An approach to coded structured light to obtain three dimensional information

Texto completo

(1)AN APPROACH TO CODED STRUCTURED LIGHT TO OBTAIN THREE DIMENSIONAL INFORMATION. Joaquim SALVI MAS. ISBN: 84-8458-107-1 Dipòsit legal: GI-1228-2001.

(2) PhD. Thesis UNIVERSITY OF GIRONA Department of Electronics, Informatics and Automation to obtain the degree of:. European PhD in Industrial Engineering by:. Joaquim Salvi Title of the thesis:. An Approach to Coded Structured Light to Obtain Three Dimensional Information Submitted and presented in Girona, december 1997 and february 1998 respectively:. Supervisor: Dr. Joan Batlle (Universitat de Girona – Spain) Supervisor: Dr. El Mustapha Mouaddib (Université de Picardie Jules Verne – France) Referee: Referee:. Professor Bernard Dubuisson (Université de Compiègne – France) Dr. Geoff Roberts (University of Wales College Newport – Great Britain). President: Examiner: Examiner: Examiner: Examiner:. Professor Josep Amat (Universitat Politècnica de Catalunya – Spain) Professor Juan Jose Villanueva (Universitat Autònoma de Barcelona – Spain) Professor Antonio Perez (Universidad Politécnica de Madrid – Spain) Professor Emilio Zapata (Universidad de Málaga – Spain) Dr. Geoff Roberts (University of Wales College Newport – Great Britain).

(3) 7. Judwlwxghv Lq wkh sdvw ihz |hduv L kdyh ehhq oxfn| wr zrun zlwk d orw ri shrsoh zklfk kdyh khoshg ph/ lq rqh ru dqrwkhu zd|/ zlwk wkh kdug dqg vwlpxodwlqj zrun ri uhvhdufklqj1 Wklv vkhhw ri sdshu wulhv wr wkdqn wkhp1 Vxuho|/ lw lv udwkhu gl!fxow wr h{sodlq wkh judwlwxgh L ihho vd|lqj mxvw zrugv/ exw L zloo wu|111 Iluvwo|/ L kdyh wr wkdqn Mrdq Edwooh/ wkh gluhfwru ri wklv wkhvlv dqg dovr rqh ri p| ehvw iulhqgv1 Wkh khos kh jdyh ph lq d orw ri glhuhqw zd|v= olnh zulwlqj grzq wkdw duwlfoh zlwk d ghdgolqh lq yh gd|v/ olnh vhdufklqj vwd|v lq iruhljq xqlyhuvlwlhv/ dqg wkh vhyhudo krxuv kh vshqw wr frqylqfh ph ri vrph gdun dvshfwv ri wklv zrun/ kdv vxuho| ohg ph wr suhvhqw wrgd| wklv wkhvlv1 L zrxog olnh wr wkdqn wkh juhdw khos jlyhq e| p| iulhqg dqg fr0gluhfwru ri wklv wkhvlv Ho Pxvwdskd Prxdggle/ zkr zhofrphg ph lq klv uhvhdufk jurxs Htxlsh Shufhswlrq hq Urerwltxh dw wkh Xqlyhuvlw| ri Slfduglh lq Dplhqv +Iudqfh,1 Wkhuh/ L ohduqw doo wkrvh dvshfwv ri fdoleudwlrq gxulqj wzr wkuhh0 prqwk vwd|v1 L uhphpehu wkh ehhuv zh kdg dw wkh Elvvds Fdih glvfxvvlqj wkrvh frpsolfdwhg htxdwlrqv/ dqg klv sdwlhqfh zkhq L prqrsrolvhg klv frp0 sxwhu gd| diwhu gd|1 L dovr zdqw wr phqwlrq klv zlih Ydohulh dqg gdxjkwhu Qlvvd1 L zdqw wr wkdqn doo wkh iulhqgv ri wkh uhvhdufk jurxs ri Frpsxwhu Ylvlrq dqg Urerwlfv dw wkh Xqlyhuvlw| ri Jlurqd/ iru fuhdwlqj wkdw dwprvskhuh zklfk doorzv dq|rqh wr zrun zlwkrxw lqwhuuxswlrqv1 Iru doo ri wkhp/ dqg iru wkhlu iulhqgo| uhodwlrqvkls/ wkdqnv wr= [hyl Fxið/ Mrdq Pduwð/ Udid Jdufld/ Shs Iruhvw/ Mrugl Iuhl{hqhw/ Shuh Ulgdr/ Ooxlv Sdfkhfr/ Wrql Ihuqdqgh} dqg Shs Fruwdgd1 L dovr zdqw wr phqwlrq Mrugl Uhjlqföv dqg Hqultxh Wruuhv iru wkhlu iulhqgo| uhodwlrqvkls1 L zrxog olnh wr wkdqn Mrvhs Wrpdv/ Vdoydgru Vdodqryd dqg [dyl Pdq|hu iru wkhlu whfkqlfdo vxssruw/ dqg Rojd Pruhqr iru khu dgplqlvwudwlyh dvvlv0 wdqfh1 D yhu| vshfldo judwlwxgh wr doo p| iulhqgv ri wkh Htxlsh Shufhswlrq hq Urerwltxh lq Dplhqv zkr zhofrphg ph lq wkhlu ode1 Wkdqnv iru doo wkh khos L uhfhlyhg/ dqg iru wkhlu sdwlhqwh wu|lqj wr xqghuvwdqg p| dzixo iuhqfk1 L zloo.

(4) 8 qhyhu irujhw wkh prphqwv zh vkduhg1 Wkdqnv wr= Odxuhqw Ghodkrfkh/ Sdvfdo Ydvvhxu/ Euxqr Pduklf/ Fodxgh Shjdug/ Hulf Eudvvduw/ dqg lq jhqhudo hyhu| rqh ri wkh Htxlsh Shufhswlrq hq Urerwltxh1 Hvshfldoo|/ L zdqw wr wkdqn Sdwulfld H|vnhqv iru khu iulhqgo| khos zlwk p| hqjolvk/ uhylvlqj wkh sxeolvkhg duwlfohv dqg hyhq wklv wkhvlv1 L zrxog olnh wr wkdqn Shs Yhkð dqg Mrdq Pduwð iru wkh wlph zh vshqw wdonlqj derxw krz wr vroyh vrph ri wkh sureohpv lqyroyhg zlwk ODWH[ dqg Vflhqwlf ZrunSodfh1 L zdqw wr wkdqn wkh shrsoh ri wkh Frpsxwhu Ylvlrq Fhqwuh ri wkh Xqlyhu0 vlwdw Dxwrqrpd gh Edufhorqd/ zkr doorzv ph wr xvh wkhlu oleudu|1 L dp dovr judwhixo wr doo wkh shrsoh zkr jdyh ph wkhlu dgylfhv dqg uhf0 rpphqgdwlrqv derxw wkh frqwhqwv ri wklv wkhvlv1 Hvshfldoo|/ L zdqw wr wkdqn Dolfld Fdvdov ri wkh Xqlyhuvlwdw Srolwëfqlfd gh Fdwdoxq|d iru khu zruwk| uhf0 rpphqgdwlrqv1 L zrxog dovr olnh wr h{suhvv p| judwlwxgh wr wkh uhylhzhuv ri wklv wkhvlv= Jhr Urehuwv ri wkh Xqlyhuvlw| ri Zdohv Froohjh dqg Ehuqdug Gxexlvvrq ri wkh Xqlyhuvlwì gh Whfqrorjlh gh Frpslëjqh1 L kdyh wr wkdqn doo wkh shrsoh ri wkh Hohfwurqlfv/ Lqirupdwlfv dqg Dx0 wrpdwlrq Ghsduwphqw dqg wkh Lqvwlwxwh ri Lqirupdwlfv dqg Dssolfdwlrqv ri wkh Xqlyhuvlw| ri Jlurqd dv zhoo/ dqg doo wkh shrsoh ri wkh Oderudwrluh ghv V|vwëphv Dxwrpdwltxhv ri wkh Xqlyhuvlw| ri Slfduglh1 Dw odvw/ exw iru ph lv wkh prvw lpsruwdqw/ L zdqw wr wkdqn wkh vxssruw uhfhlyhg iurp p| zkroh idplo|/ hvshfldoo| iurp p| sduhqwv Mrvhs Vdoyl dqg Slodu Pdv dqg p| eurwkhu Gdylg1 L zrxog dovr olnh wr h{suhvv p| judwlwxgh dqg oryh wr Doh{dqgud iru khu sdwlhqfh zkhq d zhhnhqg diwhu dqrwkhu L kdg wr zrun/ dqg zh frxog qrw jr rxw1 L dovr kdyh wr phqwlrq khu juhdw khos uhylvlqj p| hqjolvk1 L krsh/ L kdyh qrw irujrwwhq dq|erg|1 Li wkhuh lv vrphrqh zkr fdq qrw qg klv2khu qdph lq wklv olvw/ L dsrorjl}h dqg L lqylwh klp2khu wr zulwh lw grzq lq wkh iuhh vsdfh zklfk iroorzv wklv sdudjudsk1. Mrdtxlp Vdoyl.

(5) Frqwhqwv 4 Lqwurgxfwlrq 47 414 Ghvfulelqj wkh lqwhuhvw ri wklv wkhvlv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 47 415 Vfkhpd ri wkh wkhvlv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 49 5 Fdoleudwlrq 514 Frqvlghudwlrqv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 515 Prghoolqj d fdphud 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 51514 Wkh H{wulqvlf Sdudphwhuv 1 1 1 1 1 1 1 1 1 51515 Wkh Lqwulqvlf Sdudphwhuv 1 1 1 1 1 1 1 1 1 1 51516 Vxppdu| ri Fdphud prghoolqj 1 1 1 1 1 1 516 Fdoleudwlqj d fdphud 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 51614 Fdoleudwlqj wkh olqhdu prgho 1 1 1 1 1 1 1 1 51615 Fdoleudwlqj wkh qrq0olqhdu prgho 1 1 1 1 1 517 6G Uhfrqvwuxfwlrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 51714 Uhfrqvwuxfwlqj iurp olqhdu fdoleudwlrq 1 1 51715 Uhfrqvwuxfwlqj iurp qrq0olqhdu fdoleudwlrq. 1 1 1 1 1 1 1 1 1 1 1. 1 1 1 1 1 1 1 1 1 1 1. 1 1 1 1 1 1 1 1 1 1 1. 1 1 1 1 1 1 1 1 1 1 1. 1 1 1 1 1 1 1 1 1 1 1. 1 1 1 1 1 1 1 1 1 1 1. 1 1 1 1 1 1 1 1 1 1 1. 6 Wkh Hslsrodu Jhrphwu| 614 Wkh ixqgdphqwdo pdwul{ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 615 Fkdqjlqj wkh fr0ruglqdwh v|vwhp 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 61514 Zruog fr0ruglqdwh v|vwhp {hg dw wkh vhfrqg fdphud/ rshudwlqj zlwk uhvshfw wr wkh uvw fdphud1 1 1 1 1 1 1 1 61515 Zruog fr0ruglqdwh v|vwhp {hg dw wkh uvw fdphud/ rs0 hudwlqj zlwk uhvshfw wr wkh vhfrqg fdphud1 1 1 1 1 1 1 1 61516 Uhodwlrq ehwzhhq erwk Ixqgdphqwdo Pdwulfhv 1 1 1 1 1 61517 Ihdwxuhv ri Srlqwv dqg Yhfwruv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 616 Ghwhuplqlqj wkh ixqgdphqwdo pdwul{ 1 1 1 1 1 1 1 1 1 1 1 1 1 1 61614 Wkh Hljkw0Srlqwv Phwkrg 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 61615 Frqfoxvlrqv 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 61616 Wkh dojrulwp xvhg wr lqihu wkh ixqgdphqwdo pdwul{ 1 1 61617 Wkh dojrulwkp xvhg wr rewdlq wkh hslsrodu olqhv iurp $ 9. 4; 4; 4; 53 57 64 67 68 74 7; 7< 85 89 89 96 96 9: 9< :3 :4 :5 :9 :< ;5.

(6) FRQWHQWV 7 Vwuxfwxuhg Oljkw 714 Lqwurgxfwlrq 1 1 1 1 1 1 1 1 1 1 1 1 1 715 D Vxuyh|= Frghg Vwuxfwxuhg Oljkw 71514 Srvgdphu0Dowvfkxohu 1 1 1 1 71515 Fduulkloo0Kxppho 1 1 1 1 1 1 71516 Er|hu0Ndn 1 1 1 1 1 1 1 1 1 1 71517 Oh Prljqh0Zd{pdq 1 1 1 1 71518 Prulwd0\dnlpd0Vdndwd 1 1 1 71519 Yx|ovwhnh0Rrvwhuolqfn 1 1 1 1 7151: Wdmlpd0Lzdndzd 1 1 1 1 1 1 1 7151; Zxvw0Fdsvrq 1 1 1 1 1 1 1 1 7151< Jul!q0Qdudvlpkdq0\hh 1 1 1 715143 Pdux|dpd0Deh 1 1 1 1 1 1 1 715144 Lwr0Lvkll 1 1 1 1 1 1 1 1 1 1 1 715145 Fkhq0Kxqj0Fkldqj0Zx 1 1 1 716 Vxppdu| 1 1 1 1 1 1 1 1 1 1 1 1 1 1. :. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1. ;8 ;8 <3 <5 <9 <; 434 436 438 43; 443 445 448 44: 44: 44<. 8 D Qhz Frghg Sdwwhuq Surmhfwlrq 814 Lqwurgxfwlrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 815 Sdwwhuq Ghvljq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 81514 Frorxu Dvvljqphqw 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 816 V|vwhp Fdoleudwlrq 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 81614 Ghvfulelqj wkh fdoleudwlrq dojrulwkp 1 1 1 1 1 1 81615 Fdoleudwlqj wkh fdphud dqg wkh surmhfwru prgho 817 Dffxudf| ri wkh phwkrg 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 818 Phdvxulqj wkuhh0glphqvlrqdo remhfwv 1 1 1 1 1 1 1 1 1 1 81814 H{dpsoh = Phdvxulqj jhrphwulfdo remhfwv 1 1 1 1 81815 H{dpsoh = Phdvxulqj d idfh pdvn 1 1 1 1 1 1 1 1 81816 H{dpsoh = Phdvxulqj wkh sdop ri d kdqg 1 1 1 1 81817 H{dpsoh = Phdvxulqj wkh edfn ri d kdqg 1 1 1 1. 1 1 1 1 1 1 1 1 1 1 1 1. 1 1 1 1 1 1 1 1 1 1 1 1. 1 1 1 1 1 1 1 1 1 1 1 1. 1 1 1 1 1 1 1 1 1 1 1 1. 454 454 455 455 459 45: 463 469 46; 46< 46< 474 474. 9 Frqfoxvlrqv 914 Frqfoxvlrqv 1 1 1 1 1 1 1 1 1 1 1 1 1 915 Ixuwkhu Zrun 1 1 1 1 1 1 1 1 1 1 1 1 916 Sxeolfdwlrqv uhodwhg zlwk wkh wkhvlv 917 Rwkhu Uhvhdufk Frqwulexwlrqv 1 1 1. 1 1 1 1. 1 1 1 1. 1 1 1 1. 1 1 1 1. 479 479 47; 47; 47<. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1. 1 1 1 1. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1. 1 1 1 1. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1. 1 1 1 1. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1. 1 1 1 1. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1. 1 1 1 1. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1. 1 1 1 1. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1. 1 1 1 1. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1. 1 1 1 1. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1. 1 1 1 1. 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1. 1 1 1 1. 1 1 1 1.

