Motion of an uncalibrated stereo rig : self-calibration and metric reconstruction Zhengyou Zhang, Quang-Tuan Luong, Olivier Faugeras
To cite this version: Zhengyou Zhang, Quang-Tuan Luong, Olivier Faugeras. Motion of an uncalibrated stereo rig : self-calibration and metric reconstruction. [Research Report] RR-2079, 1993.
HAL Id: inria-00074592 https://hal.inria.fr/inria-00074592 Submitted on 24 May 2006
HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destin´ee au d´epˆot et `a la diffusion de documents scientifiques de niveau recherche, publi´es ou non, ´emanant des ´etablissements d’enseignement et de recherche fran¸cais ou ´etrangers, des laboratoires publics ou priv´es.
INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
Motion of an Uncalibrated Stereo Rig: Self-Calibration and Metric Reconstruction Zhengyou Zhang Quang-Tuan Luong Olivier Faugeras
N˚ 2079 Octobre 1993
PROGRAMME 4
Robotique, image et vision
ISSN 0249-6399
apport de recherche
1993
!#" $% '&)(* +# , -. / % 1024315 6879;:=@?BAC6878DE:< F A8DE::< M1NPORQ2O 9;SUT8DEAb$Yb|{}@bVY~{}@bpv1BJ)d
U{~h[Jg@Y+ba|JJd yx];\Jb|r '||x`J1 b ]Jz@z>Y+b|rrinvh+}>isryx]`yxbVYh}@by@Y[Jg>nbV_.[;#rIb~{V]ing@Y^]`hIin[Jv ];vxz _abVhYIiR{Yb|{~[JvxrIhY+kx{~hIin[Jv¡k@yah+[#]1r+{V]nb;¢8£Y[J_¤[Ev@bk@v@¥Bv@[;¦v_a[JhIin[Jva[`4]`vak@vx{V]n iug@Y~];hb|z§rIhbVYbV[YIin\xl5]JrrIk@_¨iuv@\h}@b{~[B[JY~zEiuvx]`hb|r$[;h}@bay@YIinvx{+inyx]Uy2[;invBhp[;©b|]E{} {V];_abVY^]a]`Yb1¥>v@[;¦v¡tª}>iRr]JrrIk@_ay@h«iu[EviRrv@[Ehv@b|{~b|rr];Y+¬i%[Ev@b$_a[JYbp_a[JhIin[JviRr ]|q;]iR];g>nb;¢^#ª®}@b#bVy>iny2[;R];Y#{~[Jvxr«hY^]inv>hiRr¯xY~rIh£[JY_k>R];hb|z°[EY$ht¦[ak@vx{V]ing@Y^]`hb|z iu_];\Jb|rV$ª}@by@Y[Jg>nbV_±h}@bVvg2b|{~[J_ab|r$[Ev@b1[;b|rIhIin_o];hIinv@\k@v@¥Bv@[;¦vxr$r«kx{}h}x];h h}@b1zEiRr{~YbVyx];vx{~¬£Y[J_²h}@b$bVy>inyx[`s]`Y-{~[Jvxr«hY^]inv>h|lEiuvh+bVY_or[;%zEisr«h^];vx{~b|rg2bVht¦bVbVv