• Aucun résultat trouvé

Characterizing the Uncertainty of the Fundamental Matrix

N/A
N/A
Protected

Academic year: 2021

Partager "Characterizing the Uncertainty of the Fundamental Matrix"

Copied!
7
0
0

Texte intégral

(1)Characterizing the Uncertainty of the Fundamental Matrix Olivier Faugeras, Zhengyou Zhang, Cyril Zeller, Gabriella Csurka. To cite this version: Olivier Faugeras, Zhengyou Zhang, Cyril Zeller, Gabriella Csurka. Characterizing the Uncertainty of the Fundamental Matrix. RR-2560, INRIA. 1995. �inria-00074121�. HAL Id: inria-00074121 https://hal.inria.fr/inria-00074121 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 destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés..

(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Characterizing the Uncertainty of the Fundamental Matrix Gabriella CSURKA Cyril ZELLER Zhengyou ZHANG Olivier FAUGERAS. N˚ 2560 Juin 1995. PROGRAMME 4. ISSN 0249-6399. apport de recherche.

(3)

(4)  

(5)  

(6) 

(7)    !" 

(8) #%$& '(

(9) )* + , -  .0/ 1%2436587:9<;=;>2@?*ACBEDGF%H ?IJ587K;MLNOONPD LQ69SR6T4IUWVXLYZH%[%1 \*;K7=]^7:9_5Z`^HGB%1%ND*HA abdc<efbhgfii*jkPlnmc<oWcqpdrts_uvjxwfrti*g<efj)j8py_rtz>rtcf{ abdcx|>j8pmc<ocxphySrtz mg<}v}cfb>p~jMbhj0€‚vj0bh€‚jƒ{P„† <‡xˆ<‰Šln‹fu†rt{ŒŽ<ŽS‡Šln xŽ}vgfe<j0z ‘ ’.“8”0•x–v—q”f˜G™ †rtz }gf}j0b^~vj0gxštzC› rœp‚p‚jg<{vgqšœ†zžrtz.cqŸ4p vj¡u{v€0j0bžp gxrt{<p¢McqŸvp vjŸKuv{v~gfij0{fp gxš<i*gqp‚b>r¤£W¥ ™ vj)ovg<zžrt€)rt~vj0grtzp‚c €0c<i*}vuSp‚jp‚jŸKu{v~vg<i*j0{<p‚gqšvi*gqp‚b>r¤£Ggf{~*rœp z uv{v€0j0bžp gxrt{<p¢ rt{Gp‚jŠzdg<i*jpdrti*jx¥ ¦§jPzhvgqšœšCzhvc0›¨p¢›c~†rœ©Wj0bhj0{fp ij8p‚vc†~vz8¥ ™ vj«ª4bdz>pcf{jŠrtzgzžp gxphrtzžpdrt€0gqš^gf}v}bdc<gf€ W¥¬Šzrt{%gqšœš.zžp gxphrtzžphrt€0gxši*j8p‚c†~vz p vj«}vbdj0€8rtz>rtcf{­cqŸ.p‚vj bhj0zdu†šœp z.~vj0}j0{v~vz^cf{p vj{_uvi oWj0bJcqŸWg<{vgqšœ†®0j0~Pzhgfi}†štj0z¥C™ †rtz ij0gf{z^p‚vgqpJ›j €0g<{ gqšœ›g0†z+rti*}vbhc0yfj c<uvb bdj0zhu†šœp‚zrœŸ.›j)rt{v€0bdj0g<zdjp‚vj){Sui oWj0bcxŸ.zhgfi}†štj0zovu†p p †rtz}vbdc†€0j0zhz rtz y<j0bžGpdrtijŠ€0c<{vzduvirt{ve†¥ ¦j«}vbdc<}cfzhjg!i uv€ Zz>rti*}†štj0bij8p‚vc†~ ›Srt€‚­eqrœyfj0z bdj0zhu†šœp‚z›†rt€ ­gfbhjŠ€8štc<zdjp‚c p vjŠbhj0zdu†šœp z¡cqŸ p vjMz>p‚gqpdrtz>pdrt€0gqš4i*j8p vc†~vz¥ ¬p p‚vjj0{v~cqŸ}vg<}Wj0b)›jZzhvgxšœšzdvc›¯zhcfij!j‚£}Wj0b>rti*j0{<p‚gqš.bdj0zhu†šœp‚zc<o†p‚gqrt{vj0~› rœp z>†{fp vj8pdrt€ g<{v~­bdj0gqšC~gxp g¥ °%±<²^³ž´ µv•x¶J“x˜¸·}†rt}Wcqštg<be<j0c<i*j8p‚b>xw_¹vuv{v~gfij0{fp gxši*gqp‚b>r¤£WwSºŠ{v€0j0b>p‚gqrt{<p¢<w»c0yxg<bžrtg<{v€0j«i*gqp‚bžr¤£ ¼œ½!¿¢¾ ÀÂÁà ¿¾ WÄ ÅtÀÂƢǂÈ. 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.