(7) Olvw ri Iljxuhv 514 Wkh fdphud fr0ruglqdwh v|vwhp zlwk uhvshfw wr d zruog fr0 ruglqdwh v|vwhp1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 515 Jhrphwulf uhodwlrvkls ehwzhhq wkh fdphud fr0ruglqdwh v|vwhp dqg wkh zruog fr0ruglqdwh v|vwhp1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 516 Uhodwlrqvkls ehwzhhq wkh rulhqwdwlrq yhfwruv o dqg wkh fdp0 hud fr0ruglqdwh iudph1 Wkh jxuh vkrzv rqo| wkh uhodwlrqvkls ehwzhhq wkh of yhfwru dqg wkh fS d{lv1 1 1 1 1 1 1 1 1 1 1 1 1 1 517 Jhrphwulfdo uhodwlrqvkls ehwzhhq d 6G remhfw srlqw dqg lwv 5G lpdjh surmhfwlrq1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 518 Rewdlqlqj wkh lghdo surmhfwlrq iurp wkh shuvshfwlyh uhodwlrq1 1 519 Udgldo dqg wdqjhqwldo glvwruwlrq1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 51: Hhfw ri udgldo glvwruwlrq1 Vrolg olqhv= qr glvwruwlrq> gdvkhg olqhv= udgldo glvwruwlrq +d= qhjdwlyh/ e= srvlwlyh,1 1 1 1 1 1 1 1 1 51; Hhfw ri wdqjhqwldo glvwruwlrq1 Vrolg olqhv= qr glvwruwlrq> gdvkhg olqhv= wdqjhqwldo glvwruwlrq1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 51< Wudqvirupdwlrq iurp phwulf fr0ruglqdwhv ri wkh uhwlqdo sodqh wr wkh sl{ho fr0ruglqdwhv ri wkh lpdjh sodqh1 1 1 1 1 1 1 1 1 1 1 5143 Wudqvirupdwlrq iurp wkh fr0ruglqdwh v|vwhp ri wkh fdphud lp0 djh sodqh wr wkh fr0ruglqdwh v|vwhp ri wkh frpsxwhu lpdjh exhu1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5144 Prghoolqj d fdphud zlwkrxw ohqv glvwruwlrq1 1 1 1 1 1 1 1 1 1 1 5145 Prghoolqj d fdphud zlwk udgldo ohqv glvwruwlrq1 1 1 1 1 1 1 1 1 1 5146 Wkh jhrphwulf uhodwlrq ehwzhhq d 6G remhfw srlqw dqg lwv 5G surmhfwlrq zkhq wkh rswlfdo fhqwuh lv orfdwhg lq iurqw ri wkh lpdjh sodqh1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 5147 Wkh sulqflsoh ri wuldqjxodwlrq xvhg lq rughu wr rewdlq wkh 6G vfhqh lqirupdwlrq1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1. 4< 54 56 57 58 5: 5: 5; 63 64 65 66 68 7<. 614 Wkh hslsodu jhrphwu| ehwzhhq wzr slqkroh prghov1 1 1 1 1 1 1 8: 615 Jhrphwulfdo uhsuhvhqwdwlrqv ri wkh wkuhh olqh yhfwruv ri wkh D 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 8< pdwul{ 1 ;.

(8) OLVW RI ILJXUHV 616 Wkh uhodwlrq ri d 5G lpdjh srlqw 6 lq U dqg lwv hslsrodu olqh ,6 rq U 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 617 Jhrphwulfdo uhodwlrq ri erwk fdphud d{hv dqg d zruog fr0 ruglqdwh v|vwhp1 Zruog fr0ruglqdwh v|vwhp {hg dw wkh uvw fdphud1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 618 Jhrphwulfdo uhodwlrq ri erwk fdphud d{hv dqg d zruog fr0 ruglqdwh v|vwhp1 Zruog fr0ruglqdwh v|vwhp {hg dw wkh vhfrqg fdphud1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 619 Dq h{dpsoh ri wkh hslsrodu jhrphwu| prghoohg e| wkh ixqgd0 phqwdo pdwul{1 Qrwh wkh hslsrohv dqg wkh hslsrodu olqhv irxqg1 Wkh duhd gholplwhg e| wkh fruuhvsrqghqfh srlqwv/ kdyh ehhq pdunhg e| d vtxduh1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 61: D }rrp ri wkh vtxduh duhdv ri wkh suhylrxv lpdjhv1 Qrwh wkdw wkh fruuhvsrqghqfh srlqwv olh dozd|v rq wkh hslsrodu olqhv ghqhg e| wkhlu frxsoh srlqwv1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 . . 714 Wkh hslsrodu jhrphwu| dv d wrro wr uhgxfh wkh fruuhvsrqghqfh sureohp1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 715 Dq h{dpsoh ri d srlqw zlwkrxw fruuhvsrqghqfh gxh wr d vxuidfh rffoxvlrq ri 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 716 Dq h{dpsoh ri fruuhvsrqghqfh srlqwv lpdjhg lq wkh vdph rughu iurp wkh hslsrohv1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 717 Dq h{dpsoh ri fruuhvsrqghqfh srlqwv lpdjhg lq d glhuhqw rughu iurp wkh hslsrohv1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 718 Wkh whpsrudu| frglfdwlrq1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 719 Frglqj d ?{? pdwul{ ri odvhu ehdp grwv1 1 1 1 1 1 1 1 1 1 1 1 71: ; elwv whpsrudoo| elqdu| frghg sdwwhuq surmhfwlrq1 1 1 1 1 1 1 71; ; elwv whpsrudoo| Jud| frghg sdwwhuq surmhfwlrq1 1 1 1 1 1 1 71< Wkh Fxelvfrsh v|vwhp1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 7143 Wkh sdwwhuq sursrvhg e| Fduulkloo hw do1 1 1 1 1 1 1 1 1 1 1 1 1 7144 Wkh xvh ri wkh shulrg ohqjwk wr lpsuryh wkh uhvroxwlrq ri wkh phdvxuhphqw zlwkrxw lqfuhdvlqj wkh qxpehu ri elwv ghswk ri wkh fdswxuhg lpdjh1 Reylrxvo|/ shulrg sdwwhuqv duh olplwhg wr phdvxuh rqo| vxuidfhv zlwk d ghswk glvfrqwlqxlw| vpdoohu wkdq lwv shulrg ohqjwk1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 7145 Dq h{dpsoh ri wkh fruuhvsrqghqfh sureohp wkdw kdv wr eh vroyhg ehwzhhq surmhfwhg dqg lpdjhg volwv1 1 1 1 1 1 1 1 1 1 1 7146 D ghwdlo ri wkh sdwwhuq sursrvhg e| Er|hu1 1 1 1 1 1 1 1 1 1 1 7147 D srvvleoh uhodwlrq ehwzhhq d surmhfwhg sdwwhuq dqg wkh fds0 wxuhg rqh1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1. <. 1 94 1 97 1 9;. 1 :; 1 :; 1 ;: 1 ;: 1 ;; 1 1 1 1 1 1 1. ;; <6 <7 <8 <8 <9 <:. 1 <; 1 << 1 433 1 434.

(9) OLVW RI ILJXUHV. 43. 7148 Vrph qrw frghg julg sdwwhuqv1 Wkh sdwwhuq uhvroxwlrq lv jlyhq e| wkh qxpehu ri furvvlqj srlqwv dqg wkh olqh wklfnqhvv1 1 1 1 1 7149 Julg sdwwhuqv sduwldoo| frghg e| wkh srvlwlrq ri vrph grwv/ zklfk duh xvhg dv odqgpdunv wr lqlwldwh wkh odehoolqj surfhvv1 1 714: Wkh v|vwhp frqjxudwlrq sursrvhg e| Prulwd hw do1 zlwk wkh grw glvrughuv wkdw fdq eh rewdlqhg1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 714; Dq h{dpsoh ri wkh wzr sdwwhuqv surmhfwhg rq wkh phdvxulqj vfhqh e| Prulwd1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 714< D ghwdlo ri wkh sdwwhuq sursrvhg e| Yx|ovwhnh hw do1 1 1 1 1 1 1 7153 Wkh sdwwhuq lv pdgh e| irxu glhuhqw frpelqdwlrqv1 Wkh wrs ohiw vtxduh lv uhsuhvhqwhg e| wkh v|perov 0 dqg ./ zklfk uhsuh0 vhqw d euljkw ru gdun vtxduh uhvshfwlyho|1 Wkh elqdu| qxpehuv 3 dqg 4 duh xvhg wr frgh wkh grw dw wkh vtxduh yhuwh{/ zklfk fdq dovr eh gdun ru euljkw uhvshfwlyho|1 1 1 1 1 1 1 1 1 1 1 1 1 1 7154 Wkh frgh dvvljqphqw ghwhuplqhv wkdw dq| 5{6 zlqgrz fdq lghqwli| wkh froxpq lqgh{ Hdfk zlqgrz lv pdgh e| d wulsohw ri S ydoxhv dqg dqrwkhu wulsohw ri K 1 Lq wkh h{dpsoh/ iru wkh wzr zlqgrzv/ wkh froxpq lv jlyhq e| wkh vdph vhtxhqfh rughu EES c Se c SD c EK c Ke c KD 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 7155 Wkh sulqflsoh ri gludfwlqj zklwh oljkw lq rughu wr rewdlq d udlqerz sdwwhuq1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 7156 Wkh udlqerz udqjh sdwwhuq surmhfwhg e| Wdmlpd hw do1 pdgh e| d vhw ri yhuwlfdo volwv zklfk xvh doo wkh frorxu vshfwuxp iurp eoxh xs wr uhg1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 7157 Wkh v|vwhp sursrvhg e| Jhqj1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 7158 Wkh sulqflsoh ri ryhuodslqj wkuhh vlqxvrlgdo lqwhqvlw| sdwwhuqv1 7159 Wkh shulrglfdo sdwwhuq sursrvhg e| Zxvw dqg Fdsvrq/ pdgh e| wkh vlqxvrlgdo ryhuodsslqj ri wkh wkuhh sulpdu| frorxu frp0 srqhqwv1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 715: Grw frglfdwlrq h{dpsoh xvlqj lwv irxu qhljkerxuv dqg d edvlv htxdo wr 6/ l1h1 rqo| wkuhh glhuhqw v|perov fdq eh xvhg1 1 1 1 715; D srvvleoh frghg grw pdwul{/ rewdlqhg e| wkh phwkrg sursrvhg e| Jul!q hw do1 D edvlv htxdo wr 6 kdv ehhq vxssrvhg dqg wr hdfk v|pero d frorxuhg grw kdv ehhq dvvrfldwhg1 1 1 1 1 1 1 1 1 715< D srvvleoh jhrphwulf dvvrfldwlrq ri d pdwul{ grw frghg xvlqj d edvlv htxdo wr 81 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 7163 Wkh sdwwhuq sursrvhg e| Pdux|dpd hw do1 pdgh e| wkh sur0 mhfwlrq ri pxowlsoh volwv zlwk udqgrp fxwv1 1 1 1 1 1 1 1 1 1 1 1 7164 D pxowlsoh pdwfklqj h{dpsoh1 Lq wklv fdvh wkh hqg srlqwv ri wzr vkruw olqhv/ r dqg r ri wkh sdwwhuq/ pdwfk wkh hqg srlqwv ri d olqh surmhfwhg rq wkh lpdjh sodqh/ dorqj lwv hslsrodu olqhv1 . 435 435 437 438 439. 439. 43: 43; 43< 443 444 444 445 447 447 448. . 449.

(10) OLVW RI ILJXUHV. 44. 7165 Vsdwldo pdlq frghv vshfli|lqj qrghv f / / 2 / dqg e duh +5434,/+4345,/+4353,/+4345, dqg +3545,1 Wkh vxefrgh ri f lv +4345/4353/4345/3545,1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 44; 814 Vfkhpd ri wkh 6G phdvxulqj v|vwhp edvhg rq frghg julg sdw0 whuq surmhfwlrq1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 815 Wkh judsk rewdlqhg iurp R ' 21 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 816 Wkh frorxuhg sdwwhuq1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 817 Wkh vfhqdulr ri wkh 6G phdvxulqj v|vwhp1 1 1 1 1 1 1 1 1 1 1 818 Vfkhpd ri wkh fdoleudwlqj sdwwhuq dqg lwv srvlwlrq dqg rulhq0 wdwlrq zlwk uhvshfw wr wkh zruog fr0ruglqdwh v|vwhp1 1 1 1 1 1 819 Vfkhpd ri wkh sdwwhuq xvhg lq rughu wr fdoleudwh wkh fdphud1 81: Wkh fdoleudwlqj sdwwhuq ri vtxduhv1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 81; Wkh lpdjh ri wkh fdoleudwlqj sdwwhuq judeehg e| wkh fdphud1 81< Srvlwlrq ri wkh 5G lpdjh srlqwv ri wkh fdphud1 1 1 1 1 1 1 1 1 8143 Srvlwlrq ri wkh 6G remhfw srlqwv ri wkh fdphud1 1 1 1 1 1 1 1 1 8144 Wkh phwulf julg xvhg lq surmhfwru fdoleudwlrq1 1 1 1 1 1 1 1 1 8145 Srvlwlrq ri wkh 5G furvvlqj srlqwv ri wkh surmhfwhg sdwwhuq1 1 8146 Srvlwlrq ri wkh 6G furvvlqj srlqwv ri wkh surmhfwhg sdwwhuq1 1 8147 D vfhqh frpsrvhg e| wkuhh jhrphwulf remhfwv1 d, Wkh uhdo vfhqh +qrw xvhg wr lqihu 6G lqirupdwlrq,> e, Wkh vfhqh looxpl0 qdwhg e| wkh frorxuhg sdwwhuq1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 8148 Wkh pdwfklqj rewdlqhg1 d, Wkh Pdwfklqj srlqwv iurp wkh 8:9 { :9; fdphud lpdjh sodqh/ e, Wkh Pdwfklqj srlqwv iurp wkh 845 { 845 surmhfwhg sdwwhuq iudph1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 8149 Wkh uhfrqvwuxfwlrq ri wkh 6G remhfwv1 d, D shuvshfwlyh ylhz iurp wkh uljkw vlgh1 e, D wrs ylhz1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 814: D vfhqh frpsrvhg e| d idfh pdvn1 d, Wkh uhdo vfhqh +qrw xvhg wr lqihu 6G lqirupdwlrq,> e, Wkh vfhqh looxplqdwhg e| wkh frorxuhg sdwwhuq1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 814; Wkh pdwfklqj rewdlqhg1 d, Wkh Pdwfklqj srlqwv iurp wkh 8:9 { :9; fdphud lpdjh sodqh/ e, Wkh Pdwfklqj srlqwv iurp wkh 845 { 845 surmhfwhg sdwwhuq iudph1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 814< Wkh uhfrqvwuxfwlrq ri wkh idfh pdvn1 1 1 1 1 1 1 1 1 1 1 1 1 1 8153 Dq looxplqdwhg kxpdq sdop1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 8154 Wkh pdwfklqj rewdlqhg1 d, Wkh pdwfklqj srlqwv iurp wkh 8:9{:9; fdphud lpdjh sodqh1 e, Wkh pdwfklqj srlqwv iurp wkh 589{589 surmhfwru sdwwhuq iudph1 1 1 1 1 1 1 1 1 1 1 1 1 1 8155 Uhfrqvwuxfwlrq ri wkh kxpdq sdop1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 8156 Dq looxplqdwhg edfn kdqg1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1. 1 1 1 1. 456 457 459 45:. 1 1 1 1 1 1 1 1 1. 464 464 465 466 466 467 468 469 46:. 1 46< 1 473 1 473 1 474 1 475 1 475 1 476 1 476 1 477 1 477.

(11) OLVW RI ILJXUHV. 45. 8157 Wkh pdwfklqj rewdlqhg1 d, Wkh pdwfklqj srlqwv iurp wkh 8:9{:9; fdphud lpdjh sodqh1 e, Wkh pdwfklqj srlqwv iurp wkh 589{589 surmhfwru sdwwhuq iudph1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 478 8158 Uhfrqvwuxfwlrq ri wkh kxpdq edfn kdqg1 1 1 1 1 1 1 1 1 1 1 1 1 478.

(12) Olvw ri Wdeohv 814 Lqwulqvlf dqg h{wulqvlf sdudphwhuv ri wkh fdphud vhqvru1 1 1 1 1 468 815 Lqwulqvlf dqg h{wulqvlf sdudphwhuv ri wkh sdwwhuq surmhfwru1 1 1 46: 816 D vdpsoh ri wkh dqdo|vhg remhfw srlqwv1 1 1 1 1 1 1 1 1 1 1 1 1 1 46;. 46.

