Luận văn Nghiên cứu và giải bài toán khôi phục cấu trúc xạ ảnh từ ba ảnh trong thị giác máy tính

21 ? , , , , Chtio'ng2.CO SOHINH HQC TRONG THf GIAC MA Y TINH 2.1.IDnhhQcx~anh Hinh hQcxc;!anh [3], [8] Ia congcl) cd band€ nghienCUllnhil'ngb~liloan trangthi giac may tinh.Trangphffnnay se trlnhbay cachxay dvng'khonggian quaphepchie'uph6icanh,cacdinhly lienquailde'nanhxc;!,cachbi€.udi€n cdsa xC;!anh va ynghIabi€u di€n tQadQthuffnnhffttrongkhong gian xC;!anh. 2.1.1.Phepchie'uph6icanh D€ giOi thi~uv€ hlnhhQcxC;!anhtru'oche'tquailsatphepchie'uph6icanh (xuyentam)tranghlnh2.1,ta thffydi€m anhdu'<;Jct ;!ora mQtcachtv nhientren m~tph~nganh. M l 11"0 IDnh2.1.Phepbie'nd6iph6icanhxuyentam. Dfnhnghia2.1.Phepchie'uph6icanhxuyentamla anhxC;!cuadi€m M trenm~tph~ng(d6itu'<;Jng)1I"0,leudi€m m trenm~tph~ng(anh)1I"i'm chinh Ia giaodi€m cuadu'ongth~ngquatam chie'uE va M voi m~tph~nganh1I"i 22 MQts6Hnhcha"trongphepchie'uph6icanh. Tat ca nhil'ngdi€m lIen m~tph~ng1foanhx~leu nhil'ngdi€m lIen m~t ph~ng1fitrltnhil'ngdi€m lIen du'ongth~ngl, voi l Ia giaotuye'ncua 1fovoi m~t ph~ngquaE va songsongvoi 1fi . Tat canhil'ngdi€m lIen m~tph~ng1fila anhcuanhil'ngdi~mlIen m~t ph~ng1f0 trlt nhil'ngdi€m lIen du'ongth~ngh (du'ongchantroi), voi h Ia giao tuye'ncua1fi voim~tph~ngquaE vasongsongvOi1fo. Nhil'ngdu'ongth~ngtrongm~tph~ng1fodu'Qcanhx~leunhil'ngdu'ong th~nglIen m~tph~ng1fi'A.nhcuanhil'ngdu'ongth~ngsongsonggiaonhaut~i mQtdi€m lIendu'ongchantroih, nhu'du'ongth~ngl1va l2tronghlnh2.1. Khi dichuy€ndi€m E ragfinvocungthlphepchie'uph6icanhtrdthanh phepchie'usongsong. Nh!;1nxet. Ne'udangnhatm~tph~ng1fova 1fi voi 1R2,dungmQth~tQadQf)~cac chuftnOele2 lfin Iu'QtlIen 1fova 1fi' thl phepbie'nd6i ph6icanhg~nnhli phep bi€n d6i songanhgiil'a1fova 1fi'nghlala tlt 1R2vao 1R2. Ne'ulo~ibo du'ongth~ngl E 1fova h E 1fi thl taseco mQtphepbie'n d6i songanhtlt [J~2\ l vao IR.2\ h . Tuynhien,ne'utamdrQngm6i 1R2b~ngcachthemdu'ongth~ngial?p l' vao m6i m~tph~ngkhi ay se co mQtphepbie'nd6i ph6icanhsonganhtu (IR2 + l') vao (IR2+ l'), trongdo l' du'<;1Cxacdinhtuanhcuanhil'ngdi€m lIen h va nhil'ngdi€m do anhx~vao l (duytrl tinhlien tl;lC).Nhil'ngdu'ongth~ngmd rQngm~cnhientu'onglingvoinhil'ngphu'onglIenm~tph~ngkia.Ch~ngh~nanh 23 cuaduongthdng~ va l2giaot~imQtdi€m trenh cungphuongvoi ~va~. Giaodi€mtrenh coth€ duqcnhlntha'yb~ngcachgiOih~n hungdi€manhcua mQtdi€m trem~khinodichuy€nravocung.Tu nhungnh~nxettrentacodinh nghiasau: Djnhnghia2.2.T~P L g6mnhungduongthdngsongsongvoiduongthdng l trong[R2segiaonhaut~imQtdi€m d vo cung,ky hi~umoo'di€m nayxem nhuduqcgallvaom6it~pL nhutronghlnh2.2. moo CD ~ ey IDnh2.2.Di€m t~ivocungtuongungvoit~pduongthdngL 2.1.2.Khonggianx~anh Djnh nghia2.3.Khonggianx~anh2-chi€uduqcgQila m~tphdngx~anh haym~tphdnganhduqcdinhnghiabdi: p2 =[R2 U {nhii'ngdi~mvocung} Djnhnghia2.4.Duongthdngt~ivocungtrongp2 la t~pnhungdi€m t~ivo cung, ky hi~u loo={nhii'ngdi~mvocung} Thi dQ.Duongthdngl duqcxacdinhbdi di€m blnhthuongva di€m vo cung.Khi do l diquadi€m thuongcophuongduqcchobdidi€m m=. Nhuv~y, duangth~ngx~anhchuamQtdi€m d vocung,kyhi~upI =[RI U{moo}. 24 r:r D~dinhnghlakhonggianX';lanh3-chi~u,tu'ongtl,l'taxettren 1R3va gall rnQtdi~rnt';livaclingvaorn6it~pdu'ongth~ngsongsong. Dfnh nghla2.5.KhonggianX';lanh3-chi~up3 du'<;5cdinhnghlanhu'sau: p3 =1R3U {nhungdi~m{jvocimg} Nh~nxet.Nhungdi~rnidealtrongp3 t';lOthanhmQtdat';lP2-chi~u.Tlt do tacodinhnghlasau. Dfnhnghla2.6.Nhungdi~rnd va clingtrongp3 t';lOnenrn~tph~ngdu'<;5c gQilam~tph~ngt';livaclingvachuadu'ongth~ngt';livacling. 