yx[;inv>h^r$];vxzh+}@b~iuY1{~[EYYb|rIy2[JvxzEinv@\³bVy>inyx[;R];Yinv@b|rVl´isr_¨iuv>in_inµVb|zª}@bpinv>iuh«is]inµ|]; hIin[Jva[;h}@bk@v@¥>v@[¦vxr5iRrgx]Jr«b|z[Jvh}@b¦[JY¥$[`4¶]¬>gx];v@¥lJ·´k@[Jv@\#];vxz¸2];k@\JbVY^]Jr [Jv rIb~{V]ing@Y^];h«iu[Ev [;®]r¹inv@\;nba_a[;qEinv@\°{V];_bVY^]@l¦}>iR{+}ºY+b|jJk>inYb|rh[°r«[;nqJb#]rIbVh [;rI[J{V]nb|z³»Yk@y@yx]pb|jEkx];hIin[JvxrVeb|z>k@vxz@];vx{~¬[;h}@binv>£[JY_o]`hIin[Jv°{~[EvBh^]iuv@b|zinv ]1rIb|jEk@bVvx{~bU[`4r«hbVYbV[$in_o]`\Jb|r_o]`¥Jb|r5h}>iRr_abVh}@[@z_a[JYbY[Jg@kxrIhh}x];vakxr¹inv@\]1rIbV jJk@bVvx{~b[;8_a[Jv@[@{~k>R];Yiu_];\Jb|rVeb|]8z@];h^]}x]|qEbgxbVbVvkxrIb|zh[$hb|rIhh}@by@Y[Jy2[BrIb|z _abVh}@[@zlx];vxzh+}@b$Yb|rIk>nh^r[Jg@h~]inv@b|z³];Yb1jEk>inhb1\E[B[@z ¼½>¾8¿IÀ#Á;ÂEé];_abVY^]é]ing@Y^]`hIin[Jv8l>ÄJhbVYbV[;qEisrtiu[Ev8leb|{~[EvxrIhYkx{~hIin[Jv8l>ÄJb~£{V]ing@Y^];hIin[Jv
Unite´ de recherche INRIA Sophia-Antipolis 2004 route des Lucioles, BP 93, 06902 SOPHIA-ANTIPOLIS Cedex (France) Te´le´phone : (33) 93 65 77 77 – Te´le´copie : (33) 93 65 77 65
"!# %$&'( )* ,+-. */
0
132564 798: 6 ; =?79@3AB
'( ' $
3U I3 0102 ² $ 02% U02"!$#p31 1$$&J " +$ 3$% ®$#p31 %1024315& $#p31 ')½@( +*-,.½§ ( Ã%bVh];Yh«is{+nbz0b|/ {~YIinhk@v@bp_1bV/ h}@[@z>b$yx[Ek@YR]a{V]ing@Y^];hIin[Jv ];k@h[E_o];hIiRjEk@b z>b|r1{V];_2bV/ Y^]Jrz3 k@v rI¬@rIh5bV4 _ab#r«h6bV/ Y7bV/ [Br{~[Jy>iRjEk@bJlxbVhR] Yb|{~[JvxrIhY+kx{~hIin[Jv _2bV/ hYIiRjJk@b¡8] 4 k@v1]J{~hbVk@Y5z39b|/ {+}@b~uby@Y5b|4 r~¢]4 yx]`YhIinY z3 k@v1_a[Jk@qEbV_abVvBh4invx{~[Jv@v>k¨z>b{~brI¬)r«h:bV4 _abJl~bVv rIk@y@yx[Br+];vBhjEk@bnb|r{~[>[JY^z>[Ev@v;bV/ b|r8z>kpyx[;inv>hy@YIinvx{+inyx]z>b{}x]JjEk@b%{V];_2bV/ Y^]rI[Jv>h{~[Jv@ vBk@b|r¡t{~bVhh+b}B¬>yx[Eh}8bV/ _abVvBh~]inYb;¢^·8]{~[Jv>hY^]inv>hb>bV/ y>iuy2[;R]inYbb|rIh1z3 ];g2[JY^z°[JY+_#k>?bV/ b1yx[Ek@Y-z>bVkA@ iu_];\Jb|r v@[Jv@{V]ing@Y6bV/ b|rV2·b1y@Y[Eg>=bV4 _ab$z>bVqJinbVv>h-{~b~nk>i z>bB3 b|rIh«iu_];hIin[Jv³z>b|rinvx{~[Jv@v>k@b|ry2[Jk@Y ub|r+jJk@b~nb|rpB3Cb|/ {V]`YhUz>bR]{~[Jv>hY^]inv>hbDbV/ y>inyx[`s]iuY+bJl8jJkx];v>hIi¯b/ yx];Ynb|r$zEiRrIh^]`vx{~b|r1z>b|r yx[;inv>h^r] 4 nbVk@Y^riu\Ev@b|rDbV/ y>inyx[`s]iuY+b|r#{~[EYYb|rIy2[Jvxz@];v>hb|rVl8b|r«h1_inv>in_o]%·3 iuv>inhIiR]iRr]; hIin[Jvaz>b|rinvx{~[Jv@v>k@b|rb|rIh4£[Jvxz0bV/ b%rIk@Y5nb|r4hY^]q]`kA# @ z>b®¶]¬>gx];v@¥l·k@[Jv@\bVh ¸2];k@\JbVY^]Jr rIk@YB3 ];k@h[E{V]iug@Y~];hIin[Jv z3 k@v@b#{V]`_1bV/ Y^]pbVv_a[Ek@qJbV_abVv>h|lxjEk>iyx]JrrIbVv>hyx];Y®s]Y7b|/ rI[J uk@h«iu[Ev°z3 k@vbVvxrIbV_g>nb$z39b|/ jEkx];hIin[JvxrzEinhb|rz>b®»Y+k@y@yx]@FE {V];kxrIbz>bR]Yb|z>[Jvxz@];vx{~b z>b$B3 inv>[EY_o];hIin[Jv³z@]`vxrk@v@brb|/ jEk@bVvx{~b1z3 in_o];\Jb|rrIh7bV/ Y7bV/ [Br{~[Ey>isjEk@bJl@{~bVhhb$_2bV/ h}@[@z>b z>[Jv@v@boz>b|rY7b|/ rIk>nh^];h^ry>nkxrY[Jg@kxrIhb|rjEk@b1B3 k@h«iiRr];h«iu[Ev z3 k@v@barAb|/ jJk@bVvx{~b_a[Ev@[){~k@ s]iuY+bJ G b|rz>[Jv@vbV/ b|rY7bV/ b~nb|r[Jv>hHbV/ h7bU/ k@h«iiRrAbV/ b|ryx[Jk@Yhb|rIhbVY®=3 ]n\J[JYIinh}@_aby@Y[Eyx[BrAbJ/ l bVhUz>b1g2[JvxrY7b|/ rIk>nh^];h^r[Jv>hbV/ h7b/ [Jg@h+bVvBkxrV I Áx|V¿FJ ½ ( é]ing@Y^];hIin[Jvz>bU{V];_2bV/ Y^]Jr'l5KiRr¹in[Jv³rIh7bV/ Y6bV/ [Br+{~[Jy>iRjJk@bJl;eb|{~[JvxrIh+Ykx{~hIin[Jv8l
é]ing@Y^];hIin[Jv];k@h[J_o]`hIiRjJk@b
"!# %$&'( )* ,+-. */
0
(®0
|Á8Â*8;sÁ
Á-Js½ , V@V½,º½ a  Á2V@nÁ @n é];_abVY~]#¶ [@z>b~ @ XY[Eg>ubV_ ÄJh^];hbV_abVv>h
Á JsxpÁ V |
Á-Js½ , ÁJ "! $# cxn ¸@[JY+_#k>R];h«iu[Ev% | cx &_ay>nbV_abVv>h^];h«iu[Evºz>bVh^]iR' r |
ss (J *)JxPÁ
/ 3
Á,+U.-½
x,½BV½@;pVÁ 5½
V=,º@V½@Â
102
½).=,º½ V J'½) *<Js|
43
1Á
FJ=*´nÁ
"
'( ' $
02 !30
&£nkxrIhY^]`hIin[Jv³[;h}@b$y@Y[Eg>ubV_
h[ag2b#rIh+kxzEiub|z