(13) Fkdswhu 4 Lqwurgxfwlrq 414. Ghvfulelqj wkh lqwhuhvw ri wklv wkhvlv. Wkh kxpdq ylvxdo delolw| wr shufhlyh ghswk orrnv olnh d sx}}oh1 Zh shufhlyh wkuhh0glphqvlrqdo vsdwldo lqirupdwlrq txlfno| dqg h!flhqwo| e| xvlqj wkh elqrfxodu vwhuhrsvlv ri rxu h|hv dqg/ zkdw lv pruh lpsruwdqw/ wkh ohduqlqj ri wkh prvw frpprq remhfwv zklfk zh dfklhyhg wkurxjk olylqj1 Qrzdgd|v/ prghoolqj wkh ehkdylrxu ri rxu eudlq lv d fwlrq/ wkdw lv zk| wkh kxjh sureohp ri 6G shufhswlrq dqg ixuwkhu lqwhusuhwdwlrq lv vsolw lqwr d vhtxhqfh ri hdvlhu sureohpv1 D orw ri uhvhdufk lv lqyroyhg lq urerw ylvlrq lq rughu wr rewdlq 6G lqirupdwlrq ri wkh vxuurxqghg vfhqh1 Prvw ri wklv uhvhdufk lv edvhg rq prghoolqj wkh vwhuhrsvlv ri kxpdqv e| xvlqj wzr fdphudv dv li wkh| zhuh wzr h|hv1 Wklv phwkrg lv nqrzq dv vwhuhr ylvlrq dqg kdv ehhq zlgho| vwxglhg lq wkh sdvw dqg lv ehlqj vwxglhg dw suhvhqw/ dqg d orw ri zrun zloo eh vxuho| grqh lq wkh ixwxuh1 Wklv idfw doorzv xv wr d!up wkdw wklv wrslf lv rqh ri wkh prvw lqwhuhvwlqj rqhv lq frpsxwhu ylvlrq1 Wkh vwhuhr ylvlrq sulqflsoh lv edvhg rq rewdlqlqj wkh wkuhh glphqvlrqdo srvlwlrq ri dq remhfw srlqw iurp wkh srvlwlrq ri lwv surmhfwlyh srlqwv lq erwk fdphud lpdjh sodqhv1 Krzhyhu/ ehiruh lqihuulqj 6G lqirupdwlrq/ wkh pdwk0 hpdwlfdo prghov ri erwk fdphudv kdyh wr eh nqrzq1 Wklv vwhs lv nqrzq dv fdphud fdoleudwlrq dqg zloo eh eurdgo| ghvfulehg lq wkh iroorzlqj fkdswhu1 Shukdsv wkh prvw lpsruwdqw sureohp lq vwhuhr ylvlrq lv wkh ghwhuplqdwlrq ri wkh sdlu ri krprorjxh srlqwv lq wkh wzr lpdjhv/ nqrzq dv wkh fruuhvsrqghqfh sureohp/ dqg lw lv dovr rqh ri wkh prvw gl!fxow sureohpv wr eh vroyhg zklfk lv fxuuhqwo| lqyhvwljdwhg e| d orw ri uhvhdufkhuv1 Wkh hslsrodu jhrphwu| do0 orzv xv wr uhgxfh wkh fruuhvsrqghqfh sureohp1 Dq dssurdfk wr wkh hslsrodu jhrphwu| lv ghvfulehg lq fkdswhu 61 Qhyhuwkhohvv/ lw grhv qrw vroyh lw dw doo dv d orw ri frqvlghudwlrqv kdyh wr eh wdnhq lqwr dffrxqw1 Dv dq h{hpsoh zh 47.

(14) FKDSWHU 41 LQWURGXFWLRQ. 48. kdyh wr frqvlghu srlqwv zlwkrxw fruuhvsrqghqfh gxh wr d vxuidfh rffoxvlrq ru vlpso| gxh wr d surmhfwlrq rxw ri wkh fdphud vfrsh1 Wkh lqwhuhvw ri wklv wkhvlv lv irfxvhg rq vwuxfwxuhg oljkw zklfk kdv ehhq frqvlghuhg dv rqh ri wkh prvw iuhtxhqwo| xvhg whfkqltxhv lq rughu wr uhgxfh wkh sureohpv uhodwhg wr vwhuhr ylvlrq1 Vwuxfwxuhg oljkw lv edvhg rq wkh uhod0 wlrqvkls ehwzhhq d surmhfwhg oljkw sdwwhuq lwv surmhfwlrq dqg dq lpdjh vhqvru1 Wkh ghirupdwlrqv ehwzhhq wkh sdwwhuq surmhfwhg lqwr wkh vfhqh dqg wkh rqh fdswxuhg e| wkh fdphud/ shuplwv wr rewdlq wkuhh glphqvlrqdo lqirupdwlrq ri wkh looxplqdwhg vfhqh1 Wklv whfkqltxh kdv ehhq zlgho| xvhg lq vxfk dssolfd0 wlrqv dv= 6G remhfw uhfrqvwuxfwlrq/ urerw qdyljdwlrq/ txdolw| frqwuro/ dqg vr rq1 Dowkrxjk wkh surmhfwlrq ri uhjxodu sdwwhuqv vroyh wkh sureohp ri srlqwv zlwkrxw pdwfk/ lw grhv qrw vroyh wkh sureohp ri pxowlsoh pdwfklqj/ zklfk ohdgv xv wr xvh kdug frpsxwlqj dojrulwkpv lq rughu wr vhdufk wkh fruuhfw pdwfkhv1 Lq uhfhqw |hduv/ dqrwkhu vwuxfwxuhg oljkw whfkqltxh kdv lqfuhdvhg lq lp0 sruwdqfh1 Wklv whfkqltxh lv edvhg rq wkh frglfdwlrq ri wkh oljkw surmhfwhg rq wkh vfhqh lq rughu wr eh xvhg dv d wrro wr rewdlq dq xqltxh pdwfk1 Hdfk wrnhq ri oljkw surmhfwhg rq wkh vfhqh fduulhv d odeho lqglfdwlqj iurp zklfk srvlwlrq lq wkh surmhfwru iudph lw frphv1 Zkhq wkh wrnhq lv lpdjhg e| wkh fdphud/ zh kdyh wr uhdg wkh odeho +ghfrgh wkh sdwwhuq, lq rughu wr vroyh wkh fruuhvsrqghqfh sureohp1 Wkh dgydqwdjhv dqg glvdgydqwdjhv ri vwhuhr ylvlrq djdlqvw vwuxfwxuhg oljkw dqg d vxuyh| rq frghg vwuxfwxuhg oljkw duh uhodwhg lq fkdswhu 71 Wkh zrun fduulhg rxw lq wkh iudph ri wklv wkhvlv kdv shuplwwhg wr suhvhqw d qhz frghg vwuxfwxuhg oljkw sdwwhuq zklfk vroyhv wkh fruuhvsrqghqfh sureohp xqltxho| dqg urexvwo|1 Xqltxh/ dv hdfk wrnhq ri oljkw lv frghg e| d glhuhqw zrug zklfk uhpryhv wkh sureohp ri pxowlsoh pdwfklqj1 Urexvw/ vlqfh wkh sdwwhuq kdv ehhq frghg xvlqj wkh srvlwlrq ri hdfk wrnhq ri oljkw zlwk uhvshfw wr erwk fr0ruglqdwh d{lv B dqg J 1 Dojrulwkpv dqg h{shulphq0 wdo uhvxowv duh lqfoxghg lq wkh wkhvlv1 Wkh uhdghu fdq vhh h{dpsohv ri 6G phdvxuhphqw ri vwdwlf remhfwv/ dqg wkh pruh frpsolfdwhg phdvxuhphqw ri prylqj remhfwv1 Wkh whfkqltxh fdq eh xvhg lq erwk fdvhv dv wkh sdwwhuq lv frghg e| d vlqjoh surmhfwlrq vkrw1 Wkhq lw fdq eh xvhg lq vhyhudo dssolfdwlrqv ri urerw ylvlrq1 Rxu lqwhuhvw lv irfxvhg rq wkh pdwkhpdwlfdo vwxg| ri wkh fdphud dqg sdw0 whuq surmhfwru prghov1 Zh duh dovr lqwhuhvwhg lq krz wkhvh prghov fdq eh rewdlqhg e| fdoleudwlrq/ dqg krz wkh| fdq eh xvhg wr rewdlqhg wkuhh glphq0 vlrqdo lqirupdwlrq iurp wzr fruuhvsrqghqfh srlqwv1 Ixuwkhupruh/ zh kdyh vwxglhg vwuxfwxuhg oljkw dqg frghg vwuxfwxuhg oljkw/ dqg zh kdyh suhvhqwhg d qhz frghg vwuxfwxuhg oljkw sdwwhuq1 Krzhyhu/ lq wklv wkhvlv zh vwduwhg iurp wkh dvvxpswlrq wkdw wkh fruuhvsrqghqfh srlqwv frxog eh zhoo0vhjphqwhg iurp wkh fdswxuhg lpdjh1 Frpsxwhu ylvlrq frqvwlwxwhv d kxjh sureohp dqg d orw.

(15) FKDSWHU 41 LQWURGXFWLRQ. 49. ri zrun lv ehlqj grqh dw doo ohyhov ri kxpdq ylvlrq prghoolqj/ vwduwlqj iurp d, lpdjh dftxlvlwlrq> e, ixuwkhu lpdjh hqkdqfhphqw/ owhulqj dqg surfhvvlqj/ f, lpdjh vhjphqwdwlrq zklfk lqyroyhv wkuhvkroglqj/ wklqqlqj/ frqwrxu ghwhf0 wlrq/ wh{wxuh dqg frorxu dqdo|vlv/ dqg vr rq1 Wkh lqwhuhvw ri wklv wkhvlv vwduwv lq wkh qh{w vwhs/ xvxdoo| nqrzq dv ghswk shufhswlrq ru 6G phdvxuhphqw1 Vr/ zh glg qrw wdnh lqwr dffrxqw wkh orzhu ohyhov1. 415. Vfkhpd ri wkh wkhvlv. Wklv wkhvlv lv irfxvhg rq wkh vwxg| ri vwuxfwxuhg oljkw dqg/ lq sduwlfxodu/ rq krz oljkw fdq eh frghg lq rughu wr vroyh wkh fruuhvsrqghqfh sureohp1 Ixu0 wkhupruh/ d qhz frghg vwuxfwxuhg oljkw sdwwhuq lv suhvhqwhg/ zklfk shuplwv d uhfrqvwuxfwlrq ri vwdwlf dqg prylqj remhfwv1 Wkh v|vwhp fdq eh xvhg dv d vhqvru doorzlqj xv wr phdvxuh 6G g|qdplf vfhqhv1 Lq fkdswhu 5 wkh uhdghu lv hqjdjhg lq wkh vwxg| ri fdphud dqg sdwwhuq fdoleudwlrq1 Wkh fkdswhu ghvfulehv wkh lqwulqvlf sdudphwhuv zklfk prgho wkh rswlfdo fkdudfwhulvwlfv dqg wkh lqwhuqdo jhrphwu| ri wkh vhqvru/ dqg wkh h{wulqvlf sdudphwhuv zklfk prgho wkh srvlwlrq dqg rulhqwdwlrq ri wkh vhqvru zlwk uhvshfw wr d zruog fr0ruglqdwhg v|vwhp1 Ixuwkhupruh/ wzr dojrulwkpv ri fdphud fdoleudwlrq duh ghvfulehg lq ghwdlo1 Wkh uvw rqh lv edvhg rq d olqhdu fdoleudwlrq/ wkdw lv zlwkrxw frqvlghulqj ohqv glvwruwlrq1 Wkh vhfrqg rqh lv edvhg rq d qrq0olqhdu fdoleudwlrq ri d zkroh fdphud prgho/ lqfoxglqj ohqv glvwruwlrq zklfk ohdgv wr d ixqfwlrq plqlpl}dwlrq e| lwhudwlrq1 Ilqdoo|/ wkh zd| krz 6G lqirupdwlrq ri dq remhfw srlqw fdq eh rewdlqhg nqrzlqj rqo| lwv wzr fruuhvsrqghqfh srlqwv lv ghvfulehg1 Fkdswhu 6 ghdov zlwk wkh vwxg| ri wkh ixqgdphqwdo pdwul{1 Wkh ixqgd0 phqwdo pdwul{ lv frpsxwhg e| wkh uhodwlrqvkls ehwzhhq wzr slqkroh prghov/ dqg lqfoxghv wkh lqwulqvlf sdudphwhuv ri erwk slqkroh prghov dqg wkh uhodwlyh srvlwlrq dqg rulhqwdwlrq iurp rqh wr wkh rwkhu1 Wklv pdwul{ kdv ehhq zlgho| xvhg lq rughu wr rewdlq wkh hslsrodu jhrphwu| ri wkh v|vwhp wkdw vlpsolhv frqvlghudeo| wkh vhdufk ri wkh fruuhvsrqghqfh srlqwv1 Fkdswhu 7 lv irfxvhg rq wkh vwxg| ri vwuxfwxuhg oljkw1 Vwuxfwxuhg oljkw uhgxfhv frqvlghudeo| wkh sureohp ri pdwfklqj/ suhvhqw lq vwhuhr ylvlrq v|v0 whpv1 Krzhyhu/ wkh surmhfwlrq ri uhjxodu sdwwhuqv grhv qrw vroyh wkh sureohp ri pxowlsoh pdwfklqj1 Wkh fruuhvsrqghqfh sureohp fdq eh vroyhg e| frglqj wkh oljkw surmhfwhg rq wkh vfhqh1 Wkdw lv/ hdfk wrnhq ri oljkw surmhfwhg rq wkh vfhqh fduulhv zlwk lw d odeho lqglfdwlqj zkhuh lw frphv iurp1 Zkhq wkh wrnhq lv lpdjhg e| wkh fdphud zh rqo| kdyh wr uhdg wkh odeho wr vroyh wkh fru0 uhvsrqghqfh sureohp1 Wklv fkdswhu vxuyh|v wkh prvw lqwhuhvwlqj whfkqltxhv edvhg rq frghg vwuxfwxuhg oljkw surmhfwlrq1.

(16) FKDSWHU 41 LQWURGXFWLRQ. 4:. Fkdswhu 8 suhvhqwv d qhz frghg vwuxfwxuhg oljkw sdwwhuq1 Wkh sdwwhuq lv xqltxho| frghg dqg frqvhtxhqwo| lw lv pruh urexvw wkdq shulrglfdo frg0 lqj/ dqg lw lv frghg zlwk uhvshfw wr erwk fr0ruglqdwh d{hv1 Wklv grxeoh frglqj kdv ehhq ghprqvwudwhg wr eh pruh dffxudwh wkdq vlqjoh d{h frglfd0 wlrq1 Fdoleudwlrq lv h{sodlqhg dqg wkh lqwulqvlf dqg h{wulqvlf sdudphwhuv duh vkrzq1 Ixuwhupruh/ vrph h{shulphqwdo h{dpsohv ri 6G remhfw phdvxuhphqw duh vkrzq dqg glvfxvvhg1 Ilqdoo|/ fkdswhu 9 suhvhqwv wkh frqfoxvlrqv ri wkh wkhvlv1 Ixuwkhu zrun lv dovr lqfoxghg lq vrph vxemhfwv zkhuh wkh uhvhdufk ri wklv wkhvlv fdq frqwlqxh1.

(17) Fkdswhu 5 Fdoleudwlrq Lqihuulqj wkuhh glphqvlrqdo lqirupdwlrq ohdgv xv wr d suhylrxv vwhs ri v|vwhp fdoleudwlrq1 Wklv fkdswhu ghvfulehv wkh wzr pdmruv pdwwhuv ri fdoleudwlrq1 Iluvw/ krz wkh prgho ri d v|vwhp lv ghqhg e| d vhw ri sdudphwhuv lq rughu wr dssur{lpdwh wkh sk|vlfdo ehkdylrxu ri wkh v|vwhp wr d jhrphwulfdo rqh1 Vhf0 rqgo|/ krz wkhvh sdudphwhuv duh rewdlqhg e| phdqv ri d fdoleudwlrq dojrulwkp1 Ilqdoo|/ wkh fkdswhu zloo ghvfuleh krz 6G uhfrqvwuxfwlrq fdq eh dfklhyhg e| phdqv ri wkh uhodwlrqvkls ehwzhhq wzr fdoleudwhg prghov1. 514. Frqvlghudwlrqv. Vlqfh rxu 6G lpdjlqj v|vwhp lv edvhg rq d fdphud vhqvru dqg d sdwwhuq surmhfwru v|vwhp/ wzr glhuhqw vhwv ri sdudphwhuv kdyh wr eh frpsxwhg e| fdoleudwlrq1 Iluvwo|/ zh kdyh wr ghqh wkh prgho ri d fdphud vhqvru dqg/ vhfrqgo|/ wkh prgho ri wkh sdwwhuq surmhfwru v|vwhp1 Krzhyhu/ qrwh wkdw/ erwk ghylfhv duh ghqhg xvlqj wkh vdph prgho1 Wkdw lv/ wkh fdphud dqg wkh surmhfwru sdwwhuq duh edvhg erwk rq wkh slqkroh sulqflsoh1 Wkhq/ wkh zkroh fkdswhu zloo eh irfxvhg rq wkh pdmru hruw ri prghoolqj dqg fdoleudw0 lqj d fdphud1 Wkh uhdghu pljkw qrwh wkdw doo wkh pdwkhpdwlfdo htxdwlrqv lqwurgxfhg lq wklv fkdswhu duh dovr xvhg wr prgho dqg fdoleudwh wkh sdwwhuq surmhfwru v|vwhp1. 515. Prghoolqj d fdphud. Prghoolqj d fdphud lv edvhg rq wkh ghqlwlrq ri wkh vhw ri sdudphwhuv wkdw dssur{lpdwh wkh ehkdylrxu ri wkh sk|vlfdo vhqvru wr d jhrphwulfdo prgho/ nqrzq dv wkh slqkroh prgho Îdxj <6` ^D|df <4e`1 Wkh jhrphwulfdo prgho ri wkh slqkroh frqvlvwv ri d sodqh fdoohg wkh uhwlqdo sodqh lq zklfk wkh 4;.