2.1.3.H~to~dQthuftnnha't ey l: y = 1 lm : (xu x2)t, t E IR ex Hlnh 2.3.Xaydl,l'ngtQadQthuftnha"ttrongpI Day Ia h~tQadQdu'<;5cdlingd~bi~udi~ncac d6i tu'<;5nghlnhhQctrong khonggianX';lanh.Xeth~tQadQD~cacOexeyvaduangth~ngl: y =1 trong 1R2. Quansathlnh2.3taconh~nxetdongiansau: B6 d~2.1.NhungvectO(XUx2)va (Yl'Y2) xacdinhclingrnQtdu'ongth~ng quag6ctQadQkhivachIkhi (Xl,X2)=A(Yl'Y2)' A 7: 0 M~nh d~2.1.MQi du'ongth~nglm =(xuX2)t, t E IR quag6ctQadQ(trlt tn;tcx) d~uca:tdu'ongth~ngl t';lidi~mm. 25 Bjnh nghia2.7.C~pghitri (Xl'X2)va (YI'Y2)du'<JcgQila tu'dngtu'dngne'u (Xl' X2)=A(YI' Y2)' A 7:0 hay vie'tl<:lid<:lng(xu X2) rv (YI' Y2)' kyhi~urv laquailh~tu'dngdu'dng. Nh~nxet.d daycostjtu'dngling1-1 giii'anhii'ngdu'ongth~ngdiquag6c tQadQvoi nhii'ngdi~mtrendu'ongth~ngl va mQtdi~mmarQng~£)i~mnay tu'dnglingvOidu'ongth~ngY =0,nghiaIatheophu'dng(1,0),dayl~di~mavo clingky hi~umoo.B~ngeachd6ngnha'tdu'ongth~ngl vadi~mmoovoi pI ta codjnhnghiasau: Bjnh nghia 2.8. Khong gian X<:lanh l-chi€u pI baa g6m nhii'ngc~p (Xu X2) * (0,0) thoa rv nhu'trongdjnhnghia2.7.C~p(xl' X2) du'<JCgQila tQadQ thu§nnha'tcua di~mtrongpl. Khi do khonggianX<:lanh pI du'<JCvie'tl<:lid<:lng t~ph<Jpnhu'saupI ={(Xl'X2)E 1R2\ (O,O)}. Bjnh ly 2.1.KhonggianX<:lanh pI du'<JCchiathanhhait~pconroi nhau pI ={(Xl' 1)E pI }U{(4' 0) E pI }, tu'dnglingvOinhii'ngdi~mblnhthu'ongvanhii'ngdi~mavocling. r:r Tu'dngttjnhu'trongpI tadlingbaho~cb6ntQadQthlseco bi~udi€n thu§nnha'ttu'dngling cho p2 va p3. Thtjc v~y,trongtru'ongh<Jpp2 ta xet h~ tQadQB€ cae Oexeyeztrong1R3vam~tph~ng1C:Z =1nhu'hlnh2.4. B6 d~2.2.Nhii'ngvectd(xl'X2'X3)va (YI'Y2'Y3)xacdjnhclingmQtdu'ong th~ngquag6ctQadQkhi vaemkhi (xUx2,x3) =A(YI'Y2'Y3)' A 7: 0 M~nhd~2.2.MQidu'ongth~nglmquag6ctQadQ(tritnhii'ngdu'ongth~ng trenm~txOy) giaovoi m~tph~ng1Ct<:limQtdi~mm. 26 ez ey lm : (xl' x2,x3)t,t E IR Hlnh 2.4.Xay dvngtQadQthuffnnha'trongrp2 Nh!;1nxet.MQtIffnnfi'ata tha'yco sVtu'ongling 1-1 gifi'anhfi'ngdu'ong th£ngquag6ctQadQvoinhfi'ngdi€m trenm~tph£ng7rvamQtdu'ongth£ngma rQng.Du'ongth£ngnaytu'ongling voi nhfi'ngdu'ongtrenm~tz =0, nghlala du'ongth£ngvoclingl=. BaygiGtad6ngnha'tm~tph£ng7rva l= vOirp2taco dinhnghlasau: Dfnhnghi'a2.9.C~p(xux2'X3)va (YuY2'Y3)du'QCgQila tu'ongtu'ongne"u (XUX2,X3) =)..(YI'Y2'Y3)' )..7 0 hay (xl' X2'X3)rv (YI' Y2'Y3) , ky hi~urv la quanh~tu'ongdu'ong. Dfnh nghi'a2.10.Khonggianx~anh rp2baag6mta'tca nhfi'ngbQba (xuX2'X3)"#-(0,0,0) voi quail h~tu'ongdu'ongnhu'dinh nghi'a2.9. BQ ba (Xl' X2'X3) du'QCgQi la bi€u di~nthuffnnha'tcua di€m trong rp2. Khi do rp2 du'Qcvie"tl~itheod~ngt~phQprp2={(xu X2' X3) E 1R3\ (O,O,O)}. Dfnhly 2.2.Khonggianx~anhrp2du'Qchiathanhhait~pconroinhau rp2={(xux2'1)E rp2}U {(xuX2'0)E rp2}, tu'ongling vOinhfi'ngdi€m blnhthu'ongva nhfi'ngdi€m avocling. 28 Hai du'ongth&ng l =(a,b,ef, l' = (a',b',e')T giao t~imQtdi€m m = l X l' = (be'- eb',ea'- ae',ab'- ba')T. Haidu'ongth&ngsongsongc~tnhaut~idi€m avacling(xuX2'0). - Di€m m n~mtrendu'ongth&ngl khi vachI khi mTl =0 Du'ongth&ngt~iva cling loo,c6 tQadQthuffnnha't(0,0,Ih la t~phQp nhii'ngdi€m t~ivacling. (ir Tu'dngtv tac6 th€ xaydvngchotraCinghQprp3(ngayca cho rpn VOl n E ~)nhu'ngseg~Pkh6khankhith€ hi~nynghIahlnhhQc. Dinh nghla2.11.Khonggianx~anh3-chi€u rp3du'Qcdinhnghlabai t~p nhii'ngkhonggiancontuy€n tinhl-chi€u trongkhonggianvectd 4-chi€u 1R4. Nhii'ng di€m trong rp3 du'Qcbi€u di€n thuffn nha't bai cac vectd (xu X2'X3'x4) * (0,0,0,0),trongd6 hatvectdbi€u di€n clingmQtdi€m