AN APPROACH TO CODED STRUCTURED LIGHT TO OBTAIN THREE DIMENSIONAL INFORMATION Joaquim SALVI MAS ISBN: Dipòsit legal: GI

Kích thước: px
Bắt đầu hiển thị từ trang:

Download "AN APPROACH TO CODED STRUCTURED LIGHT TO OBTAIN THREE DIMENSIONAL INFORMATION Joaquim SALVI MAS ISBN: Dipòsit legal: GI"

Bản ghi

1 AN APPROACH TO CODED STRUCTURED LIGHT TO OBTAIN THREE DIMENSIONAL INFORMATION Joaquim SALVI MAS ISBN: Dipòsit legal: GI

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: President: Examiner: Examiner: Examiner: Examiner: Professor Bernard Dubuisson (Université de Compiègne France) Dr. Geoff Roberts (University of Wales College Newport Great Britain) 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 gdundvshfwvriwklvzrun/kdvvxuho ohgphwrsuhvhqwwrgd wklvwkhvlv1 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 O D WH[ 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 Ghvfulelqjwkhlqwhuhvwriwklvwkhvlv Vfkhpdriwkhwkhvlv Fdoleudwlrq 4; 514 Frqvlghudwlrqv ; 515 Prghoolqjdfdphud ; WkhH{wulqvlfSdudphwhuv WkhLqwulqvlfSdudphwhuv Vxppdu ri Fdphud prghoolqj Fdoleudwlqjdfdphud Fdoleudwlqjwkholqhduprgho Fdoleudwlqjwkhqrq0olqhduprgho GUhfrqvwuxfwlrq ; Uhfrqvwuxfwlqjiurpolqhdufdoleudwlrq < Uhfrqvwuxfwlqjiurpqrq0olqhdufdoleudwlrq Wkh Hslsrodu Jhrphwu Wkh ixqgdphqwdo pdwul{ Fkdqjlqjwkhfr0ruglqdwhv vwhp Zruog fr0ruglqdwh v vwhp {hg dw wkh vhfrqg fdphud/ rshudwlqjzlwkuhvshfwwrwkhuvwfdphud Zruog fr0ruglqdwh v vwhp {hg dw wkh uvw fdphud/ rs0 hudwlqjzlwkuhvshfwwrwkhvhfrqgfdphud : Uhodwlrq ehwzhhq erwk Ixqgdphqwdo Pdwulfhv < IhdwxuhvriSrlqwvdqgYhfwruv :3 616 Ghwhuplqlqj wkh ixqgdphqwdo pdwul{ : WkhHljkw0SrlqwvPhwkrg : Frqfoxvlrqv : Wkh dojrulwp xvhg wr lqihu wkh ixqgdphqwdo pdwul{ 1 1 :< Wkh dojrulwkp xvhg wr rewdlq wkh hslsrodu olqhv iurp 6 ;5 9

6 FRQWHQWV : 7 Vwuxfwxuhg Olj kw ;8 714 Lqwurgxfwlrq ;8 715 DVxuyh =FrghgVwuxfwxuhgOljkw < Srvgdphu0Dowvfkxohu < Fduulkloo0Kxppho < Er hu0ndn <; OhPrljqh0Zd{pdq Prulwd0\dnlpd0Vdndwd Yx ovwhnh0rrvwhuolqfn : Wdmlpd0Lzdndzd ; 7151; Zxvw0Fdsvrq < Jul!q0Qdudvlpkdq0\hh Pdux dpd0deh Lwr0Lvkll : Fkhq0Kxqj0Fkldqj0Zx : 716 Vxppdu < 8 D Qhz Frghg Sdwwhuq Surmhfwlrq Lqwurgxfwlrq SdwwhuqGhvljq FrorxuDvvljqphqw V vwhpfdoleudwlrq Ghvfulelqjwkhfdoleudwlrqdojrulwkp : Fdoleudwlqj wkh fdphud dqg wkh surmhfwru prgho Dffxudf riwkhphwkrg Phdvxulqjwkuhh0glphqvlrqdoremhfwv ; H{dpsoh=Phdvxulqjjhrphwulfdoremhfwv < H{dpsoh=Phdvxulqjdidfhpdvn < H{dpsoh=Phdvxulqjwkhsdopridkdqg H{dpsoh=Phdvxulqjwkhedfnridkdqg Frqfoxvlrqv Frqfoxvlrqv IxuwkhuZrun ; 916 Sxeolfdwlrqvuhodwhgzlwkwkhwkhvlv ; 917 RwkhuUhvhdufkFrqwulexwlrqv <

7 Olvw ri Iljxuhv 514 Wkh fdphud fr0ruglqdwh v vwhp zlwk uhvshfw wr d zruog fr0 ruglqdwhv vwhp < 515 Jhrphwulf uhodwlrvkls ehwzhhq wkh fdphud fr0ruglqdwh v vwhp dqgwkhzruogfr0ruglqdwhv vwhp Uhodwlrqvkls ehwzhhq wkh rulhqwdwlrq yhfwruv o dqg wkh fdp0 hud fr0ruglqdwh iudph1 Wkh jxuh vkrzv rqo wkh uhodwlrqvkls ehwzhhq wkh o f yhfwru dqg wkh f S d{lv Jhrphwulfdo uhodwlrqvkls ehwzhhq d 6G remhfw srlqw dqg lwv 5Glpdjhsurmhfwlrq Rewdlqlqj wkh lghdo surmhfwlrq iurp wkh shuvshfwlyh uhodwlrq Udgldodqgwdqjhqwldoglvwruwlrq : 51: Hhfw ri udgldo glvwruwlrq1 Vrolg olqhv= qr glvwruwlrq> gdvkhg olqhv=udgldoglvwruwlrq+d=qhjdwlyh/e=srvlwlyh, : 51; Hhfw ri wdqjhqwldo glvwruwlrq1 Vrolg olqhv= qr glvwruwlrq> gdvkhg olqhv=wdqjhqwldoglvwruwlrq ; 51< Wudqvirupdwlrq iurp phwulf fr0ruglqdwhv ri wkh uhwlqdo sodqh wrwkhsl{hofr0ruglqdwhvriwkhlpdjhsodqh Wudqvirupdwlrq iurp wkh fr0ruglqdwh v vwhp ri wkh fdphud lp0 djh sodqh wr wkh fr0ruglqdwh v vwhp ri wkh frpsxwhu lpdjh exhu Prghoolqjdfdphudzlwkrxwohqvglvwruwlrq Prghoolqjdfdphudzlwkudgldoohqvglvwruwlrq Wkh jhrphwulf uhodwlrq ehwzhhq d 6G remhfw srlqw dqg lwv 5G surmhfwlrq zkhq wkh rswlfdo fhqwuh lv orfdwhg lq iurqw ri wkh lpdjhsodqh Wkh sulqflsoh ri wuldqjxodwlrq xvhg lq rughu wr rewdlq wkh 6G vfhqhlqirupdwlrq < 614 Wkhhslsodujhrphwu ehwzhhqwzrslqkrohprghov : 615 Jhrphwulfdo uhsuhvhqwdwlrqv ri wkh wkuhh olqh yhfwruv ri wkh pdwul{ h < ;

8 OLVW RI ILJXUHV < 616 Wkh uhodwlrq ri d 5G lpdjh srlqw 6 lq U dqg lwv hslsrodu olqh, 6 rq U Jhrphwulfdo uhodwlrq ri erwk fdphud d{hv dqg d zruog fr0 ruglqdwh v vwhp1 Zruog fr0ruglqdwh v vwhp {hg dw wkh uvw fdphud Jhrphwulfdo uhodwlrq ri erwk fdphud d{hv dqg d zruog fr0 ruglqdwh v vwhp1 Zruog fr0ruglqdwh v vwhp {hg dw wkh vhfrqg fdphud ; 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 pdunhge dvtxduh :; 61: D }rrp ri wkh vtxduh duhdv ri wkh suhylrxv lpdjhv1 Qrwh wkdw wkh fruuhvsrqghqfh srlqwv olh dozd v rq wkh hslsrodu olqhv ghqhge wkhlufrxsohsrlqwv :; 714 Wkh hslsrodu jhrphwu dv d wrro wr uhgxfh wkh fruuhvsrqghqfh sureohp ;: 715 Dq h{dpsoh ri d srlqw zlwkrxw fruuhvsrqghqfh gxh wr d vxuidfh rffoxvlrq ri ;: 716 Dq h{dpsoh ri fruuhvsrqghqfh srlqwv lpdjhg lq wkh vdph rughu iurpwkhhslsrohv ;; 717 Dq h{dpsoh ri fruuhvsrqghqfh srlqwv lpdjhg lq d glhuhqw rughuiurpwkhhslsrohv ;; 718 Wkhwhpsrudu frglfdwlrq <6 719 Frglqj d?{? pdwul{riodvhuehdpgrwv <7 71: ;elwvwhpsrudoo elqdu frghgsdwwhuqsurmhfwlrq <8 71; ;elwvwhpsrudoo Jud frghgsdwwhuqsurmhfwlrq <8 71< WkhFxelvfrshv vwhp <9 7143Wkhsdwwhuqsursrvhge Fduulkloohwdo <: 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 lwvshulrgohqjwk <; 7145 Dq h{dpsoh ri wkh fruuhvsrqghqfh sureohp wkdw kdv wr eh vroyhgehwzhhqsurmhfwhgdqglpdjhgvolwv << 7146Dghwdloriwkhsdwwhuqsursrvhge Er hu D srvvleoh uhodwlrq ehwzhhq d surmhfwhg sdwwhuq dqg wkh fds0 wxuhgrqh

9 OLVW RI ILJXUHV Vrph qrw frghg julg sdwwhuqv1 Wkh sdwwhuq uhvroxwlrq lv jlyhq e wkh qxpehu ri furvvlqj srlqwv dqg wkh olqh wklfnqhvv Julg sdwwhuqv sduwldoo frghg e wkh srvlwlrq ri vrph grwv/ zklfk duh xvhg dv odqgpdunv wr lqlwldwh wkh odehoolqj surfhvv : Wkh v vwhp frqjxudwlrq sursrvhg e Prulwd hw do1 zlwk wkh grwglvrughuvwkdwfdqehrewdlqhg ; Dq h{dpsoh ri wkh wzr sdwwhuqv surmhfwhg rq wkh phdvxulqj vfhqhe Prulwd <Dghwdloriwkhsdwwhuqsursrvhge Yx ovwhnhhwdo 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 fdqdovrehgdunrueuljkwuhvshfwlyho 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 cs e cs D c EK ck e ck D : 7155 Wkh sulqflsoh ri gludfwlqj zklwh oljkw lq rughu wr rewdlq d udlqerzsdwwhuq ; 7156 Wkh udlqerz udqjh sdwwhuq surmhfwhg e Wdmlpd hw do1 pdgh e d vhw ri yhuwlfdo volwv zklfk xvh doo wkh frorxu vshfwuxp iurp eoxhxswruhg < 7157Wkhv vwhpsursrvhge Jhqj Wkh sulqflsoh ri ryhuodslqj wkuhh vlqxvrlgdo lqwhqvlw sdwwhuqv Wkh shulrglfdo sdwwhuq sursrvhg e Zxvw dqg Fdsvrq/ pdgh e wkh vlqxvrlgdo ryhuodsslqj ri wkh wkuhh sulpdu frorxu frp0 srqhqwv : Grw frglfdwlrq h{dpsoh xvlqj lwv irxu qhljkerxuv dqg d edvlv htxdo wr 6/ l1h1 rqo wkuhh glhuhqw v perov fdq eh xvhg ; 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 hdfkv perodfrorxuhggrwkdvehhqdvvrfldwhg < D srvvleoh jhrphwulf dvvrfldwlrq ri d pdwul{ grw frghg xvlqj dedvlvhtxdowr Wkh sdwwhuq sursrvhg e Pdux dpd hw do1 pdgh e wkh sur0 mhfwlrqripxowlsohvolwvzlwkudqgrpfxwv 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 449

10 OLVW RI ILJXUHV 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, ; 814 Vfkhpd ri wkh 6G phdvxulqj v vwhp edvhg rq frghg julg sdw0 whuqsurmhfwlrq Wkh judsk rewdlqhg iurp R ' Wkhfrorxuhgsdwwhuq Wkhvfhqdulrriwkh6Gphdvxulqjv vwhp : 818 Vfkhpd ri wkh fdoleudwlqj sdwwhuq dqg lwv srvlwlrq dqg rulhq0 wdwlrqzlwkuhvshfwwrwkhzruogfr0ruglqdwhv vwhp Vfkhpd ri wkh sdwwhuq xvhg lq rughu wr fdoleudwh wkh fdphud : Wkhfdoleudwlqjsdwwhuqrivtxduhv ; Wkh lpdjh ri wkh fdoleudwlqj sdwwhuq judeehg e wkh fdphud < Srvlwlrqriwkh5Glpdjhsrlqwvriwkhfdphud Srvlwlrqriwkh6Gremhfwsrlqwvriwkhfdphud Wkhphwulfjulgxvhglqsurmhfwrufdoleudwlrq Srvlwlrq ri wkh 5G furvvlqj srlqwv ri wkh surmhfwhg sdwwhuq Srvlwlrq ri wkh 6G furvvlqj srlqwv ri wkh surmhfwhg sdwwhuq : 8147 D vfhqh frpsrvhg e wkuhh jhrphwulf remhfwv1 d, Wkh uhdo vfhqh +qrw xvhg wr lqihu 6G lqirupdwlrq,> e, Wkh vfhqh looxpl0 qdwhge wkhfrorxuhgsdwwhuq < 8148 Wkh pdwfklqj rewdlqhg1 d, Wkh Pdwfklqj srlqwv iurp wkh 8:9 { :9; fdphud lpdjh sodqh/ e, Wkh Pdwfklqj srlqwv iurp wkh 845{845surmhfwhgsdwwhuqiudph Wkh uhfrqvwuxfwlrq ri wkh 6G remhfwv1 d, D shuvshfwlyh ylhz iurpwkhuljkwvlgh1e,dwrsylhz : D vfhqh frpsrvhg e d idfh pdvn1 d, Wkh uhdo vfhqh +qrw xvhg wr lqihu 6G lqirupdwlrq,> e, Wkh vfhqh looxplqdwhg e wkhfrorxuhgsdwwhuq ; Wkh pdwfklqj rewdlqhg1 d, Wkh Pdwfklqj srlqwv iurp wkh 8:9 { :9; fdphud lpdjh sodqh/ e, Wkh Pdwfklqj srlqwv iurp wkh 845{845surmhfwhgsdwwhuqiudph <Wkhuhfrqvwuxfwlrqriwkhidfhpdvn Dq looxplqdwhg kxpdq sdop Wkh pdwfklqj rewdlqhg1 d, Wkh pdwfklqj srlqwv iurp wkh 8:9{:9; fdphud lpdjh sodqh1 e, Wkh pdwfklqj srlqwv iurp wkh589{589surmhfwrusdwwhuqiudph Uhfrqvwuxfwlrqriwkhkxpdqsdop Dq looxplqdwhg edfn kdqg

11 OLVW RI ILJXUHV Wkh pdwfklqj rewdlqhg1 d, Wkh pdwfklqj srlqwv iurp wkh 8:9{:9; fdphud lpdjh sodqh1 e, Wkh pdwfklqj srlqwv iurp wkh589{589surmhfwrusdwwhuqiudph Uhfrqvwuxfwlrqriwkhkxpdqedfnkdqg

12 Olvw ri Wdeohv 814 Lqwulqvlf dqg h{wulqvlf sdudphwhuv ri wkh fdphud vhqvru Lqwulqvlf dqg h{wulqvlf sdudphwhuv ri wkh sdwwhuq surmhfwru : 816 Dvdpsohriwkhdqdo vhgremhfwsrlqwv ; 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/zhkdyhwruhdgwkhodeho+ghfrghwkhsdwwhuq,lqrughuwrvroyhwkh 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 f dqg t 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 lwvwzrfruuhvsrqghqfhsrlqwvlvghvfulehg1 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 ^Idxj <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< Z c Y w y X w Z w x Π M X c y C Y c f x y m Retinal plane x m y x f 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 ^Eh 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/ zklfklv

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 suhvhqwhg Wkh H{wulqvlf Sdudphwhuv Wkh vxuidfh srlqwv ri wkh vfhqh duh uhihuuhg wr d zruog fr0ruglqdwh v vwhp/ zlwk dq ruljlq /dqgfr0ruglqdwhd{lvef ct c~ 1 Jlyhq dq remhfw srlqw S/ wklv srlqw lv h{suhvvhg zlwk uhvshfw wr R / wkdw lv zlwk wkh fr0ruglqdwhv E% c+ c5 1 Frqvlghu d vhfrqg fr0ruglqdwh v vwhp/ zlwk dq ruljlq ' S / dqg fr0ruglqdwh d{lv Ef U ct U 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 vkrzqlqhtxdwlrq A ' 7 6 f t , ~ Wkh rulhqwdwlrq ri wkh fr0ruglqdwh d{lv ri wkh fdphud zlwk uhvshfw wr wkh

20 FKDSWHU 51 FDOLEUDWLRQ 54 Retinal Plane Y c P X c Z c O c Camera co-ordinate system K Y w O w X w Z w World co-ordinate system Iljxuh 515= Jhrphwulf uhodwlrvkls ehwzhhq wkh fdphud fr0ruglqdwh v vwhp dqgwkhzruogfr0ruglqdwhv 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 Durwdwlrqri ghjuhhv durxqg dq d{h y zloo eh h{suhvvhg dv -J Eyc1 Wkhq/ wkuhh vlqjoh d{h urwdwlrqv fdq eh h{suhvvhg dv vkrzq e wkh iroorzlqj htxdwlrqv1 -J Efc % -J Etc + -J E~c 5 ' ' ' f f f S % r % f r % S % 7 S + f r + f f r + f S S 5 r 5 f r 5 S 5 f 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 E 5 c + c % '-J E~c 5 -J Etc + -J Efc % +518, Wkh Hxohu uhsuhvhqwdwlrq + '.,eoec wc '-J E~c -J Etcw -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 Etcw -J E~c ' 5 SS rswr Sr rsws rrw 6 7 SS n SSwr rr n SSwS Srw 8 +51:, rwr rws Sw 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 + ' 7 o f o t o ~ ;, Qrwh wkdw/ o f / o t 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 o f,/ 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/ f S t S ~ S 8 ' + 7 f t ~ 8 n A +51<,

22 FKDSWHU 51 FDOLEUDWLRQ 56 Z w Y w X c r Xy r Xx r Xz X w Iljxuh 516= Uhodwlrqvkls ehwzhhq wkh rulhqwdwlrq yhfwruv o dqg wkh fdphud fr0ruglqdwh iudph1 Wkh jxuh vkrzv rqo wkh uhodwlrqvkls ehwzhhq wkh o f yhfwru dqg wkh f S d{lv1 vr wkdw/ jlyhq dq remhfw srlqw / zlwk wkh frpsrqhqwve f c t c ~ zlwk uhvshfw wr / rqh fdq frpsxwh lwv uhsuhvhqwdwlrq zlwk uhvshfw wr S zlwk wkh frpsrqhqwv E fs c ts c ~S / fs ts ~S 8 ' + 7 f t ~ 8 n A +5143, Htxdwlrq 5143 lv xvxdoo hvsuhvvhg e d vlqjoh pdwul{ uhodwlrqvkls zlwk krpjhqhrxv fr0ruglqdwhv/ dv vkrzq e htxdwlrq 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 57 image co-ordinate system X r O i v w u Y r Z r retinal co-ordinate system (u 0, v 0 ) P Y c f P d X c Z c P u Image plane Retinal plane O c Camera co-ordinate system Iljxuh 517= Jhrphwulfdo uhodwlrqvkls ehwzhhq d 6G remhfw srlqw dqg lwv 5G lpdjh surmhfwlrq 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 lghdo5gsrlqwdqgwkhrevhuyhg5gsrlqw1 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 P Y c P Yc P Xc P u X u f P Zc Y u Image Plane X c Z c O c 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 / zlwkwkh~ 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 / dqglw lv sdudooho wr wkh sodqh ghqhg e wkh fr0ruglqdwh d{lv f S dqg t S 1 Vwduwlqj iurp wkhvh dvvxpswlrqv/ jlyhq dq remhfw srlqw uhihuuhg wr S /lwvlghdo 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 s ' f S ~S Vr wkdw/ t s ' t S ~S +5146, f ' s f S ~S

25 FKDSWHU 51 FDOLEUDWLRQ 59 t ' s t S ~S +5147, Zklfk fdq eh h{suhvvhg lq d pdwulfldo zd dsso lqj/ 5 7 rf f f rt 8 ' 7 f f fs ts , r f f s ~S 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 ct lv wkh lghdo surmhfwlrq dqg Ef _ ct _ duh wkh fr0ruglqdwhv ri wkh revhuyhg srlqw rq wkh lpdjh sodqh1 f ' f _ n ( % t ' t _ n ( , 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: Y r dr Ideal P projection u dt P d Observed position X r 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; Y r Axe of a maximum tangential distortion X r Axe of a minimum tangential distortion Iljxuh 51;= Hhfw ri wdqjhqwldo glvwruwlrq1 Vrolg olqhv= qr glvwruwlrq> gdvkhg olqhv= wdqjhqwldo glvwruwlrq1 ( % ' f _ E& o 2 n & 2 o e n ( + ' _ E& o 2 n & 2 o e n t +514:, o ' f_ 2 n t _ 2 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 _ & o 2 ( + ' _ & o ;, t o ' f_ 2 n t _ 2 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 _ ct _ lv wkh uhdo surmhfwlrq zlwk ohqv glvwruwlrq lq phwulf fr0ruglqdwhv/ dqg Ef R ct R lv wkh vdph srlqw h{suhvvhg lq sl{hov1 f R ' & f _ t R ' & t _ +514<, Zklfk fdq eh prghoohg lq d pdwulfldo zd dsso lqj wkh iroorzlqj pdwulfldo htxdwlrq/ fr & f f_ ' +5153, t R 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 E f 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 Ef o ct o / dqg lw kdv dovr ehhq xvhg dv wkh ruljlq iru prghoolqj wkh ohqv glvwruwlrq ri wkh fdphud1 Krzhyhu/ wkh

29 FKDSWHU 51 FDOLEUDWLRQ 63 Y r pixel Image Plane X r (0, 0) (X d, Y d ) (X p, Y p ) (0, 0) 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 E f 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 f o d{h dqg wkh t o d{h ri wkh uhwlqdo lpdjh sodqh1 Jlyhq d srlqw Ef R ct R 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 ct / zklfk lv wkh rqh revhuyhg e wkh xvhu lq sl{hov rq wkh lpdjh sodqh ri wkh frpsxwhu1 f ' f R n f t ' t R n f +5154, Zklfk fdq eh prghoohg lq d pdwulfldo zd dsso lqj wkh iroorzlqj htxdwlrq/ 5 6 f f f R f ' 7 t t f R , f

30 FKDSWHU 51 FDOLEUDWLRQ 64 (0, 0) X r V U Computer image co-ordinate system (u 0,v 0 ) Y r 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 E f c f kdv wr eh prghoohg 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 (X w, Y w, Z w ) 3D object point with respect to world co-ordinate system Affine transformation. Modelled parameters: R, T (X c, Y c, Z c ) 3D object point with respect to camera co-ordinate system Perspective transformation. Modelled parameter: f (X u, Y u ) Ideal projection on the retinal plane Pixel adjustment Modelled parameters: k u, k v (X p,y p ) Real projection on the image plane Adaptation to the computer image buffer Modelled parameters: u 0, v 0 (X i, Y i ) 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/ f ' & s fs ~S n f t ' & s ts n f +5156, ~S zklfkfdqehh{suhvvhglqdpdwulfldozd dv/ zkhuh/ rf 6 5 rt 8 ' 7 k 6 f f f f k f f r f f f fs ts ~S 6 : , k ' ' 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 (X w, Y w, Z w ) 3D object point with respect to world co-ordinate system Affine transformation. Modelled parameters: R, T (X c, Y c, Z c ) 3D object point with respect to camera co-ordinate system Perspective transformation. Modelled parameter: f (X u, Y u ) Ideal projection on the retinal plane Radial lens distortion. Modelled parameter: k 1 (X d, Y d ) Real projection on the retinal plane Pixel adjustment Modelled parameters: k u, k v (X p,y p ) Real projection on the image plane Adaptation to the computer image buffer Modelled parameters: u 0, v 0 (X i, Y i ) Real projection on the image plane Iljxuh 5145= Prghoolqj d fdphud zlwk udgldo ohqv glvwruwlrq1 5 7 rf rt r ' 7 k f f f f k f f f f f o o 2 o % o 2 o 22 o 2 + o o 2 o 5 f f f 6 5 : f t ~ 6 : , 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 zkhuh/ s fs ~S ' f _ n & o 2 f _ +515:, s ts ~S ' t _ n & o 2 t _ +515;, f _ ' Ef f &

33 FKDSWHU 51 FDOLEUDWLRQ 67 t _ ' Et f & +515<, dqg/ 5 7 fs ts ~S ' + A 9 7 f t ~ 6 : , 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 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 f S dqg t S 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 f o 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& +5164, Wkhq/ sodflqj wkh irfdo srlqw ehklqg ru lq iurqw ri wkh lpdjh sodqh zloo rqo surgxfh dq lqyhuvlrq ri wkh k sdudphwhu1 516Fdoleudwlqj 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 68 image co-ordinate system w u Y r O i v (u 0, v 0 ) P u P d f X r Y c O c C Camera co-ordinate system X c Z c P Image Plane Y w X w World co-ordinate system Z w O w 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 sodqh 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 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{ ^Eruw <4` ^Eudv <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 Qrwh wkdw/ 5 ' 7 k o n f o k % n f 5 k o 2 n f o k + n f 5 o ' r 6 5 r 8 ' 7 6 e 2 2e 8 r e 2 e 2e e , +5166,

35 FKDSWHU 51 FDOLEUDWLRQ 69 zkhuh 'E fs c ts c ~S lvd6gremhfwsrlqwzlwkuhvshfwwrwkhzruog 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 nn 2 n ULt k +5167, zkhuh wkh qrup ri d yhfwru lv frpsxwhg e wkh iroorzlqj htxdwlrq1 nn ' 2 % n 2 + n 52 * , Ixuwkhupruh/ lw fdq eh d!uphg wkdw wkh rulhqwdwlrq yhfwruv nhhs wkh iroorzlqj uhodwlrq/ o o ' f 9' o o ' ' Wkhq/ wkh irxu lqwulqvlf sdudphwhuv ri wkh fdphud +k / k / f / f,/ dqg wkh vl{ h{wulqvlf sdudphwhuv +o /o 2 / o / % / + / 5,/ fdq eh rewdlqhg iurp 5165/ dv/ f ' f ' 2 k ' E 2 f *2 k ' E 2 2 f 2 *2 o ' k E f % ' k E e f e o 2 ' k E 2 f + ' k E 2e f e o ' 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{ pxvw eh frpsxwhg lq rughu wr h{suhvv wkh uhodwlrq e olqhdu htxdwlrqv/ dqg/ n e E n e 'f +516:, 2 n 2e E n e 'f +516;, Wkhvh htxdwlrqv pd eh duudqjhg lq wkh iroorzlqj zd /

36 FKDSWHU 51 FDOLEUDWLRQ 6: dqg ' e n e e e +516<, ' 2 n 2e +5173, e e e zklfk fdq eh uhodwhg dv d fohdu vlpsolfdwlrq e / dqg ' A n A , ' A n 2 A , Zkhuh/ iurp wkh htxdwlrqv 5169/ 5174 dqg 5175/ wkh iroorzlqj htxdwlrqv duh rewdlqhg/ A ' o 5 f n o 5 k ' f n % 5 k A 2 ' o 5 A ' o 5 f n o 2 5 k 2 ' f n + 5 k +5176, Zkhuh hdfk A / A 2 dqg A duh wkuhh yhfwruv ri wkuhh frpsrqhqwv1 Dqg dqg 2 duh wzr vlqjoh ydoxhv1 Vr/ zh fdq ghqh d qhz yhfwru f/ rihohyhq frpsrqhqwv1 5 f ' 9 7 A A 2 A 2 6 : , 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 fdqehghqhgdv/ wkhq/ dqg/ " ' ' "f +5178, f % f f % f ' : , : :, Wkhq/ wkh f frpsrqhqwv fdq eh ghwhuplqhg xvlqj wkh iroorzlqj htxdwlrq/ f ' " " 3 " +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/ wkhq/ no n ' +517<, 5 ' na 2 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/ dqg 2 ' n nn 2 n ULt k a 2 ' n nn 2 n t? k +5183,

38 FKDSWHU 51 FDOLEUDWLRQ 6< wkhq/ nn ' 2 % n 2 + n 2 5 * , o o ' f 9' o o ' ' o a o ' 9' o a o ' f ' Wkh lqwulqvlf sdudphwhuv fdq eh rewdlqhg iurp/ f ' A A 2 na 2 n , dqg wkh h{wulqvlf rqhv/ f ' A 2A na 2 n , k ' na a A 2n na 2 n , k ' na 2 a A n na 2 n , o o 2 ' ' na 2 n A na a A 2n A A 2 na 2 n na 2 a A n na 2 n A 2 2 A A 2A na 2 n A , +518:, dqg o ' A 2 na 2 n +518;, % ' + ' 5 ' na 2 n na a A 2n A A 2 na 2 n 2 na 2 n na 2 a A n na 2 n 2 A 2A na 2 n <, +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 + ' 7 SS rswr Sr rsws rrw SS n SSwr rr n SSwS Srw rwr rws Sw wkh c w dqg dqjohv fdq eh frpsxwhg dsso Dojrulwkp ri fdphud fdoleudwlrq o o ' 5 7 o 6 o 2 o o 2 o 22 o 2 8 o o 2 o +5195, +5196, w Eo +5197, , Lq wkh iroorzlqj/ wkh zkroh dojrulwkp xvhg lq wkh fdoleudwlrq ri d olqhdu fdphud prgho lv suhvhqwhg1 Wkh dojrulwkp kdv ehhq surjudpphg xvlqj wkh Pdwode U hqylurqphqw1 fohdu doo> forvh doo> irshq+*vhwbribsrlqwv1suq*/*u*,> ivfdqi+g/*(i*/^8/qxpbribfrxsohv`,> iforvh+g,> vl}h+s/5,> S+4=5/=,> S+6=8/=,> S6G+4/=,> S6G+5/=,> S6G+6/=,> S5G+4/=,> S5G+5/=,> iru ^D > {n+l, n+l, }n+l, 0xn+l,-{n+l, 0xn+l,- n+l, 0xn+l,-}n+l, > yn+l,-{n+l, 0yn+l,- n+l, 0yn+l,-}n+l, {n+l, n+l, }n+l, 3 4`> ^E > xn+l,> yn+l,`> hqg +lqy+d*-d,,-d*-e> [+4=6/4,*> [+7=9/4,*> [+:=</4,*> [+43/4,> [+44/4,> +[5+4/4, - [5+4/4,,. +[5+4/5,-[5+4/5,,. +[5+4/6,-[5+4/6,,> o 2

40 FKDSWHU 51 FDOLEUDWLRQ 74 furvv+[4+4/=,*/[5+4/=,*,> vxp+sy+4,a5.sy+5,a5.sy+6,a5,> furvv+[5+4/=,*/[6+4/=,*,> vxp+sy+4,a5.sy+5,a5.sy+6,a5,> +[4 - [5*, 2 Qruph[5> +[5 - [6*, 2 Qruph[5> vtuw+[4w[5w, 2Qruph[5> vtuw+[5w[6w, 2Qruph[5> +vtuw+qruph[5,2vtuw+[4w[5w,, - +[40++[4-[5*,2Qruph[5,-[5,> +vtuw+qruph[5,2vtuw+[5w[6w,, - +[60++[5-[6*,2Qruph[5,-[5,> [52vtuw+Qruph[5,> +vtuw+qruph[5,2vtuw+[4w[5w,, - +F40++[4-[5*,2Qruph[5,,> +vtuw+qruph[5,2vtuw+[5w[6w,, - +F50++[5-[6*,2Qruph[5,,> 42vtuw+Qruph[5,> ^ u4 w{> u5 w > u6 w}> `> 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 cc% 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 'flwhudwlqj 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 Zh nqrz dovr wkdw wkh vroxwlrq zh zdqw wr qg lv wkh yhfwru f ri xqnqrzqv zklfk vroyh 519:1 Vr/ CEf & CEf &3 na Ef &3 {f & +5199, CEf & ' f +519:, CEf &3 na 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 +519<, {f & 0 +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 cc% 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 cc 2 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 & / dvvkrzq lq 51:5/ 3 4 aec Ef &3 aecef &3 ' E aec 2 Ef &3 F C D +51:5, aec? Ef &3