(18) FKDSWHU 51 FDOLEUDWLRQ. 4<. Zc y. Yw Zw. Xw. Π. x. M. Yc f. Xc y x. m y. x. C. y. m Retinal plane. f. x. Focal plane Π. Retinal plane. Iljxuh 514= Wkh fdphud fr0ruglqdwh v|vwhp zlwk uhvshfw wr d zruog fr0 ruglqdwh v|vwhp1 lpdjh lv iruphg wkurxjk d shuvshfwlyh surmhfwlrq1 D srlqw lv sodfhg dw d {hg glvwdqfh iurp wkh uhwlqdo sodqh1 Wklv srlqw lv nqrzq dv wkh rswlfdo fhqwuh +dovr irfdo srlqw, dqg lw lv ghqhg dv wkh slqkroh wkurxjk zkhuh vrph ri wkh oljkw ehdpv uh hfwhg e| wkh remhfw sdvv/ iruplqj d shuvshfwlyh lpdjh ri wkh vfhqh lq wkh uhwlqdo sodqh1 Dw wklv srlqw/ wzr frqvlghudwlrqv pxvw eh wdnhq lqwr dffrxqw1 Ri frxuvh/ wkh rswlfdo fhqwuh fdq eh sodfhg lq iurqw ri ru ehklqg wkh fdphud +vhh jxuh 514,1 Lq wkh uvw fdvh/ d surmhfwlyh lqyhuwhg lpdjh ri wkh vfhqh lv iruphg lq wkh uhwlqdo sodqh1 Lq wkh vhfrqg rqh/ d vlqjoh surmhfwlyh lpdjh lv suhvhqwhg1 Wklv vhfwlrq ghvfulehv wkh prgho ri d fdphud vhqvru zlwk d edfn0sodfhg rswlfdo fhqwuh1 Krzhyhu/ lw zloo eh vkrzq wkdw erwk fdphud frqjxudwlrqv duh prghoohg e| wkh vdph sdudphwhuv exw frqvlghulqj d glhuhqw vljq ri wkh irfdo glvwdqfh1 Wkhuh duh wzr nlqgv ri sdudphwhu vhwv zklfk prgho wkh jlyhq fdphud dv d slqkroh prgho Êh|h <5` ^Lwr <4`1 Rqh lv wkh lqwulqvlf sdudphwhu vhw/ dv wkh slqkroh srlqw/ zklfk prghov wkh lqwhuqdo jhrphwu| dqg rswlfdo fkdudfwhulv0 wlfv ri wkh lpdjh vhqvru1 Wkh rwkhu lv wkh h{wulqvlf sdudphwhu vhw/ zklfk prghov wkh srvlwlrq dqg rulhqwdwlrq ri wkh vhqvru zlwk uhvshfw wr wkh zruog fr0ruglqdwh v|vwhp1 Wkh lqwulqvlf sdudphwhu vhw frqwdlqv= d, wkh irfdo glvwdqfh s/ zklfk lv.

(19) FKDSWHU 51 FDOLEUDWLRQ. 53. wkh glvwdqfh +lq pp1, iurp wkh uhwlqdo sodqh wr wkh rswlfdo fhqwuh F > e, wkh frqyhuvlrq sdudphwhuv & dqg & zklfk uhodwh wkh krul}rqwdo dqg yhuwlfdo dgmxvwphqw wr jr iurp wkh fdphud fr0ruglqdwh v|vwhp +lq pp1, orfdwhg lq wkh uhwlqdo sodqh wr wkh lpdjh fr0ruglqdwh v|vwhp orfdwhg lq wkh lpdjh sodqh dqg h{suhvvhg lq sl{hov> dqg f, wkh srvlwlrq +f / f , ri wkh sulqflsdo srlqw +dovr lpdjh fhqwuh, ghqhg e| wkh surmhfwlrq ri wkh sulqflsdo srlqw rq wkh lpdjh sodqh/ h{suhvvhg lq sl{hov1 Wkh h{wulqvlf sdudphwhuv vhw ghqhv wkh uhodwlrq ehwzhhq wkh fdphud fr0 ruglqdwh v|vwhp orfdwhg rq wkh uhwlqdo sodqh zlwk uhvshfw wr wkh zruog fr0 ruglqdwh v|vwhp orfdwhg dw dq| srlqw lq wkh vfhqh1 Wklv uhodwlrqvkls lv ghqhg e| wkh srvlwlrq dqg wkh rulhqwdwlrq ri wkh fdphud fr0ruglqdwh v|vwhp zlwk uhvshfw wr wkh zruog fr0ruglqdwh v|vwhp1 Wkh srvlwlrq lv jlyhq e| d 6{4 wudqvodwlrq yhfwru A dqg wkh rulhqwdwlrq lv jlyhq e| wkh 6{6 urwdwlrq pdwul{ +1 Wkhq/ xs wr 44 sdudphwhuv kdyh wr eh xvhg wr prgho wkh fdphud vhqvru1 Wklv vhfwlrq jrhv wkurxjk hdfk sdudphwhu lq rughu wr rewdlq wkh pdwkhpdwlfdo htxdwlrqv zklfk uhodwh dq remhfw srlqw zlwk lwv surmhfwlyh srlqw1 Ilqdoo|/ wklv olqhdu uhodwlrqvkls zloo eh prghoohg e| d 6{7 wudqvirupdwlrq pdwul{1 Krzhyhu/ dv d uhvxow ri vrph w|shv ri lpshuihfwlrqv lq wkh ghvljq dqg dvvhpeo| ri wkh ohqv frpsrvlqj wkh rswlfdo v|vwhp/ d olqhdu uhodwlrqvkls grhv qrw krog wuxh1 Wkhvh nlqgv ri lpshuihfwlrqv/ nqrzq dv ohqv glvwruwlrq/ zloo eh glvfxvvhg1 Wkh qrq0olqhdu pdwkhpdwlfdo uhodwlrq ehwzhhq erwk srlqwv lv dovr suhvhqwhg1. 51514. Wkh H{wulqvlf Sdudphwhuv. Wkh vxuidfh srlqwv ri wkh vfhqh duh uhihuuhg wr d zruog fr0ruglqdwh v|vwhp/ zlwk dq ruljlq / dqg fr0ruglqdwh d{lv Ef c t c ~ 1 Jlyhq dq remhfw srlqw S/ wklv srlqw lv h{suhvvhg zlwk uhvshfw wr R / wkdw lv zlwk wkh fr0ruglqdwhv E % c + c 5 1 Frqvlghu d vhfrqg fr0ruglqdwh v|vwhp/ zlwk dq ruljlq ' S / dqg fr0ruglqdwh d{lv EfU c tU c ~U / sodfhg lq wkh rswlfdo fhqwuh ri wkh fdphud vhqvru dv vkrzq lq jxuh 5151 Wklv jhrphwulfdo uhodwlrqvkls shuplwv wr ghqh zklfk duh wkh h{wulqvlf sdudphwhuv ri wkh prgho/ dv wkhvh sdudphwhuv zloo ghwhuplqh wkh srvlwlrq dqg rulhqwdwlrq ri wkh fdphud fr0ruglqdwh v|vwhp zlwk uhvshfw wr wkh zruog fr0ruglqdwh v|vwhp1 Wkh srvlwlrq ri wkh rswlfdo fhqwuh S ' zlwk uhvshfw wr lv jlyhq e| wkh wudqvodwlrq yhfwru A vkrzq lq htxdwlrq 5141 5 6 |f A ' 7 |t 8 +514, |~ Wkh rulhqwdwlrq ri wkh fr0ruglqdwh d{lv ri wkh fdphud zlwk uhvshfw wr wkh.

(20) FKDSWHU 51 FDOLEUDWLRQ. 54. Retinal Plane. P Yc Zc. Xc. Yw. Oc. Camera co-ordinate system. K. Xw Zw Ow World co-ordinate system. Iljxuh 515= Jhrphwulf uhodwlrvkls ehwzhhq wkh fdphud fr0ruglqdwh v|vwhp dqg wkh zruog fr0ruglqdwh v|vwhp1 zruog fr0ruglqdwh v|vwhp lv jlyhq e| d 6{6 urwdwlrq pdwul{ +1 Wkh urwdwlrq pdwul{ fdq eh rewdlqhg e| wkh pdwul{ surgxfw ri wkuhh vlqjoh d{h urwdwlrqv1 D urwdwlrq ri ghjuhhv durxqg dq d{h O zloo eh h{suhvvhg dv -J|EOc 1 Wkhq/ wkuhh vlqjoh d{h urwdwlrqv fdq eh h{suhvvhg dv vkrzq e| wkh iroorzlqj htxdwlrqv1 5 6 f f -J|Efc % ' 7 f S% r% 8 +515, f r% S% 5 6 S+ f r+ f 8 -J|Etc + ' 7 f +516, r+ f S+ 5 6 S5 r5 f -J|E~c 5 ' 7 r5 S5 f 8 +517, f f Zkhuh S uhsuhvhqwv wkh frvlqxv ri wkh dqjoh/ dqg r uhsuhvhqwv wkh vlqxv1 Lq wkh iroorzlqj/ wzr rulhqwdwlrq uhsuhvhqwdwlrqv duh suhvhqwhg lq rughu wr rewdlq wkh urwdwlrq pdwul{ +1.

(21) FKDSWHU 51 FDOLEUDWLRQ. 55. Wkh US\ uhsuhvhqwdwlrq. + ' - t E5 c + c % ' -J|E~c 5 -J|Etc + -J|Efc % . +518,. Wkh Hxohu uhsuhvhqwdwlrq. + ' .,eoEc wc ' -J|E~c -J|Etc w -J|E~c . +519,. Erwk uhsuhvhqwdwlrqv duh srvvleoh1 Krzhyhu/ rqo| wkh Hxohu uhsuhvhqwdwlrq zloo eh frqvlghuhg1 Wkhq wkh urwdwlrq pdwul{ + lv uhsuhvhqwhg e| wkh iroorzlqj pdwul{1 + ' -J|E~c -J|Etc w -J|E~c ' 5 6 SS rSwr Sr rSwS rrw 7 SS n SSwr rr n SSwS Srw 8 rwr rwS Sw. +51:,. Lq rughu wr vlpsoli| wkh uhsuhvhqwdwlrq ri wkh + pdwul{/ lw lv pruh lqwhu0 hvwlqj wr uhsuhvhqw wkh pdwul{ zlwk uhvshfw wr lwv wkuhh rulhqwdwlrq yhfwru o / zkhuh hdfk rulhqwdwlrq yhfwru lv pdgh e| wkuhh frpsrqhqwv1 5 6 of + ' 7 ot 8 +51;, o~ Qrwh wkdw/ of / ot dqg o~ uhsuhvhqw wkh rulhqwdwlrq ri wkh wkuhh d{lv ri wkh fdphud fr0ruglqdwh v|vwhp zlwk uhvshfw wr wkh zruog fr0ruglqdwh v|vwhp1 Lq wkh jxuh 516/ wkh uhodwlrqvkls ri wkh frpsrqhqwv ri d yhfwru +h1j1 fS fdphud fr0ruglqdwh d{h, zlwk uhvshfw wr wkh zruog fr0ruglqdwh d{h +jlyhq e| of ,/ lv vkrzq1 Qrwh wkdw dq| yhfwru frxog eh sodfhg zlwk wkh ruljlq dw wkh zruog fr0ruglqdwh v|vwhp zlwkrxw fkdqjlqj lwv uhsuhvhqwdwlrq/ dv doo wkh yhfwruv duh vsdfh0iuhh hohphqwv1 Wkhq/ erwk fr0ruglqdwh v|vwhpv duh uhodwhg e| wkh iroorzlqj htxdwlrq/ 5 6 5 6 fS f 7 tS 8 ' + 7 t 8 n A +51<, ~S ~.

(22) FKDSWHU 51 FDOLEUDWLRQ. Zw. 56. Yw. Xc rXy rXx. rXz. Xw. Iljxuh 516= Uhodwlrqvkls ehwzhhq wkh rulhqwdwlrq yhfwruv o dqg wkh fdphud fr0ruglqdwh iudph1 Wkh jxuh vkrzv rqo| wkh uhodwlrqvkls ehwzhhq wkh of yhfwru dqg wkh fS d{lv1 vr wkdw/ jlyhq dq remhfw srlqw / zlwk wkh frpsrqhqwvEf c t c ~ zlwk uhvshfw wr / rqh fdq frpsxwh lwv uhsuhvhqwdwlrq zlwk uhvshfw wr S zlwk wkh frpsrqhqwv EfS c t S c ~S / 5 6 5 6 fS f 7 t S 8 ' + 7 t 8 n A +5143, ~S ~ Htxdwlrq 5143 lv xvxdoo| hvsuhvvhg e| d vlqjoh pdwul{ uhodwlrqvkls zlwk krpjhqhrxv fr0ruglqdwhv/ dv vkrzq e| htxdwlrq 5144 +vhh dovr wkh jxuh 515,/ S 'k +5144, + % A % k' +5145, f% Vxppdulvlqj wkh vhfwlrq/ vl{ h{wulqvlf sdudphwhuv duh rewdlqhg lq rughu wr prgho wkh wudqvodwlrq dqg rulhqwdwlrq ri wkh fdphud fr0ruglqdwh v|vwhp zlwk uhvshfw wr wkh zruog fr0ruglqdwh v|vwhp1 Wkh vl{ sdudphwhuv duh= wkh wkuhh frpsrqhqwv ri wkh wudqvodwlrq yhfwru A ' E|f c |t c |~ dqg wkh wkuhh dqjohv Ec wc zklfk ghqh wkh urwdwlrq pdwul{ +1.

(23) FKDSWHU 51 FDOLEUDWLRQ image co-ordinate system Oi. w. 57. Yr. Zr. retinal co-ordinate system. u. v Xr. (u0, v0). f. Yc Zc. Xc. P. Pd Pu. Image plane Retinal plane. Oc Camera co-ordinate system. Iljxuh 517= Jhrphwulfdo uhodwlrqvkls ehwzhhq d 6G remhfw srlqw dqg lwv 5G lpdjh surmhfwlrq1. 51515. Wkh Lqwulqvlf Sdudphwhuv. Dw wklv srlqw/ wkh vhfwlrq zloo irfxv rq wkh prghoolqj ri wkh lqwulqvlf sdud0 phwhuv ri wkh fdphud vhqvru1 Wkdw lv/ lw zloo irfxv rq wkh ghwhuplqdwlrq ri wkh sdudphwhuv zklfk prgho wkh lqwhuqdo jhrphwu| dqg rswlfdo fkdudfwhulv0 wlfv ri wkh lpdjh vhqvru1 Wkh lqwulqvlf fdphud prgho zloo uhodwh d 6G remhfw srlqw zlwk uhvshfw wr lwv revhuydeoh 5G surmhfwlrq rq wkh lpdjh sodqh1 Wkh 6G remhfw srlqw pxvw eh h{suhvvhg zlwk uhvshfw wr wkh fdphud fr0ruglqdwh v|vwhp/ dqg wkh 5G lpdjh srlqw zloo eh h{suhvvhg zlwk uhvshfw wr wkh lpdjh fr0ruglqdwh v|vwhp/ lq sl{hov1 Wkh jhrphwulfdo prgho wkdw kdv wr eh ghqhg lv looxvwudwhg lq jxuh 5171 Wkh prghoolqj ri wkh lqwulqvlf sdudphwhuv zloo eh vsolw lq irxu vwhsv= Lghdo surmhfwlrq1 Prghov wkh surmhfwlyh uhodwlrqvkls ehwzhhq d 6G remhfw srlqw dqg lwv 5G lpdjhg srlqw1 Ohqv glvwruwlrq1 Wdnhv lqwr dffrxqw wkh glvfuhsdqf| ehwzhhq wkh lghdo 5G srlqw dqg wkh revhuyhg 5G srlqw1 Sl{ho dgmxvwphqw1 Wudqvirupv wkh 5G revhuyhg srlqw iurp phwulf fr0ruglqdwhv wr sl{hov1 Sulqflsdo srlqw1 Jlyhv wkh 5G revhuyhg srlqw fr0ruglqdwhv zlwk uhvshfw wr wkh frpsxwhu fr0ruglqdwh v|vwhp1.