n€u chungsatkhacmQth~ngs6d l~.Khi d6 rp3du'Qcvi€t l~inhu'sau: rp3={(XUX2,X3,X4) E 1R4\ (O,O,O,O)} (X,Y, Z)T ~ (xu X2' X3' x4f (X,Y, Z)T ~ (Xl' X2' X3' X4)T d~nhtlmIan Z vai x4trongX(janh m6idiim nhQ.nmQttia4-chi€u /l>/., X2X2 BJ (X,Y, Z) ~ X3 ffinh 2.6.Y nghIahlnhhQccuabi€u di€n thuffnnha'tchodi€m trongrp3 29 Nhi)nxet.KhonggianxC;!anh3-chi€uchinhla khonggianthu'ongcuacae vectdkhac0 trong1R4theoquailh~tu'ongdu'ongt'..J.Ta co th€ ky hi~ukhong gian xC;!anh 3-chi€u: p3 =p(1R4) =(1R4\ 0)/ t'..J. Djnh ly 2.3.Khonggian p3 du'<;5Cphanthanhhai t~pconroi nhau p3 ={(Xl'X2'x3'1)E p3}U{(xl'X2'X3'0)E p~}, Iftnlu'<;5tla nhungdi€m blnhthu'ongvanhungdi€m d vaclIng. H~qua2.3. (Xl' X2' X3' X4)T t'..J(Xl / X4'X2/ X4'X3/ X4'l)T la cling bi€u di€n thuftn nhf(tchodi€m M trongp3 va (X, Y, Z) =(Xl / X4'X2/ X4'X3/ X4)T, X4~ 0 la bi€u di€n cuadi€m M tu'onglingtrong1R3,xemhlnh2.6. (Xl' X2' X3' O)T, X4 =0 la bi€u di€n thuftn hf(tcuadi€m d voclIng. - MQtdi€m trongthe'giOithlfc3-chi€ucotQadQ(X, Y, Z)T trong1R3se du'<;5cbi€u di€n thuftnhf(tbdi (X, Y, Z,l)T . (a,b,c,df labi€u di€n thuftn hf(tcuaIT: aX +bY +cZ +d =O. (O,O,O,l)Tlabi€udi€n thuftnhf(tcuam~tph£ngdvaclIng. H~qua2.4.Dol ngdugifi'adi~m- du'ongthiing- m~tphiingtrongp3 Quabadi€m khongth£nghang MI, M2 ,M3 taxtiydlfngdu'<;5cmQtm~t ph£ng7fva du'<;5cxacdinhtu 7fTMi =0, i =1,3. - Bi€m M n~mtrenm~tph£ng7fkhi vachikhi 7fTM =o. - Bam~t ph£ngkhacnhau7fI' 7f2'7f3giaonhautC;!imQtdi€m M duynhf(t vadu'<;5cxa dinhbdih~phu'ongtrlnh IT;M = 0, i = 1,3. 30 - Qua hai di€m M} va M2 trong p3 ta xac dinh du'<Jcdu'ongth~ng l ={M =(Xl' X2' X3' X4) E p3 I M =~MI +t2M2,(~,t2)E pI}, con gQi la d~ngthams6 va (~,t2) du'<Jcxemla bi€u di~nthu§nnha'tcuadu'ong th~ng(hlnh2.7.a). - Giaocuahaim~tph~ng7Cva lp trongp3 (xemhlnh2.7.b)la du'ong th~ngdu'<Jcbi€u di~n bdi l : {S7C+tlpI (s,t) E pI}, congQila d~nggiao cuadu'ongth~ng. a) b) mnh 2.7.B6i ngftugifi'adi€m - du'ongth~ng- m~tph~ngtrongp3 , ? ?? 2.1.4.Anh x~x~anhva cdsox~anh , ? 2.1.4.1.Anh x~x~anh Dinh nghi'a2.12.Chokh6nggianx~anhP(lRn+1)lienke'tvoi kh6nggian vectdIRn+1.Phepchie'uchinht~cla anhx~p: IRn+1\ {O}-7 P(lRn+1) bie'n mQtvectdME IRn+1\ {O}thanhdi€m x~anhd~idi~nbdivectddo. Nh~nxet.V€ m~thlnhhQcthl ME IRn+1\ {O}d~idi~ndi€m trongkh6ng gian,p(M) la tiaxuyentamdiquadi€m M vanh~n olamvectdchiphu'dng. 31 Djnh nghla 2.13.Anh X£;lI : p(lRnH) --+p(lRk+l) du'QcgQila ant X£;lX£;l ant neu~I E L(lRnH,IRkH)thoalop =p '0Ivai p va p' la haiphep chieuchinht~ctu IRnH\ {O}--+p(lRnH) vatuIRkH\ {O}--+p(lRkH). Tinh cha'tcuaanhX~. I la antX£;ltuyentint 10 p(AM) =AI 0 p(M) dap tuyentint. Khi I la sangantthl f condu'QcgQila d&ngcfiuX£;lant. 2.1.4.2.Cdsi'ix~anh GQi c = {El"",cnH} la cd sa cuakh6nggianvectdIRn+lva xetkh6ng gianX£;lant pn tu'dngling.ChQnt~pdi~mmu ...,mnH' mn+2E pn saDcha: { mi = p(Ei)'i = 1,...,n +1 mn+2= p(El+E2+... +EnH) (2.1) Khi d6t~pn +2 di~mml' ...,mn+umn+2du'QcgQila cd saX£;lant cua pn tu'dnglingvoi cdsachatru'ocuakh6nggianvectdIRnH. Ngaaifa, di~m mn+2du'QCxacdintdlfatrenmu...,mn+l. Djnh nghla2.14.Cd saX£;lant cuapn la t~pn +2 di~mdu'Qcxaydlfng nhu'(2.1)trongd6bfitky n +1 di~mnaaclIngdQcl~ptuyentint,nghIalakh6ng clingn~mtrensieuph&ng. Djnhly 2.4.Lienh~giuacdsaX£;lantcuap(lRnH) vacdsacuaIRn+1: - Caccdsatll~trongkh6nggianvectdIRnH xacdintduynhfitmQtcdsa X£;lantcuakh6nggianx£;lant p(lRnH). - Chatru'occd sa X£;lant {ml"'.' mnH' mn+2}cuakh6nggianx£;lant p(lRnH) tac6th~tlmdu'QcduynhfitmQtlOpcdsatll~nhautrongIRn+1nh~n 32 {Tnt,...,mn+l'mn+2}lamcosatu'ongling. 