42 FKDSWHU 51 FDOLEUDWLRQ 76 zkhuh/ aec Ef &3 ' YC Ef &3 c YC Ef &3 c c YC Ef &3 Y% Y% 2 Y% 6 +51:6, dqg f lv wkh yhfwru ri xqnqrzqv dv vkrzv 51:71 wkhq/ f 'E% c% 2 cc% 6 +51:7, 3 E C {f ' E{% c {% 2 c c {% 6 +51:8, Vr/ htxdwlrq 519; fdq eh h{suhvvhg dv 51:91 C Ef &3 C 2 Ef &3 C? Ef &3 4 F D n 3 E C YC Ef &3 YC Ef &3 Y% YC Ef &3 Y% 6 Y% 2 YC 2 Ef &3 Y% YC?Ef&3 Y% YC?Ef&3 Y% F E D C {% {% 2 {% 6 4 F D 'f?% +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/liwkhhtxdwlrqC 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 {f & ' a Ef &3 aef &3 3 a Ef &3 CEf &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 f f 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/ wkhq/ {f & ' 2 3 a 3 Ef &3 CEf &3 +51;3, f & ' f &3 2 3 a 3 Ef &3 CEf &3 +51;4, 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& 2 3 _ & ncef & n +51;6, zkhuh/ v [ ncef & n ' % 2 [ % 2 +51;7, 6 6 Wkh Prglhg Qhzwrq0Udskvrqv dojrulwkp lv wkh iroorzlqj/ 41 Il{ dq lqlwldo vroxwlrq f f 1 51 Lwhudwh/ 5141 FrpsxwhCEf & dqg aef & Frpsxwh _ & Lwhudwh iurp f xs wr ncef & 2 3 _ & n ncef & n1

44 FKDSWHU 51 FDOLEUDWLRQ Frpsxwh{f & ' 2 3 a 3 Ef &3 CEf & Frpsxwhf & ' f &3 n{f & 1 61 xqwlo {f & 0lv 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 Yuhohdvh7 U hqylurqphqw1 Wkh h{dpsoh vhwv iruwk wkh surjudpplqj ri d 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 ydoxhv1 Dfwxdoo / wkh dojrulwp fdq eh hdvlo prglhg wr rewdlq rqo wkh sdudphwhuv1 A zlwk+olqdoj,= A &&&&&&&&&&&&&&&&&&&&&&&&&&&& A & Rewdlqlqj wkh fdphud prgho A &&&&&&&&&&&&&&&&&&&&&&&&&&&& A +^^4/3/3/3`/^3/frv+w4,/vlq+w4,/3`/^3/0vlq+w4,/frv+w4,/3`/^3/3/3/4``,> A U +^^frv+w5,/3/0vlq+w5,/3`/^3/4/3/3`/^vlq+w5,/3/frv+w5,/3`/^3/3/3/4``,> A +^^frv+w6,/vlq+w6,/3/3`/^0vlq+w6,/frv+w6,/3/3`/^3/3/4/3`/^3/3/3/4``,> A +^^4/3/3/W{`/^3/4/3/W `/^3/3/4/W}`/^3/3/3/4``,> A +pxowlso +U}/U,/U{,> A +W/U,> A +^^{u`/^ u`/^}u`/^4``,> A +DD/P,> A A A A A A A &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& A & Rewdlqlqj wkh sduwldo ghulydwlyh A &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& A A A

45 FKDSWHU 51 FDOLEUDWLRQ 79 A A gxw gyw A A A A A A A A &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& A & Uhdglqj wkh vhw ri srlqwv frxsohv A &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& A & Uhdglqj wkh P Pdwul{ iurp dq DVFLL oh1 A uhdgole+uhdggdwd,> A A A iru l iurp 4 e 4 wr urzglp+ss, gr A A A rg= A &&&&&&&&&&&&&&&&&& A & Lqlwldo vroxwlrq A &&&&&&&&&&&&&&&&&& A A A A A A yw > A A A A A A A A &&&&&&&&&&&&&&&&&&&&&&&&&& A & Plqlpl}lqj e lwhudwlrqv A &&&&&&&&&&&&&&&&&&&&&&&&&& A A

46 FKDSWHU 51 FDOLEUDWLRQ 7: A iru m iurp 4 e 4 zkloh 53gr A iru l iurp 4 e 4 zkloh urzglp+ss, gr A A y A u/ / X,,> A u/ / Y,,> A u/ / u/ / gyw4,,> A u/ / u/ / gyw5,,> A u/ / u/ / A u/ / u/ / A u/ / u/ / A u/ / u/ /

47 FKDSWHU 51 FDOLEUDWLRQ 7; A u/ / u/ / A u/ / u/ / A u/ / u/ / A u/ / u/ / A u/ / u/ / A rg> A YL - pxowlso +pxowlso +lqyhuvh+pxowlso +wudqvsrvh+m,/m,,/ wudqv0 srvh+m,,/yj,,> A A yw A A A 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

48 FKDSWHU 51 FDOLEUDWLRQ 7< Y w P w X w Z w Z 1 Z 2 C 1 C 2 f f 2 1 P 2 π 1 P 1 O O 2 1 Y 1 π 2 Y 2 X 1 Projected Image X 2 Grabbed Image Iljxuh 5147= Wkh sulqflsoh ri wuldqjxodwlrq xvhg lq rughu wr rewdlq wkh 6G vfhqh lqirupdwlrq1 d 6G uhfrqvwuxfwlrq ri wkh phdvxulqj vfhqh1 Vr wkdw/ jlyhq wkh fr0ruglqdwhv riwkhsurmhfwlyhsrlqwrqwkhlpdjhsodqh 2 dqg jlyhq wkh fr0ruglqdwhv ri wkh 5G srvlwlrq iurp zkhuh wklv srlqw kdv ehhq surmhfwhg rq wkh vfhqh / wkh 6G fr0ruglqdwhv ri wkh remhfw srlqw zkhuh wkh surmhfwlqj srlqw kdv ehhq uh hfwhg rq wkh lpdjh vhqvru fdq eh frpsxwhg e wuldqjxodwlrq1 Iljxuh 5147 looxvwudwhv wklv sulqflsoh1 Dv zh kdyh h{sodlqhg wzr glhuhqw fdoleudwlqj phwkrgv/ wkdw lv wkh olqhdu dqg wkh qrq0olqhdu rqh/ zh kdyh dovr wr h{sodlq wzr glhuhqw phwkrgv ri 6G uhfrqvwuxfwlrq dv d uhvxow ri wkh phwkrg ri fdoleudwlrq xvhg Uhfrqvwuxfwlqj iurp olqhdu fdoleudwlrq Jlyhq wkh wudqvirupdwlrq pdwul{/ zklfk uhodwhv dq remhfw srlqw zlwk lwv sur0 mhfwlrq rq wkh fdswxuhg lpdjh dqg rq wkh surmhfwru lpdjh/ uhvshfwlyho / dqg wkh 5G fr0ruglqdwhv ri wkh wzr surmhfwlrqv ri wkh vdph 6G remhfw srlqw/ wkh fr0ruglqdwhv ri wklv remhfw srlqw fdq eh ghwhuplqhg1 Vxppdulvlqj wkh htxdwlrqv rewdlqhg lq wkh fdoleudwlrq surfhvv1 5 7 % % ' 7 2 e e 2 e % : 8 +51;8,

49 FKDSWHU 51 FDOLEUDWLRQ 83 dqg/ % 2 2 % ' e e e Rshudwlqj/ dqg uh0duudqjlqj wkh yduldeohv/ % : 8 +51;9, E % R % R ne 2 2 % R + R ne % R 5 R ' e % R e +51;:, E 2 + R % R n E R + R n E 2 + R 5 R ' e + R 2e +51;;, dqg/ E 2 2 % R2 % R n E % R2 + R n E 2 2 % R2 5 R ' 2e % R2 2 e +51;<, E R2 % R n E R2 + R n E R2 5 R ' 2e + R2 22e +51<3, Vr/ duudqjlqj wkh 5G srlqw fr0ruglqdwhv dqg wkh remhfw srlqw fr0ruglqdwhv lq pdwulfldo irup/ wkh uhodwlrq fdq eh h{suhvvhg dv/ zkhuh/ wkh pdwulfhv duh/ ' 8 +51<4, ' % R 2 2 % R % R 2 + R R 2 + R 2 2 % R % R2 2 2 % R R R R2 6 : 8 +51<5, 5 ' 7 % 6 R + R 8 +51<6, 5 R 5 6 e % R e 8 ' 9 e + R 2e : 7 2e % R2 2 e 8 +51<7, 2e + R2 22e

50 FKDSWHU 51 FDOLEUDWLRQ 84 Ilqdoo / wkh 6G remhfw fr0ruglqdwhv fdq eh rewdlqhg e / ' <8, Dv wkh remhfw fr0ruglqdwhv ghshqg rq wkh fruuhfw dvvrfldwlrq ri wkh sur0 mhfwhg srlqw E% R c+ R dqg wkh fdswxuhg lpdjh E% R2 c+ R2 / dq plvwdnh lq wkh fruuhvsrqghqfh hvwdeolvkphqw ohdgv wr dq huuru lq wkh remhfw srlqw fr0 ruglqdwhv ghwhuplqdwlrq1 Dojrulwkp ri olqhdu 6G uhfrqvwuxfwlrq ( fohdu doo> foi> foj> forvh doo> ( Fdphud fdoleudwhg e olqhdu phwkrg ( fdpbfdo ( W4-W5 ( vdyh D41pdw D4> ( Surmhfwru fdoleudwhg e olqhdu phwkrg ( surbfdo5 ( W4-W5 ( vdyh D51pdw D5> ( 6G uhfrqvwuxfwlrq iurp 5G surmhfwlrqv ordg D41pdw> ordg D51pdw> qxpbribfrxsohv> irshq+*vhwbribfrxsohv1suq*/*u*,> ivfdqi+g/*(g*/^7/qsrlqwv`,> iforvh+g,> S+4=5/=,> S+6=7/=,> iru ^D4+4/4,0D4+6/4,-S4+4/l,/D4+4/5,0D4+6/5,-S4+4/l,/D4+4/6,0D4+6/6,-S4+4/l,> D4+5/4,0D4+6/4,-S4+5/l,/D4+5/5,0D4+6/5,-S4+5/l,/D4+5/6,0D4+6/6,-S4+5/l,> D5+4/4,0D5+6/4,-S5+4/l,/D5+4/5,0D5+6/5,-S5+4/l,/D5+4/6,0D5+6/6,-S5+4/l,> D5+5/4,0D5+6/4,-S5+5/l,/D5+5/5,0D5+6/5,-S5+5/l,/D5+5/6,0D5+6/6,-S5+5/l,>`> ^D4+6/7,-S4+4/l,0D4+4/7,> D4+6/7,-S4+5/l,0D4+5/7,> D5+6/7,-S5+4/l,0D5+4/7,> D5+6/7,-S5+5/l,0D5+5/7,`> lqy+s*-s,-s*-i> hqg>

51 FKDSWHU 51 FDOLEUDWLRQ Uhfrqvwuxfwlqj iurp qrq0olqhdu fdoleudwlrq Iurp wkh vhfwlrq ghglfdwhg wr fdphud prghoolqj zlwk ohqv glvwruwlrq/ wkh iroorzlqj uhodwlrq kdv ehhq rewdlqhg/ L T ' ' s fs ' f _ n & o 2 f _ ~S s ts ' t _ n & o 2 t _ ~S +51<9, zkhuh/ dqg/ f _ ' Ef f & t _ ' Et f & +51<:, fs ts 8 ' + A 9 7 ~S f t ~ 6 : 8 +51<;, zkhuh + lv rewdlqhg iurp 518 dqg A lv wkh wudqvodwlrq yhfwru A ' E % c + c 5 1 Zh kdyh qrw wdnhq lqwr dffrxqw wkh & dqg & vljq dv wkh ghshqg rq wkh fdphud lpdjh sodqh ru surmhfwru sodqh srvlwlrq1 Dfwxdoo / wkh fdq eh frqvlghuhg srvlwlyh lq prghoolqj dv wkh duh ixuwkhu ghwhuplqhg lq fdoleudwlrq1 Frqvlghu wkdw wkh iroorzlqj vhw ri sdudphwhuv 7 S dqg 7 R duh ghwhuplqhg iurp wkh fdphud fdoleudwlrq dqg wkh surmhfwru fdoleudwlrq/ uhvshfwlyho 1 7 S ' S %c S +c S 5c S %c S +c S 5cs S c& c& S c S S fcfc& S S 7 R ' R %c R +c R 5c R %c R +c R 5cs R c& c& R c R R fcfc& R R +51<<, , Frqvlghu Ef S ct S dqg Ef R ctr wkh surmhfwlrq ri wkh 6G remhfw srlqw E f c t c ~ rq wkh fdphud lpdjh sodqh dqg wkh surmhfwru iudph/ uhvshf0 wlyho 1 Wkhq/ vxevwlwxwlqj wkh sdudphwhu ydoxhv ri wkh vhw iurp wkh jhqhulf rqhv/ wkh iroorzlqj htxdwlrqv duh rewdlqhg1

52 FKDSWHU 51 FDOLEUDWLRQ 86 3 L S ' rkr C 3 T S ' rkr C 3 L R ' rkr C 3 T R ' rkr C Lc f ' f S ct ' t S c % ' S %c + ' S +c 5 ' S 5c % ' S %c + ' S +c 5 ' S 5cs ' s S c& ' & S c& ' & S c f ' S fc f ' S fc& ' & S Tcf ' f S ct ' t S c % ' S %c + ' S +c 5 ' S 5c % ' S %c + ' S +c 5 ' S 5cs ' s S c& ' & S c& ' & S c f ' S fc f ' S fc& ' & S Lc f ' f R ct ' t R c % ' R %c + ' R +c 5 ' R 5c % ' R %c + ' R +c 5 ' R 5cs ' s R c& ' & R c& ' & R c f ' R fc f ' R fc& ' & R Tcf ' f R ct ' t R c % ' R %c + ' R +c 5 ' R 5c % ' R %c + ' R +c 5 ' R 5cs ' s R c& ' & R c& ' & R c f ' R fc f ' R fc& ' & R 4 D , 4 D , 4 D , 4 D , Wkhvh irxu htxdwlrqv shuplw wr rewdlq wkh E f c t c ~ xqnqrzqv xvlqj d ohdvw vtxduhv uhodwlrq vlplodu wr wkh rqh ghvfulehg lq htxdwlrq 51<81 Dojrulwkp ri qrq0olqhdu 6G uhfrqvwuxfwlrq A zlwk+olqdoj,=zlwk+olqdoj/ohdvwvtuv,= A &&&&&&&&&&&&&&&&&&&&&&&& A & Rewdlqlqj wkh htxdwlrqv1 A &&&&&&&&&&&&&&&&&&&&&&&& A +^^4/3/3/3`/^3/frv+w4,/vlq+w4,/3`/^3/0vlq+w4,/frv+w4,/3`/^3/3/3/4``,> A U +^^frv+w5,/3/0vlq+w5,/3`/^3/4/3/3`/^vlq+w5,/3/frv+w5,/3`/^3/3/3/4``,> A +^^frv+w6,/vlq+w6,/3/3`/^0vlq+w6,/frv+w6,/3/3`/^3/3/4/3`/^3/3/3/4``,> A +^^4/3/3/W{`/^3/4/3/W `/^3/3/4/W}`/^3/3/3/4``,> A +pxowlso +U}/U,/U{,> A +W/U,> A +^^{u`/^ u`/^}u`/^4``,> A +DD/P,> A A A A

53 FKDSWHU 51 FDOLEUDWLRQ 87 A A A A A &Fdphudprgho A A A A A yw f> A A A A A A A A / A XF,> A / A YF,> A & Surmhfwru prgho A A A A A yw s> A A A A A A A A / A XS,> A / A YS,> A &&&&&&&&&&&&&&&&&&&&&&&& A & 6G Uhfrqvwuxfwlrq A &&&&&&&&&&&&&&&&&&&&&&&&

54 FKDSWHU 51 FDOLEUDWLRQ 88 A & Uhdg vhw ri frxsohv 5G A uhdgole+uhdggdwd,> A A A urzglp+ss,/6/3,= A iru l iurp 4 e 4 wr urzglp+ss, gr A A A rg= A A iru l iurp 4 e 4 zkloh urzglp+ss, gr A A XF,= A A YF,= A A XS,= A A YS,= A A A S^4/4` frqyhuw+d^4`/olvw,^4`= S^4/5` frqyhuw+d^5`/olvw,^4`= A S^4/6` frqyhuw+d^6`/olvw,^4`= 0D^7`= A A S^5/4` frqyhuw+d^4`/olvw,^4`= S^5/5` frqyhuw+d^5`/olvw,^4`= A S^5/6` frqyhuw+d^6`/olvw,^4`= 0D^7`= A A S^6/4` frqyhuw+d^4`/olvw,^4`= S^6/5` frqyhuw+d^5`/olvw,^4`= A S^6/6` frqyhuw+d^6`/olvw,^4`= 0D^7`= A A S^7/4` frqyhuw+d^4`/olvw,^4`= S^7/5` frqyhuw+d^5`/olvw,^4`= A S^7/6` frqyhuw+d^6`/olvw,^4`= 0D^7`= A +pxowlso +lqyhuvh+pxowlso +wudqvsrvh+s,/s,,/wudqvsrvh+s,,/i,= A A rg= A & Uhfrqvwuxfwlrq rewdlqhg lq S6GV^l/4116`

55 Fkdswhu 6 Wkh Hslsrodu Jhrphwu Lq wklv fkdswhu zh zloo surylgh d ghvfulswlrq ri wkh hslsrodu jhrphwu ri d vwhuhrvfrslf v vwhp pdgh e wkh uhodwlrq ehwzhhq wzr slqkroh prghov1 Zh zloo dovr h{sodlq krz wklv jhrphwu lv prghoohg wkurxjk wkh ixqgdphqwdo pdwul{ dqg d phwkrg wr hvwlpdwh wklv pdwul{1 Wkh hslsrodu jhrphwu lv d srzhuixo frqfhsw zklfk kdv ehhq zlgho xvhg lq rughu wr uhgxfh wkh fruuhvsrqghqfh sureohp iurp d wzr0glphqvlrqdo wr d rqh0glphqvlrqdo vhdufk1 614 Wkh ixqgdphqwdo pdwul{ Wkh hslsrodu jhrphwu wdnhv lqwr dffrxqw wkh uhodwlrqvkls wkdw h{lvwv ehwzhhq wzr slqkroh prghov sodfhg lq wkh vsdfh ^]kdq <9f`1 Edvlfdoo / wkh hslsrodu jhrphwu prghov wkh ehkdylrxu rihslsrohv dqg hslsrodu olqhv1 Ixuwkhupruh/ wkh Ixqgdphqwdo pdwul{ lv ghqhg e wkh uhodwlrq ridq lpdjh srlqw rq dq lpdjh sodqh dqg lwv hslsrodu olqh rq wkh rwkhu sodqh ^Oxrq <5`1 Lq wkh iroorzlqj/ doo wkhvh frqfhswv duh ghvfulehg lq ghwdlo1 Frqvlghu d vwhuhrvfrslf v vwhp pdgh e wkh uhodwlrq riwzr slqkroh prghov zklfk surmhfw wkh vdph vfhqh/ dv vkrzq lq jxuh 6141 Hdfk prgho kdv dq dvvrfldwhg fr0ruglqdwh v vwhp f dqg f / orfdwhg lq wkhlu irfdo srlqw dqg uhvshfwlyho 1 Wkh lqwhuvhfwlrq riwkh olqh Ec zlwk hdfk lpdjh sodqh/ U dqg U surgxfhv wzr hslsrohv/ qdphg e dqg e 1 Frqvlghu dq lpdjh srlqw 6 rq U/ wkhq/ wkh hslsrodu jhrphwu doorzv xv wr vhdufk lwv fruuhvsrqghqfh srlqw 6 rqwkhhslsroduolqhri6 rq U / l1h1 rq wkh olqh, 6 zkhuh lw lv ghqhg e wkh lqwhuvhfwlrq riwkh sodqh U dqg wkh sodqh ghqhg e wkh wkuhh srlqwv / dqg 61 Qrwh wkdw zh gr qrw qhhg dq wkuhh glphqvlrqdo lqirupdwlrq iurp wr lqihu 6 1 Khuhdiwhu erwk fr0ruglqdwh v vwhpv f dqg f duh uhodwhg1 Frqvlghu wkdw f lv {hg dv wkh zruog fr0ruglqdwh v vwhp 1 60G phdvxuhv rid vfhqh duh 89

56 FKDSWHU 61 WKH HSLSRODU JHRPHWU\ 8: M Π e m x C l m x m C l m e I I Iljxuh 614= Wkh hslsodu jhrphwu ehwzhhq wzr slqkroh prghov1 dozd v phdvxuhg zlwk uhvshfw wr wkh zruog fr0ruglqdwh v vwhp1 Wkhq/ wkh vhfrqg fr0ruglqdwh v vwhp f lv uhodwhg wr f e d 6{6 urwdwlrq pdwul{ + dqg d wudqvodwlrq yhfwru 1 f ' f ' W f +614, f ' + f f ' , Jlyhq dq remhfw srlqw / lw zloo eh surmhfwhg rq erwk lpdjh sodqhv wkurxjkwkhirfdosrlqw1 Hdfksurmhfwhgsrlqwfdqehghqhgzlwkdpdwul{ zklfk prghov wkh rswlfdo dqg lqwhuqdo jhrphwu riwkh vhqvru/ dqg d pdwul{ zklfk sodfhv wkh vhqvru lq wkh vsdfh1 Wkh uhdghu lv uhihuuhg wr fkdswhu wzr wr xqghuvwdqg zkdw wkhvh wzr pdwulfhv orrn olnh1 Wkh surgxfw rierwk pdwulfhv ghwhuplqhv wkh surmhfwlrq ridq remhfw srlqw rq wkh lpdjh sodqh1 Frqvlghu dqg wkh lqwulqvlf pdwulfhv/ dqg dqg wkh h{wulqvlf pdwulfhv/ rivhqvruv U dqg U / uhvshfwlyho 1 Wkhq wkh surmhfwlrqv ri fdq eh h{suhvvhg dv/ r6 ' r 6 ' +616, +617,

57 FKDSWHU 61 WKH HSLSRODU JHRPHWU\ 8; Dv r dqg r duh dozd v frqvwdqw ydoxhv jlyhq e wkh krprjhqhrxv fr0 ruglqdwhv/ zh fdq h{suhvv wkh uhodwlrq zlwk dqg erwk surmhfwlrqv 6 dqg 6 e 618 dqg 6191 zkhuh/ 6 ' h +618, 6 ' h +619, h ' r +61:, h ' r +61;, dqg/ 6 ' ' % : 8 +61<, Lq wkh iroorzlqj/ frqvlghu wkh vlqjoh wudqvirupdwlrq pdwul{ 1 h Zh fdq jlyh d jhrphwulfdo lqwhusuhwdwlrq riwkh urzv ri h h{suhvvhglqhtxdwlrq6143/ h ' 7 ' ' 2 ' 8 ' 7 ^ ^ e ^ 2 ^2e ^ ^e , zkhuh hdfk yhfwru ' uhsuhvhqwv d surmhfwlyh sodqh zlwk d srlqw htxdwlrq ' 'f1 Dv fdq eh ghgxfhg iurp htxdwlrq 616/ 618/ 61:/ ' 'ffruuh0 vsrqgv wr srlqwv rq wkh lpdjh sodqh vxfk dv r 'f/ wkdw lv srlqwv dw lqqlwh1 Wkdw lv zk ' 'flv wkh irfdo sodqh1 Wkh rwkhu wzr sodqhv ' 'f dqg ' 2 'ffruuhvsrqg wr srlqwv rq wkh lpdjh sodqh vxfk dv 'fdqg 'fuhvshfwlyho 1 Wkh lqwhuvhfwlrq riwkhvh wzr sodqhv lv wkh olqh jrlqj wkurxjk wkh irfdo srlqw / dqg wkh ruljlq riwkh fr0ruglqdwh v vwhp riwkh fdphud lpdjh sodqh1 Qrwh wkdw wkh surmhfwlrq S riwkh irfdo srlqw grhv qrw fruuhvsrqg wr wkh ruljlq riwkh lpdjh fr0ruglqdwh v vwhp 1Lqidfw/lw lv dozd v orfdwhg dw d srlqw E f c f iurp +vhh fkdswhu wzr wr suryh wklv idfw,1 Dv lw lv vkrzq lq jxuh 615/ wkh irfdo srlqw ri wkh fdphud lv ghqhg e wkh lqwhuvhfwlrq riwkh wkuhh sodqhv ' 'f/ l1h1 riwkh sodqhv ' 'f/ ' 2 'fdqg ' 'f1 Wkhq/ lw fdq eh rewdlqhg iurp wkh htxdwlrq 61441

58 FKDSWHU 61 WKH HSLSRODU JHRPHWU\ 8< X y C X z X x t 3 0 QM= Focal Plane v c QM t 2 = 0 O i u Image Plane QM t 1 = 0 Iljxuh 615= Jhrphwulfdo uhsuhvhqwdwlrqv riwkh wkuhh olqh yhfwruv riwkh pdwul{ h ' ' ' % : 8 ' 5 7 f f f 6 8 h 'f +6144, Ohw xv h{suhvv wkh 6{7 pdwul{ h dv ghvfulehg e irupxod 6145/ h ' R +6145, zkhuh lv d 6{6 pdwul{ dqg R lv d 6{4 yhfwru/ erwk uhodwhg e wkh iroorzlqj htxdwlrqv/ ' 7 R R 2 R R 2 R 22 R 2 8 R ' R R 2 R Wkhq/ htxdwlrq 6144 fdq eh h{suhvvhg dv/ R 7 R e R 2e R e , 'f +6147, Rshudwlqj/ lw lv hdv wr rewdlq htxdwlrq Wkh surfhvv lv ghvfulehg lq wkh iroorzlqj htxdwlrqv=

59 FKDSWHU 61 WKH HSLSRODU JHRPHWU\ 93 n R ' f ' R 3 ' 3 R ' 3 R +6148, Wkh vdph uhodwlrq lv dssolhg iru rewdlqlqj wkh irfdo srlqw ri wkh rwkhu fdphud xvlqj htxdwlrq ' 3 R +6149, Wkhuhdghupxvwqrwhwkdwzhkdyhfrpsxwhgwkhirfdosrlqwiurpd pdwul{ h ghqhgxswrdvfdohidfwru1 Exwqrwhwkdwli h lv uhsodfhg e bc j wkhq R lv uhsodfhg e br dqg 3 e b 3 / uhvxowlqj wkdw srlqw grhv qrw fkdqjh/ dv lv uhtxluhg1 Ohw xv h{sodlq krz wkh hslsrodu olqh rid jlyhq lpdjh srlqw fdq eh frp0 sxwhg1 Lq rughu wr frpsxwh d olqh/ zh qhhg wzr srlqwv1 Rqh lv wkh hslsroh zklfk lv jlyhq e wkh surmhfwlrq riwkh irfdo srlqw rid fdphud rq wkh lpdjh sodqh riwkh rwkhu fdphud1 Wkh rwkhu lpdjh srlqw lv wkh surmhfwlrq ridq remhfw srlqw sodfhg dw wkh lqqlwh1 Wkh hslsroh e lv jlyhq e wkh surmhfwlrq ri rq U/ dqgfdqehfrpsxwhg iurp 618 dqg Wkh rwkhu hslsroh e lv jlyhq e wkh surmhfwlrq ri rq U / dqg fdq eh frpsxwhg iurp 619 dqg Wkhq/ erwk hslsrohv duh jlyhq e 614: dqg 614; uhvshfwlyho 1 e ' h 3 R e ' h 3 R +614:, +614;, Wkh rwkhu srlqw rilqwhuhvw lv wkh surmhfwlrq rid srlqw sodfhg dw wkh lqqlwh ghqhg dv " 1 Lq idfw/ wklv srlqw vdwlvhv wkh htxdwlrq 618/ zklfk fdq eh h{suhvvhg dv ' h " +614<, " ' h , Wkhq/ wkh rwkhu srlqw zklfk ghqhv wkh hslsrodu olqh ri 6 lv wkh surmhfwlrq ri " rq U / vhh jxuh 616/ zklfk fdq eh frpsxwhg wkurxjk wkh htxdwlrq 6154/

60 FKDSWHU 61 WKH HSLSRODU JHRPHWU\ 94 P 1 m C C e m I m l m I Iljxuh 616= Wkh uhodwlrq rid 5G lpdjh srlqw 6 lq U dqg lwv hslsrodu olqh, 6 rq U 1 6 " ' h , Wkh uhdghu lv uhihuuhg wr vhfwlrq wr yhuli wkdw 6 lv qrw lq xhqfhg e wkh wudqvodwlrq yhfwru R/ wkxv htxdwlrq 6154 fdq eh vlpsolhg e htxdwlrq " ' , Qrz zh duh deoh wr ghqh wkh hslsrodu olqh iurp erwk hqg srlqwv1 Vlqfh wkh hslsrodu olqh olhv rq dq lpdjh sodqh/ lw fdq eh uhsuhvhqwhg iurp wkh furvv0 surgxfw rierwk srlqwv/ zklfk doorzv xv wr frpsxwh wkh ruwkrjrqdo yhfwru ri wkh olqh1 Wkhq/ wkh hslsrodu olqh, 6 zloo eh ghqhg iurp wkh furvv0surgxfw ehwzhhq e dqg 6 / uhodwhg lq htxdwlrq Frpsxwlqj/ zh fdq h{suhvv, 6 e htxdwlrq 6158/, 6 ' e a 6 " +6156,, 6 ' e a ,, 6 ' de o % ,

61 FKDSWHU 61 WKH HSLSRODU JHRPHWU\ 95 Ohw xv mxvwli krz d furvv0surgxfw ehwzhhq wzr yhfwruv fdq eh h{suhvvhg lq d pdwulfldo zd xvlqj wkh dqwlv pphwulf pdwul{1 Frqvlghu wzr yhfwruv R dqg R / wkhq wkh furvv0surgxfw R a R fdq eh h{suhvvhg dv wkh surgxfw riwkh dqwlv pphwulf pdwul{ ri R ghqhg dv dro % e R / dv kdv ehhq suryhg lq 615:1 R ' % + 5 R ' % , 5, ' R a R ' 7 & 6 5 % ' 5 % ' 7 f 5 5 f % 8 + % f 7 % % %5 %+ +% 6 8 ' 8 ' dro % R +615:, Dw wklv srlqw/ wkh ixqgdphqwdo pdwul{ fdq eh frpsxwhg dv wkh uhodwlrq riwkh hslsrodu olqh, 6 ridq lpdjh srlqw 6 dqg lwv fruuhvsrqghqfh srlqw 6 1 Jlyhq d fruuhvsrqghqfh srlqw 6 /ri6/ lw pxvw olh rq olqh, 61Wkhq/ vlqfh/ 6, 6 'f +615;, RR ' nrnnr n ULt E) +615<, Lqidfw615;h{suhvvhvwkhsurgxfwri wzryhfwruv1 Wkhuvwrqhlvwkh yhfwru ghqhg e wkh irfdo srlqw dqg wkh fruuhvsrqghqfh srlqw 6 1 Wkh vhfrqg rqh/, 6 lv ghqhg e wkh furvv0surgxfw ehwzhhq yhfwru e dqg yhfwru 6 "1 Liwklv surgxfw lv htxdo wr f / wkhq wkh wkuhh yhfwruv duh fr0sodqdu1 Rshudwlqj/ 615; fdq eh h{suhvvhg dv/ 6 de o % 3 6 'f +6163, Wkhq/ zh fdq ghqh wkh ixqgdphqwdo pdwul{ 6/ dvwkh % pdwul{/ zklfk uhodwhv wkh hslsrodu olqh, 6 ridq lpdjh srlqw 6 dqg lwv fruuhvsrqghqfh srlqw 6 lq wkh iroorzlqj zd / 6 66 ' f +6164, 6 ' de o % , Zlwkrxw frpsoh{lw wkh uhdghu fdq xqghuvwdqg wkdw dqrwkhu ixqgdphq0 wdo pdwul{ fdq eh h{wudfwhg iurp wkh uhodwlrq ri erwk fdphudv1 6 zloo eh