(24) FKDSWHU 51 FDOLEUDWLRQ. 58. Yc. P PZc. Pu PYc. PXc e Imag. Xu. f Zc. Yu Plane Xc. Oc C. Iljxuh 518= Rewdlqlqj wkh lghdo surmhfwlrq iurp wkh shuvshfwlyh uhodwlrq1 Hdfk vwhs lv h{sodlqhg lq ghwdlo khuhdiwhu1 Vwhs 41 Prghoolqj wkh lghdo surmhfwlrq Frqvlghu wkdw dq| rswlfdo vhqvru fdq eh prghoohg dv d slqkroh fdphud1 Qrwh wkdw wkh fdphud fr0ruglqdwh v|vwhp kdv ehhq orfdwhg dw wkh rswlfdo fhqwuh / zlwk wkh ~S d{h dorqj wkh rswlfdo d{lv ri wkh lpdjh vhqvru1 Frqvlghu wkdw wkh lpdjh sodqh lv orfdwhg dw d glvwdqfh s iurp wkh rswlfdo fhqwuh / dqg lw lv sdudooho wr wkh sodqh ghqhg e| wkh fr0ruglqdwh d{lv fS dqg tS 1 Vwduwlqj iurp wkhvh dvvxpswlrqv/ jlyhq dq remhfw srlqw uhihuuhg wr S / lwv lghdo surmhfwlrq rq wkh lpdjh sodqh pxvw olh rq wkh olqh sdvvlqj wkurxjk wkh remhfw srlqw dqg wkh rswlfdo fhqwuh 1 Lq rughu wr rewdlq wkh htxdwlrqv zklfk uhodwh wkh 5G srlqw zlwk uhvshfw wr dqg s / wkh jhrphwulf odz ri wkh shuvshfwlyh uhodwlrq zloo eh xvhg ^Ndqd <4` ^Kdud <6`1 Wklv uhodwlrq lv vkrzq lq jxuh 5181 Wkhq/ wkh shuvshfwlyh uhodwlrq lv jlyhq e| wkh iroorzlqj htxdwlrqv/ f fS ' s ~S t tS ' s ~S Vr wkdw/ f ' s. fS ~S. +5146,.

(25) FKDSWHU 51 FDOLEUDWLRQ t ' s. 59 tS ~S. Zklfk fdq eh h{suhvvhg lq d pdwulfldo zd| dsso|lqj/ 5 6 5 65 6 f f rf fS 7 rt 8 ' 7 f f 8 7 t S 8 f f s r ~S. +5147,. +5148,. Vxppdulvlqj wkh vwhs/ d vlqjoh lqwulqvlf sdudphwhu kdv wr eh frqvlghuhg lq rughu wr prgho wkh lghdo surmhfwlrq1 Wklv sdudphwhu lv wkh irfdo glvwdqfh s1 Vwhs 51 Prghoolqj wkh ohqv glvwruwlrq Dv d uhvxow ri vrph w|shv ri lpshuihfwlrqv lq wkh ghvljq dqg dvvhpeo| ri wkh ohqv frpsrvlqj wkh rswlfdo v|vwhp/ wkh olqhdu uhodwlrqvkls ri wkh shuvshfwlyh surmhfwlrq grhv qrw krog wuxh ^Zhqj <5`1 Wkhvh lpshuihfwlrqv lpso| wkdw wkh revhuyhg surmhfwlrq rq wkh lpdjh sodqh glhuv iurp wkh lghdo rqh ghqhg lq wkh suhylrxv vwhs ^Jrvk ;<`1 Wkhq/ dqrwkhu wzr htxdwlrqv zklfk h{solflwo| wdnh lqwr dffrxqw wkh srvlwlrqdo huuru pxvw eh lqwurgxfhg/ zkhuh Ef c t lv wkh lghdo surmhfwlrq dqg Ef_ c t_ duh wkh fr0ruglqdwhv ri wkh revhuyhg srlqw rq wkh lpdjh sodqh1 f ' f_ n (% t ' t_ n (+. +5149,. Ohqv glvwruwlrq fdq eh prghoohg e| d udgldo dqg wdqjhqwldo dssur{lpdwlrq/ vkrzq lq jxuh 5191 Udgldo glvwruwlrq= Udgldo glvwruwlrq fdxvhv dq lqzdug ru rxwzdug glvsodfh0 phqw ri d jlyhq lpdjh srlqw iurp lwv lghdo surmhfwlrq1 Wklv w|sh ri glvwruwlrq lv pdlqo| fdxvhg e| dzhg udgldo fxuydwxuh ri wkh ohqv1 D qhjdwlyh udgldo glvsodfhphqw ri wkh lpdjh srlqwv lv uhihuuhg wr dv eduuho glvwruwlrq1 Lw fdxvhv rxwhu srlqwv wr furzg lqfuhdvlqjo| wrjhwkhu dqg wkh vfdoh wr ghfuhdvh1 D srvlwlyh udgldo glvsodfhphqw lv uhihuuhg wr dv slqfxvklrq glvwruwlrq1 Lw fdxvhv rxwhu srlqwv wr vsuhdg dqg wkh vfdoh wr lqfuhdvh1 Wklv w|sh ri glvwruwlrq lv vwulfwo| v|pphwulf derxw wkh rswlfdo d{lv1 Iljxuh 51: looxvwudwhv wkh hhfw ri udgldo glvwruwlrq1 Wkh glvsodfhphqw jlyhq e| wkh udgldo glvwruwlrq fdq eh prghoohg e| wkh iroorzlqj htxdwlrqv/ zkhuh o lv wkh udgldo glvwdqfh ri wkh revhuyhg surmhfwlrq iurp wkh surmhfwlrq ri wkh irfdo srlqw rq wkh lpdjh sodqh +nqrzq dv sulqflsdo srlqw,/ dqg & / &2 / 111 duh wkh frh!flhqwv ri wkh udgldo glvwruwlrq1.

(26) FKDSWHU 51 FDOLEUDWLRQ. 5:. Yr dr Ideal P projection u. dt. Pd Observed position. Xr. dr: radial distortion dt: tangential distortion. Iljxuh 519= Udgldo dqg wdqjhqwldo glvwruwlrq1. a b. Iljxuh 51:= Hhfw ri udgldo glvwruwlrq1 Vrolg olqhv= qr glvwruwlrq> gdvkhg olqhv= udgldo glvwruwlrq +d= qhjdwlyh/ e= srvlwlyh,1.

(27) FKDSWHU 51 FDOLEUDWLRQ. 5; Yr Axe of a maximum tangential distortion. Xr. Axe of a minimum tangential distortion. Iljxuh 51;= Hhfw ri wdqjhqwldo glvwruwlrq1 Vrolg olqhv= qr glvwruwlrq> gdvkhg olqhv= wdqjhqwldo glvwruwlrq1. (% ' f_ E& o2 n &2 oe n (+ ' t_ E& o2 n &2 oe n t o ' f_2 n t_2. +514:,. Wdqjhqwldo glvwruwlrq= Wdqjhqwldo glvwruwlrq lv lqwurgxfhg lq rughu wr prgho wkh dfwxdo rswlfdo v|vwhpv vxemhfwhg wr ydulrxv ghjuhhv ri ghfhqwhu0 lqj/ wkdw lv/ wkh rswlfdo fhqwhuv ri wkh ohqv hohphqwv duh qrw vwulfwo| froolqhdu1 Wkhvh nlqgv ri lpshuihfwlrqv dulvh iurp lpshuihfwlrqv lq wkh ohqv ghvljq dqg pdqxidfwxulqj dv zhoo dv iurp fdphud dvvhpeo|1 Wkh hhfw surgxfhg e| wdqjhqwldo glvwruwlrq lv vkrzq lq jxuh 51;1 Wdqjhqwldo glvwruwlrq lv dovr prghoohg e| dqrwkhu lqqlwh vhulh ^Zhqj <5`1 Krzhyhu/ prvw ri wkh dxwkruv kdyh d!uphg wkdw udgldo glvwruwlrq lv wkh prvw lpsruwdqw ohqv glvwruwlrq dqg wkhuhiruh/ lw lv wkh rqo| rqh wkdw kdv wr eh frqvlghuhg1 Dq| pruh hoderudwh prghoolqj zrxog qrw rqo| qrw jlyh pruh dffxudwh uhvxowv/ exw dovr zrxog fdxvh qxphulfdo lqvwdelolw| lq fdoleudwlrq ^Wvdl ;:`1 Wklv vhfwlrq wdnhv rqo| lqwr dffrxqw wkh uvw frh!flhqw & ri udgldo glv0 wruwlrq/ wkdw lv wkh frh!flhqw prghoolqj wkh pdmru frqwulexwlrq ri udgldo glv0 wruwlrq/ zklfk kdv ehhq frqvlghuhg wkh zlghvw xvhg prgho ri ohqv glvwruwlrq1.

(28) FKDSWHU 51 FDOLEUDWLRQ. 5<. Lw vhhpv wkdw yhu| olwwoh lpsuryhphqw lv jlyhq prghoolqj pruh frh!flhqwv1 Wkhq/ htxdwlrqv 514: duh vlpsolhg/ rewdlqlqj/ (% ' f_ & o2 (+ ' t_ & o2 t o ' f_2 n t_2. +514;,. Vxppdulvlqj wkh vwhs/ d vlqjoh sdudphwhu & dv wkh uvw frh!flhqw ri wkh vhulh zklfk prghov wkh udgldo glvwruwlrq kdv wr eh frqvlghuhg1 Vwhs 61 Prghoolqj wkh sl{ho dgmxvwphqw Wklv vwhs ri fdphud prghoolqj lv edvhg rq wkh wudqvirupdwlrq ri wkh uhdo surmhfwlrq rq wkh uhwlqdo sodqh iurp phwulf fr0ruglqdwhv wr wkh fr0ruglqdwhv ri wkh frpsxwhu exhu +lpdjh sodqh, lq sl{hov1 Wkh uhodwlrq ri erwk sodqhv lv vkrzq lq jxuh 51<1 Dfwxdoo|/ wklv wudqvirupdwlrq lv edvhg rq d vlpsoh vfdoh dgmxvwphqw ri erwk d{hv/ vkrzq lq htxdwlrq 514<1 Qrwh wkdw Ef_ c t_ lv wkh uhdo surmhfwlrq zlwk ohqv glvwruwlrq lq phwulf fr0ruglqdwhv/ dqg EfR c tR lv wkh vdph srlqw h{suhvvhg lq sl{hov1 fR ' & f_ tR ' & t_. +514<,. Zklfk fdq eh prghoohg lq d pdwulfldo zd| dsso|lqj wkh iroorzlqj pdwulfldo htxdwlrq/ fR & f f_ ' +5153, tR f & t_ Vxppdulvlqj wkh vwhs/ wzr fruuhfwhg frh!flhqwv kdyh wr eh prghoohg/ wkdw lv sdudphwhuv & dqg & 1 Vwhs 71 Rewdlqlqj wkh sulqflsdo srlqw Wkh sulqflsdo srlqw +dovr lpdjh fhqwuh, lv ghqhg dv d srlqw Ef c f rq wkh lpdjh sodqh jlyhq e| wkh lqwhuvhfwlrq ri wkh rswlfdo d{lv ri wkh fdphud dqg wkh lpdjh sodqh1 Wkdw lv/ wkh sulqflsdo srlqw lv wkh surmhfwlrq ri wkh irfdo srlqw rq wkh lpdjh sodqh zklfk lv jlyhq lq sl{hov1 Wkh sulqflsdo srlqw lv frqvlghuhg dv wkh ruljlq ri wkh uhwlqdo lpdjh sodqh Efo c to / dqg lw kdv dovr ehhq xvhg dv wkh ruljlq iru prghoolqj wkh ohqv glvwruwlrq ri wkh fdphud1 Krzhyhu/ wkh.

(29) FKDSWHU 51 FDOLEUDWLRQ. 63 Yr. pixel. Image Plane Xr. (0, 0) (0, 0). (Xd, Yd) (Xp, Yp). Retinal plane. Iljxuh 51<= Wudqvirupdwlrq iurp phwulf fr0ruglqdwhv ri wkh uhwlqdo sodqh wr wkh sl{ho fr0ruglqdwhv ri wkh lpdjh sodqh1 ruljlq ri wkh fr0ruglqdwh v|vwhp lq wkh frpsxwhu lpdjh sodqh lv orfdwhg dw wkh qruwk0zhvw fruqhu ri wkh lpdjh1 Iljxuh 5143 looxvwudwhv wklv uhodwlrq1 Wkhq/ d wudqvodwlrq lv qhhghg lq rughu wr sodfh wkh sulqflsdo srlqw iurp wkh fruqhu ri wkh lpdjh wr lwv uhdo orfdwlrq rq wkh lpdjh sodqh1 Wklv wudqvodwlrq lv jlyhq e| wkh ydoxhv Ef c f / zlwk uhvshfw wr hdfk d{h ri wkh fr0ruglqdwh v|vwhp1 Qrwh dovr wkdw wkh L d{h dqg T d{h ri wkh frpsxwhu lpdjh sodqh duh lqyhuwhg iurp wkh fo d{h dqg wkh to d{h ri wkh uhwlqdo lpdjh sodqh1 Jlyhq d srlqw EfR c tR h{suhvvhg zlwk uhvshfw wr wkh fdphud fr0ruglqdwh v|vwhp/ wkh vdph srlqw zloo eh h{suhvvhg zlwk uhvshfw wr wkh frpsxwhu lpdjh fr0ruglqdwh v|vwhp xvlqj wkh iroorzlqj htxdwlrqv1 Wkh uhvxowlqj srlqw uhsuh0 vhqwdwlrq lv fdoohg Ef c t / zklfk lv wkh rqh revhuyhg e| wkh xvhu lq sl{hov rq wkh lpdjh sodqh ri wkh frpsxwhu1 f ' fR n f t ' tR n f. +5154,. Zklfk fdq eh prghoohg lq d pdwulfldo zd| dsso|lqj wkh iroorzlqj htxdwlrq/ 5 6 fR f f f 7 tR 8 ' +5155, t f f .

(30) FKDSWHU 51 FDOLEUDWLRQ. 64 Yr. (0, 0) V. Xr. U Computer image co-ordinate system. (u0,v0). Camera co-ordinate system. Principal point. Iljxuh 5143= Wudqvirupdwlrq iurp wkh fr0ruglqdwh v|vwhp ri wkh fdphud lpdjh sodqh wr wkh fr0ruglqdwh v|vwhp ri wkh frpsxwhu lpdjh exhu1 Vxppdulvlqj wkh vwhs/ wkh srvlwlrq ri wkh sulqflsdo srlqw rq wkh lpdjh sodqh Ef c f kdv wr eh prghoohg1. 51516. Vxppdu| ri Fdphud prghoolqj. Wklv vhfwlrq looxvwudwhv judsklfdoo| wkh vwhsv ri fdphud prghoolqj lq rughu wr surylgh d eulhi vxppdu| iru wkh uhdghu1 Wzr glhuhqw fdphud prghov zloo eh jlyhq1 Iluvwo|/ d fdphud prgho zlwkrxw ohqv glvwruwlrq zklfk doorzv xv wr h{suhvv wkh uhodwlrq ehwzhhq wkh 6G remhfw srlqw dqg wkh 5G revhuyhg surmhfwlrq xvlqj olqhdu htxdwlrqv1 Vhfrqgo|/ d fdphud prgho zlwk udgldo ohqv glvwruwlrq/ zklfk zloo eh prghoohg xvlqj qrq0olqhdu htxdwlrqv1 Qrwh wkdw wkh 6G remhfw srlqw lv uhihuuhg wr d zruog fr0ruglqdwh v|vwhp dqg wkh revhuyhg 5G srlqw rq wkh lpdjh sodqh ri wkh frpsxwhu lv h{suhvvhg lq sl{hov1 Wkh zkroh fdphud prgho zlwkrxw ohqv glvwruwlrq Lq rughu wr h{suhvv wkh uhodwlrq ehwzhhq wkh phwulf 6G remhfw srlqw uhihuuhg wr d zruog fr0ruglqdwh v|vwhp dqg lwv revhuyhg 5G surmhfwlrq rq wkh frpsxwhu lpdjh sodqh/ h{suhvvhg lq sl{hov/ wkh pdwulfldo htxdwlrqv zloo eh h{sdqghg1 Wklv uhodwlrq lv xvxdoo| h{suhvvhg dv wkh surgxfw ri wzr pdwulfhv1 Wkh uvw pdwul{ h{suhvvhv wkh hqwluh lqwulqvlf sdudphwhuv ri wkh fdphud vhqvru/ dqg wkh.