2.1.4.3.Cdsdx~anhehinhtile Dinh nghia 2.15.T~p veeto e={el=(1,O,...,O)T,e2=(O,I,...,O)T,... ...,en+!= (0,0,...,I)T,en+2= (I,I,...,I)T}, trongdotQadQthli i euaei co ghi ~ trib~ng1, i =1,n +1, du'<JegQila cosaehinhtileeuakhonggianx~anhpn Nh~nxet. - Trongdoei voi i =1,n Ia nhii'ngdi~mt~ivoclingtheem6i~e, en+! di~mg6e,en+2di~mdonvi. - Bfftky di~mm E pn d~uco th~du'<Jebi~udi€n nhu'bai t6h<Jptuyen n+l tinheuan +1 veetcrtrongcosaehu§n,vietdu'oid~ngm =L xiei . i=l Thid\l.Cosax~anhehinhtileeuap2 vap3. p2: e ={(I,0,of, (0,1,of, (0,0,I)T, (1,1,I)T} (2.2) p3: E ={(I, 0,0,O)T,(0,1,0,O)T,(0,0,1,O)T,(0,0,0,I)T, (1,1,1,I)T} (2.3) 2.1.5.Phepbie'nd6ix~anh Phepbiend6iph6ieanha hlnh2.1Ia mQthid\leuaphepbiend6ix~anh. Trongph§nnayehuyeutrlnhbayv~phepbiend6ix~anhtu p2 -+ p2 va tu p3 -+ p3, conphepbiend6ix~anhtu p3 -+ p2 sedu'<Jetrlnhbaytrong2.2.3, tru'oehettadinhnghiat6ngquatv~phepbiend6ix~anh Dinh nghia2.16.Phepbiend6i x~anhtU m E pn vao m' E pk du'<Je dinhnghianhu'phepbiend6i tuyentinhtronghc$tQadQthu§nnhfft,nghiaIa x' r-..JHx trongdo x va x' bi~udi€n tQadQthu§nnhffteuahaidi~mm va m', H la matr~nkhongsuybieneffp(k+1)x (n +1). 33 2.1.5.1.Phep d6nganh trong rp2 Dfnhnghia2.17.Phepd6nganhtrongrp2hayphepdi6nd6iph~ng(phep cQngtuye'n)la mN anhX'.lkhanghichh tu rp2vaochinhno,nghIala badi6m ml 'm2'm3 n~mlIen mQtdu'ongth~ngkhi va chi khi h(ml) ,h(m2)'h(m3) clingn~mlIen mQtdu'ongth~ng. DfnhIy 2.5.AnhX'.lh : rp2-+ rp2la mQtphepbie'nd6iX'.lanhne'uvachi ne'ut6nt'.limQtmatr~nH3x3kh6ngsuybie'nsacchovoiba"tky di6mm nao trongrp2thl h(m) =Hm. Chungminh. Gia sa ml' m2' m3 n~mlIen du'ongth~ngl thl ITmi =0, i =1,3 . GQi H la ma tr~nca"p3x3 kh6ng suy bie'n, khi a"y H thoa: ITH-I Hmi =0, i =1,...,3.Dodota"tcacacdi6mHmi n~mlIendu'ongth~ng H-Tl vatinhcQngtuye'nbaaloan quaphepbie'nd6i. X2 (Xl / X2'Xl / X3) m m' (X~/ X;, X~/ X~) Hinh 2.8.Phepd6nganhtrongrp2 34 Phepbie"nd6inayconduQcgQila phepd6nganhtrongrp2,hlnh2.8,va matr~nH la matr~nthuffnnha'tco 9phfintu, nhungchungsaikhacmQth~ng s6dl~nenco8bi;ictQ' do.H va AH, A 7:0, clingbi€u di€n phepbie"nd6i. Vie"tl~id~ngmatr~n: (2.4) d' ( ) T I ( I I I )T ff)2trongom=xl,x2,x3 ,m=xI,x2,x3 Er. Do do,cffnitnha't4c~pdiemtuonglinglIenhaianhd€ xacdinhphepd6ng anhH (xemphVIvcB). 2.1.5.2.Phepd6nganhtrongrp3. Phepd6nganhtrongkhanggianx~anh3-chi€uduQcmatanhux~tuye"n tinhtu rp3--+rp3va noduQcbi€u di€n boi matr~nH4x4khangsuybie"n.Ma tr~nH va AH, A 7:0, clingbi€u di€n phepbie"nd6i.Ma tr~nthuffn ha'tH co 16phfintd'nhungchungsaikhacnhaumQth~ngs6d l~nenchIco15bi;ictQ' do. (Xl' X2' X3' X4f ~H ~ ( I I I I ) T Xl' X2' X3' X4 X2 X2 Xl Xl Hlnh 2.9.Phepd6nganhtrongrp3 X I IhH 2 3 Xl I l 2 3 x21haym' =Hmx2 - I h31 h32 h33 X3X3 35 Bi€u di€n d'.lngmatr~nM' =HM, trongd6 M =(xl' X2'X3'X4)T va M ' ( ' , , , )T tn3= X1'X2'X3'X4 E r . 2.2.Mo hinhCameravarangbuQcepipolar QuaCameramQtdi€m trongthe'giOithlfc3-chi€utathuduqcmQtdi€m anh trenm(Ltph~nganh2-chi€u.TrongphftnnaysetrlnhbaymahlnhCameracoban trongthigiacmaytinh.Phantichvaltrc>,ynghIahlnhh9Ccuamatr~nCamera X'.lanhvaquailh~giuahatCamerabairangbuQcepipolar. 2.2.1.Mo hinhPin-holeCamera Day la h~th6ngthigiacdongiannhfftdungd€ mahlnhh6aCameraduqc g9i la Pin-holeCamera.N6 duqcmatanhumQtCathQpVOlmQt16nho(Pin- hole)a m(LtbellvamQtffmkinhch\lpanha m(Ltd6idi~n,nhuhlnh2.10,trong d6g6ct9adQd(Lt'.litamchitu C duqcg9i la tieudi€m vatr\lcz trungVOl tr\lcthigiac.KhoangcachtITtieudi€m de'nanhg9ila tieuclfkyhi~uIa f . (X,Y, Z) xl X e O---muu (xo,Yo}..