62 FKDSWHU 61 WKH HSLSRODU JHRPHWU\ 96 ghqhg dv wkh ixqgdphqwdo pdwul{ zklfk uhodwhv wkh hslsrodu olqh, 6 ridq remhfw srlqw 6 dqg lwv fruuhvsrqghqfh srlqw 61 6 lv frpsxwhg wkurxjk wkh iroorzlqj htxdwlrqv/, 6 ' e a 6 " +6166,, 6 ' e a ,, 6 ' deo % , 6, 6 ' f +6169, 6 deo % 3 6 ' f +616:, ' f +616;, 6 ' deo % <, Zkhuh lw fdq eh suryhg wkdw 6 ' 6 /vr/ 6 66 ' f +6173, ' f +6174, Wkh uhdghu lv uhihuuhg wr vhfwlrq zkhuh lw lv vhw iruwk dqg suryhg wkdw 6 ' Fkdqjlqj wkh fr0ruglqdwh v vwhp Lq wkh iroorzlqj/ zh lqglfdwh zklfk vlpsolfdwlrqv fdq eh frqvlghuhg lq wkh ghqlwlrq riwkh ixqgdphqwdo pdwul{ lizh dvvxph wkdw zh duh xvlqj wkh qrupdolvhg fdphud prgho dqg/ liwkh zruog uhihuhqfh fr0ruglqdwh v vwhp fr0 lqflghv zlwk wkh fr0ruglqdwh v vwhp rirqh fdphud1 Dv d uvw dssurdfk/ frqvlghu rqo wkh uhodwlrq zlwk erwk fdphud fr0ruglqdwh v vwhpv/ zlwkrxw wdnlqj lqwr dffrxqw wkh lqwulqvlf pdwulfhv ru 1 Wklv idfw zloo vlpsoli wkh htxdwlrqv1 Krzhyhu/ wkh lqwulqvlf pdwulfhv zloo eh xvhg odwhu Zruog fr0ruglqdwh v vwhp {hg dw wkh vhfrqg fdphud/ rshudwlqj zlwk uhvshfw wr wkh uvw fdp0 hud1 Ohw xv hvwdeolvk wkh zruog fr0ruglqdwh v vwhp ` dw wkh vhfrqg fdphud/ zlwk dq ruljlq dqg d fr0ruglqdwh v vwhp f 1 Wkhq/ wkh fr0ruglqdwh v vwhp f riwkh uvw fdphud fdq eh uhodwhg wr wkh vhfrqg rqh e d urwdwlrqdo dqg d

63 FKDSWHU 61 WKH HSLSRODU JHRPHWU\ 97 W p P P X X C C K Iljxuh 617= Jhrphwulfdo uhodwlrq rierwk fdphud d{hv dqg d zruog fr0ruglqdwh v vwhp1 Zruog fr0ruglqdwh v vwhp {hg dw wkh uvw fdphud1 wudqvodwlrqdo pryhphqw/ zklfk doorz xv wr h{suhvv wkh ixqgdphqwdo pdwul{ iurp erwk pryhphqwv zlwkrxw frqvlghulqj wkh jhqhudo dqg pdwulfhv1 Frqvlghu wkh jhrphwulfdo uhodwlrq vkrzq lq jxuh 6171 Liwkh zruog fr0ruglqdwh v vwhp ` lv sodfhg dw / zlwk wkh vdph rulhqwd0 wlrq ri f/ wkh wudqvirupdwlrq pdwul{ ri wkh uvw fdphud/ douhdg lqwurgxfhg lq wkh suhylrxv vhfwlrq/ lv jlyhq e / h ' W f +6175, zklfk doorzv xv wr ghqh wkh vhfrqg fr0ruglqdwh v vwhp f /e dpdwul{ ghqhg e d urwdwlrq pdwul{ + dqg d wudqvodwlrq yhfwru / h ' k h + 3 k ' f +6176, Wkhq/ wkh htxdwlrq 6176 fdq eh rewdlqhg iurp jxuh 617 dsso lqj wkh iroorzlqj uhodwlrqv/ Rm ' k 3 Rm +6177, h 3 Rm` ' k 3 h 3 Rm` +6178, h 3 ' k 3 h , h ' h k +617:, hk 3 ' h kk ;,

64 FKDSWHU 61 WKH HSLSRODU JHRPHWU\ 98 h ' h k <, Rshudwlqj wkh htxdwlrq 6176 zh rewdlq wkh pdwul{ h h{suhvvhg lq htxd0 wlrq Dv/ h ' , k 3 ' ' ' 3 + W + f ' f f f W + f f ' f + + f 3 f ' f W f +6184, Qh{w/ zh zloo vlpsoli wkh iroorzlqj htxdwlrqv zklfk ghvfuleh wkh hslsrohv/ e ' h e ' h +6185, +6186, Zh fdq uhodwh wkh irfdo srlqw zlwk uhvshfw wr f/ dvwkhsrlqworfdwhg dw d glvwdqfh iurp wkh ruljlq 1 Wkhq/ wkh hslsroh e fdq eh vlpsolhg lq wkh iroorzlqj zd / e ' h ' h ' +6187, Lq wkh vdph zd / xvlqj htxdwlrq 6183 dqg wkh idfw wkdw lv wkh ruljlq ri wkh zruog fr0ruglqdwh v vwhp/ zh rewdlq/ e ' h f ' + + f ' , Wkh hslsrodu olqh, 6 fdq eh vlpsolhg e wkh iroorzlqj htxdwlrq/ dv/, 6 ' de o % 3 6 ' + a ,

65 FKDSWHU 61 WKH HSLSRODU JHRPHWU\ 99 3 ' + W ' :, wkhq/ iurp wkh ihdwxuhv ri wkh furvv0surgxfw/ zh nqrz wkdw htxdwlrq 6189 fdq eh h{suhvvhg dv/, 6 ' + E a 6 '+ d o % ;, Dv 6 lv dozd v ghqhg iurp d vfdoh idfwru zh fdq ljqruh wkh vljq/ wkhq/ Dqg wkh rwkhu hslsrodu olqh, 6, 6 ' + d o % <, fdq eh h{suhvvhg e /, 6 ' e a 3 6 ' a +6 ' d o % , wkdw lv ehfdxvh 3 ' E+ 3 ' +/ dv iru dq urwdwlrq pdwul{ + lw lv dozd v wuxh wkdw + 3 ' + 1 Qrz/ zh zdqw wr frqvlghu wkh lqwulqvlf pdwulfhv1 Wkhq/ jlyhq dq lghdo surmhfwlrq 6 dqg 6 lq phwulf xqlwv/ lwv uhdo surmhfwlrq h6 dqg h6 lq sl{hov/ duh jlyhq e wkh iroorzlqj htxdwlrqv/ h6 ' 6 6 ' 3 h6 h6 ' 6 6 ' 3 h , Wkhq/ zh rewdlq wkh iroorzlqj htxdwlrq/ zkhuh/ h6 6 h6 'f +6195, 6 ' 3 + d o % , Wkh iroorzlqj h{sodlqv krz zh kdyh rewdlqhg htxdwlrq Zh ghqh dq hvvhqwldo pdwul{, zklfk grhv qrw frqvlghu wkh lqwulqvlf pdwulfhv olnh wkh rog ixqgdphqwdo pdwul{/ wkhq/, ' + d o % +6197, 6,6 'f +6198, xvlqj wkh htxdwlrqv ghqhg lq 6194 zh rewdlq/ 3 h6, 3 h6 'f +6199,

66 FKDSWHU 61 WKH HSLSRODU JHRPHWU\ 9: vlqfh zh nqrz wkdw/ E ' 1 Wkhq zh fdq rewdlq htxdwlrq 619: zklfk lv vlplodu wr 6196 lizh vxevwlwxwh, e htxdwlrq h6 3, 3 h6 'f +619:, Wkh rwkhu ixqgdphqwdo pdwul{ fdq eh fdofxodwhg lq wkh vdph zd / rewdlq0 lqj/ zkhuh/ 'f +619;, 6 ' 3 d o % <, Lqwkhvdphzd /zhfdqsuryhkrzzhkdyhrewdlqhghtxdwlrq619<ghq0 lqj dqrwkhu hvvhqwldo pdwul{, c dv uhodwhg lq wkh iroorzlqj htxdwlrq1 xvlqj 6194 zh rewdlq/, ' d o % + +61:3, 3 h6, 3 h6 'f +61:4, zklfk doorzv xv wr jlyh wkh iroorzlqj htxdwlrq/ zklfk lv vlplodu wr 619< li zh vxevwlwxwh, e 61:3/ h6 3, 3 h6 'f +61:5, Zruog fr0ruglqdwh v vwhp {hg dw wkh uvw fdp0 hud/ rshudwlqj zlwk uhvshfw wr wkh vhfrqg fdp0 hud1 Vrph dxwkruv suhihu wr hvwdeolvk wkh zruog fr0ruglqdwh v vwhp dw wkh uvw fdphud1 Wkhq/ wkh zkroh jhrphwu lv frpsxwhg frqvlghulqj wkdw zh duh sodfhg dw wkh vhfrqg fr0ruglqdwh v vwhp1 Ohw xv hvwdeolvk wkh zruog fr0ruglqdwh v vwhp ` dw wkh uvw fdphud/ zlwk dq ruljlq dqg d fr0ruglqdwh v vwhp f1 Wkhq/ wkh fr0ruglqdwh v vwhp f riwkh vhfrqg fdphud fdq eh uhodwhg zlwk uhvshfw wr wkh uvw rqh e d urwd0 wlrq dqg d wudqvodwlrq1 Wklv uhodwlrq/ dv kdv ehhq h{sodlqhg lq wkh suhylrxv vhfwlrq/ doorzv xv wr h{suhvv wkh ixqgdphqwdo pdwul{ zlwkrxw frqvlghulqj wkh jhqhudo dqg pdwulfhv1 Frqvlghu wkh jhrphwulfdo uhodwlrq vkrzq lq jxuh 6181

67 FKDSWHU 61 WKH HSLSRODU JHRPHWU\ 9; W p P P X X C C K Iljxuh 618= Jhrphwulfdo uhodwlrq rierwk fdphud d{hv dqg d zruog fr0ruglqdwh v vwhp1 Zruog fr0ruglqdwh v vwhp {hg dw wkh vhfrqg fdphud1 Ohw xv sodfh wkh zruog fr0ruglqdwh v vwhp ` dw / zlwk wkh vdph rulhq0 wdwlrq riwkh fr0ruglqdwh v vwhp f 1 Wkhq/ wkh wudqvirupdwlrq pdwul{ h ri wkh vhfrqg fdphud lv jlyhq e / h ' W f +61:6, zklfk doorzv xv wr ghqh wkh uvw fr0ruglqdwh v vwhp f/ e dpdwul{ h ghqhg e d urwdwlrq pdwul{ + dqg d wudqvodwlrq yhfwru / h ' h + k 3 k ' f +61:7, Rshudwlqj/ Wkhq/ wkh wzr hslsrohv fdq eh frpsxwhg/ e ' h e ' h h ' :8, ' + + f ' + +61:9, ' h ' W f ' +61::, Wkh hslsrodu olqh, 6 fdq eh vlpsolhg e wkh iroorzlqj htxdwlrq/

68 FKDSWHU 61 WKH HSLSRODU JHRPHWU\ 9<, 6 'de o % 3 6 ' a +6 'd o % :;, dqg wkh rwkhu hslsrodu olqh, 6 e /, 6 ' e a 3 6 ' + a + 6 ' + d o % 6 ' + d o % 6 +61:<, Frqvlghulqj wkh vdph pdwkhpdwlfdo surfhgxuh h{sodlqhg lq htxdwlrqv :5 zh fdq lqfoxgh wkh lqwulqvlf pdwulfhv1 Ilqdoo /d ixqgdphqwdo pdwul{ fdqehrewdlqhgiurphdfkhslsroduolqh/ h6, 6 ' h6 6 h6 'f +61;3, 6 ' 3, 3 +61;4,, ' d o % + +61;5, dqg/ h6, 6 ' h6 6 h6 'f +61;6, 6 ' 3, 3 +61;7,, ' + d o % +61;8, Uhodwlrq ehwzhhq erwk Ixqgdphqwdo Pdwulfhv Wklv vhfwlrq zloo suryh wkdw erwk ixqgdphqwdo pdwulfhv 6 dqg 6 duh uhodwhg e d wudqvsrvh/ wkdw lv/ 6 ' 6 +61;9, 6 ' 6 +61;:, Iluvwo / zh kdyh wr uhphpehu vrph xvhixo ihdwxuhv ri wkh pdwulfldo dojheud= E 3 ' 3 3 E ' E 3 ' E 3 ' 3 d o % ' d o %

69 FKDSWHU 61 WKH HSLSRODU JHRPHWU\ :3 Zh zloo ghprqvwudwh wkh wudqvsrvh uhodwlrq ehwzhhq wkh ixqgdphqwdo pd0 wulfhv rivhfwlrq Wkh uhdghu pd qrwh wkdw wklv ghprqvwudwlrq dovr yhulhv wkh ixqgdphqwdo pdwulfhv ri vhfwlrq ' 3 + d o % 3 +61;;, 6 ' 3 + d o % 3 ' + d o % 3 3 ' +61;<, ' d o % ' 3 d o % + 3 ' 6 +61<3, dqg/ 6 ' 3 d o % <4, 6 ' 3 d o % + 3 ' d o % ' +61<5, ' + 3 d o% 3 ' 3 + d o % 3 ' 6 +61<6, Vlqfh 6 dqg 6 duh dozd v ghqhg iurp d vfdoh idfwru/ zh kdyh ghprq0 vwudwhg wkh uhodwlrq/ dv wkh vljq fdq eh ljqruhg Ihdwxuhv ri Srlqwv dqg Yhfwruv Zh kdyh ghflghg wr uhodwh erwk fr0ruglqdwh v whpv e d urwdwlrq pdwul{ + dqg d wudqvodwlrq yhfwru 1 Wkh vhfrqg v vwhp fdq eh uhodwhg iurp wkh uvw e wkh iroorzlqj ' + sor n +61<7, Dq lpdjh srlqw 6 R lv ghqhg dv yhfwru 6 iurp wkh irfdo srlqw dqg wkh srlqw rq wkh lpdjh sodqh 6 R 1 Iru d frqyhqlhqfh/ zh kdyh xvhg 6 dv d srlqw/ dowkrxjk zh kdyh wr ehdu lq plqg wkdw lw lv dfwxdoo d yhfwru1 Dw wklv srlqw zh kdyh wr uhphpehu wzr ihdwxuhv = 6 ' 6 R +61<8, 4, Jlyhq d srlqw lq wkh uvw lpdjh zklfk kdv ehhq {hg dv wkh ruljlq riwkh fr0ruglqdwh v vwhp1 Lwv fruuhvsrqghqfh srlqw lq wkh vhfrqg fr0ruglqdwh v vwhp lv jlyhq rqo e wkh wudqvodwlrq yhfwru 1

70 FKDSWHU 61 WKH HSLSRODU JHRPHWU\ :4 ' + sor n ' + 7 f f f 6 8 n ' +61<9, sor +61<:, 5, Jlyhq d yhfwru uhodwhg iurp wkh uvw fr0ruglqdwh v vwhp/ lwv wudqviruphg yhfwru iurp wkh vhfrqg fr0ruglqdwh v vwhp lv ghqhg rqo e wkh urwdwlrq pdwul{ +/ dv d yhfwru lv rqo ghqhg e lwv rulhqwdwlrq zlwk uhvshfw wr hdfk d{h/ qrw zlwk uhvshfw wr lwv srvlwlrq lq wkh ' + sor n ' + sor +61<;, Erwk ihdwxuhv kdyh ehhq eurdgo xvhg lq wkh suhylrxv vhfwlrqv lq rughu wr ghgxfh wkh ixqgdphqwdo pdwul{1 616 Ghwhuplqlqj wkh ixqgdphqwdo pdwul{ Wkh sureohp wr frqvlghu lv wkh hvwlpdwlrq ri wkh ixqgdphqwdo pdwul{ iurp d vhw rilpdjh srlqw fruuhvsrqghqfhv i h6 c h6 j1 Zh duh qrw lqwhuhvwhg lq krz wkh fruuhvsrqghqfhv ehwzhhq wzr lpdjhv fdq eh hvwdeolvkhg1 Krzhyhu/ wkh uhdghu pxvw nqrz wkdw wkh ixqgdphqwdo pdwul{ zloo eh fdofxodwhg iurp wkhvh fruuhvsrqghqfhv1 Wkh dffxudf riwkh uhvxowv zloo ghshqg gluhfwo rq wkh dffxudf riwkh whfkqltxh xvhg wr hvwlpdwh wkh pdwul{/ dqg lq d juhdw pdjqlwxgh/ lw dovr ghshqgv rq wkh whfkqltxh xvhg wr ghwhuplqh wkh fruuh0 vsrqghqfhv ehwzhhq erwk lpdjhv1 Wkh fruuhvsrqghqfhv ehwzhhq wzr lpdjhv fdq eh hvwdeolvkhg xvlqj dq riwkh whfkqltxhv ghvfulehg lq ^]kdq <6e`1 Lq idfw/ wkh ghwhuplqdwlrq ri wkh ixqgdphqwdo pdwul{ lv wkh sureohp ri vroylqj htxdwlrq 61<<1 Lq rwkhu zrugv/ lw lv wkh sureohp riqglqj rxw wkh frh!flhqwv ripdwul{ 6 zklfk vroyh wkh iroorzlqj htxdwlrq/ h6 6 h6 'f +61<<, Vhyhudo phwkrgv kdyh ehhq sursrvhg lq uhfhqw hduv1 Wkh uhdghu lv uh0 ihuuhg wr ^Oxrq <6e` wr kdyh dq ryhuylhz ri wkh sureohpv lqyroyhg lq gh0 whuplqlqj wkh ixqgdphqwdo pdwul{1 Wzr phwkrgv kdyh ehhq vwxglhg/ wkh uvw rqh lv edvhg rq wkh lwhudwlyh whfkqltxh riixqfwlrq plqlplvdwlrq sur0 srvhg e Qhzwrq0Udskvrq zklfk kdv ehhq lqwurgxfhg lq fkdswhu wzr/ dqg

71 FKDSWHU 61 WKH HSLSRODU JHRPHWU\ :5 wkhuhiruh/ qrw h{sodlqhg lq wklv vhfwlrq1 Krzhyhu/ lwhudwlyh phwkrgv xvx0 doo vshqg d orw rifrpsxwlqj wlph/ frqwudu wr qrq0lwhudwlyh phwkrgv1 Vr/ wkh zlgho nqrzq qrq0lwhudwlyh Hljkw0Srlqwv Phwkrg ^]kdq <9f` zklfk xvhv hljkw ru pruh pdwfklqj srlqwv lq rughu wr rewdlq wkh Ixqgdphqwdo Pdwul{/ lv suhvhqwhg dqg dqdo vhg lq wkh iroorzlqj vhfwlrq Wkh Hljkw0Srlqwv Phwkrg Wklv phwkrg grhv qrw dgguhvvhv wkh rewhqwlrq riwkh xqnqrzqv sdudphwhuv ri wkh fdphud/ exw wkh rewdlqlqj riwkh frh!flhqwv riwkh ixqgdphqwdo pdwul{1 Vr/ wkh phwkrg zloo doorz xv wr frpsxwh wkh hslsrodu jhrphwu / exw/ lw grhv qrw ohdg xv wr h{wudfw wkh fdphud sdudphwhuv riwkh v vwhp1 Lq sudfwlfh/ pruh wkdq hljkw srlqwv zloo eh xvhg wr frpsxwh wkh ixqgdphq0 wdo pdwul{ wr dvvxuh dq dffxudwh uhvxow1 Wkh phwkrg lv fdoohg wkh Hljkw0Srlqwv Phwkrg ehfdxvh d plqlpxp rihljkw srlqwv duh qhhghg wr xvh wklv phwkrg/ dv wkhuh duh rqo hljkw xqnqrzqv wr ghwhuplqh wkh pdwul{ lizh frqvlghu wkdw wkhpdwul{lvghqhgxswrdvfdohidfwru1 Htxdwlrq 61<< fdq eh zulwwhq lq wkh iroorzlqj zd d% c+ c o Rshudwlqj/ zh rewdlq wkdw/ 8 7 % + 8 'f , zkhuh/ L? s 'f , L? ' E c 2 c c? , dqg/ s ' E8 c8 2 c8 c8 2 c8 22 c8 2 c8 c8 2 c , ' E% % c+ % c% c% + c+ + c+ c% c+ c , Wkh vroxwlrq rihtxdwlrq lv s 'f/ zklfk lv qrw zdqwhg1 Wr dyrlg wklv/ zh qhhg wr lpsrvh vrph frqvwudlqwv rq wkh frh!flhqwv riwkh Ixqgdphq0 wdo Pdwul{1 Krzhyhu/ vhyhudo phwkrgv kdyh ehhq suhvhqwhg lq uhfhqw hduv ^Oxrq <6e`1 Zh h{sodlq wkh hdvlhvw rqh/ wkdw vhwv rqh riwkh frh!flhqwv ri 6 wr d nqrzq ydoxh1 Vlqfh wkh ixqgdphqwdo pdwul{ lv dozd v ghqhg iurp d vfdoh idfwru/ zh fdq vhw rqh ri wkh frh!flhqwv ri 6 wr 1 Wklv lv rqh ri

72 FKDSWHU 61 WKH HSLSRODU JHRPHWU\ :6 wkh frqvwudlqwv riwhq xvhg lq wkh fdofxodwlrq ri wkh ixqgdphqwdo pdwul{1 Wkh surfhvv rizklfk lv ghvfulehg lq wkh iroorzlqj vhfwlrq1 Qrwh wkdw htxdwlrq kdv d xqltxh vroxwlrq li 6? i?c 6j/ wkhq/ dqrwkhu phwkrg lv dovr ghvfulehg zklfk lv edvhg rq dfklhylqj wkh vlqjxodu ydoxhv ri L1 Wkh hljkw0srlqw phwkrg lv d vwdqgdug dqg lw kdv ehhq suryhg wkdw lv yhu vhqvlwlyh wr qrlvh dqg txlwh xqvwdeoh1 Frqvlghu wkh iroorzlqj lpsuryhphqw sursrvhg e vrph dxwkruv ^]kdq <9f` ^Prku <9`1 D w slfdo lpdjh fr0ruglqdwh iudph lq d 845{845 lpdjh pljkw eh derxw 2ff1 Vrphriwkhhqwulhvlqdurz ri duh %% derxw 2ff 2 /rwkhuvduh% derxw 2ff dqg wkh odvw hqwu lv 1 Vr/ wkhuh lv d yduldwlrq lq vl}h ri 2ff 2 dprqj wkh hqwulhv ri L? /dqgvrphri2ff e lq wkh hqwulhv ri L?L? 1WklvphdqvwkdwL?L? lv h{wuhpho loo0frqglwlrqhg1 D vlpsoh vroxwlrq wr wklv lv wr qrupdolvh wkh sl{hov fr0ruglqdwhv iurp dfc D 2o wr d c o ehiruh surfhhglqj wr wkh fdofxodwlrq ri wkh ixqgdphqwdo pdwul{1 Wklv surylghv d zhoo0edodqfhg pdwul{ dqg pxfk pruh vwdeoh dqg dffxudwh uhvxowv iru 61 Vroylqj 6 iurp frqvwudlqw 8 ' Lizh lpsrvh wkdw frh!flhqw 8 ' /wkhqiurphtxdwlrq61433zhfdquhodwh htxdwlrq lq wkh iroorzlqj pdqqhu/ zkhuh/ L?s '? , L? 'E c 2c c? , dqg/ s ' E8 c8 2 c8 c8 2 c8 22 c8 2 c8 c :, ' E% % c+ % c% c% + c+ + c+ c% c ;, Rshudwlqj htxdwlrq zh fdq rewdlq wkh iroorzlqj htxdwlrq/ zklfk doorzv xv wr rewdlq wkh ; xqnqrzq frh!flhqwv riwkh ixqgdphqwdo pdwul{1 L? 3 L?s ' L? 3? +6143<, s ' L? 3? , s ' L? L? 3 L?? , Wkh sureohp riwklv phwkrg lv wkdw zh vhw d ghwhuplqhg frh!flhqw wr eh nqrzq dqg htxdo wr 1 Liwklv frh!flhqw lv qrw htxdo wr f qr sureohpv

73 FKDSWHU 61 WKH HSLSRODU JHRPHWU\ :7 zloo eh frph xs/ exw liwklv lv qrw wkh fdvh/ wkhq wkh uhvxow zloo eh uhdoo loo0 frqglwlrqhg1 D vroxwlrq lv wr wu doo qlqh srvvlelolwlhv {lqj hdfk 8 frh!flhqw ri 8 wr dqg uhwdlqlj wkh ehvw pdwul{1 Wkh ehvw pdwul{ zloo eh wkh rqh zklfk uhgxfhv wkh glvwdqfh riwkh fruuhvsrqghqfh srlqwv wr wkhlu hslsrodu olqhv1 Wkh iroorzlqj vhfwlrq ghvfulehv d phwkrg zklfk doorzv xv wr rewdlq wkh < xqnqrzqv ri 6 zlwkrxw frqvlghulqj wkh dvvxpswlrq wkdw d frh!flhqw ri 6 kdv wr eh {hg1 Vroylqj 6 iurp wkh hljhq ydoxhv1 H{suhvvlqj wkh htxdwlrq lq d pdwulfldo zd / s : 9 s 2 : 'f ,??2?6 s 6 Zkhuh L lv d?{6 nqrzq pdwul{ dqg s lv wkh yhfwru wr ghwhuplqh1 Lq idfw/ zh nqrz wkdw iurp dq?{6 pdwul{/ ;L?%6 6? i?c 6j , wkdw lv/ wkh udqn rid pdwul{ lv wkh qxpehu rifroxpqv ru urzv olqhduo lqghshqghqw1 Frqvlghu wkdw?:6dv zh zloo dozd v kdyh pruh htxdwlrqv +fruuhvsrqghqfh srlqwv, wkdq xqnqrzqv wr ghwhuplqh1 ;L?% , Wkhq/ iurp wkh pdwul{ surshuwlhv/ zh fdq ghwhuplqh s li 6/ l1h/ ' 6 ', s 'f , 6 ', s fdq eh ghwhuplqhg , Dw wklv srlqw/ zh fdq d!up wkdw lihtxdwlrq 6144: lv vdwlvhg/ wkhq wkh htxdwlrq 6144; lv dovr vdwlvhg ^Urwh <8`1 Ls ' f +6144:, L Ls ' f +6144;, Iru hdfk hljhq ydoxh b f b b 2 br ri L L zhfdqrewdlqlwvhljhq yhfwru f c c 2 R iurp wkh iroorzlqj htxdwlrq/

74 FKDSWHU 61 WKH HSLSRODU JHRPHWU\ :8 b WL L 'f +6144<, zkhuh W lv d 6 % 6 lghqwlw pdwul{1 Lw lv lqwhuhvwlqj wr frpsxwh wkh qru0 pdolvhg hljhq yhfwru/ vr/ zh zloo fdofxodwh wkh iroorzlqj htxdwlrq iurp hdfk yhfwru1 ' E%c , *2 E% n + Lihtxdwlrq lv vdwlvhg wkhq wkh uvw hljhq ydoxh pxvw eh htxdo wr f/ l1h1 b f 'f1 Lizh vxevwlwxwh lw lqwr 6144< zh rewdlq dq htxdwlrq htxdo wr 6144;/ wkhq zh duh deoh wr d!up wkdw s lv htxdo wr wkh hljhq yhfwru dvvrfldwhg wr wkh vpdoohvw hljhq ydoxh ri L L Ghwhuplqdwlrq ri wkh hljhq ydoxhv dqg wkh hljhq yhfwruv ri d pdwul{ Wkhruhp = Dq?{? pdwul{ kdv d vhtxhqfh rihljhq ydoxhv jlyhq e wkh urrwv riwkh htxdwlrq 61454/ zkhuh htxdwlrq fdq eh fdofxodwhg iurp htxdwlrq nqrzq dv wkh fkdudfwhulvwlf sro qrp rid pdwul{ ^Urwh <8`1 { Eb ' f , { Eb ' _i EbW , Htxdwlrq fdq eh vroyhg/ iurp wkh uxohv ri wkh ghwhuplqdqwv/ xvlqj zh fdq rewdlq _i E '?[ E _i E , ' { Eb ' b? b?3 n b f , Wkhq/ wkh fkdudfwhulvwlf sro qrp ri D eh fdq h{suhvvhg iurp lwv urrwv lq wkh iroorzlqj irup jlyhq e htxdwlrq zkhuh/ { Eb 'Eb b f Eb b Eb b? , duhnqrzqdvwkhhljhqydoxhvri1 b f b b 2 b? ,

75 FKDSWHU 61 WKH HSLSRODU JHRPHWU\ :9 Dw wklv srlqw/ wkh hljhq yhfwru dvvrfldwhg dw hdfk hljhq ydoxh fdq eh fdofxodwhg e wkh iroorzlqj htxdwlrq Frqfoxvlrqv Eb W 'f +6145:, Wkh hljkw0srlqwv phwkrg fdq eh dssolhg iru rewdlqlqj wkh ixqgdphqwdo pd0 wul{1 Dowkrxjk wkh uhdghu pxvw uhdolvh wkdw wkh lqwulqvlf dqg h{wulqvlf fdphud sdudphwhuv fdq eh dfklhyhg iurp wkh wudqvirupdwlrq pdwul{/ wkh surfhvv wr rewdlq wkhp iurp wkh ixqgdphqwdo pdwul{ lv qrw vr hdv +wkh uhdghu lv dg0 guhvvhg wr fkdswhu wzr wr kdyh d ghvfulswlrq riwkh wudqvirupdwlrq pdwul{ ri d vlqjoh fdphud,1 Wkh sureohp lv edvhg rq wkh idfw wkdw wkh sdudphwhuv ri erwkfdphudvkdyhehhqmrlqhglqdvlqjoh{ pdwul{/ vr zh kdyh e lqwulqvlf sdudphwhuv iru hdfk fdphud dqg S h{wulqvlf sdudphwhuv/ h{suhvvlqj wkh rulhq0 wdwlrq dqg wudqvodwlrq rid fdphud iurp wkh rwkhu rqh/ wr ghwhuplqh1 Wkhq/ zh zloo kdyh e xqnqrzqv wr eh ghwhuplqhg iurp rqo 2 htxdwlrqv1 Vrph dxwkruv/ dv ^Oxrq <5` kdyh frqvlghuhg wkdw wkh vwhuhr ylvlrq v vwhp lv pdgh e wzr ylhzv wdnhq iurp wkh vdph fdphud dw wzr glhuhqw srvlwlrqv/ wkhq wkh lqwulqvlf fdphud sdudphwhuv fdq eh ghwhuplqhg iurp wkh Nuxssd htxdwlrqv dv rqo e yduldeohv duh uhtxluhg1 Zkhq wkh lqwulqvlf sdudphwhuv duh nqrzq/ wkh uhodwlyh pryhphqw rid vlqjoh fdphud/ ru wkh wudqvodwlrq ehwzhhq wzr ru pruh fdphudv/ fdq eh fdofxodwhg1 Wkh idfw wkdw wkh phwkrgv sursrvhg kdyh dozd v wkh frqvwudlqw ri xvlqj d vlqjoh fdphud/ ru pruh wkdq rqh fdphud zlwk vrph rswlfdo fkdudfwhulvwlfv/ kdv pdgh xv ghflgh wr xvh wkh ixqgdphqwdo pdwul{ rqo wr rewdlq wkh hslsr0 odu jhrphwu rirxu vwhuhr v vwhp pdgh e d vlqjoh fdphud dqg wkh oljkw surmhfwru1 Qr sdudphwhuv riwkh fdphud dqg wkh surmhfwru duh rewdlqhg iurp wkh ixqgdphqwdo pdwul{1 Lq rughu wr whvw wkh htxdwlrqv/ d vwhuhrvfrslf v vwhp pdgh e wkh uhodwlrq0 vkls ehwzhhq wzr slqkroh prghov kdv ehhq vlpxodwhg1 Erwk slqkroh prghov kdyh ehhq sodfhg lq d yluwxdo 6G vfhqdulr1 D vhw ri6g remhfw srlqwv kdv dovr ehhq sodfhg lq wkhuh1 Lqwulqvlf sdudphwhuv kdyh ehhq {hg1 Wkhq/ 5G fruuhvsrqghqfhv rq erwk lpdjh sodqhv/ rihdfk 6G remhfw srlqw/ kdyh ehhq frpsxwhg1 Wkh vhtxhqfh risrlqwv lv vkrzq lq wkh iroorzlqj pdwul{ / zkhuh hdfk urz lv rewdlqhg iurp wkh surmhfwlrq ri d 6G srlqw rq erwk lpdjh sodqhv/ wkdw lv iurp % c+ c% 2 c+ 2 1