(31) FKDSWHU 51 FDOLEUDWLRQ. 65. (Xw, Yw, Zw) 3D object point with respect to world co-ordinate system. Affine transformation. Modelled parameters: R, T (Xc, Yc, Zc) 3D object point with respect to camera co-ordinate system. Perspective transformation. Modelled parameter: f (Xu, Yu) Ideal projection on the retinal plane. Pixel adjustment Modelled parameters: ku, kv (Xp, Yp) Real projection on the image plane. Adaptation to the computer image buffer Modelled parameters: u0, v0 (Xi, Yi) Real projection on the image plane. Iljxuh 5144= Prghoolqj d fdphud zlwkrxw ohqv glvwruwlrq1 vhfrqg rqh wkh h{wulqvlf sdudphwhuv1 Vr wkdw/ wkh lqwulqvlf pdwul{ lv rewdlqhg vxevwlwxwlqj 5147 lqwr 514</ dqg wkh uhvxow lqwr 5154/ rewdlqlqj wkh iroorzlqj htxdwlrq/ fS n f ~S t S t ' & s n f ~S. f ' & s. +5156,. zklfk fdq eh h{suhvvhg lq d pdwulfldo zd| dv/ 6 fS rf k f f f 9 : 7 rt 8 ' 7 f k f f 8 9 t S : 7 ~S 8 r f f f 5. 6. 5. 6. 5. +5157,. zkhuh/ k ' s& @ ' s&. +5158,. Wkhq/ dsso|lqj 5144 lq rughu wr jlyh wkh remhfw srlqw zlwk uhvshfw wr wkh zruog fr0ruglqdwh v|vwhp/ wkh iroorzlqj pdwulfldo uhodwlrq lv rewdlqhg/.

(32) FKDSWHU 51 FDOLEUDWLRQ. 66. (Xw, Yw, Zw) 3D object point with respect to world co-ordinate system. Affine transformation. Modelled parameters: R, T (Xc, Yc, Zc) 3D object point with respect to camera co-ordinate system. Perspective transformation. Modelled parameter: f (Xu, Yu) Ideal projection on the retinal plane. Radial lens distortion. Modelled parameter: k1 (Xd, Yd) Real projection on the retinal plane. Pixel adjustment Modelled parameters: ku, kv (Xp, Yp ) Real projection on the image plane. Adaptation to the computer image buffer Modelled parameters: u0, v0 (Xi, Yi ) Real projection on the image plane. Iljxuh 5145= Prghoolqj d fdphud zlwk udgldo ohqv glvwruwlrq1. 5. 6. 5. 6. 5. o rf k f f f 9 7 rt 8 ' 7 f k f f 8 9 o2 7 o r f f f f. o2 o22 o2 f. o o2 o f. 65 |% f : 9 |+ : 9 t |5 8 7 ~ . 6 : : 8. +5159,. Wkh zkroh fdphud prgho zlwk udgldo ohqv glvwruwlrq Lq wklv fdvh/ d olqhdu uhodwlrq grhv qrw krog wuxh1 Dfwxdoo|/ wkh htxdwlrqv lqwurgxfhg e| wkh prghoolqj ri ohqv glvwruwlrq grhv qrw shuplw wkh xvh ri pdwulfhv lq rughu wr h{suhvv wkh 5G shuvshfwlyh srlqw zlwk uhvshfw wr wkh 6G remhfw srlqw1 Wkxv/ frpsxwlqj dqg duudqjlqj wkh htxdwlrqv suhvhqwhg lq wkh vwhsv ri fdphud prghoolqj/ wkh iroorzlqj uhodwlrq lv rewdlqhg1 fS ' f_ n & o2 f_ ~S t S s ' t_ n & o2 t_ ~S. s. zkhuh/ f_ '. Ef f &. +515:, +515;,.

(33) FKDSWHU 51 FDOLEUDWLRQ t_ '. 67 Et f &. +515<,. dqg/ 5. 6 f fS : 9 7 t S 8 ' + A 9 t : 7 ~ 8 ~S 5. 6. +5163,. Uhodwlrq ehwzhhq edfn dqg iurqw lpdjh sodqh Vrph dxwkruv frqvlghu wkdw wkh rswlfdo fhqwuh lv prghoohg lq iurqw ri wkh lpdjh sodqh dv vkrzq lq jxuh 51461 Wkh lpdjh sodqh lv sodfhg dw d glvwdqfh s ri wkh rswlfdo fhqwuh/ qrwh dovr wkdw erwk d{hv/ L dqg T duh lqyhuwhg zlwk uhvshfw wr fS dqg tS 1 Doo wkh pdwkhpdwlfdo htxdwlrqv lqwurgxfhg lq rughu wr prgho d fdphud zlwk wkh lpdjh sodqh lq iurqw ri wkh rswlfdo fhqwuh duh dovr ydolg wr prgho vxfk fdphud frqjxudwlrq1 Krzhyhu/ d vljq kdv fkdqjhg erwk dv d uhvxow ri wkh qrq0lqyhuvlrq ri wkh L lpdjh fr0ruglqdwh d{h zlwk uhvshfw wr wkh fo uhwlqdo d{h/ dqg wkh qhjdwlyh srvlwlrq ri wkh lpdjh sodqh dorqj wkh ~S d{h1 Frpsxwlqj/ wkh vdph pdwul{ uhodwlrq vkrzq lq htxdwlrq 5159 lv ydolg/ exw/ k ' s& @ ' s &. +5164,. Wkhq/ sodflqj wkh irfdo srlqw ehklqg ru lq iurqw ri wkh lpdjh sodqh zloo rqo| surgxfh dq lqyhuvlrq ri wkh k sdudphwhu1. 516. Fdoleudwlqj d fdphud. Fdoleudwlrq lv edvhg rq rewdlqlqj wkh lqwulqvlf dqg h{wulqvlf sdudphwhuv ri wkh fdphud prgho nqrzlqj wkh frpsrqhqwv ri d vhw ri 6G remhfw srlqwv dqg lwv 5G surmhfwlyh srlqwv1 Lq wklv vhfwlrq/ wzr fdoleudwlqj dojrulwkpv duh vkrzq1 Iluvw/ d phkrg edvhg rq pdwul{ frpsxwdwlrq zklfk shuplwv wr rewdlq wkh fdphud sdudphwhuv zlwkrxw frqvlghulqj ohqv glvwruwlrq1 Vhfrqgo|/ d pruh jhqhulf phwkrg edvhg rq ixqfwlrq plqlplvdwlrq zklfk shuplwv wr rewdlq wkh sdudphwhuv ri dq| fdphud prgho/ hyhq frqvlghulqj ohqv glvwruwlrq1.

(34) FKDSWHU 51 FDOLEUDWLRQ image co-ordinate system. 68. Yr. u. w Oi v. Xr Yc. (u0, v0). f Pd. Pu. C. Camera co-ordinate system. Zc. Oc. Plane Image. P. Xc. Yw World co-ordinate system. Xw Zw Ow. Iljxuh 5146= Wkh jhrphwulf uhodwlrq ehwzhhq d 6G remhfw srlqw dqg lwv 5G surmhfwlrq zkhq wkh rswlfdo fhqwuh lv orfdwhg lq iurqw ri wkh lpdjh sodqh1. 51614. Fdoleudwlqj wkh olqhdu prgho. Wkh olqhdu fdoleudwlrq phwkrg lv edvhg rq rewdlqlqj wkh lqwulqvlf dqg h{wulqvlf sdudphwhuv ri d fdphud zklfk kdv ehhq prghoohg zlwkrxw frqvlghulqj ohqv glvwruwlrq ^Wrvf ;:`1 Wkhq/ wkh uhodwlrq ehwzhhq d 6G remhfw srlqw dqg lwv 5G surmhfwlrq lv jlyhq e| wkh wudqvirupdwlrq pdwul{ ri wkh htxdwlrq 51591 Lq wklv fdvh/ fdoleudwlrq lv uhgxfhg wr wkh frpsxwdwlrq ri wklv wudqvirupdwlrq pdwul{/ zklfk lv fdoohg / dqg wkh ixuwkhu h{wudfwlrq ri wkh fdphud sdudphwhuv iurp wkh rewdlqhg pdwul{ Êruw <4` Êudv <8`1 Zkhq frpsxwlqj wkh pdwul{ surgxfw ri htxdwlrq 5159/ wkhq wkh iroorzlqj 6{7 wudqvirupdwlrq pdwul{ lv rewdlqhg1 Wkh pdwul{ fdq eh h{suhvvhg e| wkuhh rulhqwdwlrq yhfwruv dqg wkuhh wudqvodwlrq frpsrqhqwv e 1 5 6 5 6 k o n f o k |% n f |5 e ' 7 k o2 n f o k |+ n f |5 8 ' 7 2 2e 8 +5165, o |5 e Qrwh wkdw/ 5. 6 5 6 r e 7 r 8 ' 7 2 2e 8 r e. +5166,.

(35) FKDSWHU 51 FDOLEUDWLRQ. 69. zkhuh ' EfS c t S c ~S | lv d 6G remhfw srlqw zlwk uhvshfw wr wkh zruog fr0ruglqdwh v|vwhp/ h{suhvvhg lq pp1/ dqg Ec lv lwv 5G surmhfwlyh srlqw rq wkh lpdjh sodqh h{suhvvhg lq sl{hov1 Nqrzlqj wkdw wkh rulhqwdwlrq yhfwruv o duh xqlwduldq dqg ruwkrjrqdo/ dqg nqrzlqj wkdw wkh vfdodu surgxfw ri wzr yhfwruv lv htxdo wr wkh pxowlsolfdwlrq ri wkhlu qrupv zlwk wkh frvlqxv ri wkh dqjoh ehwzhhq wkhp/ dv 2 ' n n n2 n ULt k. +5167,. zkhuh wkh qrup ri d yhfwru lv frpsxwhg e| wkh iroorzlqj htxdwlrq1 *2 nn ' %2 n +2 n 5 2. +5168,. Ixuwkhupruh/ lw fdq eh d!uphg wkdw wkh rulhqwdwlrq yhfwruv nhhs wkh iroorzlqj uhodwlrq/ o o| ' f o o| ' . 9' '. Wkhq/ wkh irxu lqwulqvlf sdudphwhuv ri wkh fdphud +k / k / f / f ,/ dqg wkh vl{ h{wulqvlf sdudphwhuv +o /o2 / o / |% / |+ / |5 ,/ fdq eh rewdlqhg iurp 5165/ dv/ f ' | *2 k ' E | 2f o ' k E f o2 ' k E2 f o '. f ' 2 | *2 k ' E2 |2 f2 |% ' k Ee f e |+ ' k E2e f e |5 ' e. +5169,. Vr/ lw kdv ehhq suryhg wkdw/ jlyhq wkh wudqvirupdwlrq pdwul{ / zklfk uhodwhv wkh frpsxwhu lpdjh fr0ruglqdwh v|vwhp/ zlwk uhvshfw wr wkh zruog fr0ruglqdwh v|vwhp/ wkh lqwulqvlf dqg h{wulqvlf sdudphwhuv fdq eh rewdlqhg1 Wkhq/ zh kdyh wr h{sodlq wkh phwkrg xvhg wr rewdlq wkh wudqvirupdwlrq pdwul{ 1 5166 pxvw eh frpsxwhg lq rughu wr h{suhvv wkh uhodwlrq e| olqhdu htxdwlrqv/ n e E n e ' f. +516:,. 2 n 2e E n e ' f. +516;,. dqg/. Wkhvh htxdwlrqv pd| eh duudqjhg lq wkh iroorzlqj zd|/.

(36) FKDSWHU 51 FDOLEUDWLRQ. '. 6:. e n e e e. +516<,. dqg 2 2e n e e e zklfk fdq eh uhodwhg dv d fohdu vlpsolfdwlrq e|/ '. +5173,. ' A n A2 . +5174,. ' A n 2 A2 . +5175,. dqg. Zkhuh/ iurp wkh htxdwlrqv 5169/ 5174 dqg 5175/ wkh iroorzlqj htxdwlrqv duh rewdlqhg/ A ' o|5 f n o|5 k A2 ' o|5 A ' o|5 f n o|52 k. ' f n ||%5 k 2 ' f n. |+ |5 k . +5176,. Zkhuh hdfk A / A2 dqg A duh wkuhh yhfwruv ri wkuhh frpsrqhqwv1 Dqg dqg 2 duh wzr vlqjoh ydoxhv1 Vr/ zh fdq ghqh d qhz yhfwru f/ ri hohyhq frpsrqhqwv1 5 6 A 9 A2 : 9 : : A f'9 +5177, 9 : 7 8 2 Lq rughu wr lqihu wkh f yhfwru/ lw lv qhfhvvdu| wr rewdlq d vhw ri frxsohv ri srlqwv1 Hdfk frxsoh lv frpsrvhg e| d phwulf 6G remhfw srlqw uhihuuhg wr d zruog fr0ruglqdwh v|vwhp/ dqg lwv 5G surmhfwlyh srlqw rq wkh lpdjh sodqh/ h{suhvvhg lq sl{hov1 Wkh vhw ri srlqwv kdv wr eh uhsuhvhqwdwlyh/ l1h1 pxvw suhvhqw d glvfuhsdqf| dprqj wkh 6G remhfw srlqwv dorqj wkh wkuhh d{lv ri wkh zruog fr0ruglqdwh v|vwhp/ wkdw lv wkh| kdyh wr eh qrq0frsodqdu1 Lw kdv ehhq ghprqvwudwhg wkhrulfdoo| wkdw dw ohdvw vl{ qrq0frsodqdu srlqwv duh qhhghg lq rughu wr lqihu f dffxudwho|1 Wkdw*v ehfdxvh vl{ srlqwv ghvfulehv wzhoyh htxdwlrqv/ dqg hohyhq htxdwlrqv lv wkh plqlpxp qxpehu ri htxdwlrqv wkdw doorz xv wr rewdlq wkh xqnqrzqv ri f1.

(37) FKDSWHU 51 FDOLEUDWLRQ. 6;. Frqvlghu wkh vhw ri srlqw frxsohv dv 7 ' ^?'f E| c E c 1 Iurp wkh uhodwlrq lqwurgxfhg lq htxdwlrq 5174 dqg 5175/ wzr qhz pdwulfhv " dqg fdq eh ghqhg dv/ ' "f. +5178,. wkhq/ 5 9 | "'9 7 f%. | f% | | . 6 f : : f 8. +5179,. dqg/ 5. 6 9 : : '9 7 8 . +517:,. Wkhq/ wkh f frpsrqhqwv fdq eh ghwhuplqhg xvlqj wkh iroorzlqj htxdwlrq/ 3 | f ' "| " ". +517;,. Wr rewdlq wkh lqwulqvlf dqg h{wulqvlf sdudphwhuv iurp wkh ydoxhv ri wkh f frpsrqhqwv/ zh fdq xvh wkh rulhqwdwlrq yhfwru fkdudfwhulvwlfv/ ri zklfk lw lv nqrzq wkdw wkh| duh xqlwduldq/ l1h1 wkhlu qrup lv htxdo wr 41 Wkhq/ iurp 5176 wkh h{wulqvlf |5 1 sdudphwhu fdq eh rewdlqhg1 Vr wkdw/ li/ no n ' . +517<,. wkhq/ |5 '. nA2 n. dqg/ wkh rwkhu sdudphwhuv fdq eh rewdlqhg xvlqj wkh uhodwlrqvkls ehwzhhq wzr rulhqwdwlrq yhfwruv lq uhvshfw ri wkh vfdodu dqg wkh yhfwru surgxfw1 Vr/ vlqfh lw lv nqrzq wkdw/ 2 ' n n n2 n ULt k a 2 ' n n n2 n t? k dqg. +5183,.

(38) FKDSWHU 51 FDOLEUDWLRQ. 6<. *2 nn ' %2 n +2 n 52. +5184,. wkhq/ o o| o o| o a o o a o. ' ' ' '. f f. 9' ' 9' '. Wkh lqwulqvlf sdudphwhuv fdq eh rewdlqhg iurp/ A A2| nA2 n2 A2 A | f ' nA2 n2 nA| a A2| n k ' nA2 n2 nA2| a A | n k ' nA2 n2 f '. +5185, +5186, +5187, +5188,. dqg wkh h{wulqvlf rqhv/ o o2 o. nA2 n ' A nA| a A2| n nA2 n ' A nA2| a A | n A2 ' nA2 n. A A2| A2 nA2 n2 A2 A | A2 nA2 n2. +5189, +518:, +518;,. dqg nA2 n |% ' nA| a A2| n nA2 n |+ ' 2 nA2| a A | n |5 ' nA2 n. A A2| nA2 n2 A2 A | nA2 n2. +518<, +5193, +5194,.

