{"id":"https://openalex.org/W2097232929","doi":"https://doi.org/10.1109/isvd.2007.39","title":"Two-dimensional line space Voronoi Diagram","display_name":"Two-dimensional line space Voronoi Diagram","publication_year":2007,"publication_date":"2007-07-01","ids":{"openalex":"https://openalex.org/W2097232929","doi":"https://doi.org/10.1109/isvd.2007.39","mag":"2097232929"},"language":"en","primary_location":{"id":"doi:10.1109/isvd.2007.39","is_oa":false,"landing_page_url":"https://doi.org/10.1109/isvd.2007.39","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"4th International Symposium on Voronoi Diagrams in Science and Engineering (ISVD 2007)","raw_type":"proceedings-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/A5111619493","display_name":"St\u00e9phane Rivi\u00e8re","orcid":null},"institutions":[{"id":"https://openalex.org/I2800745255","display_name":"Universit\u00e9 de Haute-Alsace","ror":"https://ror.org/04k8k6n84","country_code":"FR","type":"education","lineage":["https://openalex.org/I2800745255"]}],"countries":["FR"],"is_corresponding":true,"raw_author_name":"Stephane Riviere","raw_affiliation_strings":["\u00c9quipe Mage-LMIA-Universit\u00e9 de Haute-Alsace, Mulhouse, France","Univ. de Haute-Alsace, Mulhouse"],"affiliations":[{"raw_affiliation_string":"\u00c9quipe Mage-LMIA-Universit\u00e9 de Haute-Alsace, Mulhouse, France","institution_ids":["https://openalex.org/I2800745255"]},{"raw_affiliation_string":"Univ. de Haute-Alsace, Mulhouse","institution_ids":["https://openalex.org/I2800745255"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5033062463","display_name":"Dominique Schmitt","orcid":"https://orcid.org/0000-0002-1606-8013"},"institutions":[{"id":"https://openalex.org/I2800745255","display_name":"Universit\u00e9 de Haute-Alsace","ror":"https://ror.org/04k8k6n84","country_code":"FR","type":"education","lineage":["https://openalex.org/I2800745255"]}],"countries":["FR"],"is_corresponding":false,"raw_author_name":"Dominique Schmitt","raw_affiliation_strings":["\u00c9quipe Mage-LMIA-Universit\u00e9 de Haute-Alsace, Mulhouse, France","Univ. de Haute-Alsace, Mulhouse"],"affiliations":[{"raw_affiliation_string":"\u00c9quipe Mage-LMIA-Universit\u00e9 de Haute-Alsace, Mulhouse, France","institution_ids":["https://openalex.org/I2800745255"]},{"raw_affiliation_string":"Univ. de Haute-Alsace, Mulhouse","institution_ids":["https://openalex.org/I2800745255"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":2,"corresponding_author_ids":["https://openalex.org/A5111619493"],"corresponding_institution_ids":["https://openalex.org/I2800745255"],"apc_list":null,"apc_paid":null,"fwci":0.0,"has_fulltext":false,"cited_by_count":10,"citation_normalized_percentile":{"value":0.12653752,"is_in_top_1_percent":false,"is_in_top_10_percent":false},"cited_by_percentile_year":{"min":89,"max":96},"biblio":{"volume":"25","issue":null,"first_page":"168","last_page":"175"},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/1704","display_name":"Computer Graphics and Computer-Aided Design"},"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/T10996","display_name":"Computational Geometry and Mesh Generation","score":0.9998999834060669,"subfield":{"id":"https://openalex.org/subfields/1704","display_name":"Computer Graphics and Computer-Aided Design"},"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/T11164","display_name":"Remote Sensing and LiDAR Applications","score":0.9994999766349792,"subfield":{"id":"https://openalex.org/subfields/2305","display_name":"Environmental Engineering"},"field":{"id":"https://openalex.org/fields/23","display_name":"Environmental Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12923","display_name":"Digital Image Processing Techniques","score":0.9958000183105469,"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/voronoi-diagram","display_name":"Voronoi diagram","score":0.9721066951751709},{"id":"https://openalex.org/keywords/weighted-voronoi-diagram","display_name":"Weighted Voronoi