76 FKDSWHU 61 WKH HSLSRODU JHRPHWU\ :: 5 ' 9 7 e2d.ff.fh f22b HDf 2ff fd.s e.d efs2 2Sff fffs 2 f Db.D.ff S.b fe.. DDDf 2ff 2 2 ebe 2 2 H. b DD f.d e2.d 2.D 2SHb bef ee S Df fd 2DH. eeh. H. fdh2 2b2e SH2D 2.D e.h HHH 2H SDf f bh f2h S. S 2 b.fs 2.H D 2D.2D He 2D ebh SDf HD 2HeD eb 2 S 2 f2b 2 e2d Dff 22bf e S HDf e.ff fb.. e fh efs2 eeff f 2 efh Db.D Dff e H.. DDDf e.ff 2eSf.bH 6 : ;, Rqo wkhvh frxsohv ri5g srlqwv kdyh ehhq xvhg wr rewdlq wkh ixqgdphqwdo pdwul{ 61 Wkh lwhudwlyh Qhzwrq0Udskvrq phwkrg kdv ehhq xvhg/ zklfk kdv ehhq suryhg wr eh pruh dffxudwh wkdq wkh dqdo wlfdo phwkrgv zh kdyh h{sodlqhg1 Wkh rewdlqhg pdwul{ lv wkh iroorzlqj/ 5 6 ' 7 f Db22SD b f 2eDSSSSf ed.2.fbbf fs2.s2beeb fffs.22e.f.. 2 fbbdhhd eeee2desf DS2. Sf. 2 2f.2. H <, Wkh hslsrodu jhrphwu kdv ehhq uhfryhuhg1 Wkh iroorzlqj jxuhv vkrz wkh hslsrodu olqhv dqg wkh fruuhvsrqghqfhv srlqwv rq erwk lpdjh sodqhv1 Qrwh/ wkdw doo wkh fruuhvsrqghqfh srlqwv olh rq wkh hslsrodu olqh ghqhg e wkh ixqgdphqwdo pdwul{ dqg lwv srlqw frxsoh1 Qr glvfuhsdqf lv suhvhqwhg dv wkh v vwhp kdv ehhq vlpxodwhg dqg grhv qrw dgmxvw wr d uhdo vfhqdulr1

77 FKDSWHU 61 WKH HSLSRODU JHRPHWU\ :; Iljxuh 619= Dq h{dpsoh riwkh 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 Iljxuh 61:= D }rrp riwkh vtxduh duhdv riwkh suhylrxv lpdjhv1 Qrwh wkdw wkh fruuhvsrqghqfh srlqwv olh dozd v rq wkh hslsrodu olqhv ghqhg e wkhlu frxsoh srlqwv1

78 FKDSWHU 61 WKH HSLSRODU JHRPHWU\ :< Wkh dojrulwp xvhg wr lqihu wkh ixqgdphqwdo pd0 wul{ Wkh dojrulwp kdv ehhq surjudpphg xvlqj wkh Pdsoh Y Uhohdvh 7 U hqylurq0 phqw1 A zlwk+olqdoj,= A &&&&&&&&&&&&&&&&&&&&&&&&&& A & Rewdlqlqj wkh ixqgdphqwdo pdwul{ A &&&&&&&&&&&&&&&&&&&&&&&&&& A +^^4/3/3`/^3/frv+d,/vlq+d,`/^3/0vlq+d,/frv+d,``,> A U +^^frv+e,/3/0vlq+e,`/^3/4/3`/^vlq+e,/3/frv+e,``,> A +^^frv+f,/vlq+f,/3`/^0vlq+f,/frv+f,/3`/^3/3/4``,> A +^^3/0w}/w `/^w}/3/0w{`/^0w /w{/3``,> +U{/pxowlso +U /U},,> A +^^dx4/3/xr4`/^3/dy4/yr4`/^3/3/4``,> A +^^dx5/3/xr5`/^3/dy5/yr5`/^3/3/4``,> A A +D5lw/pxowlso +W/pxowlso +U/D4l,,,> A +^^i44/i45/i46`/^i54/i55/i56`/^i64/i65/i66``,> A A A +p5w/pxowlso +I/p4,,> A &&&&&&&&&&&&&&&&&&&&&&&&& A & Rewdlqlqj wkh sduwldo ghulydwlyhv A &&&&&&&&&&&&&&&&&&&&&&&&& A A A A A & Uhdg wkh fruuhvsrqghqfh srlqwv A uhdgole+uhdggdwd,> A A A iru l iurp 4 e 4 wr urzglp+ps, gr A A A A A rg= A hydo+p,> A & Lqlwldo ydoxhv ri wkh sdudphwhuv

79 FKDSWHU 61 WKH HSLSRODU JHRPHWU\ ;3 A A yw A A A / IL^4/4`,,> A / IL^4/5`,,> A / IL^4/6`,,> A / IL^5/4`,,> A / IL^5/5`,,> A / IL^5/6`,,> A / IL^6/4`,,> A / IL^6/5`,,> A / IL^6/6`,,> A &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& A & Fdoleudwlqj e wkh Prglhg Qhzwrq Udskvrq A &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& A A A A A A iru n iurp 4 e 4 wr 48 gr A iru l iurp 4 e 4 wr urzglp+p, gr

80 FKDSWHU 61 WKH HSLSRODU JHRPHWU\ ;4 A A J,,= A gji44,,= A gji45,,= A gji46,,= A gji54,,= A gji55,,= A gji56,,= A gji64,,= A gji65,,= A gji66,,= A rg> A &mi+{n,m A pxowlso +wudqvsrvh+yj,/yj,> A &gn A MW wudqvsrvh+m,= MWM pxowlso +MW/M,= A +LMWM/MW,= A +LMWMMW/YJ,,= A &{n A A & exvfdqw m A A iru m iurp 3 e 4 zkloh dfdedu? 4gr A 0 5a+0m,-gn,> A & mi+{n 0 5a+0m,-gn,m A iru l iurp 4 e 4 zkloh urzglp+p, gr A A J,,= A rg=

81 FKDSWHU 61 WKH HSLSRODU JHRPHWU\ ;5 A +wudqvsrvh+yj5,/yj5,> A &mi+{n05a+0l,-gn,m? mi+{n,m A li qyj5n? qyjn wkhq > A rg= A & {n.4 A A 0 5a+0m,-gn,> A A A A A A li qyjn? 43a+043, wkhq euhdn> > A rg= A FY+4,> A & Wkh pdwul{ I lv rewdlqhg lq yhfwru FY+4, Wkh dojrulwkp xvhg wr rewdlq wkh hslsrodu olqhv iurp 6 Wkh dojrulwp kdv ehhq surjudpphg xvlqj wkh Pdwode U hqylurqphqw1 fohdu doo> forvh doo> ( Jhqhudwlrq ri wkh fruuhvsrqghqfh srlqwv ( prghoolqj Fdphud 41 Ruljlq ri R zruog fr0ruglqdwh v vwhp p+*^dx43xr43>3dy4yr43>3343>3334`*,> ;8> <3> 463> 468> D4> hydo+dduw4+4/=,,> hydo+dduw4+5/=,,> hydo+dduw4+6/=,,> hydo+dduw4+7/=,,> ( prghoolqj Fdphud 51 Pryhg iurp R1 v p+*^ > 3 frv+d, vlq+d, 3>3 0vlq+d, frv+d, 3> `*,> v p+*^frv+e, 3 0vlq+e, 3> >vlq+e, 3 frv+e, 3> `*,> v p+*^frv+f, vlq+f, 3 3>0vlq+f, frv+f, 3 3> > `*,> v ppxo+u{/v ppxo+u /U},,> v p+*^4 3 3 w{>3 4 3 w >3 3 4 w}> `*,> v ppxo+w5/u5,> p+*^dx53xr53>3dy5yr53>3343>3334`*,> v ppxo+d5/uw5,>

82 FKDSWHU 61 WKH HSLSRODU JHRPHWU\ ;6 314> 31:> 315> 4333> 0733> 563> ;8> <3> 463> 468> hydo+dduw5+4/=,,> hydo+dduw5+5/=,,> hydo+dduw5+6/=,,> hydo+dduw5+7/=,,> ( Jhqhudwlqj wkh 6G zruog srlqw yhfwru ^433>0733>5333>4`> ^633>0733>6333>4`> ^833>0733>7333>4`> ^:33>0733>5333>4`> ^<33>0733>6333>4`> ^433>083>7333>4`> ^633>083>5333>4`> ^833>083>6333>4`> ^:33>083>7333>4`> ^<33>083>5333>4`> ^433>83>6333>4`> ^633>83>7333>4`> ^833>83>5333>4`> ^:33>83>6333>4`> ^<33>83>7333>4`> ^433>733>5333>4`> ^633>733>6333>4`> ^833>733>7333>4`> ^:33>733>5333>4`> ^<33>733>6333>4`> ( Rewhqlqj wkh 5G srlqwv lq erwk lpdjh sodqhv1 iru DUW4-Y+=/l,> S4+4/l,2S4+6/l,> S4+5/l,2S4+6/l,> S4+6/l,2S4+6/l,> DUW5-Y+=/l,> S5+4/l,2S5+6/l,> S5+5/l,2S5+6/l,> S5+6/l,2S5+6/l,> hqg ( Ixqgdphqwdo Pdwul{ jlyhq e Pdsoh1 ^013648<55984< :5:3<<3> :95<77< 13339:5574:3:: 05143<<8;;84> :493: :5:4;`> ( Hslsrodu Jhrphwu ri Fdphud 51 jxuh+4,> krog rq> d{lv+^0845/845/0433/845`,> wlwoh+*fdphud 5*,> iru sorw+s5+4/l,/s5+5/l,/*j-*,>

83 FKDSWHU 61 WKH HSLSRODU JHRPHWU\ ;7 sorw+^0845>845` / ^ op5+4/l,. op5+6/l,, 2 +0op5+5/l,, / op5+4/l,. op5+6/l,, 2 +0op5+5/l,,`,> hqg> }rrp rq> ( Hslsrodu Jhrphwu ri Fdphud 41 jxuh+5,> krog rq> d{lv+^0845/845/0433/845`,> wlwoh+*fdphud 4*,> iru sorw+s4+4/l,/s4+5/l,/*j-*,> sorw+^0845>845` / ^ op4+4/l,. op4+6/l,, 2 +0op4+5/l,, / op4+4/l,. op4+6/l,, 2 +0op4+5/l,,`,> hqg> }rrp rq>

84 Fkdswhu 7 Vwuxfwxuhg Oljkw Vwhuhrylvlrq lv dq dwwudfwlyh dqg zlgho xvhg phwkrg/ exw/ lw lv udwkhu olplwhg wr exlog 6G vxuidfh pdsv/ gxh wr wkh fruuhvsrqghqfh sureohp1 Wkh fruuh0 vsrqghqfh sureohp fdq eh uhgxfhg xvlqj d phwkrg edvhg rq wkh vwuxfwxuhg oljkw frqfhsw/ surmhfwlqj d jlyhq sdwwhuq rq wkh phdvxulqj vxuidfhv1 Krz0 hyhu/ vrph uhodwlrqv ehwzhhq wkh surmhfwhg sdwwhuq dqg wkh uh hfwhg rqh pxvw eh vroyhg1 Wklv uhodwlrqvkls fdq eh gluhfwo irxqg frgli lqj wkh surmhfwhg oljkw/ vr wkdw/ hdfk lpdjhg uhjlrq ri wkh surmhfwhg sdwwhuq surylghv wkh qhhghg lqiru0 pdwlrq wr vroyh wkh fruuhvsrqghqfh sureohp1 Wklv fkdswhu zloo dqdo vh vwhuhr ylvlrq sureohpv jrlqj wkurxjk vwuxfwxuhg oljkw1 Ixuwkhupruh/ frghg vwuxfwxuhg oljkw lv suhvhqwhg/ dqg d vxuyh ri wkh prvw lpsruwdqw whfkqltxhv lv uhodwhg1 714 Lqwurgxfwlrq Zkhq 6G lqirupdwlrq ri d jlyhq vxuidfh lv qhhghg/ zh kdyh wr fkrrvh ehwzhhq d sdvvlyh phwkrg dqg dq dfwlyh rqh ^Dkoh ;<`1 Wkh prvw zlgho nqrzq sdv0 vlyh phwkrg lv vwhuhrylvlrq ^Eduq ;5` zklfk fdq eh dfklhyhg lq wzr glhuhqw zd v1 Lq wkh uvw zd / dq rswlfdo vhqvru lv pryhg wr nqrzq uhodwlyh srvlwlrqv lq wkh vfhqh1 Lq wkh vhfrqg zd / wzr ru pruh rswlfdo vhqvruv duh suhylrxvo {hg lq nqrzq srvlwlrqv ^D df <4`1 Wkh vxuidfh wr eh phdvxuhg lv surmhfwhg rq wkh lpdjh sodqh ri hdfk vhqvru wkurxjk hdfk irfdo srlqw1 Dv ghvfulehg lq fkdswhu 5/ 6G fr0ruglqdwhv ri wkh remhfw srlqw fdq eh rewdlqhg e wuljrqrp0 hwu ^Kdud <6` ^Kdoo ;5` ^Idxj <6` iurp wkh nqrzq surmhfwlrqv ri dq remhfw srlqw dqg/ ixuwkhupruh/ iurp wkh uhodwlrqvkls ehwzhhq wkh rswlfdo vhqvruv1 Exw zh kdyh wr nqrz/ zlwk fruuhfwqhvv/ iru hdfk remhfw srlqw/ lwv surmhfwlrqv rq wkh rswlfdo vhqvru lpdjh sodqhv1 Lq idfw/ lq rughu wr rewdlq wkh 6G fr0 ruglqdwhv ri d jlyhq srlqw iurp? jlyhq surmhfwlrqv +rqh iurp hdfk vhqvru,/ wkhvh surmhfwlrqv pxvw qhfhvvdulo frph iurp wkh vdph remhfw srlqw1 Wklv ;8

85 FKDSWHU 71 VWUXFWXUHG OLJKW ;9 sureohp lv nqrzq dv wkh fruuhvsrqghqfh sureohp1 Wkhuh duh vrph hvvhqwldo sureohpv lq vwhuhr ylvlrq zklfk gl!fxow wr vroyh wkh fruuhvsrqghqfh sureohp ehwzhhq erwk lpdjh sodqhv ^]kdq <6e`1 Vrph dxwkruv frqvlghu wr vroyh wkh sureohp ri pdwfklqj iurp vlqjxodu srlqwv/ l1h1 qglqj srlqwv lq erwk lpdjh sodqhv zlwk vlplodu qhljkerxukrrg fkdudfwhu0 lvwlfv/ fodvvli lqj wkhp dv fruqhuv dqg yhuwh{hv1 Vlqfh erwk 5G lpdjhv duh surmhfwlrqv ri wkh vdph vfhqh/ lpdjhv wdnhq iurp glhuhqw srvlwlrqv pd eh txlwh glhuhqw rqh iurp wkh rwkhu1 Frqvhtxhqwo / wklv k srwkhvlv lv udwkhu srru dqg ehfrphv hyhq srruhu li zh wdnh lqwr dffrxqw wkdw wkh suhflvlrq ri wkh 6G phdvxuhphqw ghshqgv rq wkh glvwdqfh ehwzhhq erwk fdphudv1 Wkh suhflvlrq ri wkh phdvxuhphqw lpsuryhv lqfuhdvlqj wklv glvwdqfh1 Krzhyhu/ zkhqrqhlqfuhdvhvwkhglvwdqfhehwzhhqerwkfdphudv/wkhsurmhfwlyhlp0 djhv ri wkh 6G vfhqhv ehfrph pruh glhuhqw/ frqvwudlqlqj frqvlghudeo wkhlu pdwfk xvlqj vlplodulw 1 Ri frxuvh/ zh pxvw dovr frqvlghu wkdw lq wkh vdph 6G vfhqh pruh wkdq rqh remhfw zlwk dssur{lpdwho wkh vdph vkdsh dqg vl}h frxog eh suhvhqwhg1 Lq wklv fdvh/ lw ehfrphv udwkhu gl!fxow wr pdwfk wkh lpdjhv iurp wkh phuh frqfhsw ri vlqjxodu srlqwv1 Krzhyhu/ vrph jhrphwul0 fdo frqvwudlqwv fdq eh xvhg wr uhgxfh wkh sureohp ri pdwfklqj1 Rqh ri wkh prvw srsxodu frqvwudlqwv lv nqrzq dv wkh hslsrodu frqvwudlqw lpsrvhg e wkh jhrphwulfdo uhodwlrqvkls ehwzhhq erwk fdphudv +^Idxj <6`/ s 49<,/ vkrzq lq jxuh 7141 Wkh hslsrodu uhodwlrq lv ghqhg dv iroorzv1 Wkh lqwhuvhfwlrq ri wkh olqh E c 2 zklfk hdfk lpdjh sodqh/ Z dqg Z 2 / surgxfhv wzr hslsrohv/ qdphg. dqg. 2 1 Frqvlghu dq lpdjh srlqw rq Z / wkh hslsrodu jhrphwu doorzv xv wr vhdufk lwv fruuhvsrqghqfh srlqw 2 rq wkh hslsrodu olqh ri rq Z 2 / l1h1 rq wkh olqh e R /zkhuhe R lv ghqhg e wkh lqwhuvhfwlrq ri wkh sodqh Z 2 dqg wkh sodqh ghqhg e wkh wkuhh srlqwv / 2 dqg 1 Qrwh wkdw zh gr qrw qhhg dq wkuhh glphqvlrqdo lqirupdwlrq iurp wr lqihu 2 1 Wkh hslsrodu frqvwudlqw doorzv xv wr uhgxfh wkh vhdufklqj ri wkh fruuh0 vsrqghqfh iurp wkh wzr0glphqvlrqdo vsdfh ri wkh lpdjh sodqh lqwr wkh rqh0 glphqvlrqdo rqh ri wkh hslsrodu olqh1 Krzhyhu/ zh pxvw eh dzduh wkdw vrph srlqwv lq dq lpdjh sodqh pljkw qrw kdyh d fruuhvsrqghqfh rq wkh rwkhu rqh gxh wr d vxuidfh rffoxvlrq ru vlpso ehfdxvh lw kdv ehhq surmhfwhg rxw ri wkh vfrsh ri wkh fdphud/ vhh jxuh 7151 Hyhq/ lpdjhg srlqwv pd qrw olh rq wkh hslsrodu olqhv lq wkh vdph rughu +vhh jxuh 716 dqg 717,1 Qrwh wkdw zh duh xqdeoh wr nqrz d sulru zkhwkhu d srlqw kdv d fruuhvsrqghqfh ru qrw/ zklfk gl!fxowv hqruprxvo wkh pdwfklqj dqg lpsrvhv wkh xvh ri d srvwhulru vwhs wr uhpryh idovh pdwfklqjv prvwo e xvlqj d uhjlrq jurzlqj dojrulwkp1 Dw wklv srlqw dqrwkhu frqvwudlqw zklfk dulvhv iurp wkh vwuxfwxuh ri wkh remhfwv ri wkh vfhqh/ fdq eh xvhg1 Lw frxog eh dvvxphg wkdw wkh vfhqh vxuidfhv ghswk0ydu vprrwko doprvw hyhu zkhuh1 Wklv frqvwudlqw lv nqrzq dv wkh glvsdulw judglhqw1 Dowkrxjk wklv frqvwudlqw fdq eh xvhg wr uhgxfh idovh

86 FKDSWHU 71 VWUXFWXUHG OLJKW ;: P w Π E1 C 1 C 2 E 2 P 1 e P2 e P1 P 2 π 1 π 2 Iljxuh 714= Wkh hslsrodu jhrphwu dv d wrro wr uhgxfh wkh fruuhvsrqghqfh sureohp1 P w Q w E 1 C 1 C 2 E 2 π 1 q 1 p 1? p q ep2 ep π 2 Iljxuh 715= Dq h{dpsoh ri d srlqw zlwkrxw fruuhvsrqghqfh gxh wr d vxuidfh rffoxvlrq ri 1

87 FKDSWHU 71 VWUXFWXUHG OLJKW ;; P w Q w E 1 C 1 C 2 E 2 π 1 q 1 p 1 p q ep2 ep π 2 Iljxuh 716= Dq h{dpsoh ri fruuhvsrqghqfh srlqwv lpdjhg lq wkh vdph rughu iurp wkh hslsrohv1 Q w P w E 1 C 1 C 2 E 2 π 1 p 1 q 1 ep p 2 q 2 ep 1 2 π 2 Iljxuh 717= Dq h{dpsoh ri fruuhvsrqghqfh srlqwv lpdjhg lq d glhuhqw rughu iurp wkh hslsrohv1

88 FKDSWHU 71 VWUXFWXUHG OLJKW ;< pdwfkhv rewdlqhg iurp wkh jhrphwulfdo surshuwlhv ri wkh hslsrodu olqh/ wkh uhdghu pxvw nqrz wkdw lw fdq qrw eh xvhg dw ghswk glvfrqwlqxlwlhv1 Qrwh wkdw ghswk glvfrqwlqxlwlhv pdlqo surgxfh wkh hgjhv dqg yhuwh{hv wkdw zloo eh xvhg dv wrnhqv wr wkh pdwfklqj surfhvv1 Zh pxvw dovr qrwh wkdw doprvw doo vwhuhr ylvlrq v vwhpv uhvwulfw wkh rewdlqlqj ri 6G lqirupdwlrq iurp wkh yhuwh{hv dqg fruqhuv ri wkh remhfwv1 Khqfh/ wkh glvsdulw judglhqw frxog qrw eh xvhg h{fhsw li zh rewdlq 6G lqirupdwlrq iroorzlqj wkh hgjhv/ idfw wkdw kdugo frpsolfdwhv wkh pdwfklqj surfhvv lqfuhdvlqj wkh frpsxwlqj wlph1 Lw lv nqrzq wkdw wkh fruuhvsrqghqfh sureohp fdq eh doohyldwhg ohdylqj r vwhuhr ylvlrq/ dqg jrlqj wr dq dfwlyh phwkrg ^Mduy ;6` ^Mduy <6` ^Eudg ;;` ^Urfn :8` ^Ehvo ;;`1 Rqh ri wkh prvw zlgho xvhg dfwlyh phwkrgv lv edvhg rq vwuxfwxuhg oljkw surmhfwlrq ^Zloo :4`1 Khuh/ wkh vhfrqg vwhuhr fdphud lv uhsodfhg e d oljkw vrxufh/ zklfk surmhfwv d nqrzq sdwwhuq ri oljkw rq wkh phdvxulqj vfhqh1 Wkh uvw vwhuhr fdphud lpdjhv wkh looxplqdwhg vfhqh dqg/ dqdo vlqj wkh ghirupdwlrqv ri wkh lpdjhg sdwwhuq zlwk uhvshfw wr wkh sur0 mhfwhg rqh/ wkh ghvluhg 6G lqirupdwlrq fdq eh rewdlqhg1 Ri frxuvh/ ghshqglqj rq wkh fkrvhq sdwwhuq/ vrph fruuhvsrqghqfhv ehwzhhq wkh surmhfwhg sdwwhuq dqg wkh lpdjhg rqh/ vkrxog eh vroyhg1 Prvw ri wkh sursrvhg vwuxfwxuhg oljkw whfkqltxhv duh edvhg rq wkh surmhfwlrq ri uhjxodu sdwwhuqv rq wkh phdvxu0 lqj vfhqh1 Li d vlqjoh oljkw grw ru d volw olqh lv surmhfwhg rq wkh vfhqh/ wkhq/ wkhuh lv qr fruuhvsrqghqfh sureohp wr eh vroyhg/ exw zkroh wkh vfhqh kdv wr eh vfdqqhg wr rewdlq wkh 6G pds1 Vkludl hw do1/ lq 4<:4/ sursrvhg d volw olqh surmhfwlrq wr uhfrjqlvh srolkhgulf remhfwv ^Vklu :4e`1 Lq 4<:6/ Djlq hw do1 jhqhudolvhg wklv lghd wr uhfrjqlvh fxuylolqhdu remhfwv ^Djlq :6`1 Wzr hduv odwhu/ zlwk wkh jrdo wr uhfrjqlvh hlwkhu srolkhgulf ru fxuylolqhdu remhfwv/ Srssohvwrqh hw do1 sursrvhg d pruh jhqhudo v vwhp ^Srss :8`1 Lq 4<;9/ \d0 pdprwr hw do1^\dpd ;9` sursrvhg d kdoi sodqh looxplqdwlrq v vwhp lqvwhdg ri d volw olqh1 Lq idfw/ elqdulvlqj dq lpdjh ri d vfhqh looxplqdwhg e d kdoi sodqh sdwwhuq frxog eh frpsduhg wr wkh erxqgdu hgjh ghwhfwlrq ehwzhhq wkh looxplqdwhg dqg wkh revfxuhg duhd1 Uhfhqwo / vrph v vwhpv wr rewdlq 6G pdsv ri d vfhqh kdyh ehhq suhvhqwhg1 Wkh prvw frpprq v vwhpv duh vlplodu wr wkh kljk0vshhg phwkrg suhvhqwhg e R}hnl hw do1^r}hn ;9`/ dqg wkh ylghr udwh phwkrg suhvhqwhg e \rnr dpd hw do1^\rnr <7`1 Wkhuh duh dovr vrph dxwkruv/ dv Vdwr hw do1 ^Vdwr ;5`/ zkr xvh d phwkrg edvhg rq wkh surmhfwlrq ri wzr volw olqhv zlwk glhuhqw rulhqwdwlrq dqg srvlwlrq lq wkh 6G fr0ruglqdwhv v vwhp1 Lq d vlplodu zd / Nhpprwvx dqg Ndqdgh ^Nhpp <7` ^Nhpp <8` fkrvh wr surmhfw wkuhh olqhv rq wkh phdvxulqj vfhqh1 Lq rughu wr lpsuryh wkh dffxudf ri wkh v vwhp/ dq dowhuqdwlyh zd lv wr surmhfw d julg ri grwv ru olqhv rq wkh vfhqh wr fryhu wkh hqwluh vfrsh ri wkh fdphud1 Dvdgd hw do1 sursrvhg wr xvh d sdwwhuq pdgh e d vhw ri yhuwlfdo/ sdudooho dqg htxlglvwdqw/ vwulsh olqhv ^Dvdg ;8` ^Dvdg ;9` ^Dvdg ;;`1 Zdqj

89 FKDSWHU 71 VWUXFWXUHG OLJKW <3 hw do1/ kdyh h{whqghg Dvdgd*v lghd zlwk wkh vhtxhqwldo surmhfwlrq ri wzr ru0 wkrjrqdo vwulsh sdwwhuqv ^Zdqj ;:` ^Zdqj ;<` ^Zdqj <4`1 Ixuwkhupruh/ lq rughu wr rewdlq 6G vxuidfh surshuwlhv/ Jx hw do1 ^Kx ;9` ^Kx ;<`/ Vwrfnpdq hw do1 ^Vwrf ;9` dqg Vkulnkdqgh hw do1 ^Vkul ;<` kdyh sursrvhg wkh zlgho nqrzq phwkrg edvhg rq wkh surmhfwlrq ri d julg1 Rwkhu dxwkruv dovr xvh wkh lqirupdwlrq jlyhq e dq lqwhqvlw lpdjh wr rewdlq/ zlwk d ehwwhu dffx0 udf / wkh erxqgdu hgjhv ri wkh vfhqh ^Lnhx <4` ^Zdqj <5` ^Kx ;<e` ^Yhpx ;9` ^Krvk <3`1 Wkhq/ dq hdvlhu fruuhvsrqghqfh sureohp kdv wr eh vroyhg= iru hdfk srlqw ri wkh lpdjhg sdwwhuq/ wkh fruuhvsrqglqj srlqw ri wkh surmhfwhg sdwwhuq kdv wr eh lghqwlhg1 Doo wkhvh phwkrgv doorz xv wr rewdlq 6G lqirupdwlrq iurp wkh jhrphwulf frqvwudlqw sursdjdwlrq/ hvshfldoo iurp wkh hslsrodu frq0 vwudlqw/ dqg vrph ri wkhp duh udwkhu olplwhg wr phdvxuh vxuidfhv zlwk ghswk glvfrqwlqxlwlhv1 Lq uhfhqw hduv/ d qhz vwuxfwxuhg oljkw whfkqltxh kdv lqfuhdvhg lq lp0 sruwdqfh1 Wklv whfkqltxh lv edvhg rq dq xqltxh frglfdwlrq ri hdfk wrnhq ri oljkw surmhfwhg rq wkh vfhqh1 Zkhq wkh wrnhq lv lpdjhg e wkh fdphud/ wklv frglfdwlrq doorzv xv wr rewdlq wkh fruuhvsrqghqfh l1h1 wr nqrz zkhuh lw frphv iurp1 Wkhq/ 6G phdvxuhphqwv duh gluhfwo rewdlqhg dv zh gr qrw kdyh wr xvh kdug frpsxwdwlrqdo jhrphwulf frqvwudlqwv1 Wklv whfkqltxh lv edvlfdoo nqrzq dv frghg vwuxfwxuhg oljkw1 Vhyhudo frghg vwuxfwxuhg oljkw whfkqltxhv kdyh ehhq sursrvhg lq wkh odvw hduv/ zklfk kdyh ehhq glvfxvvhg dqg frpsduhg lq wkh iroorzlqj vxuyh D Vxuyh = Frghg Vwuxfwxuhg Oljkw Frghg vwuxfwxuhg oljkw lv d whfkqltxh edvhg rq wkh surmhfwlrq ri oljkw sdwwhuqv rq wkh phdvxulqj vfhqh1 Wkh surmhfwhg sdwwhuqv duh frqyhqlhqwo frghg lq rughu wr vroyh wkh fruuhvsrqghqfh sureohp1 Hdfk wrnhq ri oljkw lv surmhfwhg rq wkh vfhqh fduu lqj d odeho zklfk lqglfdwhv zkhuh lw frphv iurp1 Zkhq wkh wrnhq lv lpdjhg e wkh fdphud/ zh kdyh wr uhdg wklv odeho wr rewdlq dq xqltxh pdwfk1 Dv lw fdq eh ghgxfhg iurp wkh vhfwlrq ghglfdwhg wr 6G uhfrqvwuxfwlrq lq fkdswhu 5 +vhh dovr jxuh 5147,/ zh fdq xvh irxu htxdwlrqv lq rughu wr fdofxodwh wkh wkuhh xqnqrzqv wkdw ghwhuplqh wkh fr0ruglqdwhv ri wkh remhfw srlqw 1 Lq idfw/ rqh ri wkh irxu htxdwlrqv lv olqhduo ghshqghqw rq wkh rwkhu rqhv/ vr/ zlwk wkh dlp wr ghwhuplqh wkh wkuhh xqnqrzqv/ rqo wkuhh ri wkh irxu htxdwlrqv kdyh wr eh xvhg1 Dv dq lpdjh ri wkh vfhqh kdv wr eh fdswxuhg wr ghgxfh ghswk vfhqh lqirupdwlrq/ wkh lpdjh srlqw fr0ruglqdwhv E% R2 c+ R2 duh nqrzq/ dv lw lv wkh surmhfwlrq ri wkh remhfw srlqw zlwk fr0 ruglqdwhv E% R c+ R c5 R 1 Wkhq/ rqo rqh htxdwlrq ri wkh wzr/ zklfk uhodwhv