-- .----- {x,y) Z f IDnh2.10.Ma hlnhPin-holeCameravOlh~t9adQdclit D€ ti~ntrongcachbi€u di€n tad(Ltm(Ltph~nganhlentru'octamCameraC khi d6mahlnhPin-holeCameratronghlnh2.10duqcbi€u di€n tu'onglingnhu' hlnh2.11,b~ngcachd(Ltamchie'uC t'.lig6ccuah~t9adQdclit vam(Ltph~ng 36 ant Z =f. R6rangday laphepchie'uph6icantxuyentam(nhu'trong2.1.1)cua mQtdi€mtrongkhong ianlenm~tph~ngant. Y M c z p z f tam ~ Camera mijtphiinganh ffinh 2.11.D';lnghinthQccuaPin-holeCamera QuamohintPin-holeCamerasebie'ndi€m M trongkhonggiancotQadQ (X, Y, Z)T len di€m m co tQadQ (x,y)T trenm~tph~ngant, m la giao di€m cuadu'ongth~ngdiquadi€m M vatamchie'uC voim~tph~ngant.Xet h . ., d):, d h' h 2 11 ,? 1" fX, fY V ".!' al tamglac ong ';lngtrong III . , tacot1vx =Z va y =Z' let l';li du'oid';lngthu~nnha'tnhu'sau: x xllfXI If 0 0 oilY AlyilfYI =10 f ° 011z lllZI 10 ° 10" 1 (2.5) trongdo A =Z la chi€u saucuadi€m. Congthuc(2.5)la bi€u di~nphepchie'u xuyentamdungh~tQadQthu~nha't.VaCameradu'Qcxacdintnhu'sau: - Tamchie'uC la tamcuaCamerahaycongQila tieudi€m. 37 M<}tph~ngquaHimCamerasongsongvdi m<}tph~nganh IT gQila m<}t ph~ngchinhcuaCamera. Duongth~ngtitHimC tn;(cgiaovdim<}tph~nganhgQiIatr\lcchlnh. Giaocuatr\lCchlnhvam<}tph~nganhgQiIadi€m chinh,kYhi~ula p. 2.2.2.Mo hlnhCCD Camera TrongmahlnhPin-holeCamera,tQadQanhgiasaduQCdotrongh~tQadQ dclit vdi cackhoangchiadonvi trenhaitr\lcb~ng nhau.Tuynhien,trongth1!cte cacCameraky thu~t s6haymayquayphimgQichungla CCD Camerasa d\lngthanhphffncambiend€ t~onendi€m anh.TQa dQanhduQctinhtheedonvi pixelnhungtrenmoi tr\lct11~d6khanggi6ngnhaudonhii'ngdi~manhbi biend~ng(khangvuong). Dod6,phaichuy€ntQadQaOOtittQadQdclitsangtQadQtinhtheepixel. x Y t IYo t " 8 .t- I '" Pixel I pY ! ~ ..( ~ px ..( "~YXo Hlnh2.12.H~tQaanhvadi~manh(pixel) GQi Px va Py la chi~urQngva chi~ucaocuapixeltheephuongx va y tuongling. p =(xo,Yo,l)T Ia tQadQthuffnnha'tcua di€m chinhduQcdo trong tQadQpixel(c6th€ khacHimcuam<}tph~nganh).a: la g6cl~ch(moo2.12)ph\l 38 thuQcvaonhasanxua'tbQph?ncamlingtrongcacCCD Camera.ax =f / Px ' ay =f /Py bi6udi€n tieuql'cuaCameratrongdonvi pixeltheohaiphu'ongx, Y tu'ongling.Ngoaira,anhhu'dngcuanhungpixelkhongvuongdu'c;cxemnhu' thams6d l~s =(f / Py) tana, thu'ongthlthams6 s cuacacCamerab~ngO. Khi d6d~ngt6ngquatcuamatrq,nainhciJCameradu'c;cxacdi~hnhu'sau: ax s xo K=!O 0 ay YO 0 1 (2.6) matr?nK condu'C;cgQila thamsf{nQicuaCamera.Khi d6phepbie"nd6idi6m trongkhonggiandclit lenm~tph~nganhchobdi: x fX+ZPxl lax s xolll 0 0 oilY fY +ZPy = 0 ay Yo 0 1 0 0II Z zoo 1 0 0 1 0" 1 (2.7) Dinh nghi'a2.18.MQtCameradu'c;cgQila dinhco du'c;cne"ubie"tru'ocma tr?nK. Ngu'C;cl~i,n6du'c;cgQila Camerakhongdinhco. 2.2.3.Camerax~anh 2.2.3.1.Ma tr~nCamerax~anh Trongphfin2.2.1va 2.2.2,HimCameradu'c;cd~tt~ig6ccuakhungtQadQ the"gioithvc.Tuynhien,thu'ongthlkhungtQadQCamerakhongtrungvoikhung tQadQthe"gioithvc.VI the"nhungdi6mtrongkhonggian3-chi~udu'C;cbi6udi€n trongkhungtQadQthi gicJi th1!Cphai du'C;cbie"nd6i sangkhungtQadQCamera tru'ockhi X<:lanh len m~tph~nganh.Hai khungtQadQnay c6 m6i lien h~voi nhauthongquamatr(inquayRax3vavectottnhtiin t. 39 Ma tr~n(tn,tcgiao)quay R bi€u di€n phuongcuakhungtQadQCamerava vectdtinhtien t bi€u di€n tQadQcua HimCameratrongkhungtQadQthegidi thtfc.Dodo,R va t duQcgQilathamsifngo{licuaCamera. z Mcam y ~ x llinh 2.13.Phepbiend6idclitgiil'akhungtQadQthegioith\icvaCamera Ta dungkyhi~uMworld= (Xw,Yw'Zw)Tbi€u di€n cuadi€m trongkhung tQadQthe gidi th\icva Mcam= (Xc,Yc,Zc)T Ia bi€u di€n