(10)   

(11)     &

(12) ,d

(13)  0 $&

(14) &

(15) ,  'n . 

(16) <!& 'n

(17) Š  ,

(18) ±_ “

(19) ± ˜%»j }vg<}†rtj0b}bdc<}Wc<zdj«uv{vj g<{vgqšœzhj ~vjš  rt{v€0j0b>pdrœp‚u~vj«šœrj0 jgZ štgGi*gqp‚b>rt€0j«ŸKc<{v~vg<i*j0{<p‚gqštjx¥  rt~j0 j~vj!ovg<zdj j0z>pM~vj!€0gqšt€0u†štj0bštgEigxp bžrt€0j«ŸKc<{v~vg<i*j0{<p gxštj«j8pPzhcf{Ert{v€0j0bžphrœp‚uv~vjj0{ i j0 i*j«p j0i*}vz8¥ )cfuvzicf{<p‚bhcf{z¡~vj0u_£­ i j8 p vc†~vj0z+~†rœ© j0 bhj0{fp j0z¥ Jg­}vbdj0ir j0 bdjj0zžpuv{vj  i j8 p‚vc†~vj«zžp gxphrtzžphrtsSujx¥^· šœštj{vj!~vc<{v{vj ~vc<{v€ ~vjb j0 zduSšœp‚gqp‚zŠ}b j0 €8rtzŠs_u  g<u }vb>r¤£G~vjš  gf{vgqšœ†zdj)~  uv{Gefbhgf{v~{vc<i ovbdj ~vj)b j0 gqšœrtzdgqpdrtc<{vzwf€0jŠs_u†rWj0{ŸKgqrœp uv{vji j8 p‚vc†~j€0

(20) c u† p j0uvzdjj0{ p j0i*}vz8¥ Jg%zdj0€0c<{v~vj j0z>pPuv{j* i j8 p vc†~vjg<{vgxšœ_phrtsSujx¥^· šœštjGj0zžp«p‚b j0 zM}j0u €0

(21) c u† p j0uvzdjj0{ p‚j0i}vzŠigxrtz«gf}  }vbhc£Srti*gqpdrœy<jx¥ Šc<uvz icf{<p‚bhcf{zJ~vgf{z €0j }vg<}†rtj0bw g }vg<bžphrtb.~  rtigfe<j0z.zž†{<p‚ j8 pdrts_uvj0z €0c<i*ij¡!~  rti*g<efj0z b j0 j8šœštj0zws_u  j8šœštj ~vc<{v{vj€0j0}j0{v~vg<{fp~j0zb j0 zhu†šœp‚gqp‚z zh#u "*zdg<i*ij0{<p}bdc†€ vj0z ~vj €0j0uS£E~vjštgG i j8 p‚c†~vj z>p‚gqpdrtzžphrts_uvjŠ}cfu$b j8 p bdj)u†pdrœšœrtz j0 jŠ~vg<{vzštg!}vbhgxphrts_uvjx¥ %ƒµ”“8³d'— &±Z )˜ (j0 cf i j8 p‚b>rt*j j0 }†rt}cxštgqrtbdjx,w +gqp‚b>rt€0j Ÿ=cf{~vgfij0{<p‚gqštjx.w -ž{v€0j0b>pdrœp‚u~vjx w +gqp‚b>rt€0j¡~j€0cyxgfb>rtgf{€0j.