90 FKDSWHU 71 VWUXFWXUHG OLJKW <4 wkh 6G remhfw srlqw zlwk uhvshfw wr wkh 5G surmhfwhg wrnhq rq wkh surmhfwru iudph/ kdv wr eh xvhg1 Wkdw lv ehfdxvh zlwk rqo rqh htxdwlrq/ wkhuh lv vx!flhqw lqirupdwlrq wr ghgxfh wkh 6G remhfw srlqwv fr0ruglqdwhv1 Wkhuhiruh/ rqo rqh ri wkh wzr fr0ruglqdwhv E% R c+ R ri wkh surmhfwhg srlqw kdv wr eh nqrzq1 Li wkh % fr0ruglqdwh % R lv nqrzq/ wkhq wkh sdwwhuq lv fdoohg d froxpq frghg sdwwhuq1 Rwkhuzlvh/ li lw lv + R wkh rqo nqrzq fr0ruglqdwh/ wkhq wkh sdwwhuq lv fdoohg d urz frghg sdwwhuq1 Wklv lghd doorzv wkh surmhfwhg sdwwhuq wr eh frghg mxvw dorqj rqh frpsrqhqw fr0ruglqdwh/ vr wkh fdswxuhg oljkw/ dw wkh srlqw E% R2 c+ R2 rq wkh lpdjh sodqh/ fduulhv lqirupdwlrq iurp urz % R /ru froxpq + R / iurp zklfk lw kdv ehhq hplwwhg1 Reylrxvo / d sdwwhuq edvhg rq erwk d{hv frglfdwlrq lv pruh urexvw dqg wkh uhvxowv ri wkh 6G phdvxulqj srlqwv pruh dffxudwh wkdq vlqjoh d{h frglfdwlrq1 Wkh vxuyh grqh doorzhg xv wr fodvvli wkh glhuhqw whfkqltxhv frglqj wkh surmhfwhg sdwwhuq/ lqwr wkuhh fodvvlfdwlrqv1 Orrnlqj rqo dw wkh whpsrudo ghshqghqfh/ sdwwhuqv fdq eh fodvvlhg lqwr= Vwdwlf= Wkh sdwwhuq lv olplwhg wr vwdwlf vfhqhv zlwk prwlrqohvv remhfwv1 Wklv lv dozd v gxh wr wkh qhfhvvlw wr surmhfw d vhw ri glhuhqw sdwwhuqv wr rewdlq/ lq d frghg zd / wkh odeho iru hdfk froxpq +ru urz, ri wkh sdwwhuq1 Dq pryhphqw lqvlgh wkh vfhqh ehwzhhq wzr sdwwhuq surmhfwlrq iudphv dozd v surgxfhv d fruuh0 vsrqghqfh huuru1 G qdplf= Wkh sdwwhuq lv qrw olplwhg wr vwdwlf vfhqhv1 Wkhq/ li wkh vfhqh remhfwv fdq pryh/ wkh froxpq ru urz ri wkh surmhfwru lpdjh kdv wr eh frghg zlwk d vlqjoh sdwwhuq surmhfwlrq1 Orrnlqj rqo dw wkh oljkw surmhfwhg/ sdwwhuqv fdq eh fodvvlhg lqwr= Elqdu = Dq ri wkh E% R c+ R srlqwv ri wkh sdwwhuq fdq rqo kdyh rqh ri wzr srvvleoh ydoxhv/ zklfk duh frghg zlwk 3 dqg 4 uhvshfwlyho 1 Wklv elqdu ydoxh qrupdoo uhsuhvhqwv rsdflw dqg wudqvsduhqf / l1h1 surmhfwhg oljkw devhqfh ru suhvhqfh rq wkh re0 mhfw1 Juh ohyho= Hdfk sdwwhuq srlqw fdq kdyh dq dvvrfldwhg juh ydoxh/ zklfk uhsuhvhqwv wkh wudqvsduhqf +ru rsdflw, ohyho ri wkh srlqw djdlqvw wkh surmhfwhg oljkw1 Vlqfh wkh lqirupdwlrq lv frghg dv d juh oljkw ohyho/ qrupdoo wzr vwhsv duh qhfhvvdu lq rughu wr qg rxw 6G lqirupdwlrq1 Iluvwo / zh kdyh wr rewdlq dq lpdjh ri wkh vfhqh looxplqdwhg zlwk wkh vdph oljkw lqwhqvlw iru hdfk srlqw +zlwkrxw frglqj,1 Vhfrqgo / zh pxvw rewdlq wkh uhihuhqfh oljkw

91 FKDSWHU 71 VWUXFWXUHG OLJKW <5 qhhghg wr fdqfho wkh vxuidfh uh hfwlrq hhfw/ zklfk ghshqgv rq wkh nlqg ri vxuidfhv zkhuh wkh oljkw lv uh hfwhg1 Wklv olplwdwlrq phdqv wkdw wkh sdwwhuq kdv wr eh dovr fodvvlhg dv d vwdwlf sdwwhuq1 Frorxu= Hdfk sdwwhuq srlqw kdv wr eh dvvrfldwhg zlwk d kxh ydoxh1 Lq rughu wr xvh wkh frorxu frqvwdqf surshuw / wkh kxh ydoxhv xvhg kdyh wr eh txlwh glhuhqw iurp hdfk rwkhu1 Wkh pdlq jrdo lv wr jhw dq h!flhqw dqg dffxudwh vhjphqwdwlrq1 Vlqfh wkh v vwhp surmhfwv frorxu rq wkh vfhqh/ lwv xvh lv olplwhg wr d qhx0 wudo frorxu vfhqh/ dv kljko vdwxudwhg frorxu remhfwv fdq surgxfh orrvlqj ri sdwwhuq uhjlrqv lq wkh vhjphqwdwlrq vwhs dqg srvwhulru ghfrglfdwlrq1 Hyhq sdoh frorxuhg remhfwv pd surgxfh d frorxu iuhtxhqf vkliwlqj dv d uhvxow ri wkh lqwulqvlf frorxu ri wkh phd0 vxulqj remhfwv1 Reylrxvo / wkh glvfuhsdqf vxhuhg e wkh frorxu fdswxuhg e wkh fdphud zlwk uhvshfw wr wkh rqh surmhfwhg e wkh sdwwhuq/ udwkhu frpsolfdwhv wkh vhjphqwdwlrq vwhs1 Ixuwkhupruh/ dqrwkhu fodvvlfdwlrq fdq eh sursrvhg/ zklfk fodvvlhv wkh sdwwhuqv iurp lwv vxuidfh ghswk glvfrqwlqxlwlhv ghshqghqfh1 Wkhq/ orrnlqj dw wklv ghshqghqfh/ wkh sdwwhuqv fdq eh fodvvlhg dv= Shulrglfdo= Wkh frglfdwlrq lv shulrglfdoo uhshdwhg dorqj wkh sdwwhuq1 Qrupdoo / wklv whfkqltxh lv xvhg wr uhgxfh wkh qxpehu ri elwv wkdw frgh wkh sdwwhuq/ exw surgxfh d olplwdwlrq ri wkh ghswk glvfrqwlqxlw zklfk fdq qrw eh odujhu wkdq kdoi ri wkh shulrg ohqjwk1 Devroxwh= Hdfk froxpq ru urz ri wkh surmhfwhg sdwwhuq kdv d xqltxh frglfdwlrq1 Vr/ lw grhv qrw kdyh dq ghswk glvfrqwlqxlw ghshqghqfh1 Khuhdiwhu iroorzv d vxuyh ri wkh frghg sdwwhuqv suhvhqwhg lq uhfhqw hduv/ dqdo vlqj iru hdfk rqh lwv dgydqwdjhv dqg glvdgydqwdjhv Srvgdphu0Dowvfkxohu Dowvfkxohu hw do1 ^Dowv ;4` ^Dowv ;:` dqg Srvgdphu hw do1 ^Srvg ;5` sursrvhg d vwdwlf elqdu frglfdwlrq ri d vwulshg sdwwhuq1 Wkh vdph sulqflsoh lv dovr sursrvhg e Pxqg dqg Sruwhu ^Pxqg ;:` dqg Plqrx hw do1 ^Plqr ;4`1 Wkh v vwhp/ vkrzq lq jxuh 718/ lv edvhg rq wkh xwlolvdwlrq ri d sdwwhuq vwuxfwxuhg dv d grw pdwul{ ri?{? elqdu oljkw ehdpv1 Hdfk? froxpq ri wkh sdwwhuq fdq eh lqghshqghqwo frqwuroohg/ vr lw fdq eh oljkwhg ru revfxuhg1 Wkhq/ vhyhudo

92 FKDSWHU 71 VWUXFWXUHG OLJKW <6 Projector Camera z 2 1 y 2 y 1 x 1 1 x Projected Patterns 0 Scene 0 O z 1 Captured Image y w xw z w Iljxuh 718= Wkh whpsrudu frglfdwlrq1 pdvnv fdq eh xvhg wr doorz frglqj dq sdwwhuq grw/ lq d whpsrudo zd / dv d vhtxhqwldo surmhfwlrq ri glhuhqw sdwwhuqv/ dv vkrzq lq jxuh 7191 Wkh qxpehu ri sdwwhuqv wr eh surmhfwhg lv ghwhuplqhg e wkh qxpehu ri froxpqv wr eh frghg1 Eorfnlqj oljkw ehdpv/ d sdwwhuq lpdjh vhtxhqfh fdq eh surmhfwhg zklfk surmhfwv d vhtxhqfh ri f dqg iru hdfk ehdp1 Wkhq/ hdfk srvlwlrq ri wkh phdvxulqj vxuidfh lv whpsrudoo dqg vhtxhqwldoo looxplqdwhg iru wkh glhuhqw ydoxhv ri wkh vdph grw ehdp/ zklfk surgxfhv d frgh wkdw fdq glvwlqjxlvk rqh grw iurp lwv qhljkerxuv1 Lq idfw/ wkh v vwhp zrunv dv iroorzv1 Iluvwo / dq hqwluh grw looxplqdwhg sdwwhuq lv surmhfwhg rq wkh vfhqh1 Wkh fdphud lpdjhv doo wkh grwv uh hfwhg iurp wkh vxuidfhv ri wkh vfhqh/ dqg vwruhv wkhlu srvlwlrqv lq d gdwd edvh1 Lq wkh iroorzlqj vwhsv/ hdfk frghg sdwwhuq lv surmhfwhg dqg wkh v vwhp fdq dvn iru wkh oljkw lqirupdwlrq lq wkh vwruhg srvlwlrqv1 Zkhq doo wkh sdwwhuqv kdyh ehhq surmhfwhg/ hdfk vwruhg srlqw kdv d frgh zklfk doorzv xv wr ghwhuplqh iurp zklfk ehdp froxpq lw kdv ehhq surmhfwhg1 Wkh v vwhp sursrvhg e Srvgdphu hw do1 dqg Dowvfkxohu hw do1 lv olplwhg wr vwdwlf vfhqhv dv lw kdv wr fdswxuh dq lpdjh iurp hdfk surmhfwhg sdwwhuq1 Exw/ dv sursrvhg e Dowvfkxohu hw do1 ^Dowv ;4`/ wkh v vwhp fdq eh lpsuryhg wr eh xvhg lq g qdplf vfhqhv1 Lq wklv fdvh/ doo wkh sdwwhuqv duh surmhfwhg zlwk d glhuhqw iuhtxhqf zdyh dqg? fdphudv duh xvhg/ rqh iru hdfk surmhfwhg sdwwhuq1 Hdfk fdphud vkrxog kdyh dq rswlfdo owhu/ dv hdfk fdphud zloo lpdjh

93 FKDSWHU 71 VWUXFWXUHG OLJKW <7 Reference Dot Pattern Mask = Mask = OO Second Pattern Mask = First Pattern Mask = O Third Pattern Illuminated Dot Obscured Dot Iljxuh 719= Frglqj d?{? pdwul{ ri odvhu ehdp grwv1 rqo rqh sdwwhuq iuhtxhqf 1 Wkhq/ doo wkh sdwwhuqv fdq eh surmhfwhg dw wkh vdph wlph1 Wkh fdphudv vkrxog eh orfdwhg dv forvh dv srvvleoh1 Hyhq vr/ zh frxog dovr qg wkdw vrph srlqwv fdq qrw eh ghfrghg dv wkh duh qrw lpdjhg iurp doo wkh fdphudv1 Wkh lghd/ sursrvhg e Srvgdphu dqg Dowvfkxohu/ kdv ehhq zlgho vwxglhg1 Edvlfdoo / wkh frglfdwlrq dqg wkh vshhg phdvxuhphqw kdyh ehhq lpsuryhg1 Iru h{dpsoh/ Lqrnxfkl hw do1 ^Lqrn ;7` lq 4<;7 sursrvhg fkdqjlqj wkh elqdu frglfdwlrq/ vkrzq lq jxuh 71:/ wr d Jud rqh/ vkrzq lq jxuh 71;/ zklfk lv pruh huuru urexvw1 Odwhu/ lq 4<;9/ Vdwr hw do1 ^Vdwr ;9`/ dqg lq 4<;:/ Vdwr dqg Lqrnxfkl ^Vdwr ;:`/ sursrvhg wr xvh d oltxlg fu vwdo ghylfh/ douhdg vwxglhg e Lqrnxfkl hw do1 ^Lqrn :5` lq 4<:5/ zklfk shuplwv d odujhu qxpehu ri froxpqv wr eh surmhfwhg zlwk d kljk dffxudf 1 Wkh v vwhp dovr lpsuryhv wkh frglqj vshhg/ djdlqvw wkh xvh ri d volgh surmhfwru/ vlqfh wkh OFG fdq eh hohfwurqlfdoo frqwuroohg1 Li dq remhfw kdv d kljk wh{wxudo frqwudvw ru dq kljk uh hfwhg vxuidfh uhjlrqv/ wkhq/ vrph sdwwhuq vhjphqwdwlrq huuruv fdq eh surgxfhg1 Qrupdoo / wklv sureohp fdq eh vroyhg fdswxulqj wkh uvw lpdjh zlwkrxw sdwwhuq surmhf0 wlrq +ru zlwk wkh vdph oljkw surmhfwlrq iru hdfk grw, wr rewdlq d uhihuhqfh oljkw lqwhqvlw 1 Wklv lqwhqvlw jlyhv d g qdplf wkuhvkrog iru hdfk sl{ho1 Vdwr hw do1 ^Vdwr ;9`/ lq 4<;9/ dqg Vdwr dqg Lqrnxfkl ^Vdwr ;:`/ lq 4<;:/ sursrvhg d frpsohphqwdu phwkrg ri vhjphqwdwlrq edvhg rq wkh surmhfwlrq iru hdfk

94 FKDSWHU 71 VWUXFWXUHG OLJKW <8 (HSB) A B C D E F G (LSB) H Iljxuh 71:= ; elwv whpsrudoo elqdu frghg sdwwhuq surmhfwlrq1 (HSB) A B C D E F G (LSB) H Iljxuh 71;= ; elwv whpsrudoo Jud frghg sdwwhuq surmhfwlrq1

95 FKDSWHU 71 VWUXFWXUHG OLJKW <9 CCD camera Semiconductor Laser 12 faces Polygon Mirror Slit Ray Scanning Iljxuh 71<= Wkh Fxelvfrsh v vwhp1 frghg sdwwhuq/ ri lwv srvlwlyh dqg qhjdwlyh uhsuhvhqwdwlrq1 Wkhq/ frpsdulqj wkh lqwhqvlw lpdjh zlwk erwk uhsuhvhqwdwlrqv/ d ehwwhu vhjphqwdwlrq ri wkh surmhfwhg sdwwhuq fdq eh rewdlqhg1 Wkh sureohp ri d oljkw surmhfwru lv vrphwlphv d uhvxow ri khdw luudgldwlrq rqwr wkh vfhqh/ dqg ri lwv elj vl}h dqg zhljkw1 Lq 4<<8/ Kdwwrul hw do1 ^Kdww <8` sursrvhg wr uhsodfh wkh oljkw surmhfwru zlwk d vhpl0frqgxfwru odvhu/ wkdw jlyhv d kljk srzhu looxplqdwlrq zlwk orz khdw luudgldwlrq1 Wkh sursrvhg v vwhp/ qdphg Fxelvfrsh/ lv vkrzq lq jxuh 71< dqg lv frpsrvhg e d FFG fdphud/ d vhpl0frqgxfwru odvhu dqg d vfdqqhg pluuru/ v qfkurqrxvo frqwuroohg1 Wkh pdlq jrdo ri wklv v vwhp lv wr rewdlq vfhqh lpdjhv edvhg rq wkh whpsrudu frglfdwlrq sursrvhg e Srvgdphu hw do1 dqg Dowvfkxohu hw do1 Lq 4<<8/ wkh vdph whpsrudu frglfdwlrq ghvfulehg e Srvgdphu hw do1 dqg Dowvfkxohu hw do1 lq 4<;5 lv djdlq sursrvhg e Pÿoohu ^Pxoo <8`1 Lq 4<<9/ Vdwr ^Vdwr <9` sursrvhg d qhz prylqj prgxodwhg sdwwhuq oljkw dqg vsdwlr0whpsrudo lpdjh surfhvvlqj Fduulkloo0Kxppho Wkh v vwhp sursrvhg e Fduulkloo hw do1 ^Fduu ;8`/ lq 4<;8/ lv edvhg rq jhwwlqj 6G lqirupdwlrq ri dq vfhqh iurp wzr iudph lpdjhv1 Wkh uvw lpdjh lv fdswxuhg zlwk d frqvwdqw looxplqdwlrq surmhfwlrq rq wkh vfhqh1 D olqhdu zhgjh owhu lv xvhg wr surmhfw rq wkh vfhqh d sdwwhuq euljkwo looxplqdwhg rq

96 FKDSWHU 71 VWUXFWXUHG OLJKW <: Iljxuh 7143= Wkh sdwwhuq sursrvhg e Fduulkloo hw do1 rqh vlgh/ dqg kdoi euljkwo rq wkh rwkhu/ lq rughu wr vqds wkh vhfrqg lpdjh1 Wklv sdwwhuq lv vkrzq lq jxuh Iru wkh uvw surmhfwlrq/ lw lv vxssrvhg wkdw wkh looxplqdwlrq lqwhqvlw lv frqvwdqw dorqj wkh %0d{lv dqg wkh +0d{lv ri wkh surmhfwru1 Iru wkh vhfrqg surmhfwlrq/ looxplqdwlrq kdv wr eh d frqvwdqw dorqj wkh 0d{lv1 Wkhq/ iru hdfk fdswxuhg sl{ho/ wkh lqwhqvlw vxewudfwlrq ri wkh wzr lpdjhv/ fdq eh fdofxodwhg1 Wklv glhuhqfh doorzv wr rewdlq wkh froxpq ri wkh surmhfwru iurp zklfk wkh grw kdv ehhq hplwwhg1 Xvlqj d glhuhqfh udwlr iru hdfk sl{ho pdnhv lw hdvlhu wr fdqfho srvvleoh vxuidfh uh hfwlrqv ru kljkoljkwv vlqfh dq looxplqdwlrq udwlr fkdqjh lv dozd v surgxfhg e d fkdqjh lq wkh surmhfwhg looxplqdwlrq1 Xvlqj hohfwur0rswlfv owhuv/ wkh v vwhp fdq surylgh 6G lqirupdwlrq dw kljk vshhg/ exw/ olnh v vwhpv wkdw surmhfw pruh wkdq rqh lpdjh/ lw lv olplwhg wr vwdwlf vfhqhv1 Reylrxvo / dq prylqj remhfw lq wzr frqvhfxwlyh iudphv zrxog surgxfh d fruuhvsrqghqfh huuru1 Fduulkloo hw do1 d!up wkdw wkh v vwhp surylghv dq dffxudwh phdvxuhphqw zlwk sl{hov ri ; elwv ghswk +elwv shu sl{ho,1 Wkh whvw kdv ehhq olplwhg wr 45 elwv ghswk/ dowkrxjk Fduulkloo hw do1 d!up wkdw wkhuh lv d jrrg fruuhodwlrq ehwzhhq wkh vhqvru shufhswlrq dqg uhdolw 1 Lq rughu wr lpsuryh wkh dffxudf phdvxuhphqw zlwkrxw lqfuhdvlqj wkh elw ghswk ri wkh fdswxuhg lpdjh/ d qhz phwkrg edvhg rq d vdzwrrwk sdwwhuq frxogehsursrvhg1 Wzrvdzwrrwksdwwhuqvriglhuhqwshulrgohqjwkduh

97 FKDSWHU 71 VWUXFWXUHG OLJKW <; t 1 (a) t 2 (b) Iljxuh 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 vkrzq lq jxuh Reylrxvo / wkh pd{lpxp vxuidfh ghswk glvfrqwlqxlw wr eh phdvxuhg lv olplwhg wr wkh fkrvhq shulrg1 Lq 4<<6/ Kxqj ^Kxqj <6` sursrvhg d juh ohyho vlqxvrlgdo sdwwhuq1 Wkh shulrg ri wkh fdswxuhg sdwwhuq ghshqgv rq wkh ghswk ri wkh vxuidfh zkhuh lw lv uh hfwhg1 Kxqj sursrvhg wr wuldqjxodwh iurp wkh froxpq skdvh ri wkh lpdjhg srlqw1 Iru hdfk sl{ho/ wklv skdvh fdq eh dssur{lpdwho rewdlqhg iurp wkh oljkw lqwhqvlw 1 Krzhyhu/ lw vhhpv wkdw d urexvw 6G uhfrqvwuxfwlrq frxog eh rewdlqhg surmhfwlqj uvw d frqvwdqw looxplqdwhg sdwwhuq dv sursrvh Fduulklo ho do1 Wklv phwkrg/ dv wkh vdzwrrwk rqh/ dovr vxhuv wkh olplwdwlrq ri d shulrglfdo frglfdwlrq Er hu0ndn Lq 4<;:/ Er hu dqg Ndn ^Er h ;:` sursrvhg wr looxplqdwh wkh vfhqh xvlqj d vlqjoh sdwwhuq surmhfwlrq1 Wkh sdwwhuq lv frghg e yhuwlfdo frorxuhg volwv1 Wkh fruuhvsrqghqfh sureohp/ l1h1 jlyhq dq lpdjhg volw/ wr rewdlq iurp zklfk surmhfwru volw lw kdv ehhq hplwwhg/ fdq eh vroyhg dv wkh volw frglfdwlrq xvhg wr exlog wkh sdwwhuq/ lv nqrzq1 Reylrxvo / dv zlwk dq rwkhu v vwhp wkdw surmhfwv frorxu/ lwv xvh lv olplwhg wr suhgrplqdqwo frorxu qhxwudo vxuidfhv/

98 FKDSWHU 71 VWUXFWXUHG OLJKW << Back Wall 1 Target Projection Grid Image Plane Iljxuh 7145= Dq h{dpsoh ri wkh fruuhvsrqghqfh sureohp wkdw kdv wr eh vroyhg ehwzhhq surmhfwhg dqg lpdjhg volwv1 dv kljko vdwxudwhg kxhv frxog surgxfh volw lghqwlfdwlrq huuruv1 Iljxuh 7145 vkrzv rqo d vwulsh sdwwhuq surmhfwlrq xs wr zklfk wkh pdwfk0 lqj sureohp fdq eh frpsolfdwhg ehwzhhq surmhfwhg dqg lpdjhg volwv1 Qrwh wkdw/ volwv surmhfwhg rxw ri wkh vfrsh ri wkh fdphud ru uh hfwhg rq rffoxghg vxuidfhv fdq qrw eh lpdjhg1 Dovr/ wkh volwv lpdjhg e wkh fdphud fdq eh rewdlqhg lq d glhuhqw rughu1 Er hu hw do1 sursrvhg wr frgli wkh sdwwhuq xvlqj wkh wkuhh edvlf frorxu frpsrqhqwv= uhg/ juhhq dqg eoxh1 Wkh sdwwhuq zloo eh pdgh e d vhtxhqfh ri yhuwlfdo volwv frorxuhg zlwk dq ri wkh wkuhh edvlf frpsrqhqwv1 Vhh jxuh Wkh volw fdq dovr eh zklwh1 Rswlrqdoo / ehwzhhq hdfk volw/ d eodfn jds fdq eh sodfhg/ wklv jds fdq eh xvhg dv d volw vhsdudwru1 Qrwh wkdw/ zklwh fdq eh hdvlo rewdlqhg dqg lghqwlhg dv wkh frorxu zklfk kdv dssur{lpdwho wkh vdph uhg/ juhhq dqg eoxh lqwhqvlw ydoxhv1 Li lw lv vxssrvhg wkdw wkh sdwwhuq lv pdgh e q yhuwlfdo volwv/ Er hu hw do1 sursrvhg wr glylgh wklv sdwwhuq lq 6 vxesdwwhuqv/ hdfk rqh pdgh e n yhuwlfdo volwv/ dv? ' 6 &1 Lq rughu wr frgh wkh p sdwwhuqv lq d xqltxh zd / & lv d ydoxh dv odujh dv qhfhvvdu 1 Dv hdfk vxesdwwhuq lv hplwwhg zlwkrxw dq ehjlqqlqj dqg hqglqj frgh/ pxowlsoh pdwfklqjv fdq eh rewdlqhg lq uhfhswlrq1 Dfwxdoo / zh gr qrw nqrz zkhuh d vxesdwwhuq hqgv dqg wkh qh{w ehjlqv/ dqg pruhryhu/ wkh volwv kdyh ehhq uh0rughuhg dv d ixqfwlrq ri wkh phdvxuhg vxuidfh glvfrqwlqxlwlhv1 Dq h{dpsoh ri wkh sdwwhuq ghirupdwlrq fdq eh vkrzq

99 FKDSWHU 71 VWUXFWXUHG OLJKW 433 Red Green Blue Iljxuh 7146= D ghwdlo ri wkh sdwwhuq sursrvhg e Er hu1 lq jxuh 7147/ zkhuh d :5 volw sdwwhuq zdv hplwwhg dqg d 83 volw rqh kdv ehhq uhfhlyhg1 Iru frorxu volw ghwhfwlrq/ Er hu hw do1 sursrvhg d kdugzduh wkdw ghwhfwv uhg/ juhhq dqg eoxh shdnv lq wkh UJE vljqdo jlyhq e wkh fdphud1 Wkxv/ wkh vshflf kdugzduh fdq rewdlq wkh lqgh{ dqg srvlwlrq ri hdfk shdn lq uhdo0 wlph1 Lq idfw/ wkh lqgh{ fdq kdyh d surshu ydoxh iru hdfk glhuhqw frorxu +iru h{dpsoh/ wkh 4/ 5 dqg 6 lqgh{ ydoxhv fdq eh dvvrfldwhg zlwk uhg/ juhhq dqg eoxh frorxuv uhvshfwlyho,1 Frqvhtxhqwo / wkh srvlwlrq fdq eh rewdlqhg iurp wkh fdphud v qfkurqlvp1 Reylrxvo / FDJ ihdwxuh doorzv wkh htxdolvdwlrq ri wkh UJE vljqdov1 Lq rughu wr rewdlq wkh vxesdwwhuqv iurp wkh lqgh{ dqg srvlwlrq surylghg e wkh kdugzduh/ d vriwzduh dojrulwkp lv xvhg1 Lq wkh uvw vwhs/ doo wkh srvvleoh pdwfklqjv ehwzhhq wkh hplwwhg dqg uhfhlyhg vxesdwwhuqv duh irxqg1 Lq wkh vhfrqg rqh/ d uhjlrq jurzlqj dojrulwkp lv xvhg wr uhpryh doo wkh huurqhrxv pdwfklqjv1 Lq wkh wklug rqh/ iru hdfk vxesdwwhuq qrw pdwfkhg/ d khxulvwlf dojrulwkp lv xvhg wr qg lwv ehwwhu pdwfklqj1 Zkhq wkh wklug vwhs lv frpsohwhg/ d gdwd edvh/ zklfk dvvrfldwhv fdswxuhg dqg surmhfwhg vxesdwwhuqv/ kdv ehhq rewdlqhg1 Ilqdoo / d wuldqjxodwlrq surfhvv surylghv wkh 6G vfhqh lqirupdwlrq wkh zhuh orrnlqj iru1 Prqnv hw do1 ^Prqn <5` ^Prqn <6` dvvhuw wkdw wkh pdwfklqj surfhvv sur0 srvhg e Er hu lv qrw rswlpdo/ ehfdxvh li wkh pdwfkhv iru wkh uvw yhuwlfdo volw frqwdlq huuruv/ wkhvh zloo eh sursdjdwhg wkurxjk wkh uhvw ri wkh gdwd1 Prqnv

100 FKDSWHU 71 VWUXFWXUHG OLJKW 434 Projected Pattern Sequence Occlusion Camera Detected Sequence Iljxuh 7147= D srvvleoh uhodwlrq ehwzhhq d surmhfwhg sdwwhuq dqg wkh fdswxuhg rqh1 hw do1 kdyh ghyhorshg d qhz whfkqltxh wkdw holplqdwhv wklv sureohp dqg kdv wkh dgydqwdjh ri ehlqj qrq0lwhudwlyh/ hqfrglqj wkh wrsrorj ri wkh hqwluh vhw ri vwulshv dv d gluhfwhg df folf judsk ^Eudv <3`1 Dv zh vdlg ehiruh/ wkh sdwwhuq sursrvhg e Er hu hw do1/ lv olplwhg wr phd0 vxuh suhgrplqdqwo qhxwudo frorxu vxuidfhv/ dv kljko vdwxudwhg kxhv frxog surgxfh volw lghqwlfdwlrq huuruv1 Hyhq vr/ lw lv vshfldoo uhfrpphqghg lq g qdplf hqylurqphqwv vlqfh rqo rqh surmhfwlrq lv qhhghg Oh Prljqh0Zd{pdq Wkhuh duh d orw ri glhuhqw sdwwhuqv wkdw fdq eh jhqhudwhg edvhg rq wkh julg sdwwhuq frqfhsw1 Vrph h{dpsohv fdq eh vkrzq lq jxuh Zkhq d julg sdwwhuq lv frqvlghuhg/ wkh qxpehu ri furvvlqj srlqwv wr eh surmhfwhg kdv wr eh fkrvhq/ exw wkh olqh wklfnqhvv kdv dovr wr eh fkrvhq/ vlqfh lw ghshqgv gluhfwo rq wkh vprrwkqhvv ri wkh lpdjhg vxuidfh wh{wxuh1 D yhu wklfn olqh zloo jlyh d orz uhvroxwlrq/ dqg/ zlwk d yhu wklq rqh/ zh zloo rewdlq d orw ri glvfrqwlqxlwlhv/ zklfk zloo frpsolfdwh wkh pdwfklqj surfhvv1 Reylrxvo / wkh wklfnqhvv ri wkh olqh kdv wr eh fkrvhq nqrzlqj wkh nlqg ri vfhqhv wr eh phdvxuhg1 Qrwh wkdw wkh sdwwhuqv vkrzq lq jxuh 7148 duh qrw frghg1 Oh Prljqh dqg Zd{pdq ^Prlj ;7` ^Prlj ;8` ^Prlj ;;` sursrvhg wr dgg grwv rq wkh julg/ zklfk fdq eh xvhg dv odqgpdunv wr lqlwldwh wkh ghfrglfdwlrq ru odehoolqj ri wkh surmhfwhg sdwwhuq1 Lq jxuh 7149 vrph julg sdwwhuqv frghg zlwk grwv duh vkrzq1

101 FKDSWHU 71 VWUXFWXUHG OLJKW 435 (a) (b) (c) Iljxuh 7148= Vrph qrw frghg julg sdwwhuqv1 Wkh sdwwhuq uhvroxwlrq lv jlyhq e wkh qxpehu ri furvvlqj srlqwv dqg wkh olqh wklfnqhvv1 (a) (b) (c) Iljxuh 7149= Julg sdwwhuqv sduwldoo frghg e wkh srvlwlrq ri vrph grwv/ zklfk duh xvhg dv odqgpdunv wr lqlwldwh wkh odehoolqj surfhvv1

102 FKDSWHU 71 VWUXFWXUHG OLJKW 436 Wkh frqjxudwlrq xvhg iru wkh v vwhp orfdwhv wkh surmhfwru dw d {hg glvwdqfh iurp wkh fdphud/ dorqj wkh 0d{lv1 Wklv phdqv wkdw yhuwlfdo olqhv duh lpdjhg qhduo zlwkrxw ghirupdwlrq/ nhhslqj lwv qdwxudo sdudooholvp dqg frqwlqxlw / vr wkh zloo eh hdvlo ghwhfwdeoh1 D vtxduh qhljkerxukrrg rs0 hudwru lv frpsxwhg lq hdfk lpdjhg sl{ho wr uhgxfh wkh doehgr ru kljkoljkw hhfw1 Wkh v vwhp xvhv wkh nqrzq orfdwlrq ri wkh yhuwlfdo olqhv dv jxlghv wr vhdufk wkh grwv dqg krul}rqwdo olqh lqwhuvhfwlrqv1 Wkhq/ wkh krul}rqwdo olqhv duh h{soruhg wr mrlq wkhlu glvfrqwlqxlwlhv1 Qrwh wkdw/ li d orw ri glvfrqwlqx0 lwlhv duh irxqg/ lw lv gxh wr wkh idfw wkdw wkh fkrvhq olqh wklfnqhvv lv wrr wklq1 Ilqdoo / iru hdfk odqgpdun grw/ wzr glhuhqw hgjh odehoolqj surfhvvhv/ xvlqj d juhhg dojrulwkp/ duh xvhg1 Rqh surfhvv vwduwv lwv odehoolqj iurp wkh uljkw vlgh/ dqg wkh rwkhu iurp wkh ohiw vlgh ri wkh odqgpdun1 Vr/ vl{ glhuhqw odehoolqjv duh rewdlqhg iurp wkuhh odqgpdun grw sdwwhuqv1 Wkhq/ d phujlqj dojrulwkp lv xvhg wr odeho wkh zkroh sdwwhuq1 Lqlwldoo / rqo wzr odehoohg hgjhv duh frqvlghuhg1 Wkhq/ wkh wu wr odeho wkh qrw odehoohg hgjhv zlwk d orfdo lqwhusrodwlrq1 Dowkrxjk wkh odehoolqj dojrulwkp orrnv olnh udwkhu frpsolfdwhg/ krul}rqwdo olqhv fdq eh kljko eurnhq gxh wr ghswk glvfrqwlqxlwlhv ri wkh phdvxulqj vxuidfhv/ dqg wkh fdq dovr glvdsshdu sduwldoo 1 Vr/ hdvlo pdwfklqj huuruv frxog eh rewdlqhg xvlqj d vlpsohu odehoolqj surfhvv1 Wkh sdwwhuq sursrvhg e Oh Prljqh hw do1 lv vshfldoo lqglfdwhg wr zrun lq g qdplf hqylurqphqwv/ dv d vlqjoh surmhfwlrq lv qhhghg wr rewdlq 6G lqiru0 pdwlrq ri wkh vfhqh1 Exw lw lv wuxh wkdw wkh uhvroxwlrq jlyhq e wkh sdwwhuq lv udwkhu olplwhg/ edvlfdoo iru wzr uhdvrqv= uvwo / yhuwlfdo olqhv gr qrw jlyh dq ghswk lqirupdwlrq vlqfh wkh duh rqo xvhg dv vhdufklqj jxlghv1 Dqg qdoo / wkh pdwfklqj surfhvv pd eh udwkhu vorz lq ghqvlw julgv/ vr/ g qdplf kljk uhvroxwlrq lv qrw shuplwwhg1 Exw wkh xwlolvdwlrq ri wkh sursrvhg sdwwhuq dv d ylvlrq vhqvru iru preloh urerw qdyljdwlrq lq vwuxfwxuhg lqgrru hqylurqphqwv lv kljko uhfrpphqghg1 Lq vxfk hqylurqphqwv/ wkh vxuidfhv gr qrw xvxdoo kdyh kljk frqwudvw wh{wxuhv/ zklfk fdq ghirup wkh sdwwhuq/ dqg wkh vfhqh lv txlwh uhjxodu/ dqg vpdoo glhuhqfhv gr qrw jlyh lqwhuhvwlqj lqirupdwlrq Prulwd0\dnlpd0Vdndwd Lq 4<;; Prulwd hw do1 ^Prul ;;` sursrvhg d elqdu sdwwhuq ri oljkw grwv dv dq 0duud 1 Wkh 0duud kdv wkh douhdg ghqhg frglqj zlqgrz surshuw / dv dq vxesdwwhuq lqwr d zlqgrz h{lvwv rqo rqfh dqg dw rqh sodfh zlwklq d shulrg +iru pruh lqirupdwlrq vhh wkh Yx ovwhnh0rrvwhuolqfn phwkrg,1 Wr vlpsoli wkh fruuhvsrqghqfh sureohp/ Prulwd hw do1 sodfhg wkh fdphud dqg surmhfwru vr wkdw wkh { + dqg { 5 glvwdqfhv ehwzhhq wkhp duh }hur/ dv vkrzq lq jxuh 714:1 Wkhq/ wkh grw lghqwlfdwlrq surfhvv lv txlwh vlpsolhg/