ª }@b$bVy>inyx[;R];Y$\EbV[J_abVhY¬ EUvb @x];_ay>nbh[rI}@[;¦ h}@bazEi°{~k>nht¬ inv n[){V]inµ~inv@\h}@b1y@YIinvx{+inyx] yx[;inv>h[`m]{V];_abVY^]m ] ~in_o];\Jb|r g ~inv>hYIinvxr¹iR{yx];Y^]`_abVhbVY^r2b|rIhIin_o]`hb|z ¦inh}³]a{+R]Jrrtis{V]Z{V]ing@Y^];h«iu[Ev³_abVh}@[@z &_o]`\Jb|ra¦®iuh+}L[;qJbVYIR]¬C[;$h}@b³y2[;invBh^r[`$iuv>hbVYb|r«hokxrIb|z [EY°rIb~£ {V]ing@Y^];hIin[Jv eb|{~[JvxrIh+Ykx{~hIin[Jv³Y+b|rIk>nh|¡t]> ¢ ]J{+¥y@Y[JwIb|{~hIin[Jv[Jv h}@bnb~hin_o]`\Jb ]; h 5¡g¢ XY[Ewtb|{~hIin[Jv [Jv ]y>R];v@byxbVYy2bVvxzEis{~k>£];Yh[h}@b$in_o]`\Jb y>R];v@ba¡h+[JyqJinbV¦$¢ ÄJhbVYbV[J\EY^];_²[;h}@b$_abVhYIiR{Yb|{~[JvxrIhY+kx{~hIin[Jv[EYU{~Y[Brr«£bV¬Jbkxrtiu[Ev
c
| | | J
02 ³ #+%0
eb|rIk>nh^r[;h}@b1rIb~{V]ing@Y^]`hIin[Jv | YY+[JY^r5invYb|{~[EvxrIhYkx{~hIin[Jv¡IzEisr«h^];vx{~b|rinv_iiu_bVhbVY^r~¢qJbVY^rIkxry2[J r¹inhIin[Jvxr[; h+}@b$y@YIinvx{+iuyx] y2[;invBh^r @
"!# %$&'( )* ,+-. */
pa315$ n>l>;Bl>|@h}x];hrIhbVYbV[>r{~[Jy>iR{{~k@b|r5y>s]¬];vpiu_yx[JYh~];vBh5Y[;nb
&h5iRr5¦b~xY+b|{~[J\Jv>inµVb|z
iuvk@vxz>bVY^r«h^];vxzEinv@\oh+}@b#;BG bVv>qJinY[Ev@_abVvBh$rIk@YY+[Jk@vxz>b|z kxrU];vxzy@Y+[qEiRz>ba]Y[Jg@kxrIh ¦®]¬£[JY1;B G Yb|{~[Jvxr«hYkx{~hIin[Jv8"&¹v [JY^z>bVY£[JYh}@bo;BG Yb|{~[JvxrIhY+kx{~hIin[Jv h[gxbay2[BrI r¹ing>ubJl2[Jv@b$v@bVb|z h[¥Bv@[;¦*h}@b$Yb~R];hIin[Jvxr«}>iuy gxbVhI¦bVbVvh}@b1;BG ¦[JYIRz{~[B[EY^zEinvx];hb|r ];vxzCh}@b~inY{~[JYYb|r«yx[JvxzEinv@\C;B Gin_o];\Jb³{~[>[JY^zEinvx];hb|r¨£[JYab|]J{+}L{V];_abVY^]ml%];vxz§h}@b Yb~R];hIinqJb\JbV[J_abVh+Y¬gxbVhI¦bVbVvh}@bht¦[o{V]`_abVY^]JrVª}>iRriRrh}@by@k@Yyx[>rIb[;©{V];_abVY^] {V]iug@Y~];hIin[Jv8 E/¦b|]nh}C[`®¦[JY+¥ [JvC{V];_abVY^] {V]iug@Y~];hIin[JvL}x]Er1g2bVbVvL{V];YYIinb|z§[Jk@h gB¬ Yb|rIb|];Y^{+}@bVY^rb~inh}@bVYinv XZ}@[Jh[J\EY^];_a_abVh+Y ¬ nJl 5[JYinvCÃ%[E_ay@k@hbVY KiRr¹in[Jv ];vxz e[Jgx[EhIiR{V r >l8Jl5|@lx @lxJ@l2J@lxJ %¡tr«bV b > £[JY$]Yb|{~bVv>h$YbVqEiubV¦¢^xª}@bkxrIkx] _abVh}@[@z[;©{V]ing@Y^];h«iu[Ev³iRrh[{~[J_ay@k@hb{V];_abVY^]Jryx];Y^]`_abVhbVY^r£Y[J_[Jv@b1[JY_a[JY+b iu_];\Jb|r[;Z]`v[Jg@wIb|{~h[; [JY^zEinvx];hb|r%,) +*"!