(22) Œ.  

(23)   

(24) 

(25)    

(26) "!$#%& (')

(27)  *(+,-. /021. *

(28) )*  ¶ J—q(” 8Kµ 6 35476 ”0•xµ^

(29) ± > 6 ¶J– ± 6 ”– & –†”0•.8? 9;:=<  ± > 6 ¶J– ± 6 ”– & –†”0•.8? @ A µ -B ^(” 8 6C ” < D E ¥tŒ ™ jšœrt{vj0g<b €0bžrœp j0bžrtc<{GF"FHFHF"FHF"FHFHF"FHF"FHFHF"F=FHF"FHF"FHFHF"FHF"FHFHF"FHF"FHFHF"F { FHF"FHF"FHFHF"FHF"FHFHF"F=FHF"FHF"FHFHF"FHF"FHFHF"FHF"FHFHF"F E ¥ ™ jŠ{vc<{†šœrt{vj0g<b€0bžrœp‚j0b>rtcfD £ FHF"FHF"FHFHF"FHF"FHFHF"F E ¥ _¥tŒ ¬ƒ}vg<bdg<i*j8p‚j0b>rt®0gxp>rtcf{ cqŸJp‚vj)ŸKu{v~vg<i*j0{<p‚gqšvi*gqp‚b>r¤I { FHFHF"FHF"FHFHF"FHF"FHFHF"F=FHF"FHF"FHFHF"FHF"FHFHF"FHF"FHFHF"F E ¥ _¥ ™ vjŠ€0b>rœp‚j0b>rtcfJ ~ FHFHF"FHF"FHFHF"F=FHF"FHF"FHFHF"FHF"FHFHF"FHF"FHFHF"F E ¥ E ™ jŠc<u†pdšœrtj0bhz bdjž|žj0€8phrtcf{i*j8p vc†K ± –†”0• 8/? LM:=< ±—fµ Nv–S.• 8K– 6 —f k¥tŒ »c<i*}u†pdrt{ve«p vjŠ€0c0yqgfb>rtgf{v€0j)i*gqp‚b>r¤P£ F"FHF"FHFHF"F=FHF"FHF"FHFHF"FHF"FHFHF"FHF"FHFHF"F k†¥tŒq¥tŒ ™ vjŠz>p‚gqpdrtz>pdrt€0gqšWi*j8p vc†R~ FHFHF"FHF"FHFHF"F=FHF"FHF"FHFHF"FHF"FHFHF"FHF"FHFHF"F k†¥tŒq¥ ™ vjŠg<{vgqšœ_pdrt€0gqšCij8p‚vc†G~ FHFHF"FHF"FHFHF"F=FHF"FHF"FHFHF"FHF"FHFHF"FHF"FHFHF"F k¥ ™ jŠ<†}Wj0b :j8šœšœrt}vzhcxrt~­cqŸ.uv{v€0j0b>p‚gqrt{fp¢ FHF"FHFHF"F=FHF"FHF"FHFHF"FHF"FHFHF"FHF"FHFHF"F S;T ?UB¡±<.• 8 ± 6 ”–†(” 8=µ 6 ‡S¥tŒ )cxp gxphrtcf{Xz F"FHFHF"FHF"FHFHF"FHF"FHFHF"FHF"FHFHF"F=FHF"FHF"FHFHF"FHF"FHFHF"FHF"FHFHF"F ‡S¥ Yf{<p vj8pdrt€~gxp Z g F"FHF"FHFHF"FHF"FHFHF"FHF"FHFHF"F=FHF"FHF"FHFHF"FHF"FHFHF"FHF"FHFHF"F ‡_¥ _¥tŒ Y_p‚gqpdrtz>pdrt€0gqš +j8p vc†[ ~ FHF"FHFHF"FHF"FHFHF"F=FHF"FHF"FHFHF"FHF"FHFHF"FHF"FHFHF"F ‡_¥ _¥ ¬Š{vgqšœ_pdrt€0gqš +j8p vc†X~ FHF"FHFHF"FHF"FHFHF"F=FHF"FHF"FHFHF"FHF"FHFHF"FHF"FHFHF"F ‡_¥ _¥ E mj0gqšW~vgqp‚\ g FHF"FHFHF"FHF"FHFHF"FHF"FHFHF"F=FHF"FHF"FHFHF"FHF"FHFHF"FHF"FHFHF"F O A µ 6 — & J]“ 8Kµ 6. ``badc]e f7g7h. 9 9 @. k k k ˆ ˆ O Q. Q Q. Ž 3WV Œ<Œ Œ Œk ŒŽ qk 9_^.