103 FKDSWHU 71 VWUXFWXUHG OLJKW 437 object 1 object 2 object 1 object 2 object 3 object 3 x x Projector y Camera Iljxuh 714:= Wkh v vwhp frqjxudwlrq sursrvhg e Prulwd hw do1 zlwk wkh grw glvrughuv wkdw fdq eh rewdlqhg1 dv wkh grw srvlwlrq rq wkh lpdjh sodqh fdq rqo pryh krul}rqwdoo 1 Lq idfw/ dq 0duud sdwwhuq lv pdgh e d vhw ri flufxodu euljkw/ ru gdun/ grwv/ uhsuhvhqwhg e wkh elqdu qxpehu f/ ru / uhvshfwlyho 1 Iurp wkh 0 duud surshuw / wkh lqglfhv ri doo wkh hohphqwv ri wkh revhuyhg sdwwhuq fdq eh xqltxho ghwhuplqhg uhodwlyh wr wkh surmhfwhg sdwwhuq li qr sdwwhuq glvrughuv h{lvw1 Krzhyhu/ iru frpsoh{ vxuidfhv wkh iroorzlqj glvrughuv fdq eh rewdlqhg= 410 Plvvlqj grwv= Li wzr ru pruh remhfwv duh phdvxuhg/ d grw surmhfwlrq fdqehuh hfwhgrqdqrffoxghgvxuidfh/dqg/lwfdqqrwuhdfkwkhlpdjh sodqh1 Dq h{dpsoh fdq eh vhhq lq jxuh 714: zkhuh wkh srlqwv iurp r2 wr re duh surmhfwhg rq wkh edfn vxuidfh ri wkh remhfw r1 510 Grw glvsodfhphqw= Li wkh ghswk glvwdqfh ehwzhhq wzr remhfwv lv glhu0 hqw/ wkh uh hfwhg grwv zloo eh lpdjhg zlwk d krul}rqwdo glvsodfhphqw ehwzhhq wkhp1 Dv fdq eh vhhq lq jxuh 714:/ wkh grwv uh hfwhg rq remhfw 5 pryh uljkw e rqh grw rq wkh lpdjh sodqh1 610 Grw shupxwdwlrq= Li dq remhfw lv sodfhg lq iurqw ri dqrwkhu/wkh sur0 mhfwhg grwv fdq vzds srvlwlrqv1 Dv vkrzq lq jxuh 714:/ wkh surmhfwhg grwv irup rs wr rh uhdfk wkh lpdjh sodqh lq wkh srvlwlrq ri r2 wr re/ dvwkh kdyh ehhq rffoxghg e remhfw 61 Lq rughu wr rewdlq wkh lpdjh fr0ruglqdwhv ri hdfk grw/ wklv phwkrg kdv wr surmhfw d zkroh looxplqdwhg grw pdwul{1 Wkhq/ d elqdu +euljkw dqg gdun, grw pdwul{ lv surmhfwhg/ vr wkdw hdfk zlqgrz/ zlwk d {hg vl}h/ ghwhuplqhv wkh froxpq lqgh{ iurp zklfk wkh lpdjhg srlqw kdv ehhq surmhfwhg/ vhh jxuh 714;1 Wkh sursrvhg v vwhp lv txlwh vlpsoh/ vr wkdw 6G lqirupdwlrq fdq eh rewdlqhg zlwkrxw d orw ri frpsxwdwlrqdo frvw1 Krzhyhu/ wkh v vwhp lv olplwhg

104 FKDSWHU 71 VWUXFWXUHG OLJKW 438 Illuminated dot Iljxuh 714;= Dq h{dpsoh ri wkh wzr sdwwhuqv surmhfwhg rq wkh phdvxulqj vfhqh e Prulwd1 wr vwdwlf vfhqhv ehfdxvh lw qhhgv wzr surmhfwlrqv1 Dv wkh v vwhp surmhfwv lvrodwhg oljkw grwv/ qrw d orw ri wh{wxudo lqirupdwlrq fdq eh rewdlqhg/ dqg/ li wkh grw vl}h lv uhgxfhg wr jhw pruh uhvroxwlrq/ wkhq/ kljk frqwudvw wh{wxudo vxuidfhv frxog prgli wkh grw vkdsh/ pdnlqj lwv orfdolvdwlrq gl!fxow1 Uhfhqwo / lq 4<<9/ Odyrlh hw do1 ^Odyr <9` kdv sursrvhg d elqdu julg sdwwhuq zlwk d svhxgr0udqgrp grw frglfdwlrq dw hdfk julg yhuwh{1 Wkh grwv duh elqdu frghg/ wkdw lv wkh fdq eh looxplqdwhg ru revfxuhg/ dv wkh P0duud sursrvhg e Prulwd Yx ovwhnh0rrvwhuolqfn D prgho zlwk d g qdplf sdwwhuq/ froxpq frghg/ lv sursrvhg ^Yx o <3`1 Wkh edvlf vwuxfwxuh ri wkh sdwwhuq lv olnh d uhjxodu fkhvv0erdug/ dowhuqdwlqj euljkw dqg gdun vtxduhv1 Wkhq/ wkh sdwwhuq lv prgxodwhg ryhuodsslqj d euljkw ru gdun vsrw dw hyhu vtxduh yhuwh{ +jxuh 714<,/ vr wkdw/ dq vtxduh ri wkh uhjxodu fkhvv0erdug sdwwhuq fduulhv dq dgglwlrqdo lqirupdwlrq elw/ wkdw wrjhwkhu zlwk wkh qhljkerxulqj elwv/ zloo eh xvhg wr frgh hdfk froxpq1 Lq rughu wr lghqwli hdfk grw ri wkh frghg sdwwhuq/ wkh duh uhsuhvhqwhg lq elqdu 1 Wkh grw jhwv wkh ydoxh li lw frphv iurp d euljkw grw1 Rwkhuzlvh/ lw wdnhv wkh f ydoxh/ dvvrfldwhg zlwk gdun1 Dgglwlrqdoo / wkh fkhvv0erdug doorzv qglqj rxw qdwxudoo li wkh frghg grw frphv iurp d euljkw ru gdun vtxduh/ uhsuhvhqwhg e wkh v perov dqg n uhvshfwlyho 1 Wkhq/ irxu glhuhqw frpelqdwlrqv/ vkrzq lq jxuh 7153/ fdq eh rewdlqhg1 Iroorzlqj wkh fkhvv0erdug/ wkh frgh dvvljqphqw lv edvhg rq wzr elqdu vhtxhqfhv/ uhsuhvhqwhg e S & dqg K & ri 96 elwv ohqjwk1 Wkh grwv ri doo wkh &

105 FKDSWHU 71 VWUXFWXUHG OLJKW 439 Iljxuh 714<= D ghwdlo ri wkh sdwwhuq sursrvhg e Yx ovwhnh hw do Iljxuh 7153= Wkh sdwwhuq lv pdgh e irxu glhuhqw frpelqdwlrqv1 Wkh wrs ohiw vtxduh lv uhsuhvhqwhg e wkh v perov 0 dqg./ zklfk uhsuhvhqw 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

106 FKDSWHU 71 VWUXFWXUHG OLJKW 43: c 0 b 1 c 2 b 3 c 4 b 5 c 6 b 7 c 8 b 9 c 10 b 11 c 12 b 13 c 14 c 15 b 0 c 0 b 0 c 1 b 1 c 1 b 2 c 2 b 2 c 3 b 3 c 3 b 4 c 4 b 4 c 5 b 5 c 5 b 6 c 6 b 6 c 7 b 7 c 7 b 8 c 8 b 8 c 9 b 9 c 9 b 10 c 10 b 10 c 11 b 11 c 11 b 12 c 12 c 13 b 14 b 13 c 14 b 12 c 13 b 14 b 15 c 15 b 15 Iljxuh 7154= Wkh frgh dvvljqphqw ghwhuplqhv wkdw dq 5{6 zlqgrz fdq lghq0 wli 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 cs e cs D c EK ck e ck D 1 froxpq zklfk fruuhvsrqgv wr d n vtxduh w sh duh dvvljqhg wkh elqdu ydoxh S & / zkloh wkrvh ri wkh w shduhdvvljqhgwkhydoxhk & 1D9elwvohqjwkfrgh lv qhhghg wr frgh wkh 96 glhuhqw froxpqv ri wkh sdwwhuq1 Wkhq/ dq zlqgrz ri 5{6 vtxduhv doorzv frglqj dq froxpq lqgh{1 Lq idfw/ dq zlqgrz zlwk 9 vtxduhv vl}h fdq eh xvhg/ exw reylrxvo / d frpsdfw zlqgrz lv ohvv dhfwhg e vxuidfh glvfrqwlqxlwlhv wkdw dq horqjdwhg rqh1 Wklv lv wkh pdlq uhdvrq zk d 5{6 zlqgrz kdv ehhq fkrvhq lqvwhdg ri d 4{91 Hdfk 5{6 zlqgrz kdv d wulsohw ri S & frghv dqg dqrwkhu ri K & / hdvlo gli0 ihuhqwldeoh1 Wkh vruwlqj ri wkhvh frghv jlyhv d 9 elwv ohqjwk qxpehu zklfk frglhv d froxpq lqgh{ ri wkh sdwwhuq1 Dq h{dpsoh iru wkh froxpq & 'lv vkrzq lq jxuh 7154/ zkhuh wkh frgh qxpehu/ zklfk uhsuhvhqwv froxpq 6/ lv jlyhq e wkh vhtxhqfh= EES cs e cs D c EK ck e ck D 1 Wkh frgh dvvljqphqw fdq qrw eh fkrvhq lqghshqghqwo gxh wkh idfw wkdw wkh lghqwlfdwlrq zlqgrzv ryhuods1 Wkhq d nlqg ri uhfxuvlyh frgh jhqhudwlrq vhtxhqfh/ zlwk dq lqlwldo eluwk sro qrp / kdv ehhq xvhg/ zklfk doorzv wkh rewdlqlqj ri d ghvluhg ohqjwk shulrg uhshwlwlrq wr frgh lq d xqltxh zd doo wkh froxpqv ri wkh sdwwhuq1 Wkh sdwwhuq suhvhqwhg kdv edvlfdoo wzr olplwdwlrqv1 Wkh uvw rqh lv edvhg rq wkh gl!fxow wr phdvxuh kljk wh{wxudo vxuidfhv/ zklfk surgxfh sduwldo orvw ri uhjlrqv ri wkh sdwwhuq1 Dqg wkh vhfrqg rqh lv dvvrfldwhg zlwk wkh vxuidfh rulhqwdwlrq1 Li wkhvh duh qrw shushqglfxodu wr wkh rswlfdo d{lv ri wkh surmhfwru/ d ghirupdwlrq ri wkh surmhfwhg sdwwhuq lv suhvhqwhg/ zklfk iurp d ghwhuplqhg dqjoh rulhqwdwlrq/ grhv qrw doorz wkh lghqwlfdwlrq ri wkh sdwwhuq1 Krzhyhu/ wkh sdwwhuq lv zhoo0uhfrpphqghg wr phdvxuh g qdplf vxuidfhv zkhuh rqo rqh surmhfwlrq lv shuplwwhg1 Lq 4<<8/ Sdmgod ^Sdmg <8e` kdv uhlpsohphqwhg wkh vdph sdwwhuq/ h{sodlq0 lqj wkh fdoleudwlrq surfhvv1 Sdmgod kdv sursrvhg/ dv dq lpsuryhphqw/ wr xvh

107 FKDSWHU 71 VWUXFWXUHG OLJKW 43; z λ 1 P Violet λ (x p,z p ) Object Surface Red λ 2 z 0 Lens Camera x White light lamp Iljxuh 7155= Wkh sulqflsoh ri gludfwlqj zklwh oljkw lq rughu wr rewdlq d udlqerz sdwwhuq1 d kh{djrqdo frglfdwlrq lqvwhdg ri d vtxduh rqh1 Lq wklv fdvh/ wkh zlqgrzv vl}h lv uhgxfhg dqg dv d uhvxow/ wkh qxpehu ri qrw lqgh{hg froxpqv gxh wr ghswk glvfrqwlqxlwlhv lv ghfuhdvhg1 Krzhyhu/ wkh lghqwlfdwlrq vwhs lv pruh frpsolfdwhg1 7151: Wdmlpd0Lzdndzd Wkh v vwhp sursrvhg e Wdmlpd ^Wdml <3` lq 4<<3 lv edvhg rq wkh yhuwlfdo volw frglqj whfkqltxh/ zkhuh hdfk volw lv hplwwhg zlwk d glhuhqw zdyhohqjwk1 Wkh surmhfwhg sdwwhuq lv olnh d udlqerz sdwwhuq/ dv wkh zkroh frorxu vshfwuxp/ iurp eoxh xs wr uhg/ lv surmhfwhg1 Wklv fdq eh rewdlqhg gludfwlqj zklwh oljkw dv vkrzq lq jxuh Wkh sursrvhg sdwwhuq lv vkrzq lq jxuh Ghswk fdq eh rewdlqhg xvlqj wkh wuldqjxodwlrq sulqflsoh/ exw uvw wkh volw dqjoh +wkh %R fr0ruglqdwh, zklfk kdv surgxfhg wkh frorxu sl{ho lpdjhg rq wkh lpdjh sodqh/ kdv wr eh nqrzq1 Lq idfw/ wkh volw hplvvlrq dqjoh fdq dovr eh ghwhuplqhg nqrzlqj wkh volw zdyhohqjwk1 Wkhremhfwvlooxplqdwhge wkhudlqerzsdwwhuqduhlpdjhgzlwkdvlqjoh prqrfkurpdwlf fdphud lqvwhdg ri d frorxu fdphud1 Wzr glhuhqw frorxu owhuv sodfhg lq iurqw ri wkh fdphud duh xvhg/ dqg wzr lpdjhv ri wkh vfhqh duh fdswxuhg1 Iru wkh vdph srlqw/ wkh lqwhqvlw uhodwlrq ehwzhhq wkh wzr lpdjhv grhv qrw ghshqg rq wkh looxplqdwlrq/ qru rq wkh frorxu remhfw1 Wdmlpd hw do1

108 FKDSWHU 71 VWUXFWXUHG OLJKW 43< Blue Red Iljxuh 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 kdyh suryhg wkdw wklv lqwhqvlw uhodwlrq ghshqgv gluhfwo rq wkh zdyhohqjwk volw +b,1 Dv wkh v vwhp qhhgv wzr iudphv iru hdfk phdvxuhphqw/ lw lv olplwhg wr vwdwlf vfhqhv/ exw reylrxvo wkh v vwhp fdq phdvxuh g qdplf vfhqhv xvlqj d frorxu fdphud1 Lq 4<<9/ Jhqj ^Jhqj <9`/ vhhplqjo zlwkrxw nqrzlqj wkh zrun douhdg grqh e Wdmlpd hw do1/ sursrvhg wkh vdph udlqerz sdwwhuq lq rughu wr rewdlq 6G lqirupdwlrq iurp wkh phdvxulqj vxuidfh1 Krzhyhu/ wkh lqlwldo lghd sursrvhg e Wdmlpd hw do1 zdv lpsuryhg e Jhqj xvlqj d FFG frorxu fdphud dqg xvlqj d Olqhdu Yduldeoh Zdyhohqjwk Ilowhu +OYZI,1 Wdmlpd*v lghd lpsuryhg e Jhqj fdq eh vkrzq lq jxuh Lq wklv fdvh/ wkh surmhfwru hplwv d zklwh oljkw sodqh/ zklfk/ xvlqj d f olq0 gulfdo ohqv/ jhqhudwhv d frqwlqxrxv idq ri sodqhv1 Iurp wkh qdwxudo ghvljq ri wkh OYZI/ d frqwlqxrxv frorxu vshfwuxp lv rewdlqhg/ vr wkhuh duh qrw wzr oljkw sodqhv surmhfwhg rq wkh vfhqh zlwk wkh vdph zdyhohqjwk1 Reylrxvo / wkh v vwhp uhvroxwlrq ghshqgv rq wkh fdphud ihdwxuhv wr glvwlqjxlvk dprqj glhuhqw frorxuv/ ru glhuhqw zdyhohqjwkv1 Uhfhqwo / Vpxwq dqg Sdmgod ^Vpxw <9` kdyh uhlpsohphqwhg wkh v vwhp sursrvhg e Wdmlpd hw do1 Zlwk uhjdug wr wkh olplwdwlrqv ri wkh udlqerz sdw0 whuq/ wkh vdlg wkdw wkh vxuidfh fdq kdyh dq frorxu/ exw kdv wr eh rsdtxh1 Reylrxvo / qr rwkhu zdyhohqjwk qrw frplqj iurp wkh surmhfwru/ fdq eh hplw0 whg rq wkh vfhqh1 Wkh vfhqh kdv wr eh oljkw frqwuroohg dqg wkh phdvxulqj

109 FKDSWHU 71 VWUXFWXUHG OLJKW 443 y λ i Baseline Camera Scene λ 1 z λ 2 Projector x Cylindrical lens Linear variable wavelength filter Light stripes with different wavelength Iljxuh 7157= Wkh v vwhp sursrvhg e Jhqj1 remhfwv fdq qrw eh xruhvfhqw/ qru skrvskruhvfhqw1 7151; Zxvw0Fdsvrq Lq 4<<4/ Zxvw dqg Fdsvrq ^Zxvw <4` sursrvhg wkh surmhfwlrq ri d vlqxvrlgdo lqwhqvlw sdwwhuq rq wkh phdvxulqj vxuidfhv1 Wkh sursrvhg sdwwhuq lv pdgh e wkh ryhuodsslqj ri wkuhh vlqxvrlgv/ ri q shulrgv dorqj wkh %0d{lv1 Hdfk vlqxvrlg lv dvvrfldwhg zlwk hdfk sulpdu frorxu +uhg/ juhhq dqg eoxh,1 Wkh juhhqv vlqxvrlg lv vkliwhg bf zlwk uhvshfw wr wkh uhg/ dqg wkh eoxh lv vkliwhg bf zlwk uhvshfw wr wkh juhhq/ dv vkrzq lq jxuh Wkh sdwwhuq lv froxpq frghg/ vr doo wkh urzv duh lghqwlfdo/ uhvxowlqj d frorxu yhuwlfdo iulqjh sdwwhuq/ dv vkrzq lq jxuh Lqvwhdg ri rewdlqlqj wkh froxpq lqgh{ e ghfrglqj wkh lpdjhg sdwwhuq/ ZxvwdqgFdsvrqsursrvhgwrrewdlqwkhghswkgluhfwo iurpwkhzdyhskdvh vkliwlqj1 Wklv whfkqltxh lv zlgho xvhg lq Prluh phwkrgv wr phdvxuh frqwlq0 xrxv vxuidfhv ^Uhlg ;9`1 Vrph olplwdwlrqv fdq eh revhuyhg lq wkh phwkrg1 Wkh vfhqh kdv wr eh suhgrplqdqwo frorxu qhxwudo/ lq vslwh ri frorxu surmhfwlrq1 Dv wkh sdwwhuq lv pdgh e shulrglfdo iulqjhv/ lw lv olplwhg wr phdvxuh vxuidfhv zlwkrxw glvfrq0 wlqxlwlhv odujhu wkdq d iulqjh shulrg1 Dffruglqj wr Zxvw dqg Fdsvrq/ fdphud uhvsrqvh ghshqgv rq wkh iuhtxhqf ri wkh hplwwhg iulqjhv/ dqg wkh klvwrjudp htxdolvdwlrq xvhg wr frpshqvdwh iru wkh qrq0olqhdu lqwhqvlw uhvsrqvh dovr

110 FKDSWHU 71 VWUXFWXUHG OLJKW 444 Intensity 255 Red Green Blue 0 Iljxuh 7158= Wkh sulqflsoh ri ryhuodslqj wkuhh vlqxvrlgdo lqwhqvlw sdwwhuqv1 Iljxuh 7159= Wkh shulrglfdo sdwwhuq sursrvhg e Zxvw dqg Fdsvrq/ pdgh e wkh vlqxvrlgdo ryhuodsslqj ri wkh wkuhh sulpdu frorxu frpsrqhqwv1

111 FKDSWHU 71 VWUXFWXUHG OLJKW Coded Order: ( w ij,w ij-1 w i-1j w ij+1 w i+1j ) Coded Word = Iljxuh 715:= Grw frglfdwlrq h{dpsoh xvlqj lwv irxu qhljkerxuv dqg d edvlv htxdo wr 6/ l1h1 rqo wkuhh glhuhqw v perov fdq eh xvhg1 surgxfhv vrph phdvxulqj huuruv1 Lpsurylqj wkhvh dvshfwv wkh v vwhp fdq rewdlq d ehwwhu uhvroxwlrq dqg urexvwqhvv1 7151< Jul!q0Qdudvlpkdq0\hh Jul!q hw do1 ^Juli <5`/ lq 4<<5/ kdyh fduulhg rxw d pdwkhpdwlfdo vwxg derxw zklfk vkrxog eh wkh odujhvw vl}h doorzhg iru d frghg pdwul{ grw sdwwhuq1 Lw lv vxssrvhg wkdw= 410 D grw srvlwlrq lv frghg zlwk lqirupdwlrq hplwwhg e lwvhoi dqg wkh lqirupdwlrq ri lwv irxu qhljkerxuv +Qruwk/ Vrxwk/ Hdvw dqg Zhvw,1 510 Wkhuh fdq qrw eh wzr glhuhqw grw srvlwlrqv zlwk wkh vdph frgh1 610 Wkh lqirupdwlrq lv ghwhuplqhg xvlqj d {hg edvlv/ zklfk ghwhuplqhv wkh v perov xvhg wr frgh wkh pdwul{1 710 Wkh eljjhvw pdwul{/ zklfk jlyhv d ehwwhu uhvroxwlrq/ lv ghvluhg1 Li d edvlv htxdo wr 6 lv grqh/ d srvvleoh grw frglfdwlrq lv vkrzq lq jxuh 715:1 Jul!q hw do1 suryhg wkdw/ jlyhq d edvlv e/ wkh odujhvw pdwul{ + wkh eljjhvw q{p pdwul{, fdq eh rewdlqhg iurp lwv odujhvw krul}rqwdo yhfwru +T6,/ dqg lwv odujhvw yhuwlfdo yhfwru +T6,1 T6lv d yhfwru frpsrvhg e wkh vhtxhqfh ri doo wkh wulsohwv ri qxpehuv wkdw fdq eh exlog zlwkrxw uhshwlwlrq xvlqj d eedvlv1 T6 lv d yhfwru pdgh e wkh vhtxhqfh ri doo wkh sdluv ri qxpehuv wkdw fdq eh rewdlqhg zlwkrxw uhshwlwlrq1 Wkhq/ wkh uvw urz ri wkh pdwul{ lv jlyhq gluhfwo e T6/ s f ' T6 +714, dqg/ wkh rwkhu pdwul{ hohphqwv fdq eh ghwhuplqhg dsso lqj wkh htxdwlrq 715 zkhuh lv wkh urz lqgh{ dqg ydulhv iurp f wr wkh T6ohqjwk/ dqg lv wkh froxpq lqgh{ dqg ydulhv iurp f wr wkh T6ohqjwk1

112 FKDSWHU 71 VWUXFWXUHG OLJKW 446 s ' nees 3 n T6 6J_ K +715, Iru h{dpsoh/ li d edvlv htxdo wr 6 lv vxssrvhg/ wkhq lwv odujhvw yhfwruv duh= T6 ' E , T6 ' E , Vr/ wkh rewdlqhg pdwul{ lv/ Diwhu wkh frghg pdwul{ lv irxqg rxw/ d glhuhqw surmhfwlrq fdq eh dvvr0 fldwhg iru hdfk ydoxh/ wkdw lv/ iru hdfk qxpehu zklfk ehorqjv wr wkh lqwhuydo i ckj1 Iru h{dpsoh/ d frorxuhg grw sdwwhuq fdq eh rewdlqhg li wkh uhg/ juhhq dqg eoxh frorxuv duh dvvrfldwhg wr wkh 4/ 5 dqg 6 qxpehuv uhvshfwlyho / re0 wdlqlqj d sdwwhuq olnh wkh rqh vkrzq lq jxuh 715;1 Wkh uhvroxwlrq ri wkh sdwwhuq fdq eh lqfuhdvhg e vlpso lqfuhdvlqj wkh edvlv ydoxh1 Ghshqglqj rq wkh frorxu glvfulplqdwlqj fdsdelolw ri wkh v v0 whp hpsor hg wr ylvxdolvh wkh vfhqh/ doprvw dq ghjuhh ri uhvroxwlrq fdq eh rewdlqhg1 Lq pdq dssolfdwlrqv/ wkh vfhqh lv qrw exlow e frorxu qhxwudo vxuidfhv1 D vlpsoh uhdvrq frxog eh wkdw wkh lpdjlqj v vwhp xvhg lv rqo deoh wr fdswxuh prqrfkurpdwlf lpdjhv1 Wkhq/ prqrfkurpdwlf oljkw kdv wr eh surmhfwhg rq wkh vfhqh1 Lq wklv fdvh/ hdfk qxpehu ri wkh grw pdwul{ fdq eh fkdqjhg wr jhrphwulf irup1 Dq h{dpsoh lv vkrzq lq jxuh 715<1 Wkhphwkrgsursrvhge Jul!qhwdo1 lvwkhxqltxhphwkrgvwxglhge zklfk/ iurp wkh ghfrglfdwlrq ri wkh sdwwhuq fdswxuhg e wkh fdphud/ iru hdfk

113 FKDSWHU 71 VWUXFWXUHG OLJKW 447 Iljxuh 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 Iljxuh 715<= D srvvleoh jhrphwulf dvvrfldwlrq ri d pdwul{ grw frghg xvlqj d edvlv htxdo wr 81

114 FKDSWHU 71 VWUXFWXUHG OLJKW 448 Iljxuh 7163= Wkh sdwwhuq sursrvhg e Pdux dpd hw do1 pdgh e wkh surmhf0 wlrq ri pxowlsoh volwv zlwk udqgrp fxwv1 lpdjh srlqw E% R2 c+ R2 / wkh surmhfwru srvlwlrq srlqw E% R c+ R iurp zklfk lw kdv ehhq hplwwhg/ fdq eh nqrzq1 Dv vkrzq lq wkh pdwkhpdwlfdo vhfwlrq ghglfdwhg wr vxuidfh phdvxulqj/ lw lv qrw qhfhvvdu wr nqrz erwk surmhfwru fr0ruglqdwhv1 Wkhq/ wkh sdwwhuq fdq eh reylrxvo vlpsolhg wr rewdlq d vlqjoh urz frghg ru froxpq frghg sdwwhuq/ exw zh frxog srvvleo rewdlq d sdwwhuq zklfk xvhv wkh zlqgrzv sursrvhg e Yx ovwhnh hw do1 ^Yx o <3`/ ru wkh frorxuhg volwv sursrvhg e Er hu dqg Ndn ^Er h ;:`1 Lq 4<<9/ Gdylhv hw do1 ^Gdyl <9` uh0lpsohphqwhg wkh frorxuhg grw surmhf0 wlrq v vwhp sursrvhg e Jul!q hw do1 Krzhyhu/ dv dq lpsuryhphqw/ Gdylhv sursrvhv wkh xvh ri d vshfldoo ghyhorshg irupxodwlrq ri wkh Krxjk Wudqvirup lq rughu wr h{wudfw wkh lpdjhg grwv ri wkh surmhfwhg sdwwhuq Pdux dpd0deh Wkh phwkrg ghvfulehg e Pdux dpd hw do1 ^Pdux <6`/ lq 4<<6/ lv edvhg rq wkh surmhfwlrq ri pxowlsoh yhuwlfdo volwv1 Wkh volwv duh frghg iurp wkh srvlwlrq ri vrph udqgrp fxwv/ dv vkrzq lq jxuh Doo wkh udqgrp fxwv duh sodfhg wr rewdlq vkruw olqhv/ vdwlvi lqj wkh frqgl0 wlrq wkdw lwv ohqjwk lv lq wkh lqwhuydo du f (c u f n (o/ zkhuhu f lv d vwdqgdug ohqjwk dqg ( lv jlyhq e d udqgrp qxpehu1 Vhjphqw pdwfklqj lv shuiruphg rqo edvhg rq wkh fruuhvsrqghqfh ri wkh

115 FKDSWHU 71 VWUXFWXUHG OLJKW 449 s' s'' l pattern Epipolar lines image Iljxuh 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 vkruw olqh hqg srlqwv dorqj wkh hslsrodu olqhv1 Dv wkh fxwv duh udqgrpo glvwulexwhg/ pruh wkdq rqh sdwwhuq olqh fdq eh pdwfkhg zlwk d jlyhq lpdjh vhjphqw1 Dq h{dpsoh ri pxowlsoh pdwfklqj lv vkrzq lq jxuh Wkhq/ wkh lpdjhg vhjphqwv duh fodvvlhg lqwr wkuhh jurxsv= 410 Vhjphqwv zlwk qr frpshwlwruv/ dv wkh kdyh rqo rqh pdwfklqj1 510 Vhjphqwv wkdw fdq eh vlpso lghqwlhg xvlqj dgmdfhqf uhodwlrqv ehwzhhq lwvhoi dqg lwv qhljkerxuv1 Lq wklv fdvh/ wkh pdwfklqj fdq qrw eh grqh gxh wr wkh qrlvh rq wkh lpdjh ru gxh wr d vxuidfh ghswk glvfrqwlqxlw / zklfk kdv fkdqjhg wkh olqh ohqjwk1 610 Rwkhuzlvh1 Pdux dpd hw do1 frqvlghu wkdw wkh hslsrodu olqhv fdq dozd v eh krul}rq0 wdoo frpsxwhg/ vr wkdw rqo wkh %0d{lv kdv wr eh h{dplqhg wr qg rxw doo wkh olqhv rq wkh sdwwhuq zklfk pdwfk zlwk wkh lpdjhg vhjphqw1 Wklv lv qrw dozd v wuxh/ dv hslsrodu olqhv ghshqg rq wkh surmhfwru srvlwlrq dqg rulhqwd0 wlrq/ zlwk uhvshfw wr wkh fdphud d{lv/ dqg Pdux dpd hw do1 gr qrw lpsrvh dq uhvwulfwlrq1 Exw/ lw lv wuxh wkdw wkh lpdjh fdq dozd v eh wudqviruphg wr jhw krul}rqwdo hslsrodu olqhv/ dv wkh vdlg lq wkhlu zrun1 Lq rughu wr uhpryh huurqhrxv pdwfklqj/ wkh lqirupdwlrq ri qhljkerxu vhjphqwv lv wdnhq xs1 Ilqdoo / zlwk wkh dlp wr rewdlq wkh fruuhvsrqghqfh ri doo wkh vhjphqwv iurp jurxsv 5 dqg 6/ d uhjlrq jurzlqj dojrulwkp pdwfklqj iurp dgmdfhqf frqvwudlqwv lv xvhg1 Wr lpsuryh wkh v vwhp/ zh frxog wklqn ri xvlqj d sdwwhuq zlwk dq lqwho0 oljhqw fxw glvwulexwlrq zklfk pdwfkhv wkh vdph lpdjhg vhjphqw/ dorqj lwv hslsrodu olqhv +lq rughu wr qrw rewdlq pruh wkdq rqh pdwfklqj,1 Vr/ d elmhf0 wlrq ehwzhhq fdswxuhg lpdjh dqg surmhfwhg sdwwhuq lv ghwhuplqhg1 Exw/ dv