(39) FKDSWHU 51 FDOLEUDWLRQ. 73. Qrwh wkdw zh fdq h{wudfw wkh Hxohu urwdwlrq dqjohv dsso|lqj wkh htxdwlrq 51: lqwurgxfhg lq wkh vhfwlrq ghglfdwhg wr wkh h{wulqvlf sdudphwhuv1 Wkhq dsso|lqj wkh iroorzlqj hjdolvdwlrq/ 5. 6 5 6 SS rSwr Sr rSwS rrw o o2 o + ' 7 SS n SSwr rr n SSwS Srw 8 ' 7 o2 o22 o2 8 o o2 o rwr rwS Sw +5195, wkh c w dqg dqjohv fdq eh frpsxwhg dsso|lqj/ . o ' @hU|@? o2 w ' @hUULt Eo o ' @hU|@? o2. +5196, +5197, +5198,. Dojrulwkp ri fdphud fdoleudwlrq Lq wkh iroorzlqj/ wkh zkroh dojrulwkp xvhg lq wkh fdoleudwlrq ri d olqhdu fdphud prgho lv suhvhqwhg1 Wkh dojrulwkp kdv ehhq surjudpphg xvlqj wkh U hqylurqphqw1 Pdwode. fohdu doo> forvh doo> g @ irshq+*vhwbribsrlqwv1suq*/*u*,> S @ ivfdqi+g/*(i*/^8/qxpbribfrxsohv `,> iforvh+g,> qsrlqwv @ vl}h+S/5,> S5G @ S+4=5/=,> S6G @ S+6=8/=,> {n @ S6G+4/=,> |n @ S6G+5/=,> }n @ S6G+6/=,> xn @ S5G+4/=,> yn @ S5G+5/=,> D @ ^`> E @ ^`> iru l @4=qsrlqwv/ D @ ^D > {n+l, |n+l, }n+l, 0xn+l,-{n+l, 0xn+l,-|n+l, 0xn+l,-}n+l, 3 3 3 4 3> 3 3 3 0yn+l,-{n+l, 0yn+l,-|n+l, 0yn+l,-}n+l, {n+l, |n+l, }n+l, 3 4`> E @ Ê > xn+l,> yn+l,`> hqg [ @ +lqy+D*-D,,-D*-E> [4 @ [+4=6/4,*> [5 @ [+7=9/4,*> [6 @ [+:=</4,*> F4 @ [+43/4,> F5 @ [+44/4,> Qruph[5 @ +[5+4/4, - [5+4/4,, . +[5+4/5,-[5+4/5,, . +[5+4/6,-[5+4/6, ,>.

(40) FKDSWHU 51 FDOLEUDWLRQ. 74. SY @ furvv+[4+4/=,*/[5+4/=,*,> [4W[5W @ vxp+SY+4,a5.SY+5,a5.SY+6,a5,> SY @ furvv+[5+4/=,*/[6+4/=,*,> [5W[6W @ vxp+SY+4,a5.SY+5,a5.SY+6,a5,> x3 @ +[4 - [5*, 2 Qruph[5> y3 @ +[5 - [6*, 2 Qruph[5> dx @ vtuw+[4W[5W, 2Qruph[5> dy @ vtuw+[5W[6W, 2Qruph[5> u4 @ +vtuw+Qruph[5,2vtuw+[4W[5W,, - +[40++[4-[5*,2Qruph[5,-[5,> u5 @ +vtuw+Qruph[5,2vtuw+[5W[6W,, - +[60++[5-[6*,2Qruph[5,-[5,> u6 @ [52vtuw+Qruph[5,> w{ @ +vtuw+Qruph[5,2vtuw+[4W[5W,, - +F40++[4-[5*,2Qruph[5,,> w| @ +vtuw+Qruph[5,2vtuw+[5W[6W,, - +F50++[5-[6*,2Qruph[5,,> w} @ 42vtuw+Qruph[5,> W4 @ ^ dx 3 x3 3> 3 dy y3 3> 3 3 4 3> 3 3 3 4`> W5 @ ^ u4 w{> u5 w|> u6 w}> 3 3 3 4`>. 51615. Fdoleudwlqj wkh qrq0olqhdu prgho. Wkh qrq0olqhdu fdoleudwlrq ri d fdphud lv edvhg rq wkh jhqhudo sulqflsoh ri ixqfwlrq plqlplvdwlrq zklfk kdv ehhq zlgho| xvhg lq rughu wr rewdlq d vrox0 wlrq ri d qrq0olqhdu ixqfwlrq1 Wkhq/ wklv sulqflsoh fdq dovr eh xvhg wr rewdlq wkh lqwulqvlf dqg h{wulqvlf sdudphwhuv ri dq| fdphud prgho/ hyhq li ohqv glv0 wruwlrq kdv ehhq frqvlghuhg1 Wkh uhodwlrq ehwzhhq wkh 6G remhfw srlqwv dqg lwv 5G surmhfwlyh srlqw lv d qrq0olqhdu uhodwlrq zklfk srlqwv xv wr xvh dq lw0 hudwlyh phwkrg lq rughu wr uhgxfh wkh glvfuhsdqf| ehwzhhq wkh revhuyhg 5G surmhfwlrq dqg wkh prghoohg rqh/ lwhudwlrq e| lwhudwlrq1 Wkh Lwhudwlyh Dssur{lpdwlrq Phwkrg Wkh Lwhudwlyh Dssur{lpdwlrq Phwkrg lv dq dpsoldwlrq wr ? xqnqrzqv ri wkh phwkrg ri Qhzwrq0Udskvrq wr rewdlq wkh urrwv ri d ixqfwlrq1 Lw kdv ehhq zlgho| xvhg ryhu wkh sdvw ghfdghv dqg lv wkh prvw zlgho| xvhg vwdqgdug rswlplvdwlrq whfkqltxh1 Wkh xqnqrzq frh!flhqwv duh uvw fdofxodwhg xvlqj dq dssur{lpdwh lqlwldo jxhvv ri wkh fdphud sdudphwhu ydoxhv1 Wkhq/ wkh ixqfwlrq lv hydoxdwhg dqg wkh lghdo frpsrqhqwv duh rewdlqhg1 Glhuhqfhv ehwzhhq wkh lghdo dqg wkh uhdo sdudphwhuv duh hvwlpdwhg dv wkh huuru ixqfwlrq1 Wkh fdoleudwlrq surfhgxuh lv wkh lwhudwlyh rswlplvdwlrq ri fdphud sdudphwhuv e| plqlplvlqj wkh huuru ixqfwlrq1 Khuhdiwhu zloo iroorz d ghvfulswlrq ri wkh phwkrg/ dqg krz lw kdv ehhq rulhqwhg wr wkh ghwhuplqdwlrq ri wkh fdphud sdudphwhuv1 Li wkh uhdghu lv lqwhu0 hvwhg lq wkh pdwkhpdwlfdo sulqflsoh ri wkh phwkrg/ kh lv lqylwhg wr uhdg errnv.

(41) FKDSWHU 51 FDOLEUDWLRQ. 75. rq Qxphulfdo Dqdo|vlv ^Vwrh ;3`1 Krzhyhu/ wkh jhqhudo sulqflsoh lv wkh iro0 orzlqj= Frqvlghu d ixqwlrq C zklfk ghshqgv rq 6 xqnqrzqv i% c %2 c c %6 j1 Frqvlghu wkdw zh zdqw wr rewdlq wkh ydoxhv ri i% c %2 c c %6 j zklfk vroyh wkh htxdwlrq C E% c %2 c c %6 ' f lwhudwlqj iurp dq lqlwldo vroxwlrq txlwh forvh wr wkh uljkw xqnqrzq ydoxhv1 Wkhq/ iurp d ghwhuplqhg lwhudwlrq zh fdq dssur{lpdwh wkh qh{w xqnqrzqv iurp wkh odvw rqhv/ xvlqj htxdwlrq 519<1 Zh nqrz wkdw d forvhu vroxwlrq CEf& ri CEf fdq eh dssur{lpdwhg iurp CEf&3 e| 51991 Zh nqrz dovr wkdw wkh vroxwlrq zh zdqw wr qg lv wkh yhfwru f ri xqnqrzqv zklfk vroyh 519:1 CEf& CEf&3 n a Ef&3 {f& CEf& ' f. +5199, +519:,. Vr/ CEf&3 n a Ef&3 {f& ' f +519;, Wkhq/ 519< pxvw eh lwhudwhg xqwlo wkh rewdlqhg sdudphwhuv duh vpdoohu wkdq d {hg ydoxh 0/ dv vkrzv 51:31 {f& ' a 3 Ef&3 CEf&3 {f& 0. +519<, +51:3,. Wr dgdsw htxdwlrq 519< wr wkh sureohp ri plqlplvlqj dq htxdwlrq C ri 6 xqnqrzqv i% c %2 c c %6 jc1 zh vxssrvh kdylqj rewdlqhg ? frxsohv ri srlqwv 7 ' ^?'f E| c E c / zkhuh lv d 6G remhfw srlqw dqg E c lv lwv sur0 mhfwlyh srlqw rq wkh fdphud lpdjh sodqh1 Wkhq wkh yhfwru ri xqnqrzqv i% c %2 c c %6 j/ l1h1 wkh fdphud sdudphwhuv/ fdq eh rewdlqhg lq wkh iroorzlqj zd|1 Ghqh C Ef&3 dv wkh hydoxdwlrq ri ixqfwlrq C xvlqj wkh xqnqrzqv hvwlpdwhg lq wkh & lwhudwlrq dqg wkh srlqw fruuhvsrqghqfh E| c E c 1 Ghqh yhfwru CEf&3 dv lw lv uhodwhg lq htxdwlrq 51:41 CEf&3 ' EC Ef&3 c C2 Ef&3 c c C? Ef&3 | +51:4, Wkhq/ zh fdq ghqh pdwul{ aECEf&3 dv d pdwul{ ri sduwldo glhuhqwldov ri CEf hydoxdwhg zlwk wkh ydoxhv ri yhfwru f lq wkh lwhudwlrq &/ dv vkrzq lq 51:5/ 3 4 aEC Ef&3 E aEC2 Ef&3 F F aECEf&3 ' E +51:5, C D aEC? Ef&3 .

(42) FKDSWHU 51 FDOLEUDWLRQ. 76. zkhuh/ aEC Ef&3 '. YC Ef&3 YC Ef&3 YC Ef&3 c c c Y% Y%2 Y%6. +51:6,. dqg f lv wkh yhfwru ri xqnqrzqv dv vkrzv 51:71 f ' E% c %2 c c %6 . +51:7,. {f ' E{% c {%2 c c {%6 . +51:8,. wkhq/. Vr/ htxdwlrq 519; fdq eh h{suhvvhg dv 51:91 4 3 YC Ef&3 C Ef &3 Y% YC2 Ef&3 E C2 Ef F E & 3 F n E E Y% D E C C YC E f&3 ? C? Ef &3 3. Y%. YC Ef&3 Y%2. . Ef&3 YCY% 6 YC? Ef&3 Y%6. 43 4 {% FE F E {%2 F F 'f FC D ?% D {%6. +51:9, Wkhq/ zh fdq vroyh hdfk lwhudwlrq duudqjlqj htxdwlrq 51:9 lq wkh irup ri htxdwlrq 519<1 Zh zloo lwhudwh xqwlo {f& 0/ wkhq/ zh zloo kdyh wkh yhfwru ri xqnqrzqv zklfk vroyh C1 Vr/ li wkh htxdwlrq C lv wkh fdphud prgho zklfk shuplwv wr rewdlq d 5G surmhfwlyh srlqw rq wkh lpdjh sodqh iurp d jlyhq 6G remhfw srlqw/ wkh fdphud sdudphwhuv ri vxfk d prgho fdq eh frpsxwhg1 Qrwh wkdw htxdwlrq 519< pxvw eh vroyhg xvlqj wkh svhxgr0lqyhuvh pdwul{ dv lw grhv qrw kdyh wkh vdph qxpehu ri urzv dqg froxpqv/ vr htxdwlrq 519< fdq eh vroyhg xvlqj htxdwlrq 51::1 3 | a Ef&3 CEf&3 {f& ' a | Ef&3 aEf&3 . +51::,. Wkh uhdghu lv uhihuuhg wr wkh qh{w vhfwlrq iru dq lqwurgxfwlrq ri wkh Prg0 lhg Qhzwrq0Udskvrq Phwkrg/ zklfk lpsrvhv d uhgxfwlrq ri wkh huuru dw hdfk lwhudwlrq1 Wkh prglhg Qhzwrq0Udskvrq phwkrg Wkh Qhzwrq0Udskvrq wkhruhp jxdudqwhhv wkh frqyhujhqfh ri d jlyhq ixqfwlrq wr d orfdo vroxwlrq rqo| li wkh vwduwlqj vroxwlrq ff lv fkrvhq vx!flhqwo| forvh wr wkh ghvluhg vroxwlrq1 Wkhq/ wkh Qhzwrq0Udskvrq phwkrg pd| glyhujh lq vrph fdvhv1 Wkh prglhg Qhzwrq Udskvrq Phwkrg jxdudqwhhv dozd|v.

(43) FKDSWHU 51 FDOLEUDWLRQ. 77. d uhgxfwlrq ri wkh huuru lq hdfk lwhudwlrq/ l1h1 d frqyhujhqfh ri wkh phwkrg ^Vwrh ;3`1 Krzhyhu/ wkh uhdghu vkrxog qrwh wkdw wkh phwkrg pd| frqyhujh wr dq xqghvluhg vroxwlrq/ hyhq kdylqj d odujh huuru1 Iurp wkh htxdwlrq 519< ri wkh Qhzwrq0Udskvrq phwkrg/ zh nqrz wkdw wkh huuru ri hdfk lwhudwlrq vkrxog eh frpsxwhg xvlqj htxdwlrq 51:;/ {f& ' a 3 Ef&3 CEf&3 . +51:;,. Krzhyhu/ zh kdyh douhdg| vdlg wkdw htxdwlrq 51:; lv qrw dozd|v vdwlvhg/ l1h1 wkh huuru grhv qrw uhgxfh lq hdfk lwhudwlrq/ wkhq/ htxdwlrq 51:< lv qrw dozd|v vdwlvhg1 {f&. {f&3. +51:<,. Wkh prglhg Qhzwrq0Udsvrq phwkrg sursrvhv wr frpsxwh wkh huuru xv0 lqj wkh iroorzlqj htxdwlrq/ {f& ' 23 a 3 Ef&3 CEf&3 . +51;3,. f& ' f&3 23 a 3 Ef&3 CEf&3 . +51;4,. wkhq/. Frqvlghu wkh iroorzlqj vxevwlwxwlrq wr vlpsoli| wkh frpsxwdwlrq= _& ' a 3 Ef&3 CEf&3 . +51;5,. Wkhq/ wkh frpsxwdwlrq ri wkh Prglhg Qhzwrq0Udsvrq phwkrg lv uh0 gxfhg wr frpsxwh wkh yduldeoh/ l1h1 lw lv uhgxfhg wr rewdlq wkh plqlpxp srqghudwlrq ri wkh Mdfreldq pdwul{ zklfk uhgxfhv wkh huuru lq hdfk lwhudwlrq/ ' 6? f m CEf& 23 _& . nCEf& n. +51;6,. zkhuh/ nCEf& n '. v[ 6. %2 . [ 6. %2. +51;7,. Wkh Prglhg Qhzwrq0Udskvrqv dojrulwkp lv wkh iroorzlqj/ 41 Il{ dq lqlwldo vroxwlrq ff 1 51 Lwhudwh/ 5141 Frpsxwh CEf& dqg aEf& 1 5151 Frpsxwh _& 1 5161 Lwhudwh iurp f xs wr nCEf& 23 _& n. nCEf& n1.