cua di€m trong khungtQadQCamera.Khi dophepbiend6idclit tttkhungtQadQthegioith\ic vaokhungtQadQCamera,nhuhlnh2.13,duQCvietduoidC;!ngthuffnhfftnhusail: B~ngcachkethQp(2.5),(2.7)va(2.8)tadinhnghiamatr~nxc;!anhnhusail: Dinh nghIa2.19.Ma tr~nCameraxc;!anh P3X4bi€u di€n m6i lien ket giil'a tQadQthu~n hfftcua m =(x,y,If trenm~tph~nganhvoitQadQthuffnhfft M =(X, Y, Z, I)T trongkhungtQadQthegioith\icthongquaphuongtrlnh Am =PM (2.9) Xc] Xw Y R 0 13x3 -t Yw R -Rt Zc I - OT OT =>Mcam = T IM world (2.8) 1 1 Zw 13 3 03 1 I 1 40 va(2.8)du'QcgQilaphu'dngtrlnhCamera.Trongd6 P =KR[I3x3I-t] Nh!;lnxet. - Ma tr~nP c612phfintti'nhu'ngchic611b!;lctIt do(K: 5thams6,R: 3thamsO',t: 3thams6).Di€u nayc6nghIaP la matr~nthuffnnha'tc6cacphffn tli'saikhacnhaumQth~ngs6ti1~. - Ma tr~nconbelltraica'p3X 3 cua P b~ngKR la khong~uybien.Do d6 ba'tky matr~nP3X4naoc6 matr~nconbell tnii M3X3khongsuybiend€u la ma tr~n Cameracua Camerax~ anh hfi'uh~n.Voi Q =KR khi d6 P =KR[I3x3 I -t] =Q[I I Q-Ip4]' P4 la cQtcu6iclingcua P Dfnhnghla2.20.MQtCamerax~anht6ngquatla Cameradu'Qcbi~udi~n bdimQtmatr~nthuffnnha't3x4tuyy h~ngb~ng3vac6 11b~ctl,l'do. Dfnhnghla2.21.(SuytudinhnghIa2.16)Camerax~anht6ngquatla phep biend6ix~anhtu 1'3(1R4)vao1'2(1R3)nhu'sau: Xl XII IPn Pl2 Pl3 P14 X2 x21=IP21 P22 P23 P24 X3 x3I IP31 P32 P33 P34I I X4 (2.10) trongd6 (xl' X2'X3f E 1'2 va (Xl, X2,X3,x4f E 1'3 la bi~udi~ntQadQ thuffnnha'tcuadi~mm trenm~tph£nganhva M trongthegioithl,l'ctu'dngling. 2.2.3.2.YnghlahinhhQcciiamatr!;lnCamerax~anh Trongphffnnayse phantichcacthanhphffntrongCamerax~anhva y nghIahlnhhQccuan6.Giasli',mQtCamerax~anht6ngquatdu'QCphantich thanhP =[Q I P4], trongd6 Q lamatr~n3x3khongsuybien. 41 qT PH Pl2 P13 I Q =Ip21 P22 P2a =qf Pal Pa2 Paa qJ (2.11) r:r Nhii'ngvectdcQtcua ma tr~nCamera.Gia sa, d~tcaccQtcuamatr~n CameraP Ia Pi' voi i =1,4nghIalaP =[PI I P2 I Pa I P4]'Khi do,m6ivectd Pi COY nghlahinhhQcnhu'sau: PI' P2'Pa la nhii'ngdi€m tri~tlieu cuacactI1;lCX, Y, Z tu'ongungtrong h~tQadQthe'giOith\fc,donhii'ngdi€m nayla anhcuaphu'ongcuacactI1;lctQa dQnhu'hinh2.14.Ch~ngh;;tn,tI1;lcX co phu'ongD =(I,0,0,of (di€md vo cling)thidu'<JcX;;tanht;;tiPI =PD, PI chinhIa cQtdc1uliencuamatr~nP. P4 la anhcuag6ctQadQthe'giOith\fc.Th\fcv~y,voi D =(0,0,0,If chinhIa phu'ongchI vaog6ctQadQthe'giOith\fcvaco anhla P D =P4' 0 Z Y IDnh2.14.Badi€m PI' P2,Pa la nhii'ngdi€m tri~tlieucuaphu'ongcactI1;lc Cir Nhii'ngvectddongcuamatr~nCamera.VectddongcuaCameraX;;tanhIa cacvectd4-chi€u,chungcoynghlahinhhQcnhu'nhii'ngm~tph~ngd~tbi~ttrong the'gioi th\fc.Ta kY hi~u7r[, 7rf, 7rJ Ia badongcuatu'onglingcuaCamera. 42 PH PI2 PI3 PI4 T 1T"1 P =IP21 P22 P23 P24 =1T"! P3I P32 P33 P34 1T"T3 (2.12) 1T"1'1T"21ahai m~t ph~ngtr1}.c.Thl;tcv~y,nhungdi€m M thuQcm~t ph£ng 1T"1thoa 1T";M =0 va duQCx~anht~inhungdi€m anhco tQadQ PM =(0,y,wf manhungdi€m nayn~mlIen tr\lcanh y: M~tkhac, PC =0 suy ra 1T";C =0 nghlala C clingn~mlIen m~tph£ng1T"I'Do do, 1T"1lam~tph£ngduQcxacdinhbdiHimthigiacC vatf\lCy trongm~tph£ng anhnhutronghinh2.15.TudngtV,m~tph£ng1T"2duQcxacdinhbditamC va tr\lc x trongm~tph£nganh.Nhu v~y 1T"1va 1T"2la hai m~tph!ng trongthe giOithl;tcmaanhla m~tph£ngx =0 vay =o. C Zc Zc Yc Yc Xc ffinh 2.15.Hai m~tph£ngtf\lC1T"1'T"2va m~tph£ngchinh1T"s 1T"3co vai trbla m~tph~ngchinh.M~tph£ngnaychuatamCamerava songsongvoi m~tph£nganh(hinh2.15).Thl;tcv~y,nhungdi€m M thuQc m~t ph£ngnayco anh1aduongth!ngd vo cungPM =(x,y,0f . Do d6 nhungdi€m n~mlIen m~tph£ngchinhcua Cameraneu va chI neu 43 7f{ M =o.Ne"uC IatamCamerathlC thuQcm~tph&ngchinhvaPC =0 T hay 1I"3C =o. <:ir Di~m chinh. Phap tuye"ncua m~tph&ng1I"3=(PH' P12'P13'P14)T la vectd n =(PIU P12'P13f ho~cco th€ bi€u di€n bdi di€m 1T3=(PIU P12'P13'of tIeD m~tph&ngd vacung,nhu'hlnh2.16.X~anhdi€m naybdimatr~nP~taxacdinh di€mchinhP =P 1T3=Qq3'voiqf ladongthlibacuaQ . Zc "" T 1I"3=(PIU P12'P13'0) ~ IDnh 2.16.Phu'ongcuam~tph&ngchinh1I"3 2.2.3.3.Tac dQngcuaCameraleDdi~m r::r x~ anhxuoi.TrangmohlnhCamerax~anhsebie"nd6idi€m M trong khonggian x~ anh vao di€m anh thoa m =PM. Ngoai fa, nhfi'ngdi€m D =(X, Y, Z,of tIeDm~tph&ngd va cung,bi€u di€n nhfi'ngdi€m tri~ttieu trangp3 tu'onglingdi€m d =(X, Y, Z)T trongthe"giOithlfc1R3,nhu'hlnh2.17, khidonhfi'ngdi€m naydu'Qcanhx~vaodi€m anhmd: md =PD =[Q I P4]D=Qd (2.13) Nh~nxet.CQtthlitu',P4,cuamatr~nP khongtacdQngleDdi€m D. 44 ~...p (X,Y, z, O)T \............................ \d Yc Y ffinh 2.17.X<;tantxu6icuadi€m r:r x~anhngu'Q'c.Chodi€mant movamatr~nCameraP. Xacdintt~p di€m trongthe'giOithlfcco ant Ia mo.T~pdi€m nayt<;tohanhtiadi quatam Cameratrongthe'giOithlfc,xemhInt2.I8.a.TianayIa du'ongthAngdu'Qcn6ibdi haidi€m Ia tamCameraC vadi€m P+mocod<;tnghu'sau: M (J.l)=P+mo+J.lC (2.14) voi p+ Ia matr~ngiadaDcuaP. C Zc YcYc Zc C a) b) ffinh 2.18. a. X<;tant ngu'Qccua di€m, b. Chi€u saucua di€m 45 r:r Chi~usau cua di~m.Cho di~mMo trongth~gioi thtfcva matr~n CameraP. TImchi€u saucuaCamerazo.Ta co di~mMo =(X,Y,Z,T)T duQcnhlnthffytrongCameraP la di~manhMoP =mo=..\(x,y,l)T khido chi€u sau,hlnh2.18.b,cua di~mMo trenm~tph~ngtru'ocm~tph~ngcmnh CameraduQctinhtheocongthuc(xemchungminhtrong[8]): Asign(detQ) Zo= Tllq311 (2.15) 2.2.3.4.Tac dQngcuaCameralendu'ongth~ng r:r X~anhxuoi.X~anhduongth~ng(tia)trongkhonggianx~anhtenm~t ph~nganh.Duongth~ngva tamCameraxac dinhmQtm~tph~ng7r.Duong th~nganhIa giaocua7rvam~tph~ngcmnh,xemhlnh2.19.a. c Yc JlB Yc c Zc a) b) mnh 2.19.a.x~anhxuoiduongth~ng(tia).a.x~anhnguQcduongth~ng(tia) GQiA, B la haidi~mtrongkhonggian3-chi€ucoanhla haidi~ma, b thongquaCamerax~anh P. Khi do mQtdi~mM(Jl) =A +JlB n~mtren duongth~ngn6ihaidi~mA, B codi~manhIa: m(Jl) =PM(Jl) =P(A +JlB) =a +Jlb (2.16) 45 r:Jr Chi~usau cua di~m.Cho di€m Mo trongthe'giOitht!cva matr~n CameraP. TImchi€u sancuaCamerazo.Ta co di€m Mo =(X,y,Z,T)T duQcnhlnthty trongCameraP la di€m anhMoP =mo =A(X,y,l)T khido chi€u san,hlnh2.18.b,cua di€m Mo trenm~tph~ngtru'ocm~tph~ngchinh CameraduQctinhtheocongthuc(xemchungminhtrong[8]): Asign(detQ) Zo= Tllq311 (2.15) 2.2.3.4.Tac dQngcuaCameraleDdliongth~ng r:r X~anhxuoi.X~anhduongth~ng(tia)trongkhonggianx~anhlenm~t ph~nganh. Duong th~ngva tam Cameraxac dinh mQtm~tph~ng1r. Duong th~nganhla giaocua1rvam~tph~ngchinh,xemhlnh2.19.a. c yc J-lB yc c Zc a) b) Hinh2.19.a.X~anhxuoiduongth~ng(tia).a.X~anhnguQcduongth~ng(tia) GQi A, B la haidi€m trongkhonggian3-chi€uco anhla haidi€m a, b thongqua Camerax~ anh P. Khi do mQtdi€m M(J-l) =A + J-lB n~mtren duongth~ngn6ihaidi€m A, B codi€m anhIa: m(J-l) =PM(J-l) =P(A + J-lB)=a + J-lb (2.16) 46 c:r x~ anh ngu'Q'c.T~p nhii'ngdi€m du'<;5cx~ anh len du'ongthing l =(xu X2'X3)T bdimatr~nCamerax~anhP la m~tphingtrl=pTl, xem hlnh2.19.b. 2.2.4.Quanh~giiIahaiCamerax~anh- rangbuQcepipolar Trang2.2.3,x~anhdi€m M trongkhanggianvaodi€m anh'in trenm~t phinganhbdiCameraC du'<;5cth€ hi~nbdi m6iquailh~PM . m. Trong ph§nnay,sexemxetm6iquailh~cuadi€m x~anhM vahaidi€m anhm, m' cuanotrenhaim~tphinganhkhacnhau,xemhlnh2.20. M6i quail h~ giii'ahai anhph6icanh(di€m trenhai anh) cua