advanced Image and Video Retrieval Techniques","score":0.9908000230789185,"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/T10627","display_name":"Advanced Image and Video Retrieval Techniques","score":0.9908000230789185,"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/T10824","display_name":"Image Retrieval and Classification Techniques","score":0.9853000044822693,"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/T10531","display_name":"Advanced Vision and Imaging","score":0.9851999878883362,"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/zernike-polynomials","display_name":"Zernike polynomials","score":0.978995680809021},{"id":"https://openalex.org/keywords/computation","display_name":"Computation","score":0.5996101498603821},{"id":"https://openalex.org/keywords/moment","display_name":"Moment (physics)","score":0.5676225423812866},{"id":"https://openalex.org/keywords/factorial","display_name":"Factorial","score":0.5593880414962769},{"id":"https://openalex.org/keywords/factorial-experiment","display_name":"Factorial experiment","score":0.44583889842033386},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.43099120259284973},{"id":"https://openalex.org/keywords/computer-science","display_name":"Computer science","score":0.3768802285194397},{"id":"https://openalex.org/keywords/algorithm","display_name":"Algorithm","score":0.3419747054576874},{"id":"https://openalex.org/keywords/mathematical-analysis","display_name":"Mathematical analysis","score":0.28681764006614685},{"id":"https://openalex.org/keywords/physics","display_name":"Physics","score":0.23433837294578552},{"id":"https://openalex.org/keywords/optics","display_name":"Optics","score":0.19237971305847168},{"id":"https://openalex.org/keywords/statistics","display_name":"Statistics","score":0.13385143876075745},{"id":"https://openalex.org/keywords/classical-mechanics","display_name":"Classical mechanics","score":0.08591750264167786}],"concepts":[{"id":"https://openalex.org/C92423082","wikidata":"https://www.wikidata.org/wiki/Q132146","display_name":"Zernike polynomials","level":3,"score":0.978995680809021},{"id":"https://openalex.org/C45374587","wikidata":"https://www.wikidata.org/wiki/Q12525525","display_name":"Computation","level":2,"score":0.5996101498603821},{"id":"https://openalex.org/C179254644","wikidata":"https://www.wikidata.org/wiki/Q13222844","display_name":"Moment (physics)","level":2,"score":0.5676225423812866},{"id":"https://openalex.org/C183763347","wikidata":"https://www.wikidata.org/wiki/Q120976","display_name":"Factorial","level":2,"score":0.5593880414962769},{"id":"https://openalex.org/C169222746","wikidata":"https://www.wikidata.org/wiki/Q4116558","display_name":"Factorial experiment","level":2,"score":0.44583889842033386},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.43099120259284973},{"id":"https://openalex.org/C41008148","wikidata":"https://www.wikidata.org/wiki/Q21198","display_name":"Computer science","level":0,"score":0.3768802285194397},{"id":"https://openalex.org/C11413529","wikidata":"https://www.wikidata.org/wiki/Q8366","display_name":"Algorithm","level":1,"score":0.3419747054576874},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.28681764006614685},{"id":"https://openalex.org/C121332964","wikidata":"https://www.wikidata.org/wiki/Q413","display_name":"Physics","level":0,"score":0.23433837294578552},{"id":"https://openalex.org/C120665830","wikidata":"https://www.wikidata.org/wiki/Q14620","display_name":"Optics","level":1,"score":0.19237971305847168},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.13385143876075745},{"id":"https://openalex.org/C74650414","wikidata":"https://www.wikidata.org/wiki/Q11397","display_name":"Classical