diagram","score":0.7407419681549072},{"id":"https://openalex.org/keywords/power-diagram","display_name":"Power diagram","score":0.7123457193374634},{"id":"https://openalex.org/keywords/centroidal-voronoi-tessellation","display_name":"Centroidal Voronoi tessellation","score":0.5963417887687683},{"id":"https://openalex.org/keywords/diagram","display_name":"Diagram","score":0.5483940839767456},{"id":"https://openalex.org/keywords/line","display_name":"Line (geometry)","score":0.5390498042106628},{"id":"https://openalex.org/keywords/plane","display_name":"Plane (geometry)","score":0.5292669534683228},{"id":"https://openalex.org/keywords/partition","display_name":"Partition (number theory)","score":0.5182576179504395},{"id":"https://openalex.org/keywords/line-segment","display_name":"Line segment","score":0.5114089846611023},{"id":"https://openalex.org/keywords/combinatorics","display_name":"Combinatorics","score":0.5018739700317383},{"id":"https://openalex.org/keywords/mathematics","display_name":"Mathematics","score":0.492872953414917},{"id":"https://openalex.org/keywords/dual-graph","display_name":"Dual graph","score":0.4533596932888031},{"id":"https://openalex.org/keywords/euclidean-space","display_name":"Euclidean space","score":0.43311139941215515},{"id":"https://openalex.org/keywords/euclidean-geometry","display_name":"Euclidean geometry","score":0.42959436774253845},{"id":"https://openalex.org/keywords/geometry","display_name":"Geometry","score":0.35238873958587646},{"id":"https://openalex.org/keywords/graph","display_name":"Graph","score":0.28290224075317383},{"id":"https://openalex.org/keywords/planar-graph","display_name":"Planar graph","score":0.20512992143630981}],"concepts":[{"id":"https://openalex.org/C24881265","wikidata":"https://www.wikidata.org/wiki/Q757267","display_name":"Voronoi diagram","level":2,"score":0.9721066951751709},{"id":"https://openalex.org/C20468658","wikidata":"https://www.wikidata.org/wiki/Q17130624","display_name":"Weighted Voronoi diagram","level":3,"score":0.7407419681549072},{"id":"https://openalex.org/C20573064","wikidata":"https://www.wikidata.org/wiki/Q13410436","display_name":"Power diagram","level":3,"score":0.7123457193374634},{"id":"https://openalex.org/C205672865","wikidata":"https://www.wikidata.org/wiki/Q5062961","display_name":"Centroidal Voronoi tessellation","level":3,"score":0.5963417887687683},{"id":"https://openalex.org/C186399060","wikidata":"https://www.wikidata.org/wiki/Q959962","display_name":"Diagram","level":2,"score":0.5483940839767456},{"id":"https://openalex.org/C198352243","wikidata":"https://www.wikidata.org/wiki/Q37105","display_name":"Line (geometry)","level":2,"score":0.5390498042106628},{"id":"https://openalex.org/C17825722","wikidata":"https://www.wikidata.org/wiki/Q17285","display_name":"Plane (geometry)","level":2,"score":0.5292669534683228},{"id":"https://openalex.org/C42812","wikidata":"https://www.wikidata.org/wiki/Q1082910","display_name":"Partition (number theory)","level":2,"score":0.5182576179504395},{"id":"https://openalex.org/C182124507","wikidata":"https://www.wikidata.org/wiki/Q166154","display_name":"Line segment","level":2,"score":0.5114089846611023},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.5018739700317383},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.492872953414917},{"id":"https://openalex.org/C169827030","wikidata":"https://www.wikidata.org/wiki/Q2294516","display_name":"Dual graph","level":4,"score":0.4533596932888031},{"id":"https://openalex.org/C186450821","wikidata":"https://www.wikidata.org/wiki/Q17295","display_name":"Euclidean space","level":2,"score":0.43311139941215515},{"id":"https://openalex.org/C129782007","wikidata":"https://www.wikidata.org/wiki/Q162886","display_name":"Euclidean geometry","level":2,"score":0.42959436774253845},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.35238873958587646},{"id":"https://openalex.org/C132525143","wikidata":"https://www.wikidata.org/wiki/Q141488","display_name":"Graph","level":2,"score":0.28290224075317383},{"id":"https://openalex.org/C101837359","wikidata":"https://www.wikidata.org/wiki/Q547823","display_name":"Planar