-, .' ];Yb$Yb~R];hb|z gB¬ 0111 * 3+444 11110 # 3+4444 / 2 , 5 76 1 & 4 ! 25 ¦}@bVYb / iRr];v];Yg>inhY^]`Y¬ar{V]nbJlx];vxz86 isr]1 c1_];hYIi @8l){V]nb|z h}@bUy2bVY^rIy2b|{~hIinqJb y@Y[JwIb|{~hIin[Jv_o];hY«i @8GbVv@[JhIinv@\h}@b}@[J_a[J\JbVv@bV[Ekxr${~[B[JY~zEiuvx]`hb|r[;®]qEb|{~h[JY +"!-#!:9;9iuvxz h}>iRr_a[@z>b~xiRrh}x]`h "9? $ xh[B[`sr5[`´y@Y+[JwIb|{~hIinqJb\JbV[J_abVhY+¬Jl;¦}>iR{}iRrZbV_abVY\;inv@\ ]Jr$];v ];hhY~]J{~hIinqJb$£Y^];_bV¦©[EY¥a£[JY1{~[J_y@k@hbVY$qJiRr¹in[Jv n| 2Linh}h}@brIh^];hbp[;%h}@b ];Yh[;8h}@bh+b|{}@v@[;n[J\E¬JlB{V];_abVY~]1zEiRrIh[JY+hIin[JviRrYb|]JrI[Jvx]`g>u¬arI_] l>];vxzh}@by>inv@}@[;nb _a[)z>b~4isrh}>kxr]\J[>[@z];y@y@Y [ @Bin_o]`hIin[Jv8 ª}@b$_o]`hYIi @ {V];vgxbz>b|{~[J_ayx[>rIb|z³]Jr
E6
6F
' G!
¦}@bVYbU iRr%]p _o];hY«i @8l;_o];y@y>inv@\ph}@bv@[JY_]iuµVb|z¨in_o];\Jb{~[>[JY^zEinvx];hb|rZh[$h}@b YbVhIinvx]5iu_];\Jb#{~[>[JY^zEinvx];h+b|rVl8¡C' +;¢iRrh}@bzEiRrIy>R]J{~bV_abVv>ha¡Y[Eh^];hIin[Jv³];vxz hY^];vxrts]` hIin[Jv¢Y[E_ h}@b¦[JYIRz {~[>[JY^zEinvx];hb³rI¬@rIhbV_ h[³h+}@b°{V];_bVY^]³{~[>[JY^zEinvx];hbrI¬@rIhbV_³ ª}@b1_a[>rIh\JbVv@bVY^]_];hYIi @³ {V];v gxb$¦Y«iuh+hbVvº]Jr {~[Eh ¡^;¢ ¦}@bVYb
H!
0111 JLKMN KMON P I 2 JSWYRUX Z?TUV [
*Q 3 444 ,\Q 5 !
'( ' $
K
iRrh}@b£[@{V]ubVv@\Eh}³[;h}@b1{V];_bVY^]@l ]`vxz ];Yb h}@b}@[JYIinµV[Jv>h^]-]`vxzLqEbVYhIiR{V]-r{V]ub ]J{~h[JY~rVl¦}@[Br«binvBqEbVY^rIb|r {+}x];Y^]J{~hbVY«iuµVb$h+}@b1r¹inµVb#[`h}@b1y>i @@b~8inv³h}@bp¦[JYIRz{~[B[EY^zEinvx];hb$k@v>inh|l ];vxz ];Y+bah}@b{~[B[JY~zEiuvx]`hb|r$[;h}@by@YIinvx{+inyx]yx[;inv>h[;h}@bo{V];_abVY~]@li bJnl h}@binv>hbVY^rIb|{~hIin[Jv gxbVhI¦bVbVvh}@b1[Ey@hIiR{V] ] @>iRr];vxzh+}@biu_];\Jb$y>R];v@bJl];vxz ¨isrh}@ba];v@\;nbag2bVht¦bVbVvh}@bYbVhIinvx]©] @)b|rV8ª®}>isryx];Y^];_bVhbVYiRr-iuv>hY[@z>kx{~b|zh[ ]J{V{~[Jk@v>h£[JYh}@b ]E{~h%h}x];hh}@by>i @@b~\JY«isz_o]¬av@[Jhgxb$b @x]J{~hIn¬a[JY+h}@[J\J[Jvx] &v y@Y^]J{~h«is{~binhiRrqJbVY+¬o{+n[BrIb1h+ [ @