116 FKDSWHU 71 VWUXFWXUHG OLJKW 44: Pdux dpd hw do1 vdlg/ li wkh dgmdfhqf frqvwudlqwv duh xvhg/ wkh pdwfklqj fdq eh grqh xvlqj d uhjlrq jurzlqj dojrulwkp1 Krzhyhu/ vxuidfh ghswk glv0 frqwlqxlw ru qrlvh sureohpv lq wkh lpdjh frxog surylgh vhjphqwv zklfk gr qrw kdyh d pdwfklqj olqh1 Lq wklv fdvh/ d uhjlrq jurzlqj dojrulwkp kdv wr eh xvhg wr pdwfk wkhp1 Wkh phwkrg lv vxlwdeoh iru phdvxulqj 6G remhfwv zlwk uhodwlyho vprrwk vxuidfhv1 Iru remhfwv zlwk d orw ri glvfrqwlqxlwlhv dqg zlwk kljko wh{wxuhg vxuidfhv/ wkh phwkrg zloo kdyh vrph gl!fxowlhv1 Reylrxvo / olqh glvfrqwlqx0 lwlhv surgxfhg lq wkh fdswxuhg vhjphqwv frpsolfdwh wkh pdwfklqj surfhvv1 Krzhyhu/ wkh sdwwhuq lv shuihfwo xvhg wr phdvxuh g qdplf vfhqhv/ dv rqo rqh surmhfwlrq lv uhtxluhg Lwr0Lvkll Lq 4<<8/ Lwr dqg Lvkll ^Lwr <8` sursrvhg d wkuhh0ohyho fkhfnhuerdug sdwwhuq1 Hdfk 5G sdwwhuq srlqw ru qrgh kdv d vtxduh irup dqg lw lv vxuurxqghg e irxu surmhfwhg vtxduhv zklfk fdq eh frghg xvlqj rqo wkuhh glhuhqw juh ohyhov1 Wkh juh ohyhov xvhg duh= revfxuhg E? e?r + ' 2DD/ kdoi looxpl0 qdwhg E? e?r + ' 2H dqg looxplqdwhg E? e?r + ' f/ zklfk kdyh ehhq frghg dv 2/ dqg f uhvshfwlyho 1 Lwr hw do1 ghqhg d vxefrgh ri d qrgh dv wkh forfnzlvh frpelqdwlrq ri wkh frghv ri wkh irxu dgmdfhqw qrghv1 Vhh jxuh Wkh dvvhuw wkdw wkh txdqwlw ri frghv lv odujh hqrxjk wr dyrlg idovh pdwfklqjv ehwzhhq wzr vsdwldoo frghg sdwwhuqv1 Dowkrxjk sdwwhuq jhqhud0 wlrq zlwkrxw uhshdwhg dsshdudqfh ri wkh vdph ihdwxuh frgh ydoxhv lv d ixuwkhu uhvhdufk vxemhfw/ lw lv qrw d sureohp ehfdxvh wkh hslsrodu frqvwudlqw holp0 lqdwhv frqixvlrq1 Lwr hw do1 d!up wkdw wkh phwkrg surylghv kljk vsdwldo uhvroxwlrq dqg pdnhv lw srvvleoh wr wdnh kljko dffxudwh phdvxuhphqwv ri vwdeoh dqg prylqj remhfwv1 Lqghhg/ d xqltxh sdwwhuq surmhfwlrq lv qhhghg wr lqihu 6G vfhqh lqirupdwlrq1 Krzhyhu/ lw vhhpv wkdw wkh phdvxulqj ri kljko vdwxudwhg vxuidfhv pd frpsolfdwh wkh vhjphqwdwlrq ri wkh sdwwhuq/ dv lw frxog eh gl!fxow wr glvwlqjxlvk wkh wkuhh glhuhqw juh ohyhov Fkhq0Kxqj0Fkldqj0Zx Lq 4<<:/ Fkhq hw do1 ^Fkhq <:` sursrvhg wkh surmhfwlrq ri d sdwwhuq pdgh e frorxuhg vwulshv1 Wkh dssurdfk lv prvw vlplodu wr wkh rqh sursrvhg e Er hu hw do1 ^Er h ;:`1 Krzhyhu/ Fkhq sursrvhg wr frgh hdfk volw lqghshq0 ghqwo iurp wkh rwkhuv1 Iluvwo / Fkhq jhqhudwhv d vhtxhqfh ri frorxuhg volwv c 2 c c c? 1 Wkhq/ lq rughu wr lqfuhdvh wkh lqwhqvlw yduldwlrq ehwzhhq dgmdfhqw volwv/ d eodfn volw lv lqvhuwhg ehwzhhq wkhp/ rewdlqlqj wkh vhtxhqfh E cc 2 cc c c c? 1 Hdfk volw lv frgh zlwkrxw uhshwlwlrq ydu lqj lwv

117 FKDSWHU 71 VWUXFWXUHG OLJKW 44; P P 4 2 P 0 1 P P The interesting node. The four neighbours of the node. Iljxuh 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 KVL uhsuhvhqwdwlrq1 Wkh lqwhqvlw ydoxh duh udqgrpo jhqhudwhg lq wkh udqjh du 6? cu o1 Lw kdv ehhq {hg juhdwhu wkdq U 6? wr hqvxuh dq hdvlo hgjh h{0 wudfwlrq/ dqg vpdoohu wkdq U lq rughu wkdw wkh kxh ydoxh fduulhv vx!flhqw froru lqirupdwrq1 Qrwh wkdw kljk lqwhqvlw ydoxhv uhgxfh wkh frorxuhg lqiru0 pdwlrq jrlqj wr zklwh1 Dfwxdoo / wkh kxh ydoxh lv wkh prvw lpsruwdqw dqg lv jhqhudwhg dsso lqj htxdwlrq 718/ M J ' M? M n ' EM n M 6R 6J_ Sf J +718, zkhuh/ Fkhq hw do1 sursrvhg wr { M? ' fdqg M 6R ' ef1 Wkhwzrydo0 xhv kdyh ehhq irxqg e h{shulphqwdwlrq whvwlqj doo wkh srvvleoh frpelqdwlrqv dqg vhohfwlqj wkh ehvw vhw ri sdudphwhuv1 Ixuwkhupruh/ lq rughu wr { wkh vdwx0 udwlrq ydoxh iru hdfk volw/ doo wkh frorxuv MU ri wkh vhw ' i c 2 c c c? j kdyh ehhq wudqviruphg wr wkh -C uhsuhvhqwdwlrq1 Wkh odujhvw vdwxudwlrq ydoxh zklfk vdwlvhv wkdw EU cm c7 lv ydolg lq wkh -C vsdfh/ lv wkhq vhohfwhg dv wkh vdwxudwlrq ri 1 Krzhyhu/ wkh pdmru frqwulexwlrq ri Fkhq hw do1 lv wkh sursrvdo ri xvlqj wkh frorxuhg sdwwhuq rqo wr vroyh wkh fruuhvsrqghqfh sureohp ehwzhhq wzr lpdjlqj vhqvruv1 Lw lv nqrzq wkdw/ li d vlqjoh fdphud lv xvhg/ lw lv qhfhvdu wr qg frorxu fruuhvsrqghqfh ehwzhhq wkh surmhfwhg sdwwhuq dqg wkh judeehg

118 FKDSWHU 71 VWUXFWXUHG OLJKW 44< rqh1 Lq jhqhudo/ gxh wr glhuhqw uh hfwlrq surshuwlhv ri 6G vxuidfhv/ wkh frorxu uhfrughg e wkh fdphud lv glhuhqw iurp wkh rqh surmhfwhg e wkh oljkw/ zklfk gl!fxowv wr vroyh wkh fruuhvsrqghqfh1 Fkhq hw do1 sursrvhg wr xvh wzr lpdjlqj vhqvruv dv d vwhuhr ylvlrq v vwhp/ wkh sdwwhuq ri oljkw lv surmhfwhg rq wkh vfhqh dqg fdswxuhg dw wkh vdph wlph e erwk fdphudv1 Wkh frorxu lqirupdwlrq ri wkh lpdjhg volwv lv rqo xvhg lq rughu wr vroyh wkh fruuhvsrqghqfh sureohp ehwzhhq erwk lpdjlqj vhqvruv1 Wklv dssurdfk kdv wkh dgydqwdjh wkdw wkh surmhfwru grhv qrw qhhg wr eh fdoleudwhg1 Krzhyhu/ rqh ri wkh pdmru frqvwudlqwv lv wkdw erwk fdphudv kdyh wr eh v qfkurqlvhg lq lpdjlqj1 Ixuwkhupruh/ wkh srvlwlrq ri wkh fdphudv lq wkh vfhqh dqg wkh rswlfdo ehkdylrxu ri wkhp/ pd jlyh d glhuhqw lpdjhg frorxu ehwzhhq erwk fdphudv zklfk udwkhu frqvwudlqwv wkh sulqflsoh sursrvhg e Fkhq hw do1 716 Vxppdu Lq rughu wr surylgh dq hdvlhu ryhuylhz ri wkh whfkqltxhv suhvhqwhg/ zh vxp0 pdulvh wkhp lq wkh iroorzlqj fkurqrorjlfdo olvw/ uhodwlqj dovr wkh dxwkruv zkr xvhg ru lpsuryhg wkh whfkqltxhv odwho 1 4<;5 Srvgdphu 0 Dowvfkxohu1 D whpsrudo vsdfh0hqfrghg sur0 mhfwhg ehdp v vwhp ^Dowv ;4` ^Dowv ;:` ^Srvg ;5`1 Dovr sursrvhg e = Pxqg dqg Sruwhu ^Pxqg ;:` dqg Plqrx hw do1 ^Plqr ;4`1 Odwho xvhg= Lqrnxfkl hw do1^lqrn ;7` lq 4<;7/ Vdwr hw do1^vdwr ;9` ^Vdwr ;:` lq 4<;9 dqg 4<;:/ Kdwwrul hw do1^kdww <8` dqg Pÿoohu ^Pxoo <8` lq 4<<81 4<;8 Fduulkloo 0 Kxppho1 Dq lqwhqvlw udwlr sdwwhuq surmhf0 wlrq ^Fduu ;8`1 Odwho xvhg= Kxqj ^Kxqj <6` lq 4<<61 4<;: Er hu 0 Ndn1 D vwulsshg sdwwhuq frghg e frorxuhg volwv ^Er h ;:`1 Odwho xvhg= Prqnv hw do1 ^Prqn <5` ^Prqn <6` lq 4<<5 dqg 4<<61 4<;; Oh Prljqh 0 Zd{pdq1 D julg sdwwhuq zlwk vrph odqg0 pdun grwv ^Prlj ;7` ^Prlj ;8` ^Prlj ;;`1 4<;; Prulwd 0 \dnlpd 0 Vdndwd1 Dq P0duud sdwwhuq surmhf0 wlrq v vwhp ^Prul ;;`1

119 FKDSWHU 71 VWUXFWXUHG OLJKW 453 Odwho xvhg= Odyrlh ^Odyr <9` lq 4<<91 4<<3 Yx ovwhnh 0 Rrvwhuolqfn1 D vlqjoh elqdu 0hqfrghg fkhvv erdug sdwwhuq ^Yx o <3`1 Odwho xvhg= Sdmgod ^Sdmg <8e` lq 4<<81 4<<3 Wdmlpd 0 Lzdndzd1 D udlqerz sdwwhuq ^Wdml <3`1 Odwho xvhg= Jhqj ^Jhqj <9` dqg Vpxwq hw do1 ^Vpxw <9` lq 4<<91 4<<4 Zxvw 0 Fdsvrq1 D frorxu vlqxvrlgdo sdwwhuq ^Zxvw <4`1 4<<5 Jul!q 0 Qdudvlpkdq 0 \hh1 D pdwkhpdwlfdo vwxg derxw hqfrghg sdwwhuqv ^Juli <5`1 Odwho xvhg= Gdylhv hw do1 ^Gdyl <9` lq 4<<91 4<<6 Pdux dpd 0 Deh1 D pxowlsoh volwv zlwk udqgrp fxwv sdwwhuq surmhfwlrq ^Pdux <6`1 4<<8 Lwr 0 Lvkll1 D Wkuhh0Ohyho Fkhfnhuerdug Sdwwhuq1 ^Lwr <8`1 4<<: Fkhq0Kxqj0Fkldqj0Zx1 D xqltxho hqfrghg frorxuhg vwulshg sdwwhuq ^Fkhq <:`1

120 Fkdswhu 8 DQhzFrghgSdwwhuq Surmhfwlrq Wklv fkdswhu suhvhqwv d qhz urexvw frghg sdwwhuq surmhfwlrq1 6G phdvxuhphqw lv edvhg rq wkh surmhfwlrq ri d frorxuhg julg sdwwhuq rq wkh vfhqh1 Wkh sdwwhuq lv frorxuhg0frghg lq rughu wr vroyh wkh fruuhvsrqghqfh sureohp e d vlqjoh vkrw dqg zlwkrxw xvlqj jhrphwulfdo frqvwudlqwv1 Wkh whfkqltxh sursrvhg shuplwv d udslg dqg urexvw 6G vfhqh phdvxuhphqw ri g qdplf vfhqhv1 Wkh fkdswhu dovr h{sodlqv wkh fdoleudwlrq phwkrg zh kdyh xvhg dqg vkrzv vrph h{shulphqwdo uhvxowv ri 6G phdvxuhphqw1 814 Lqwurgxfwlrq Rxu dlp lv wr suhvhqw d qhz dssurdfk lq rughu wr vroyh wkh fruuhvsrqghqfh sureohp lq wkh phdvxuhphqw ri 6G gdwd lq g qdplf +l1h1 prylqj, vfhqhv1 Lw lv nqrzq wkdw udqjh lqirupdwlrq fdq eh rewdlqhg e wuldqjxodwlrq xvlqj d v vwhp edvhg rq wzr fdphudv1 Krzhyhu/ wzr sureohpv duh suhvhqwhg1 Iluvwo / wkh sureohp ri phdvxulqj prvwo frqwlqxrxv vxuidfhv/ zkhuh ihz pdwfklqj srlqwv duh suhvhqwhg/ dqg hyhq wkh sureohp ri phdvxulqj kljk wh{wxuhg vxu0 idfhv/ zkhuh lw ehfrphv qhduo lpsrvvleoh wr vroyh wkh pdwfklqj +vhh wkh suhylrxv fkdswhu iru d ghwdlohg h{sodqdwlrq,1 Vhfrqgo / wkh sureohp ri v q0 fkurqlvlqj erwk fdphudv wr vqds wkh prylqj lpdjh dw wkh vdph wlph1 Zh sursrvh d qhz frghg vwuxfwxuhg oljkw whfkqltxh wr vroyh wkh fruuhvsrqghqfh sureohp xvlqj d vlqjoh fdphud1 Wkh whfkqltxh lv edvhg rq rewdlqlqj 6G vfhqh lqirupdwlrq iurp d vlqjoh sdwwhuq vkrw surmhfwlrq/ wkhq lw frxog eh xvhg wr phdvxuh vwdwlf dqg g qdplf remhfwv1 Zh kdyh wdnhq lqwr dffrxqw wkdw wkh surmhfwlqj sdwwhuq pxvw eh vhjphqwhg lq dq hdv dqg urexvw zd / doorzlqj xv wr vroyh wkh fruuhvsrqghqfh sureohp h!flhqwo zlwkrxw kljk frpsxwlqj 454

121 FKDSWHU 81 D QHZ FRGHG SDWWHUQ SURMHFWLRQ 455 uhtxluhphqwv1 815 Sdwwhuq Ghvljq Wkh sdwwhuq lv frpsrvhg e wkh vlqjoh surmhfwlrq ri d julg pdgh e q { q ruwkrjrqdo volwv1 Wkh volwv duh frghg lq vxfk d zd wkdw hdfk volw zlwk lwv wzr qhljkerxuv dsshduv rqo rqfh lq wkh zkroh sdwwhuq1 Yhuwlfdo volwv duh frghg xvlqj glhuhqw frorxu sulplwlyhv iurp wkh krul}rqwdo rqhv1 Wklv idfw shuplwv wr vhjphqw wkh yhuwlfdo volwv dqg wkh krul}rqwdo rqhv vhsdudwho / idfw wkdw uhgxfhv frqvlghudeo wkh frpsxwlqj wlph1 Reylrxvo / yhuwlfdo volwv duh froxpq frghg +[ d{h,/ dqg wkh krul}rqwdo rqhv duh urz frghg +\ d{h,1 Wklv frglfdwlrq doorzv xv wr frgh= d, hdfk volw rq d vlqjoh d{h/ dqg e, hdfk furvvlqj srlqw rq erwk d{hv1 Lw lv nqrzq wkdw/ zlwk wkh jrdo wr rewdlq d pruh urexvw frglfdwlrq/ wkh surmhfwlrq ri volwv lv pruh lqwhuhvwlqj wkdq wkh surmhfwlrq ri grwv1 Hdfk lpdjhg furvv0srlqw fdq eh ghfrghg rewdlqlqj wkh +[/\, srvlwlrq iurp zkhuh lw frphv/ dqg wkh lpdjhg volw vhjphqwv fdq eh ghfrghg nqrzlqj zkhwkhu lw lv yhuwlfdo ru krul}rqwdo/ dqg/ zkdw lv pruh lqwhuhvwlqj/ nqrzlqj rqh ri lwv wzr fr0ruglqdwhv1 Wkh frglfdwlrq xvhg doorzv xv wr sodfh wkh sdwwhuq surmhfwlrq v vwhp lq dq rulhqwdwlrq zlwk uhvshfw wr wkh fdphud1 Lw lv nqrzq wkdw d froxpq frghg sdwwhuq kdv wr eh sodfhg zlwk dq krul}rqwdo glvsodfhphqw iurp wkh fdphud1 Qrwh wkdw urz frghg sdwwhuqv kdyh wr eh sodfhg zlwk d yhuwlfdo glvsodfhphqw1 Lq rxu fdvh/ zh gr qrw kdyh wr wdnh fduh ri wkh orfdwlrq ri wkh fdphud dqg wkh surmhfwru wr hqvxuh d ghwhuplqhg d{h glvsodfhphqw1 D vfkhpdwlf gudiw ri wkh phdvxulqj v vwhp lv vkrzq lq jxuh Frorxu Dvvljqphqw Lq wklv vhfwlrq zh ghvfuleh wkh dojrulwkp xvhg lq rughu wr dvvljq frorxu sulp0 lwlyhv wr d vhtxhqfh ri volwv1 Zh kdyh xvhg d frglfdwlrq vlplodu wr wkh rqh xvhg e Jul!q hw do1 ^Juli <5`1 Rxu lqwhuhvw lv wkdw hdfk volw zlwk lwv wzr qhljkerxuv h{lvwv rqo rqfh lq wkh zkroh vhtxhqfh/ vxfk d jurxslqj lv fdoohg wulsohw1 Iluvwo / zh kdyh wr ghqh wkh vhw ri frorxu sulplwlyhv zh kdyh lq rughu wr frorxu hdfk volw1 Li lv wkh vhw ri sulplwlyhv/ ' i c 2c c Rj/ dqg iru lqvwdqfh/ 'uhg/ 2'juhhq/ 'eoxh/ hwf1/ wkhq/ zh kdyh wr ghqh wkh qxpehu ri volwv ri wkh vhtxhqfh wr eh frghg1 Dqg 7 lv wkh vhtxhqfh ri volwv 7 ' Er cr 2 c c r? 1 Krzhyhu/ li wkh qxpehu R ri sulplwlyhv kdv ehhq {hg/ zh kdyh wr nqrz wkh pd{lpxp vhtxhqfh ri wulsohwv jlyhq e R sulplwlyhv lq rughu wr nqrz zkhwkhu wkh zkroh vhtxhqfh 7 fdq ru fdq qrw eh frorxuhg1 Dqg wkh pd{lpxp vhtxhqfh ri lv ghqhg dv T ' E c 2 c c 6 1

122 FKDSWHU 81 D QHZ FRGHG SDWWHUQ SURMHFWLRQ 456 3D scene object j i imaging pattern detail j i pattern projection detail Image Sensor Pattern Projecting System Iljxuh 814= Vfkhpd ri wkh 6G phdvxulqj v vwhp edvhg rq frghg julg sdwwhuq surmhfwlrq1 Ohw xv h{dplqh wkh sureohp pruh lq ghwdlo1 Jlyhq d vhw ri frorxu sulpl0 wlyhv / zh zlvk wr exlog d vhtxhqfh ri frorxuv vr wkdw hdfk wulsohw ri dgmdfhqw frorxuv lq wkh vhtxhqfh lv glhuhqw iurp doo wkh rwkhu wulsohwv lq rughu wr re0 wdlq d xqltxh frglfdwlrq1 Vlqfh wkhuh lv dq ryhuods ri wkh wulsohwv lq wkh vhtxhqfh/ wkh wulsohwv duh ghshqghqw1 Wkh dojrulwkp zklfk doorzv xv wr re0 wdlq wkh vhtxhqfh pxvw wdnh wkhvh uhtxlvlwhv lqwr dffrxqw1 Wkh sureohp fdq eh vroyhg xvlqj wkh judsk wkhru 1 Jlyhq d R edvlv zh fdq rewdlq d vhw qrghv/ zkhuh hdfk qrgh lv ghqhg e d wulsohw ri wkuhh hohphqwv ri 1 Dv zh kdyh wr orrn diwhu wkh rughu dqg rqo wkuhh hohphqwv fdq eh suhvhqwhg lq wkh qrgh/ zh zloo rewdlq T- R ' R qrghv1 D qrgh lv ghqhg e dwulsohw& m & " 1 Dw wklv srlqw/ zh fdq frqvwuxfw wkh judsk1 Hdfk & qrgh zloo kdyh R lqsxwv dqg R rxwsxwv/ h{fhsw wkh qrghv zkhuh ' ' & zklfk zloo kdyh R lqsxwv dqg R rxwsxwv1 Wkhq/ wkh sureohp lv uhgxfhg wr frqvwuxfw wkh vhtxhqfh ri wulsohwv e ylvlwlqj doo wkh qrghv ri wkh judsk rqo rqfh1 Qrwh wkdw zh fdq jr iurp qrgh ' & wr qrgh ' & rqo li ' dqg & ' 1 Ohw xv looxvwudwh wkh sureohp xvlqj dq h{dpsoh1 Frqvlghu R '2/zklfk jlyhv T- 2 '2 'Hqrghv/ wkdw lv +444,/ +445,/ +454,/ +455,/ +544,/ +545,/ +554,/ +555,1 Wkh iroorzlqj jxuh 815 vkrzv wkh uhodwlrq dprqj wkh qrghv1 Wkhq/ wkh dojrulwkp wr eh xvhg lv uhgxfhg wr ylvlw doo wkh qrghv ri wkh judsk rqo rqfh1 Zh fdq vwduw dw dq qrgh1 Li zh frqvlghu dv vwduwlqj

123 FKDSWHU 81 D QHZ FRGHG SDWWHUQ SURMHFWLRQ Iljxuh 815= Wkh judsk rewdlqhg iurp R '21 qrgh ' +444,/ zh rewdlq wkh sdwk ~ +444,/ +445,/ +455,/ +555,/ +554,/ +545,/ +454,/ +544, / zklfk jlyhv wkh pd{lpxp vhtxhqfh ri sulplwlyhv T ' E c c c 2c 2c 2c c 2c c 1 Wkh sureohp fdq eh vroyhg e d vlpsoh dojrulwkp edvhg rq wkh edfnwudfnlqj sulqflsoh ^Eudv <3`1 Krzhyhu/ Jul!q hw do1 vwxglhg wkh sureohp lq ghwdlo dqg wkh sursrvhg dq lwhudwlyh dojrulwkp/ zklfk lv wkh iroorzlqj/ T ' irr crer c R c ER R c c ER c c c 2R c c 2 c jc irr2crer 2c c R22c +814, ER R2c c ER 22c c c R2c c 22c 2j irrer crer ER cr jc irrrj Dv dq h{dpsoh ri d frorxuhg frglfdwlrq zh kdyh frpsxwhg T ydu lqj R1 Frqvlghu wkh iroorzlqj rxwsxwv= iru R ' wkhq T ' ~444 > iru R '2wkhq T '~554/ 544/ 4/ 555 > iru R 'wkhq T ' ~664/ 654/ 644/ 564/ 554/ 544/ 4/ 665/ 655/ 5/ 666 dqg R 'ewkhq T ' ~774/ 764/ 754/ 744/ 674/ 664/ 654/

124 FKDSWHU 81 D QHZ FRGHG SDWWHUQ SURMHFWLRQ / 574/ 564/ 554/ 544/ 4/ 775/ 765/ 755/ 675/ 665/ 655/ 5/ 776/ 766/ 6/ Uhprylqj wkh frppdv iurp T /irupv d vhtxhqfh ri frorxu sulplwlyhv1 Wkh iroorzlqj dojrulwkp doorzv xv wr rewdlq wkh wulsohwv jlyhq e htxdwlrq 814/ zh kdyh rqo wr { R1 ~ ~ Dv dq h{dpsoh ri frorxuhg frglfdwlrq zh kdyh pdgh d sdwwhuq xvlqj 9 frorxuv1 Zkdw lv lpsruwdqw lv wkdw zh pxvw fkrrvh zhoo0vsdfhg frorxuv lq wkh KVL frqh/ dv wkh zloo eh wkh ehvw vhjphqwhg rqhv1 Wkdw lv zk zh kdyh ghflghg wr frorxu krul}rqwdo volwv lq uhg/ juhhq dqg eoxh > dqg wkh yhuwlfdo volwv lq pdjhqwd/ f dq dqg hoorz1 Wkh volw vhtxhqfh lv rewdlqhg e dsso lqj htxdwlrq 814 xvlqj d R edvlv htxdo wr 6/ wkdw lv ' i c 2c j1 Wkhq/ wkh sulplwlyhv kdyh ehhq vxevwlwxwhg e uhg/ juhhq dqg eoxh wr frgh krul}rqwdo volwv> dqg pdjhqwd/ f dq dqg hoorz wr frgh wkh yhuwlfdo rqhv1 Wklv frglfdwlrq jlyhv xv d xqltxh hqfrghg julg ri 5< { 5< ruwkrjrqdo volwv1 Wkh volwv duh frqvwdqwo vsdfhg lq d 845 { 845 lpdjh zlwk d sl{ho ghswk ri ; elwv/ olnh wkh rqh vkrzq lq jxuh 8161 Wkh uhvroxwlrq ri wkh sdwwhuq pd eh lqfuhdvhg e mxvw fkdqjlqj wkh R edvlv/ l1h1 xvlqj pruh frorxu sulplwlyhv1 Zh fdq hyhq lqfuhdvh wkh uhvroxwlrq uhgxflqj wkh vsdflqj ehwzhhq volwv/ surgxflqj dovr d uhgxfwlrq ri wkh duhd looxplqdwhg lq wkh vfhqh1 Wkh vl}h ri wkh lpdjh fdq dovr eh fkdqjhg surgxflqj sdwwhuqv vxfk dv 45; { 45; sl{hov/ 589 { 589 sl{hov/ 845 { 845 sl{hov/ ru hyhq qrw vtxduh pdwul{ vxfk dv 766 { 8;71 Vr/ zh fdq wdnh dgydqwdgjh ydu lqj wkh glvwdqfh dprqj volwv ru lqfhudvlqj wkh vl}h ri wkh lpdjh lq dq d{h1 Hyhq wkh zlghqhvv lq sl{hov ri wkh volwv lv dq lpsruwdqw frqvwudlqw lq rughu wr phdvxuh wkh vfhqh dffxudwho zlwkrxw rewdlqlqj d orw ri glvfrqwlqxlwlhv1 Dv dq h{dpsoh/ zh kdyh ghflghg wr surmhfw d 845 { 845 lpdjh/ zlwk d sdwwhuq ri 5< { 5< volwv pdgh xvlqj d R edvlv R '/ dqg d frqvwdqw vsdflqj ehwzhhq vhjphqwv ri 49 sl{hov lq erwk gluhfwlrqv1 Ilqdoo / wkh zkroh sdwwhuq lv fhqwuhg lq wkh lpdjh/ wkdw lv zlwk d vsdfh iurp hdfk vlgh ri 65 sl{hov1

125 FKDSWHU 81 D QHZ FRGHG SDWWHUQ SURMHFWLRQ 459 Iljxuh 816= Wkh frorxuhg sdwwhuq1 Lq wkh dlp ri whvwlqj wkh sursrvhg sdwwhuq/ d ode vfhqdulr kdv ehhq vhw +vhh jxuh 817,1 Wkh v vwhp lv frpsrvhg e d UJE fdphud/ d frpsxwhu dqg dq hohfwurqlf volgh surmhfwru1 Lq jxuh 817/ wkh fdphud fdoleudwlqj sdwwhuq lv dovr vkrzq1 D 845 { 845 sl{ho uhvroxwlrq UJE lpdjh lv pdgh e wkh frpsxwhu1 Wkhlpdjhlvsurmhfwhgrqwkhphdvxulqjvfhqhxvlqjwkhhohfwurqlfvolgh surmhfwru zklfk wdnhv wkh lpdjh iurp wkh YJD frpsxwhu rxwsxw1 816 V vwhp Fdoleudwlrq Wkh fdoleudwlrq phwkrg lv edvhg rq rewdlqlqj wkh lqwulqvlf dqg h{wulqvlf sdudphwhuv ri wkh fdphud prgho dqg wkh surmhfwlqj v vwhp prgho nqrzlqj wkh fr0ruglqdwhv ri wkh 6G remhfw srlqwv1 Lq idfw/ wkh sureohp lv uhgxfhg wr wkh frpsxwdwlrq ri wkh htxdwlrqv zklfk prgho wkh uhodwlrq ehwzhhq wkh 6G remhfw srlqwv Ef ct c~ dqg wkhlu 5G revhuydeoh fruuhvsrqghqfh srlqw Ef ct 1 Fdphud dqg surmhfwru ghylfhv duh prghoohg xvlqj wkh vdph htxdwlrqv dv erwk kdyh wkh vdph jhrphwulfdo ehkdylrxu1 Wkhq/ zh zloo ghvfuleh wkh fdoleudwlrq dojrulwkp zlwkrxw wdnlqj lqwr dffrxqw li lw lv xvhg wr fdoleudwh wkh fdphud vhqvru ru wkh surmhfwru v vwhp1 Krzhyhu/ wkh vhfwlrq zloo ixuwkhu h{sodlq krz zh kdyh fdoleudwhg wkh fdphud dqg wkh surmhfwru vhsdudwho 1 Qrwh wkdw erwk kdyh xvhg wkh vdph dojrulwkp ri fdoleudwlrq1

126 FKDSWHU 81 D QHZ FRGHG SDWWHUQ SURMHFWLRQ 45: Iljxuh 817= Wkh vfhqdulr ri wkh 6G phdvxulqj v vwhp Ghvfulelqj wkh fdoleudwlrq dojrulwkp Wkh olqhdu uhodwlrq ehwzhhq d 6G remhfw srlqw dqg lwv 5G surmhfwlrq lv prg0 hoohg e d %e wudqvirupdwlrq pdwul{1 Wklv pdwul{ frqwdlqv wkh 9 h{wulqvlf sdudphwhuv ghqhg e wkh wkuhh urwdwlrq dqjohv Ekc qc / h{suhvvhgdvd% urwdwlrq pdwul{ +/ dqg wkh wudqvodwlrq yhfwru A ' E f c t c ~ >dqgwkh7 lqwulqvlf sdudphwhuv ghqhg e wkh surmhfwlrq ri wkh rswlfdo fhqwuh lq wkh lpdjh sodqh E f c f dqg wkh shuvshfwlyh sdudphwhuv Ek ck +wkh uhdghu lv uhihuuhgwrfkdswhuwzrirudghwdlohgh{sodqdwlrq,1 Krzhyhu/dvduhvxowri vrph w shv ri lpshuihfwlrqv lq wkh ghvljq dqg dvvhpeo ri wkh ohqv frpsrv0 lqj wkh rswlfdo v vwhp/ d olqhdu uhodwlrq grhv qrw krog wuxh1 Wkhvh nlqgv ri lpshuihfwlrqv/ nqrzq dv ohqv glvwruwlrq/ fdq eh prghoohg e d udgldo dqg wdqjhqwldo dssur{lpdwlrq1 Udgldo glvwruwlrq fdxvhv dq lqzdug ru rxwzdug glvsodfhphqw ri d jlyhq lpdjh srlqw iurp lwv lghdo orfdwlrq/ dqg lw kdv ehhq ghprqvwudwhg wr eh txlwh lpsruwdqw lq fdphud prghoolqj +vhh fkdswhu 5 iru d ghwdlohg h{sodqdwlrq,1 Udgldo ohqv glvwruwlrq fdq eh dssur{lpdwhg e qrq0 olqhdu htxdwlrqv jryhuqhg e wkh uvw frh!flhqw & 1 Wkh uhodwlrq ehwzhhq d 6G remhfw srlqw dqg lwv revhuydeoh surmhfwlyh srlqw lv wkhq prghoohg e wkh iroorzlqj htxdwlrqv 815 dqg 816 +wkh uhdghu vkrxog frpsduh wkhp zlwk htxdwlrq 515: / qrwh wkdw zh gr qrw kdyh wr wdnh fduh ri wkh vljq ri wkh sdudphwhuv dv wkh zloo eh lqihuuhg lq fdoleudwlrq,1 zkhuh/ L ' f f _ & o 2 f _ 'f +815, T ' t t _ & o 2 t _ 'f +816,