graph","level":3,"score":0.20512992143630981},{"id":"https://openalex.org/C105795698","wikidata":"https://www.wikidata.org/wiki/Q12483","display_name":"Statistics","level":1,"score":0.0}],"mesh":[],"locations_count":1,"locations":[{"id":"doi:10.1109/isvd.2007.39","is_oa":false,"landing_page_url":"https://doi.org/10.1109/isvd.2007.39","pdf_url":null,"source":null,"license":null,"license_id":null,"version":"publishedVersion","is_accepted":true,"is_published":true,"raw_source_name":"4th International Symposium on Voronoi Diagrams in Science and Engineering (ISVD 2007)","raw_type":"proceedings-article"}],"best_oa_location":null,"sustainable_development_goals":[],"awards":[],"funders":[],"has_content":{"pdf":false,"grobid_xml":false},"content_urls":null,"referenced_works_count":9,"referenced_works":["https://openalex.org/W1967005434","https://openalex.org/W1984529618","https://openalex.org/W2038730428","https://openalex.org/W2088444809","https://openalex.org/W2113438075","https://openalex.org/W2120934134","https://openalex.org/W4213074663","https://openalex.org/W4239339957","https://openalex.org/W4247939603"],"related_works":["https://openalex.org/W2015548030","https://openalex.org/W2739428663","https://openalex.org/W2383087774","https://openalex.org/W1992160534","https://openalex.org/W2116864555","https://openalex.org/W2234055990","https://openalex.org/W2397039560","https://openalex.org/W2108536469","https://openalex.org/W2120293571","https://openalex.org/W2123081526"],"abstract_inverted_index":{"Given":[0],"a":[1,11,64,127,176,191,208,233],"set":[2,92],"of":[3,13,18,27,67,90,93,100,148,186,197,204,207,229,232],"points":[4,19,54,81,110,131],"called":[5],"sites,":[6],"the":[7,14,21,28,42,68,72,88,91,96,103,112,146,183,187,202,205,227,230,236],"Voronoi":[8,29,35,46,69,152,239],"diagram":[9,30,70,119,153,181,240],"is":[10,157,215,241],"partition":[12,89],"plane":[15,97],"into":[16,98],"sets":[17,99],"having":[20,102],"same":[22,104],"closest":[23,105],"site.":[24],"Several":[25],"generalizations":[26],"have":[31],"been":[32],"studied,":[33],"mainly":[34],"diagrams":[36,47],"for":[37,48,57,154,169],"different":[38],"distances":[39],"(other":[40],"than":[41],"Euclidean":[43],"one),":[44],"and":[45,120,132,139,164,182,189],"sites":[49,108,156,188],"that":[50,84,145,201,226],"are":[51,109],"not":[52],"necessarily":[53],"(line":[55],"segments":[56],"example).":[58],"In":[59],"this":[60,118,136,149,180,222],"paper":[61],"we":[62,76,86,125,199],"present":[63],"new":[65,192],"generalization":[66],"in":[71,74,95,111,158,217,235,242],"plane,":[73],"which":[75],"shift":[77],"our":[78],"interest":[79],"from":[80],"to":[82,134,224],"lines,":[83],"is,":[85],"compute":[87],"lines":[94,101,133],"site":[106],"(where":[107],"plane).":[113],"We":[114,143,173,219],"first":[115,122],"define":[116],"formally":[117],"give":[121,140,165,190],"properties.":[123,142],"Then":[124],"use":[126],"duality":[128],"relationship":[129,178],"between":[130,179],"visualize":[135],"data":[137],"structure":[138],"more":[141],"show":[144,175,200,225],"size":[147,203,228],"line":[150,209,234,237],"space":[151,238],"n":[155],"Theta(n":[159],"<sup":[160],"xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"":[161],"xmlns:xlink=\"http://www.w3.org/1999/xlink\">2</sup>":[162],")":[163],"an":[166,195],"optimal":[167],"algorithm":[168],"its":[170,212],"explicit":[171],"computation.":[172],"then":[174],"remarkable":[177],"dual":[184],"arrangement":[185,196],"result":[193,223],"on":[194],"lines:":[198],"zone":[206,231],"augmented":[210],"with":[211],"incident":[213],"faces":[214],"still":[216],"O(n).":[218,243],"finally":[220],"apply":[221]},"counts_by_year":[{"year":2024,"cited_by_count":1},{"year":2021,"cited_by_count":2},{"year":2017,"cited_by_count":1},{"year":2016,"cited_by_count":1},{"year":2015,"cited_by_count":2},{"year":2013,"cited_by_count":2},{"year":2012,"cited_by_count":1}],"updated_date":"2025-11-06T03:46:38.306776","created_date":"2025-10-10T00:00:00"}