(44) FKDSWHU 51 FDOLEUDWLRQ. 78. 5171 Frpsxwh {f& ' 23 a 3 Ef&3 CEf&3 1 5181 Frpsxwh f& ' f&3 n {f& 1 61 xqwlo {f& 0 lv uhdfkhg1 Dojrulwkp ri fdphud fdoleudwlrq Lq wkh iroorzlqj/ wkh hqwluh dojrulwkp xvhg lq wkh fdoleudwlrq ri dq| fdphud prgho lv suhvhqwhg1 Wkh dojrulwkp kdv ehhq surjudpphg xvlqj wkh Pdsoh U hqylurqphqw1 Wkh h{dpsoh vhwv iruwk wkh surjudpplqj ri d Y uhohdvh 7 fdphud prgho frqvlghulqj wkh uvw frh!flhqw ri ohqv glvwruwlrq1 Wkh lqlwldo vroxwlrq frxog eh rewdlqhg dsso|lqj wkh phwkrg ghvfulehg lq wkh vhfwlrq zklfk ghvfulehv wkh fdoleudwlrq ri d olqhdu fdphud prgho dqg frqvlghulqj qr ohqv glvwruwlrq/ l1h1 & ' f1 Wkh irfdo glvwdqfh kdv ehhq {hg/ rwkhuzlvh wkh v|vwhp lv xqvwdeoh ehfdxvh vhyhudo frpelqdwlrqv ri Esc & c & fdq eh rewdlqhg zlwkrxw fkdqjlqj wkh E@ c @ ydoxhv1 Dfwxdoo|/ wkh dojrulwp fdq eh hdvlo| prglhg wr rewdlq rqo| wkh E@ c @ sdudphwhuv1. A zlwk+olqdoj,= A &&&&&&&&&&&&&&&&&&&&&&&&&&&& A & Rewdlqlqj wkh fdphud prgho A &&&&&&&&&&&&&&&&&&&&&&&&&&&& A U{=@duud|+^^4/3/3/3`/^3/frv+w4,/vlq+w4,/3`/^3/0vlq+w4,/frv+w4,/3`/^3/3/3/4``,> A U|=@duud|+^^frv+w5,/3/0vlq+w5,/3`/^3/4/3/3`/^vlq+w5,/3/frv+w5,/3`/^3/3/3/4``,> A U}=@duud|+^^frv+w6,/vlq+w6,/3/3`/^0vlq+w6,/frv+w6,/3/3`/^3/3/4/3`/^3/3/3/4``,> A W=@duud|+^^4/3/3/W{`/^3/4/3/W|`/^3/3/4/W}`/^3/3/3/4``,> A U=@pxowlso|+pxowlso|+U}/U|,/U{,> A DD=@pxowlso|+W/U,> A P=@duud|+^^{u`/^|u`/^}u`/^4``,> A J=@pxowlso|+DD/P,> A [=@J^4/4`> \=@J^5/4`> ]=@J^6/4`> A [x=@[-i2]> A \x=@\-i2]> A [g=@+xn0x3,2nx> \g=@+yn0y3,2ny> A X=@[x0[g0n4-+[g-[g.\g-\g,-[g> A Y=@\x0\g0n4-+[g-[g.\g-\g,-\g> A &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& A & Rewdlqlqj wkh sduwldo ghulydwlyh A &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& A gXw4=@gl+X/w4,> gYw4=@gl+Y/w4,> A gXw5=@gl+X/w5,> gYw5=@gl+Y/w5,> A gXw6=@gl+X/w6,> gYw6=@gl+Y/w6,>.

(45) FKDSWHU 51 FDOLEUDWLRQ. 79. A gXW{=@gl+X/W{,> gYW{=@gl+Y/W{,> A gXW|=@gl+X/W|,> gYW|=@gl+Y/W|,> A gXW}=@gl+X/W},> gYW}=@gl+Y/W},> A gXi=@gl+X/i,> gYi=@gl+Y/i,> A gXnx=@gl+X/nx,> gYnx=@gl+Y/nx,> A gXny=@gl+X/ny,> gYny=@gl+Y/ny,> A gXx3=@gl+X/x3,> gYx3=@gl+Y/x3,> A gXy3=@gl+X/y3,> gYy3=@gl+Y/y3,> A gXn4=@gl+X/n4,> gYn4=@gl+Y/n4,> A &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& A & Uhdglqj wkh vhw ri srlqwv frxsohv A &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& A & Uhdglqj wkh P Pdwul{ iurp dq DVFLL oh1 A uhdgole+uhdggdwd,> A Ss=@uhdggdwd+vhwbribfrxsohv/8,> A S5G=@pdwul{+urzglp+Ss,/5/3,=S6G=@pdwul{+urzglp+Ss,/6/3,= A iru l iurp 4 e| 4 wr urzglp+Ss, gr A S5G^l/4`=@rs+4/rs+l/Ss,,= S5G^l/5`=@rs+5/rs+l/Ss,,= A S6G^l/4`=@rs+6/rs+l/Ss,,= S6G^l/5`=@rs+7/rs+l/Ss,,= S6G^l/6`=@rs+8/rs+l/Ss,,= A rg= A &&&&&&&&&&&&&&&&&& A & Lqlwldo vroxwlrq A &&&&&&&&&&&&&&&&&& A SL=@614748<5987> A yw4=@lqlwldobydoxhbw4 > A yw5=@lqlwldobydoxhbw5 > A yw6=@lqlwldobydoxhbw6 > A yW{=@lqlwldobydoxhbW{ > A yW|=@lqlwldobydoxhbW| > A yW}=@lqlwldobydoxhbW} > A yx3=@lqlwldobydoxhbx3 > A yy3=@lqlwldobydoxhby3 > A yn4=@3> A ydx=@lqlwldobydoxhbdx > A ydy=@lqlwldobydoxhbdy > A yi=@{hgbydoxh> ynx=@ydx2yi> yny=@ydy2yi> A &&&&&&&&&&&&&&&&&&&&&&&&&& A & Plqlpl}lqj e| lwhudwlrqv A &&&&&&&&&&&&&&&&&&&&&&&&&& A YJ=@yhfwru+5-urzglp+Ss,/3,= A M=@pdwul{+5-urzglp+Ss,/44/3,= YL=@yhfwru+44/3,=.

(46) FKDSWHU 51 FDOLEUDWLRQ. 7:. A iru m iurp 4 e| 4 zkloh m ?@ 53 gr A iru l iurp 4 e| 4 zkloh l ?@ urzglp+Ss, gr A yxn=@S5G^l/4`> yyn=@S5G^l/5`> A y{u=@S6G^l/4`> y|u=@S6G^l/5`> y}u=@S6G^l/6`> A YJ^l-504`=@vlpsoli|+vxev+{u@y{u/ |u@y|u/ }u@y}u/ w4@yw4/ w5@yw5/ w6@yw6/. W{@yW{/ W|@yW|/ W}@yW}/ i@yi/ nx@ynx/ ny@yny/ x3@yx3/ y3@yy3/ n4@yn4/ xn@yxn/ yn@yyn/ X,,> A YJ^l-5`=@vlpsoli|+vxev+{u@y{u/ |u@y|u/ }u@y}u/ w4@yw4/ w5@yw5/ w6@yw6/ W{@yW{/ W|@yW|/ W}@yW}/ i@yi/ nx@ynx/ ny@yny/ x3@yx3/ y3@yy3/ n4@yn4/ xn@yxn/ yn@yyn/ Y,,> A M^l-504/4`=@vlpsoli|+vxev+{u@y{u/ |u@y|u/ }u@y}u/ w4@yw4/ w5@yw5/ w6@yw6/ W{@yW{/ W|@yW|/ W}@yW}/ i@yi/ nx@ynx/ ny@yny/ x3@yx3/ y3@yy3/ n4@yn4/ xn@yxn/ yn@yyn/gXw4,,> M^l-5/4`=@vlpsoli|+vxev+{u@y{u/ |u@y|u/ }u@y}u/ w4@yw4/ w5@yw5/ w6@yw6/ W{@yW{/ W|@yW|/ W}@yW}/ i@yi/ nx@ynx/ ny@yny/ x3@yx3/ y3@yy3/ n4@yn4/ xn@yxn/ yn@yyn/ gYw4,,> A M^l-504/5`=@vlpsoli|+vxev+{u@y{u/ |u@y|u/ }u@y}u/ w4@yw4/ w5@yw5/ w6@yw6/ W{@yW{/ W|@yW|/ W}@yW}/ i@yi/ nx@ynx/ ny@yny/ x3@yx3/ y3@yy3/ n4@yn4/ xn@yxn/ yn@yyn/gXw5,,> M^l-5/5`=@vlpsoli|+vxev+{u@y{u/ |u@y|u/ }u@y}u/ w4@yw4/ w5@yw5/ w6@yw6/ W{@yW{/ W|@yW|/ W}@yW}/ i@yi/ nx@ynx/ ny@yny/ x3@yx3/ y3@yy3/ n4@yn4/xn@yxn/ yn@yyn/ gYw5,,> A M^l-504/6`=@vlpsoli|+vxev+{u@y{u/ |u@y|u/ }u@y}u/ w4@yw4/ w5@yw5/ w6@yw6/ W{@yW{/ W|@yW|/ W}@yW}/ i@yi/ nx@ynx/ ny@yny/ x3@yx3/ y3@yy3/ n4@yn4/ xn@yxn/ yn@yyn/gXw6,,> M^l-5/6`=@vlpsoli|+vxev+{u@y{u/ |u@y|u/ }u@y}u/ w4@yw4/ w5@yw5/ w6@yw6/ W{@yW{/ W|@yW|/ W}@yW}/ i@yi/ nx@ynx/ ny@yny/ x3@yx3/ y3@yy3/ n4@yn4/xn@yxn/ yn@yyn/gYw6,,> A M^l-504/7`=@vlpsoli|+vxev+{u@y{u/ |u@y|u/ }u@y}u/ w4@yw4/ w5@yw5/ w6@yw6/ W{@yW{/ W|@yW|/ W}@yW}/ i@yi/ nx@ynx/ ny@yny/ x3@yx3/ y3@yy3/ n4@yn4/ xn@yxn/ yn@yyn/gXW{,,> M^l-5/7`=@vlpsoli|+vxev+{u@y{u/ |u@y|u/ }u@y}u/ w4@yw4/ w5@yw5/ w6@yw6/ W{@yW{/ W|@yW|/ W}@yW}/ i@yi/ nx@ynx/ ny@yny/ x3@yx3/ y3@yy3/ n4@yn4/xn@yxn/ yn@yyn/gYW{,,> A M^l-504/8`=@vlpsoli|+vxev+{u@y{u/ |u@y|u/ }u@y}u/ w4@yw4/ w5@yw5/ w6@yw6/ W{@yW{/ W|@yW|/ W}@yW}/ i@yi/ nx@ynx/ ny@yny/ x3@yx3/ y3@yy3/ n4@yn4/ xn@yxn/ yn@yyn/gXW|,,> M^l-5/8`=@vlpsoli|+vxev+{u@y{u/ |u@y|u/ }u@y}u/ w4@yw4/ w5@yw5/ w6@yw6/ W{@yW{/ W|@yW|/ W}@yW}/ i@yi/ nx@ynx/ ny@yny/ x3@yx3/ y3@yy3/ n4@yn4/xn@yxn/ yn@yyn/gYW|,,> A M^l-504/9`=@vlpsoli|+vxev+{u@y{u/ |u@y|u/ }u@y}u/ w4@yw4/ w5@yw5/ w6@yw6/ W{@yW{/ W|@yW|/ W}@yW}/ i@yi/ nx@ynx/ ny@yny/ x3@yx3/ y3@yy3/ n4@yn4/ xn@yxn/ yn@yyn/gXW},,> M^l-5/9`=@vlpsoli|+vxev+{u@y{u/ |u@y|u/ }u@y}u/ w4@yw4/ w5@yw5/ w6@yw6/ W{@yW{/ W|@yW|/ W}@yW}/ i@yi/ nx@ynx/ ny@yny/ x3@yx3/ y3@yy3/ n4@yn4/xn@yxn/ yn@yyn/gYW},,>.

(47) FKDSWHU 51 FDOLEUDWLRQ. 7;. A M^l-504/:`=@vlpsoli|+vxev+{u@y{u/ |u@y|u/ }u@y}u/ w4@yw4/ w5@yw5/ w6@yw6/. W{@yW{/ W|@yW|/ W}@yW}/ i@yi/ nx@ynx/ ny@yny/ x3@yx3/ y3@yy3/ n4@yn4/ xn@yxn/ yn@yyn/gXnx,,> M^l-5/:`=@vlpsoli|+vxev+{u@y{u/ |u@y|u/ }u@y}u/ w4@yw4/ w5@yw5/ w6@yw6/ W{@yW{/ W|@yW|/ W}@yW}/ i@yi/ nx@ynx/ ny@yny/ x3@yx3/ y3@yy3/ n4@yn4/xn@yxn/ yn@yyn/gYnx,,> A M^l-504/;`=@vlpsoli|+vxev+{u@y{u/ |u@y|u/ }u@y}u/ w4@yw4/ w5@yw5/ w6@yw6/ W{@yW{/ W|@yW|/ W}@yW}/ i@yi/ nx@ynx/ ny@yny/ x3@yx3/ y3@yy3/ n4@yn4/ xn@yxn/ yn@yyn/gXny,,> M^l-5/;`=@vlpsoli|+vxev+{u@y{u/ |u@y|u/ }u@y}u/ w4@yw4/ w5@yw5/ w6@yw6/ W{@yW{/ W|@yW|/ W}@yW}/ i@yi/ nx@ynx/ ny@yny/ x3@yx3/ y3@yy3/ n4@yn4/xn@yxn/ yn@yyn/gYny,,> A M^l-504/<`=@vlpsoli|+vxev+{u@y{u/ |u@y|u/ }u@y}u/ w4@yw4/ w5@yw5/ w6@yw6/ W{@yW{/ W|@yW|/ W}@yW}/ i@yi/ nx@ynx/ ny@yny/ x3@yx3/ y3@yy3/ n4@yn4/ xn@yxn/ yn@yyn/gXx3,,> M^l-5/<`=@vlpsoli|+vxev+{u@y{u/ |u@y|u/ }u@y}u/ w4@yw4/ w5@yw5/ w6@yw6/ W{@yW{/ W|@yW|/ W}@yW}/ i@yi/ nx@ynx/ ny@yny/ x3@yx3/ y3@yy3/ n4@yn4/xn@yxn/ yn@yyn/gYx3,,> A M^l-504/43`=@vlpsoli|+vxev+{u@y{u/ |u@y|u/ }u@y}u/ w4@yw4/ w5@yw5/ w6@yw6/ W{@yW{/ W|@yW|/ W}@yW}/ i@yi/ nx@ynx/ ny@yny/ x3@yx3/ y3@yy3/ n4@yn4/ xn@yxn/ yn@yyn/gXy3,,> M^l-5/43`=@vlpsoli|+vxev+{u@y{u/ |u@y|u/ }u@y}u/ w4@yw4/ w5@yw5/ w6@yw6/ W{@yW{/ W|@yW|/ W}@yW}/ i@yi/ nx@ynx/ ny@yny/ x3@yx3/ y3@yy3/ n4@yn4/xn@yxn/ yn@yyn/gYy3,,> A M^l-504/44`=@vlpsoli|+vxev+{u@y{u/ |u@y|u/ }u@y}u/ w4@yw4/ w5@yw5/ w6@yw6/ W{@yW{/ W|@yW|/ W}@yW}/ i@yi/ nx@ynx/ ny@yny/ x3@yx3/ y3@yy3/ n4@yn4/ xn@yxn/ yn@yyn/gXn4,,> M^l-5/44`=@vlpsoli|+vxev+{u@y{u/ |u@y|u/ }u@y}u/ w4@yw4/ w5@yw5/ w6@yw6/ W{@yW{/ W|@yW|/ W}@yW}/ i@yi/ nx@ynx/ ny@yny/ x3@yx3/ y3@yy3/ n4@yn4/xn@yxn/ yn@yyn/gYn4,,> A rg> A YL =@hydop+0413 - pxowlso|+pxowlso|+lqyhuvh+pxowlso|+wudqvsrvh+M,/M,,/ wudqv0 srvh+M,,/YJ,,> A yw4=@yw4.YL^4`> yw5=@yw5.YL^5`> yw6=@yw6.YL^6`> A yW{=@yW{.YL^7`> yW|=@yW|.YL^8`> yW}=@yW}.YL^9`> A ynx=@ynx.YL^:`> yny=@yny.YL^;`> A yx3=@yx3.YL^<`> yy3=@yy3.YL^43`> A yn4=@yn4.YL^44`> A rg>. 517. 6G Uhfrqvwuxfwlrq. Vlqfh rxu 6G lpdjlqj v|vwhp lv edvhg rq d fdphud vhqvru dqg d sdwwhuq surmhfwru v|vwhp/ wzr glhuhqw vhwv ri sdudphwhuv kdyh wr eh frpsxwhg e| fdoleudwlrq1 Rqfh erwk v|vwhpv kdyh ehhq fdoleudwhg/ lw lv srvvleoh wr rewdlq.