MON * Q
M
,Q
P
£Y[J_ M N ];vxz M vah+}@b£[;u[;¦inv@\xlE¦b%kxrIbh}@b K [;n[;¦inv@\ov@[Eh^];hIin[Jvxr N JLK M N ];vxz LJ K M @ b$h+}Bkxr}x]qJb¯xqJbinv>hYIinvxr¹iR{ yx];Y^];_abVh+bVY^r£[JYb|]J{}³{V]`_abVY^] N l l*%QVl,\Q];vxz >P % % E-r4iRr{+nb|];Y|l;¦b{V];v@v@[Jh%rIbVyx]`Y^];hb
&
ª }@bUy@Y+[Jg>nbV_²iRrinkxrIhY^]`hb|zinv³¸4iu\2VJ@ª}@bnb~£h];vxzaY«iu\E}Bhin_o];\Jb|r];hhIin_ab ©];Y+b Yb|rIyxb|{~h«iuqEb~u¬Cz>bVv@[Jhb|z§g>¬ & ]`vxz&|l5];vxz h+}@[BrIbo];hphIin_ab a];Y+boz>bVv@[Jhb|z gB¬ & ];vxz &; E yx[;inv>h ,invh}@bpiu_];\Jby>R];v@b &5iRrv@[Jhb|z³]Er ,l];vxz]yx[`iuv>h inv ;rIyx]J{~bab @@y@Yb|rr«b|zinv§h}@bo{~[>[JY^zEinvx];hbor«¬)rIh+bV_ ];hh^]J{+}@b|zh[h+}@ b ££h}§{V];_abVY^]iRr v@[Jhb|z]Jr Iª®}@b®r«b|{~[JvxzrIk@gxr{~YIiny@h|l`i8];v>¬Jl|¦i@invxzEiR{V];hbh}@binvxz>b @a[;xh}@byx[`iuv>h iuvL{~[Evxr¹iRz>bVY^];hIin[Jv8ª}>kxr ,!pisrph}@bin_o];\Jbyx[`iuv>hpinv &"[;h}@$ b #>£h} `B G²yx[;inv>h|l ];vxz !iRrh}@% b #B£h+}º;B G yx[;inv>hUb @@y@Yb|rrIb|z invh}@ba{~[B[EY^zEinvx];hb#r«¬)rIh+bV_ ];h+h^]J{+}@b|z h[ah}@$ b £h}³{V];_abVY^]m inh}@[Jk@hn[Brr[;$\JbVv@bVY^]inhI¬Jl%¦b{}@[>[BrIb³]Jrh}@b¦[JYIRz {~[B[JY~zEiuvx]`hbºrI¬@rIhbV_ h}@b1{~[B[EY^zEinvx];hb#r«¬)rIh+bV_ ];hh^]E{}@b|zh[h}@bnb~hU{V]`_abVY^]a];h x·´bVh19¡ ' ¢gxb$h}@b zEisr«y>s]E{~bV_abVvBhgxbVhI¦bVbVvh}@bnb~£h%];vxzpYIin\J}Bh{V];_abVY~]Jr[;xh}@brIh+bVYbV[UY«iu\2·bVhC¡ ''& ( &n¢ gxbh}@b#zEiRrIy>R]J{~bV_bVvBh$[;%h+}@b#rIhbVY+bV[aYIin\og2bVhI¦©bVbVv U];vxz ¦inh}Yb|r«yxb|{~hh[ah}@b ub~£h©{V]`_abVY^]@;·bVhC¡ '') +( )¢8g2bh}@bUzEiRrIy>R]J{~bV_abVv>h[;h+}@b®r«hbVYbV[$YIin\$gxbVhI¦bVbVv ];vxz
¦inh}Yb|rIy2b|{~hh[ph}@bUY«iu\E}Bh{V];_abVY~]@B·bVh & ];vxz ) gxb$h}@binvBhY«iuvxrtis{_o];hY«is{~b|r [;%h}@b$nb~£h-]`vxz³Y«iu\E}Bh${V];_abVY^]ErVl@Yb|rIy2b|{~hIinqJb~n¬J8ª}@by@Y[Jg>nbV_±{V];vv@[¦ gxbarIh^];h+b|z ]Jr
S
!
W ! W D!