mN khung canh (mQtdi€m trangkhanggian) du'<;5Cma ta bdi hlnh hQc epipolarmaC'l;!th€ la matr~n cosa. 2.2.4.1.Hinh hQCepipolar Hinh2.20.Anhclmdi~mM trenhaiCamera. HinhhQcepipolarchIdlfavaocacthamso'nQicuacacCameravavi tricua chung.No coth€ du'<;5Cmatabdimatr~nF cfip3X 3 coh~ngb~ng2.N€u bi€t tru'(kcacthamso'nQicuaCamerathlco th€ nh~ndu'<;5ctQadQanhdffdu'<;5cchutin hoa va khi do hlnh hQCepipolarse du'<;5cma ta bdi ma tr~nthi€t y€u [3], [8]. Ngu'<;5cl~i,hlnhhQcepipolardu'<;5Cmatabdimatr~ncosd [2],[10].Vi v~y,ma tr~ncosdth€ hit%nm6ilienh~giii'acacthamso'nQivavi tricuahaiCamera. 47 HlnhhQcepipolarla hlnhhQccuacacm~tph~nganhgiaovoi chumm~t ph~ngcoduongcdsanhutronghlnh2.21. miffphiing---- epipolar ~-- I I / I II 0 0' dlldngcdsO lfinh 2.21.Chumm~t ph~ngepipolar- 7r1,7r2,. ..,7rn dtiongcdsdIa duongdiquaHim0 va 0' cuaCamera.Nhungdiemepipolee va e' la nhunggiaodiemcuaduongcdsahaim~tph~nganh.M~tkhac,diem epipolechinhIa anhcuatamCameranaytrenm~tph~nganhkia.Nhungdiem m va m' la anhcuadiemM trongkh6nggian.M~tph~ng7r Ia m~tphiing epipolarchuam, m' , M vaduongcdsa(xemhlnh2.22). dlldngcdsO 0 lfinh 2.22.X<;lanhmQtdiemtrongkh6nggiantrenhaim~tph~nganh 48 f)i~mm trenm~tph£nganhthlinha"tIa hlnhchi€u cuadi~mbfftky trong khonggian3-chi€un~mtrentia Tl nhu'hlnh2.23.Hlnhchi€u cuabfftky di~m naotrentia Tl sen~mtrendu'ongth£ngi' trenm~tph£nganhthlihai.Tu'ongtv, d<5ivditiaT2vadu'ongth£ngi. 0 0' dllungcdsO 0 0' dllungcdsO llinh 2.23.Khaini~mdu'ongepipolar Do v~y,rangbuQcepipolardu'<;1Cmota nhu'sau:M6i di~mm trenm~t ph£ngthlinhfftsecodi~mm' tu'onglingn~mtrendu'ongepipolari' vatu'ongtv bfftky di~mm' naotrenm~tph£ngthlihaiclingco di~mm tu'onglingn~m trendu'ongepipolari. 49 2.2.4.2.Ma tr~ncdsi'i Ma tr~ncdsi'iF la bi€u di~nd~isf)cuarangbuQcepipolar.Ma tr~ncdsa chinhla anhx~bi€n nhungdi€m trenm(ltph£ngnaylendu'ongepipolartrenm(lt kia,nghlala l' =Fm va l =FTm' (2.17) dom n~mtrenl' va m' n~mtrenl (theoh~qua2.2)nentacorangbuQcsau: m,TFm=0vamTFTm'=0 (2.18) trongmatr~nF cffp3x3 du'<jcxacdinhduynhfft,co 7b~ct1;l'do,h~ngb~ng2 va detF =0. dliungcdsll Hlnh2.24.Matr~ncdsaquanh~voiphepd6nganh Th1;l'cv~y,gia sam(ltph£ng7r khongxuyenquabfftky tamCameranao nhu'hinh2.24.Ta thffytiadi quadi€m m trenm(ltph£ngthlinhfftgiaovoim(lt ph£ng7r tC;tidi€m M va di€m M du'<jcXC;tanhvao di€m m' trenm(ltph£ng thlihai.Anh x~bi€n d6i tu anhnay sanganhkhacthongquam(ltph£ng7r trong tru'ongh<jpnayla phepd6nganh(xem2.1.5.1)du'<jcmotabaimatr~nHIT va nhungdi€m tu'dnglingtrenhaianhthoaquanh~sau: 50 m' =H m7r (2.19) M~t khac,theoh~qua2.2,tac6th~xacdinhdu'ongthiingepipolarl' b~ng cachn6idi~mepipolee' vadi~mm' (xemkyhi~u[e']xtrongph\ll\lcA.1): l' , , [ ' ] , =e Xm = e xm (2.20) ke'thQpphuongtrlnh(2.17),(2.19)va(2.20)taduQcquailh~sau: F = [e']xH7r (2.21) Ngoaifa,duangth~ngepipolarl' = Fm chuadi~mepipolee' Dene' thoa phuongtrlnhe'T(Fm) = (e'TF)m = 0 (tuongtt;t'l = FTm' chuae).Suyra: e'TF =0 vaFe =0 (2.22) Vi matr~nH 7r c68b~ctt;t'doh~lllgb~ng3vamatr~nphand6ixung[e']x c6h~ngb~ng2 Dentheo(2.21)F c6h~ngb~ng2.M~tkhactheo(2.22)tac6 detF =0 dod6matr~nF chic67b~ctt;t'do. Tinh ma tr~ncd sd F tu haimatr~nCameraP, P'. Ta c6 tiax~anh nguQctUdi~mm di xuyenquadi~mp+m (giaiphuongtrlnhm =PM vdi p+ la matr~ngiadaocuaP) vatamCameraC (thoaPC =0). X~anhdi€m C va P+m leDm~tph~nganhthuhai ta duQchai di~manh p'C = e' va p' P+m. Khi d6,duangepipolardi quanhungdi~manhnayduQctinhbCiic6ng thucl' = e' X (P' P+m) = Fm. Do d6,matr~nco sciF c6th~duQcxacdinh bCii: F =[e']xp'p+ (2.23)