mechanics","level":1,"score":0.08591750264167786},{"id":"https://openalex.org/C165699331","wikidata":"https://www.wikidata.org/wiki/Q461533","display_name":"Wavefront","level":2,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/tim.2009.2015540","is_oa":false,"landing_page_url":"https://doi.org/10.1109/tim.2009.2015540","pdf_url":null,"source":{"id":"https://openalex.org/S10892749","display_name":"IEEE Transactions on Instrumentation and Measurement","issn_l":"0018-9456","issn":["0018-9456","1557-9662"],"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 Instrumentation and Measurement","raw_type":"journal-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":29,"referenced_works":["https://openalex.org/W22714044","https://openalex.org/W134722258","https://openalex.org/W1509691074","https://openalex.org/W1667237361","https://openalex.org/W1969694079","https://openalex.org/W1981471818","https://openalex.org/W1988223947","https://openalex.org/W2002374829","https://openalex.org/W2011974067","https://openalex.org/W2017851467","https://openalex.org/W2021783512","https://openalex.org/W2032991508","https://openalex.org/W2040708001","https://openalex.org/W2067788074","https://openalex.org/W2076944242","https://openalex.org/W2081944396","https://openalex.org/W2086936614","https://openalex.org/W2095355565","https://openalex.org/W2098355686","https://openalex.org/W2099381890","https://openalex.org/W2102679304","https://openalex.org/W2118376687","https://openalex.org/W2141955292","https://openalex.org/W2146880335","https://openalex.org/W2159498975","https://openalex.org/W2752061190","https://openalex.org/W3146384988","https://openalex.org/W6600941027","https://openalex.org/W6605443999"],"related_works":["https://openalex.org/W1965955745","https://openalex.org/W2017851467","https://openalex.org/W2116187845","https://openalex.org/W1973450972","https://openalex.org/W4365453583","https://openalex.org/W4229629925","https://openalex.org/W1633163079","https://openalex.org/W2792127186","https://openalex.org/W2002931749","https://openalex.org/W4283718311"],"abstract_inverted_index":{"Modified":[0],"direct":[1,23,86,103],"methods":[2,24,87],"for":[3,25,75],"the":[4,39,43,49,52,76,101,128],"computation":[5,31],"of":[6,18,42,51,54,67,112],"Zernike":[7],"and":[8,78,126],"pseudo-Zernike":[9],"moments":[10,28,93],"are":[11,94,107],"presented":[12],"in":[13,22],"this":[14,68],"paper.":[15],"The":[16,64,91],"presence":[17],"many":[19],"factorial":[20,53,77],"terms":[21],"computing":[26],"Zernike-type":[27],"makes":[29],"their":[30,116,121],"process":[32],"a":[33,55,108],"very":[34],"time":[35],"consuming":[36],"task.":[37],"Although":[38],"computational":[40],"power":[41],"modern":[44],"computers":[45],"is":[46,58,70,134],"impressively":[47],"increasing,":[48],"calculation":[50],"big":[56],"number":[57],"still":[59],"an":[60],"inaccurate":[61],"numerical":[62],"procedure.":[63],"main":[65],"concept":[66],"paper":[69],"that,":[71],"using":[72,100],"Stirling's":[73],"approximation":[74,111],"applying":[79],"some":[80],"suitable":[81],"mathematical":[82],"properties,":[83],"novel":[84],"factorial-free":[85],"can":[88],"be":[89],"developed.":[90],"resulting":[92],"not":[95,119],"equal":[96],"to":[97,123],"those":[98],"computed":[99],"original":[102],"methods,":[104],"but":[105],"they":[106,131],"sufficiently":[109],"accurate":[110],"them.":[113],"In":[114],"addition,":[115],"variability":[117],"does":[118],"affect":[120],"ability":[122],"uniquely":[124],"describe":[125],"distinguish":[127],"objects":[129],"that":[130],"represent.":[132],"This":[133],"verified":[135],"by":[136],"appropriate":[137],"pattern":[138],"recognition":[139],"experiments.":[140]},"counts_by_year":[{"year":2024,"cited_by_count":2},{"year":2022,"cited_by_count":2},{"year":2021,"cited_by_count":2},{"year":2020,"cited_by_count":1},{"year":2018,"cited_by_count":1},{"year":2016,"cited_by_count":2},{"year":2014,"cited_by_count":1},{"year":2013,"cited_by_count":1},{"year":2012,"cited_by_count":3}],"updated_date":"2026-06-11T09:08:48.828518","created_date":"2025-10-10T00:00:00"}