127 FKDSWHU 81 D QHZ FRGHG SDWWHUQ SURMHFWLRQ 45; dqg/ 5 7 f S t S ~ S ' + A 9 7 f t ~ 6 : , f ' s f S ~ S +818, t ' s t S ~ S +819, f _ ' Ef f & +81:, t _ ' Et f & +81;, o 2 ' f 2 _ n t 2 _ +81<, Ohw xv frqvlghu htxdwlrq 815 dqg 816 dv d ixqfwlrq C ' iclc CT j zklfk ghshqgv rq 44 xqnqrzqv1 Wkhvh 44 xqnqrzqv duh duudqjhg lq d yhfwru fdoohg f/ wkdwlvf ' ifc cc fj/ iru lqvwdqfh/ zh kdyh {hg [ dv yhfwru f ' ikc qc c f c t c ~ c& c& c f c f c& j1 Zh kdyh qrw lqfoxghg wkh irfdo glvwdqfh s dv dq xqnqrzq sdudphwhu/ gxh wr wkh iroorzlqj ' s& ' s& +8144, Qrwh wkdw lqqlwh frpelqdwlrqv ri Esc& c& sdudphwhuv fdq eh rewdlqhg zlwkrxw fkdqjlqj wkh ydoxhv1 Wkdw lv zk zh kdyh wr { rqh ri wkh wkuhh sdudphwhuv/ rwkhuzlvh lw ehfrphv xqvwdeoh dv wkh v vwhp kdv lqqlwh vroxwlrqv1 Wkxv zh kdyh ghflghg wr { wkh irfdo glvwdqfh s dv d frqvwdqw1 Wkh rewdlqlqj ri wkh f yhfwru lv wkh pdlq remhfwlyh ri wkh fdoleudwlqj phwkrg1 Dv d uhvxow ri wkh lqfoxvlrq ri wkh ohqv glvwruwlrq lq wkh fdphud prgho/ C kdv ehfrph d qrq0olqhdu htxdwlrq1 Wkdw lv zk zh kdyh wr xvh dq lwhudwlyh fdoleudwlqj phwkrg wr plqlplvh C lq rughu wr rewdlq f1 Zh zdqw wr fdofxodwh wkh ydoxhv ri f zklfk vroyh wkh htxdwlrq CEf ' f lwhudwlqj iurp dq lqlwldo vroxwlrq f f 1 Wkh lqlwldo vroxwlrq lv frpsxwhg xvlqj wkh olqhdu phwkrg ri Wrvfdql h{sodlqhg lq fkdswhu 5/ dvvxplqj qr ohqv glvwruwlrq E& 'f1 Lq rughu wr fdofxodwh wkh prgho lqfoxglqj ohqv glvwruwlrq zh kdyh xvhg wkh zlgho nqrzq Qhzwrq0Udskvrq lwhudwlyh phwkrg ri ixqfwlrq

128 FKDSWHU 81 D QHZ FRGHG SDWWHUQ SURMHFWLRQ 45< plqlplvdwlrq/ dovr h{sodlqhg lq fkdswhu 51 Wkhq/ iurp d ghwhuplqhg lwhudwlrq &/ zh fdq dssur{lpdwh wkh qh{w ydoxhv ri f iurp wkh odvw rqhv/ xvlqj htxdwlrq Qrwh wkdw wkh vroxwlrq zh zdqw wr qg lv yhfwru [ ri wkh xqnqrzqv zklfk vroyhv CEf & CEf &3 na Ef &3 {f & +8145, CEf & ' f +8146, Wkhq htxdwlrq 8147 lv ghulyhg/ zklfk pd eh vroyhg iurp wkh ohdvw0vtxduhv phwkrg h{suhvvhg lq htxdwlrq {f & ' a 3 Ef &3 CEf & , {f & ' a Ef 3 &3 aef &3 a Ef &3 CEf & , Li zh frqvlghu wkdw? glhuhqw fruuhvsrqghqfh pdwfklqjv duh rewdlqhg/ wkhq CEf &3 lv dq 2?0glphqvlrqdo yhfwru1 aef &3 lv dq E2?% pd0 wul{ lq zklfk hdfk froxpq lv d sduwldo ghulydwlyh ri C zlwk uhvshfw wr hdfk xqnqrzq/ dqg hdfk urz dqg hydoxdwlrq ri wkhvh ghulydwlyhv iurp hdfk fruuh0 vsrqghqfh frxsoh ri srlqwv1 {f & lv d 440glphqvlrqdo yhfwru zklfk frqwdlqv wkh hvwlpdwhg huuru ri wkh 44 xqnqrzq sdudphwhuv wr eh ghwhuplqhg1 Iro0 orzlqj/ hdfk frpsrqhqw ri htxdwlrq 8148 lv h{suhvvhg lq pruh ghwdlo1 3 aef &3 ' E C YCL f Ef&3 Yk YCT f Ef&3 Yk YCL Ef &3 3 CEf &3 ' E C YCL f Ef&3 Yq YCT f Ef&3 Yq CL f Ef &3 CT f Ef &3 CL Ef &3 CT Ef &3 CL? Ef &3 CT? Ef &3 YCL f Ef&3 Y YCT f Ef&3 Y 4 F D YCL f Ef&3 Y f Yk YCT Ef &3 Yk YCL? Ef &3 Yk YCT? Ef &3 Yk +8149, YCL f Ef&3 Y& YCT f Ef&3 Y& YCL Ef &3 Y& YCT Ef &3 Y& YCL? Ef &3 Y& YCT? Ef &3 Y& +814:, 4 F D

129 FKDSWHU 81 D QHZ FRGHG SDWWHUQ SURMHFWLRQ 463 Htxdwlrq 8148 lv frpsxwhg e wkh lwhudwlyh dojrulwkp1 Diwhu hdfk lwhud0 wlrq/ wkh huuru irxqg lq hdfk sdudphwhu lv dgghg wr lwv ydoxh rewdlqlqj wkh qhz ydoxh iru wkh qh{w lwhudwlrq1 Zh pxvw lwhudwh xqwlo htxdwlrq 814; lv uhdfkhg1 {f & ;, Fdoleudwlqj wkh fdphud dqg wkh surmhfwru prgho Lq rughu wr fdoleudwh wkh v vwhp zh rqo kdyh wr sxw wkh fdphud dqg wkh sur0 mhfwru lq wkh vfhqdulr wdnlqj lqwr dffrxqw wkdw wkh fdphud fdq vqds surshuo wkh vfhqh looxplqdwhg e wkh surmhfwru1 Zkhq wkh fdphud dqg wkh surmhfwru kdyh ehhq fruuhfwo sodfhg dqg wkhlu ohqv kdyh ehhq surshuo irfxvhg/ wkh fdoleudwlrq surfhvv vwduwv1 Iluvwo / zh kdyh wr { d 6G zruog uhihuhqfh fr0 ruglqdwh v vwhp/ wkhq/ zh kdyh wr sodfh d 6G nqrzq vxuidfh lq wkh vfhqh/ xvxdoo wzr ruwkrjrqdo sodqhv +wkh vr0fdoohg fdoleudwlqj sdwwhuq,1 Wkh sodqhv ri wkh fdoleudwlqj sdwwhuq kdyh wr eh ruwkrjrqdov lq rughu wr rewdlq frpsr0 qhqwv ri wkh 6G remhfw srlqwv zlwk uhvshfw wr wkh wkuhh d{hv1 Wklv sulqflsoh lv rqh ri wkh prvw frpprqo xvhg lq rughu wr fdoleudwh wkh v vwhp surshuo 1 Vhhdovrjxuh817wrylhzwkhvsdfhuhodwlrqehwzhhqwkhfdphud/wkhsur0 mhfwru dqg wkh fdoleudwlqj sdwwhuq1 Lq rughu wr vlpsoli wkh phdvxuhphqw ri wkh 6G remhfw srlqwv zlwk uhvshfw wr wkh 6G zruog fr0ruglqdwh v vwhp zh kdyh sodfhg lw lq wkh orzhu fruqhu ri wkh fdoleudwlqj sdwwhuq dv vkrzq lq wkh iroorzlqj vfkhpd 8181 Fdphud Fdoleudwlrq Lq rughu wr fdoleudwh wkh fdphud vhqvru zh kdyh wr { d vhw ri 6G remhfw srlqwv rq wkh vxuidfh ghqhg e wkh fdoleudwlqj sdwwhuq1 Zh kdyh ghflghg wr xvh d sdwwhuq pdgh e htxlglvwdqwv eodfn vtxduhv/ zklfk kdv ehhq xvhg e vhyhudo dxwkruv/ iru lqvwdqfh ^Idxj <6`1 Hdfk vtxduh kdv d vl}h ri 63 pp1 { 63 pp1 dqg wkh glvwdqfh ehwzhhq vtxduhv lv 53 pp1 lq erwk gluhfwlrqv1 Hdfk vtxduh ghqhv irxu yhuwh{hv/ wkdw lv irxu 6G remhfw srlqwv1 Dv rxu sdwwhuq lv pdgh e 5 sodqhv zlwk 53 vtxduhv iru sodqh/ zh rewdlqhg d wrwdo ri 493 remhfw srlqwv/ zklfk lv hqrxjk wr fdoleudwh wkh v vwhp1 Lq rughu wr vlpsoli wkh phdvxuhphqw ri wkh 6G remhfw srlqwv zh kdyh pryhg wkh zruog fr0ruglqdwh v vwhp dv vkrzq lq wkh iroorzlqj vfkhpd 819/ vkrzlqj dovr wkh srvlwlrq ri wkh vtxduhv lq wkh sdwwhuq1 Iljxuh 81: suhvhqwv wkh uhdo fdoleudwlqj sdwwhuq xvhg wr fdoleudwh wkh fdphud1 Rqfh wkh 6G fdoleudwlqj sdwwhuq kdv ehhq {hg lq wkh vfhqh/ zh kdyh wr jude lwv lpdjh1 Wkh srvlwlrq ri hdfk vtxduh yhuwh{ lq wkh 5G lpdjh kdv ehhq kdqg0vhjphqwhg lq rughu wr rewdlq lwv srvlwlrq dffxudwho lq wkh lpdjh sodqh1

130 FKDSWHU 81 D QHZ FRGHG SDWWHUQ SURMHFWLRQ 464 Calibrating Pattern Z w 3D object points O w X w 3D world co-ordinate system. Y w Iljxuh 818= Vfkhpd ri wkh fdoleudwlqj sdwwhuq dqg lwv srvlwlrq dqg rulhqwdwlrq zlwk uhvshfw wr wkh zruog fr0ruglqdwh v vwhp1 Z w Calibrating Pattern D object points X w O w 3D world co-ordinate system (in mm.) Y w Iljxuh 819= Vfkhpd ri wkh sdwwhuq xvhg lq rughu wr fdoleudwh wkh fdphud1

131 FKDSWHU 81 D QHZ FRGHG SDWWHUQ SURMHFWLRQ 465 Iljxuh 81:= Wkh fdoleudwlqj sdwwhuq ri vtxduhv1 Wkhq/ wkh dvvrfldwlrq ri wkh 5G remhfw srlqwv dqg wkh 6G remhfw srlqwv lv rewdlqhg gluhfwo / ghqlqj d pdwul{ ri 493 urzv dqg 8 froxpqv +wkh wkuhh fr0ruglqdwhv ri wkh 6G remhfw srlqw dqg wkh wzr fr0ruglqdwhv ri lwv lpdjhg srlqw,1 Iljxuh 81; vkrzv wkh lpdjh ri wkh fdoleudwlqj sdwwhuq fdswxuhg e wkh fdphud vhqvru1 Iljxuh 81< vkrzv wkh srvlwlrq ri wkh vtxduh yhuwh{hv ghwhfwhg lq wkh lpdjh sodqh+lpdjhsrlqwv,/dqgjxuh8143vkrzvwkhsrvlwlrqriwkhvtxduhyhuwh{hv phdvxuhg zlwk uhvshfw wr wkh zruog fr0ruglqdwh v vwhp +remhfw srlqwv,1 Wkh pdwul{ lv wkh lqsxw ri wkh fdoleudwlqj dojrulwkp ghqhg lq wkh suhylrxv vhfwlrq1 Zh kdyh frpsxwhg lqwulqvlf dqg h{wulqvlf sdudphwhuv xvlqj wkh phwkrg sursrvhg e Wrvfdql/ h{sodlqhg lq wkh olqhdu fdoleudwlrq vhfwlrq ri fkdswhu 51 Krzhyhu/ wkh pd{lpxp glvfuhsdqf ehwzhhq wkh prghoohg surmhfwlrqv dqg wkh revhuydeoh srlqwv lv udwkhu odujh dv lw lv durxqg 51; sl{0 hov1 Wkh pdlq sureohp lv gxh wr wkh idfw wkdw Wrvfdql grhv qrw wdnh lqwr dffrxqw ohqv ghirupdwlrq dqg wkdw lw lv d qrq0lwhudwlyh phwkrg1 Wkhq dq lw0 hudwlyh dojrulwkp lv txlwh lqwhuhvwlqj wr ghfuhdvh wklv glvfuhsdqf / uhdgmxvwlqj wkh lqwulqvlf dqg h{wulqvlf sdudphwhuv ri wkh v vwhp1 Zh xvhg wkh lwhudwlyh phwkrg zlwkrxw prghoolqj wkh ohqv ghirupdwlrq dqg rewdlqhg d pd{lpxp hu0 uru durxqg 415 sl{hov/ zklfk lpsuryhv frqvlghudeo Wrvfdql*v phwkrg1 Wkhq/ wkh ohqv ghirupdwlrq zdv prghoohg1 Wkh lwhudwlyh dojrulwkp vwduwv iurp wkh sdudphwhuv rewdlqhg e Wrvfdql dqg d qxoo udgldo glvwruwlrq1 Wkh lqlwldo vr0 oxwlrq zdv frpsduhg zlwk wkh qdo sdudphwhuv1 Wkh pd{lpxp glvfuhsdqf

132 FKDSWHU 81 D QHZ FRGHG SDWWHUQ SURMHFWLRQ 466 Iljxuh 81;= Wkh lpdjh ri wkh fdoleudwlqj sdwwhuq judeehg e wkh fdphud1 Iljxuh 81<= Srvlwlrq ri wkh 5G lpdjh srlqwv ri wkh fdphud1

133 FKDSWHU 81 D QHZ FRGHG SDWWHUQ SURMHFWLRQ 467 Iljxuh 8143= Srvlwlrq ri wkh 6G remhfw srlqwv ri wkh fdphud1 ehwzhhq wkh prghoohg surmhfwlrqv dqg wkh revhuydeoh srlqwv zlwk wkh re0 wdlqhg sdudphwhuv zdv durxqg sl{hov1 Wkh h{shulphqwv suryhg wkdw rqfh wkh qxpehu ri lwhudwlrqv h{fhhgv 53/ wkh dffxudf ri wkh v vwhp grhv qrw lpsuryh vxevwdqwldoo 1 Wkh iroorzlqj wdeoh 814 vkrzv wkh lqlwldo vroxwlrq dqg wkh uhvxowlqj lqwulqvlf dqg h{wulqvlf sdudphwhuv rewdlqhg e wkh lwhudwlyh dojrulwkp1 Krzhyhu/ lq wdeoh 814 zh kdyh vkrzq rqo wkh prvw vljqlfdwlyh ghflpdov/lqwkhprghozhkdyhxvhgxqwlowkh43 ghflpdo1 Surmhfwru Fdoleudwlrq Wkh fdoleudwlrq ri wkh sdwwhuq oljkw surmhfwru lv vlplodu wr wkh fdoleudwlrq ri wkh fdphud vhqvru1 Krzhyhu/ wkh rewdlqlqj ri wkh pdwul{ glhuv d olwwoh1 Ri frxuvh/ wkh phdvxuhphqw ri wkh 5G lpdjh srlqw lv gluhfw dv zh nqrz krz wkh frorxuhg julg sdwwhuq kdv ehhq frqvwuxfwhg1 Wkh sdwwhuq lv surmhfwhg rq wkh fdoleudwlqj vxuidfh1 Zh kdyh sodfhg wzr vkhhwv ri sdshu zlwk d plolphwulfdo julg rq wkh fdoleudwlqj vxuidfh +vhh jxuh 8144,1 Wklv phwulf julg doorzv xv wr rewdlq wkh 6G srvlwlrq ri wkh remhfw srlqwv1 Zh kdyh xvhg dv remhfw srlqwv doo wkh furvvlqj srlqwv ri wkh frorxuhg julg1 Lq rughu wr rewdlq wkh furvvlqj srlqwv zh kdyh phdvxuhg erwk 6G hqglqj srlqw srvlwlrqv ri hdfk volw/ rewdlqlqj wkh htxdwlrq ri wkh volw lq wkh vsdfh1 Wkhq/ dqdo vlqj hdfk volw zlwk wkh rwkhuv/ zh rewdlq doo wkh furvvlqj srlqwv ri hdfk volw/ dqg zlwk wklv/ wkh vhw ri 6G remhfw srlqwv1 Dvvrfldwlqj hdfk 5G remhfw

134 FKDSWHU 81 D QHZ FRGHG SDWWHUQ SURMHFWLRQ 468 Wdeoh 814= Lqwulqvlf dqg h{wulqvlf sdudphwhuv ri wkh fdphud vhqvru1 f f f f D f f f D f 2f k :5<5 41:5<7 41:5<7 41:5<7 41:5<7 q : : : : : 0613<; f 4<1: 0;718 0;71< 0;71< 0;71< 0;71< t 4771< ~ 4;5< < s & ;13: 7531; : 75413: & 66518: 76<15; f : < 73814< 73814< f 55:158 5<:189 5<:1:; 5<:1;3 5<:1;3 5<:1;3 & << < Iljxuh 8144= Wkh phwulf julg xvhg lq surmhfwru fdoleudwlrq1

135 FKDSWHU 81 D QHZ FRGHG SDWWHUQ SURMHFWLRQ 469 Iljxuh 8145= Srvlwlrq ri wkh 5G furvvlqj srlqwv ri wkh surmhfwhg sdwwhuq1 srlqw zlwk lwv 6G remhfw srlqw/ zh kdyh rewdlqhg d pdwul{ ri :;6 urzv dqg 8 froxpqv1 Wkh iroorzlqj jxuhv 8145 dqg 8146 vkrz wkh 5G lpdjh srlqwv dqg wkh 6G remhfwv srlqwv uhvshfwlyho 1 Qrwh wkdw zh glg qrw xvh wkh wzr odvw krul}rqwdo volwv ehfdxvh wkh kdyh ehhq surmhfwhg rxw ri wkh phwulf julg/ wkxv/ wkh :;6 furvvlqj srlqwv kdyh ehhq rewdlqhg e furvvlqj 5< yhuwlfdo volwv e 5: krulw}rqwdo volwv1 Wdnlqj lqwr dffrxqw wkh vdph frqvlghudwlrqv douhdg ghvfulehg lq wkh suhylrxv vhfwlrq rq fdphud fdoleudwlrq dqg xvlqj wkh vdph lwhudwlyh dojrulwkp ri qrq0olqhdu fdoleudwlrq zlwk wkh :;6{8 pdwul{/zh kdyh rewdlqhg wkh vhw ri lqwulqvlf dqg h{wulqvlf sdudphwhuv zklfk prgho wkh surmhfwru v vwhp1 Wkh iroorzlqj wdeoh vkrzv wkh lqlwldo vroxwlrq rewdlqhg dovr e wkh olqhdu phwkrg sursrvhg e Wrvfdql/ dqg wkh sdudphwhuv rewdlqhg lwhudwlqj wkh qrq0olqhdu prgho1 Zh kdyh dovr xvhg 53 lwhudwlrqv lq rughu wr plqlplvh wkh ixqfwlrq1 817 Dffxudf ri wkh phwkrg Lq rughu wr frpsduh 6G jhrphwulf lqirupdwlrq lqihuuhg e wkh sursrvhg frghg vwuxfwxuhg oljkw v vwhp dqg wkh uhdo 6G lqirupdwlrq ri wkh vfhqh/ zh kdyh surmhfwhg wkh frorxuhg sdwwhuq rq wkh fdoleudwlqj sodqh1 Wkh fdoleudwlqj sodqh kdv d glphqvlrqv ri 563 pp1 zlwk uhvshfw wr hdfk d{h f / t dqg ~ 1 Vhyhudo remhfw srlqwv kdyh ehhq sodfhg rq wkh fdoleudwlqj sodqh vxuidfh1 Zh kdyh phdvxuhg wkh 6G uhdo srlqw/ dqg zh kdyh lqihuuhg wkh 6G remhfw srlqw iurp erwk surmhfwlyh 5G srlqwv1 Wkh +[/ \/ ], huuru phdvxuhg lv

136 FKDSWHU 81 D QHZ FRGHG SDWWHUQ SURMHFWLRQ 46: Iljxuh 8146= Srvlwlrq ri wkh 6G furvvlqj srlqwv ri wkh surmhfwhg sdwwhuq1 Wdeoh 815= Lqwulqvlf dqg h{wulqvlf sdudphwhuv ri wkh sdwwhuq surmhfwru1 f f f f D f f f D f 2f k 41;3<4 41:334 41:83: 41:844 41:844 41:844 q 031:3<3 0319<<7 031: : : : << ; f 0:<1; 0751: ; 0941; 0941; t 0;51: 7<1< 0461: ~ 4::514 4;<516 4<4417 4<431; 4<431; 4<431; s & 5841<6 59<1;3 5:51;; 5:51;3 5:51;3 5:51;3 & :<1:: 5;51:7 5;5199 5;5199 5;5199 f 6: : ;5 6831;5 6831;5 f < < 79313< 79313< & ; < < <

137 FKDSWHU 81 D QHZ FRGHG SDWWHUQ SURMHFWLRQ 46; Wdeoh 816= D vdpsoh ri wkh dqdo vhg remhfw srlqwv1 S S R R % o + o 5 o % f Sb ee bs f2 f 22b f bb 2 22Hb Df ee e S 2f f Db b2 f.h DbS2 b ef.h D2 bs f b 2f D bee 22bHe 2f2 ess Sf e S bf f 22 Hbf2 ffb 2.2 wkh glvfuhsdqf ehwzhhq uhdo fr0ruglqdwhv dqg wkh lqihuuhg rqhv1 Zh kdyh phdvxuhg wkh huuru lq dssur{lpdwho 433 srlqwv1 Wdeoh 816 vkrzv d vdpsoh ri 7 remhfw srlqwv zlwk wkhlu 5G surmhfwlrqv rq wkh fdphud lpdjh sodqh E S c S dqglqwkhsdwwhuqiudphe R c R / wkhlu uhdo 6G fr0ruglqdwhv E% o c+ o c5 o dqg wkh 6G fr0ruglqdwhv rewdlqhg e rxu phwkrg E% 6 c+ 6 c5 6 1 Wkh dyhudjh ydoxhv ri wkh huuru ghyldwh 31:5; pp (, iurp wkh f d{h/ pp1 +315: (, iurp wkh t d{h/ dqg pp (, iurp wkh ~ d{h1 Krzhyhu/ wkh pd{lpxp huuru ghyldwh 4148 pp (, iurp wkh f d{h/ 4176 pp (, iurp wkh t d{h dqg 319; pp1 +315< (, iurp wkh ~ d{h1 Wkh uhvxowv duh txlwh lqwhuhvwlqj li zh wdnh lqwr dffrxqw wkdw wkh ghyldwlrq ghjuhh lv kljko lq xhqfhg e wkh vhjphqwdwlrq surfhvv/ l1h e wkh lpdjh qrlvh dqg wkh sureohp ri judeelqj wkh surmhfwhg olqhv ri wkh sdwwhuq zlwk glhuhqw wklfnqhvv gxh wr glhuhqw ghswk surmhfwlrqv1 818 Phdvxulqj wkuhh0glphqvlrqdo remhfwv Rqfh wkh fdphud dqg wkh surmhfwru kdyh ehhq fdoleudwhg/ wkh v vwhp lv do0 orzhg wr uhfrqvwuxfw 6G lqirupdwlrq ri dq remhfw srlqw iurp lwv surmhfwlyh 5G srlqwv1 Wkh dojrulwkp ri uhfrqvwuxfwlrq iurp qrq0olqhdu fdoleudwlrq h{sodlqhg lq vhfwlrq kdv ehhq xvhg1 Reylrxvo / li wkh srvlwlrq dqg rulhqwdwlrq ri wkh fdphud ru wkh surmhfwru lq wkh vsdfh lv prglhg/ wkhq/ wkh v vwhp kdv wr eh uh0fdoleudwhg/ dv wkh lqwulqvlf dqg h{wulqvlf sdudphwhuv kdyh fkdqjhg1 Wkh UJE fdphud vqdsv dq lpdjh ri wkh sdwwhuq surmhfwhg rq wkh vfhqh wkdw zh zdqw wr phdvxuh1 D UJE0SDO :9; { 8:9 { 57 elw lpdjh lv judeehg1 Wkh lpdjh lv fdswxuhg e d Pdwur{ Frphw dftxlvlwlrq fdug lqwr wkh php0 ru ri d SF Shqwlxp1 Dw wklv srlqw vwudljkw olqhv duh h{soruhg lq rughu wr ghwhfw furvv0srlqwv1 Furvv0srlqwv duh wkh wrnhqv xvhg dv lpdjh srlqwv wr rewdlq 6G lqirupdwlrq ri wkh fdswxuhg vfhqh1 Wkh furvv0srlqwv kdyh ehhq kdqg0vhjphqwhg1 Xvlqj wkh frorxuhg frglfdwlrq ri wkh volwv zklfk irup wkh furvv0srlqw lq wkh lpdjh sodqh/ wkh pdwfklqj/ l1h1 wkh ghwhuplqdwlrq ri wkh

138 FKDSWHU 81 D QHZ FRGHG SDWWHUQ SURMHFWLRQ 46< Iljxuh 8147= D vfhqh frpsrvhg e wkuhh jhrphwulf remhfwv1 d, Wkh uhdo vfhqh +qrw xvhg wr lqihu 6G lqirupdwlrq,> e, Wkh vfhqh looxplqdwhg e wkh frorxuhg sdwwhuq1 srvlwlrq ri wkh vdph srlqw lq wkh surmhfwru lpdjh sodqh/ lv fduulhg rxw1 Wkhq/ 6G remhfw srlqw fr0ruglqdwhv duh rewdlqhg iurp erwk 5G srvlwlrqv ri wkhlu surmhfwlyh srlqwv lq wkh lpdjh sodqh dqg lq wkh surmhfwhg sdwwhuq/ uhvshf0 wlyho 1 Lq wkh iroorzlqj/ wzr h{dpsohv ri 6G phdvxuhphqw ri vwdwlf remhfwv lv vkrzq dqg glvfxvvhg1 Dv h{dpsohv zh kdyh fkrvhq wkh phdvxuhphqw ri d vfhqh pdgh e wkuhh jhrphwulfdo remhfwv/ dqg wkh phdvxuhphqw ri d idfh pdvn1 Ixuwkhupruh/ wzr h{dpsohv ri g qdplf vfhqh phdvxuhphqw duh vkrzq1 Erwk h{dpsohv duh edvhg rq wkh phdvxuhphqw ri d kxpdq kdqg1 Lq wkh uvw h{dpsoh/ udqjh lqirupdwlrq ri wkh sdop ri wkh kdqg lv rewdlqhg1 Wkh vhfrqg h{dpsoh looxvwudwhv wkh rewdlqlqj ri wkh udqjh lqirupdwlrq iurp wkh edfn ri wkh vdph kdqg diwhu d pryhphqw H{dpsoh = Phdvxulqj jhrphwulfdo remhfwv Wkh uvw h{dpsoh lv frpsrvhg e d vfhqh zlwk wkuhh jhrphwulf dqg dfkur0 pdwlf remhfwv looxplqdwhg e wkh frorxuhg sdwwhuq/ dv vkrzq lq jxuh Iljxuh 8147 vkrzv wkh phdvxuhg fruuhvsrqghqfh srlqwv lq wkh surmhfwru dqg fdphud iudph/ dqg jxuh 8148 vkrzv wkh 6G uhfrqvwuxfwlrq H{dpsoh = Phdvxulqj d idfh pdvn Wkh vhfrqg h{dpsoh lv edvhg rq wkh 6G uhfrqvwuxfwlrq ri d idfh pdvn/ vhh jxuh Iljxuh 814: vkrzv wkh phdvxuhg fruuhvsrqghqfh srlqwv lq erwk

139 FKDSWHU 81 D QHZ FRGHG SDWWHUQ SURMHFWLRQ 473 Iljxuh 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 Iljxuh 8149= Wkh uhfrqvwuxfwlrq ri wkh 6G remhfwv1 d, D shuvshfwlyh ylhz iurp wkh uljkw vlgh1 e, D wrs ylhz1

140 FKDSWHU 81 D QHZ FRGHG SDWWHUQ SURMHFWLRQ 474 Iljxuh 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 iudphv dqg jxuh 814; wkh 6G uhfrqvwuxfwlrq H{dpsoh = Phdvxulqj wkh sdop ri d kdqg Wkh wklug h{dpsoh lv edvhg rq wkh 6G uhfrqvwuxfwlrq ri d kxpdq sdop/ vhh jxuh 814<1 Iljxuh 8153 vkrzv wkh phdvxuhg fruuhvsrqghqfh srlqwv lq erwk iudphv dqg jxuh 8154 wkh 6G uhfrqvwuxfwlrq1 Lq wklv fdvh/ zh kdyh uhgxfhg wkh glvwdqfhv dprqj volwv lq rughu wr lqfuhdvh wkh uhvroxwlrq ri wkh v vwhp/ wkhq/zhkdyhsurmhfwhgwkhvdphfrorxuhgjulgsdwwhuqpdghlqd589{589 lpdjh H{dpsoh = Phdvxulqj wkh edfn ri d kdqg Wkh irxuwk h{dpsoh lv edvhg rq wkh 6G uhfrqvwuxfwlrq ri d kxpdq edfn kdqg/ vhh jxuh Iljxuh 8156 vkrzv wkh phdvxuhg fruuhvsrqghqfh srlqwv lq erwk iudphv dqg jxuh 8157 wkh 6G uhfrqvwuxfwlrq1 Lq wklv fdvh/ zh kdyh dovr uhgxfhg wkh glvwdqfhv dprqj volwv lq rughu wr lqfuhdvh wkh uhvroxwlrq ri wkh v vwhp/ wkhq/ zh kdyh surmhfwhg wkh vdph frorxuhg julg sdwwhuq pdgh lq d 589 { 589 lpdjh1

141 FKDSWHU 81 D QHZ FRGHG SDWWHUQ SURMHFWLRQ 475 Iljxuh 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 Iljxuh 814<= Wkh uhfrqvwuxfwlrq ri wkh idfh pdvn1

142 FKDSWHU 81 D QHZ FRGHG SDWWHUQ SURMHFWLRQ 476 Iljxuh 8153= Dq looxplqdwhg kxpdq sdop1 Iljxuh 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

143 FKDSWHU 81 D QHZ FRGHG SDWWHUQ SURMHFWLRQ 477 Iljxuh 8155= Uhfrqvwuxfwlrq ri wkh kxpdq sdop1 Iljxuh 8156= Dq looxplqdwhg edfn kdqg1

144 FKDSWHU 81 D QHZ FRGHG SDWWHUQ SURMHFWLRQ 478 Iljxuh 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 Iljxuh 8158= Uhfrqvwuxfwlrq ri wkh kxpdq edfn kdqg1