(30) . 7

(31) +/+.    . .0/+

(32)    $

(33) . #  .  + ]*

(34) =!. .  . * 6­&$ ! ™ vj z>p‚uv~S cqŸ p vj«ŸKuv{~vgfij0{<p‚gqšCigxp bžr¤£ZrtzŠy<j0bž%bdj0€0j0{<prt{ €0cfi}vu†p‚j0by_rtzžrtc<{W¥!-¢p)›g<zrt{<p bdc†~vuv€0j0~ rt{"!$#%%'& 9)( gfzg efj0{j0bdgqšœrtzdgqpdrtc<{ cxŸp‚vj j0zhzdj0{<pdrtgqš4igxp bžr¤£~vj0zd€0b>rtoWj0~rt{"!*%+ ^3,( p‚c«uv{v€0gqšœrtovbdgqp‚j0~ rtigfe<j0z¥™ vj)rti*}cfb>p‚g<{v€0jcqŸ.p‚vj)ŸKu{v~vg<i*j0{<p‚gqš4i*gqp‚b>r¤£­oj0€0cfij0z¡j8y_rt~vj0{<p rt{­p vjM€0g<zdj«cqŸ.p¢›cp‚vj uv{€0gxšœrtovbhgxp j0~ €0gfij0bdg<z,¥ -ž{p‚†rtz€0gfzhj p‚j ŸKuv{~vgfij0{<p‚gqšigxp bžr¤£Prtzp‚vj -xj8*€0c<{v€0j0}†p0wxg<zrœp€0c<{<p‚gqrt{vz gqšœš_p‚vj+efj0c<i*j8p bžrt€0gqšrt{†ŸKc<bdigxphrtcf{)bdj8štgqpdrt{vep¢›c~†rœ©j0bdj0{<pJrti*g<e<j0z.¥ -¢p.~j8p‚j0bdirt{vj0z4g<{v~«rtzJ~vj8p j0bdi!rt{j0~ o<p vj}Wc<zžrœphrtcf{vz)cxŸ+p‚vjp¢›c­j0}†rt}cxštj0zg<{v~%p vj j0}†rt}cxštg<b p bdg<{vzžŸ=cfbhi*gqpdrtc<{Zigf}v}Srt{ve*g<{j0}†rt}cxštg<b šœrt{vj cxŸ)p‚vj­ª4bhzžp!rtigfe<j­p‚c rœp‚z€0cfbhbdj0zh}Wc<{v~†rt{ve%j0}Srt}WcqštgfbPšœrt{vj%rt{ p vj%zdj0€0c<{v~§rti*g<e<jx¥ -¢pZ€0g<{ oWj €0c<i*}vuSp‚j0~ŸKbhcfi g€0j0bžp‚gqrt{!{_uvi oj0b cqŸC}cqrt{fp€0cfbhbdj0zh}Wc<{v~vj0{v€0j0z^c<o†p‚gqrt{vj0~ Ÿ=bdc<i p vj }vgqrtb¡cqŸ^rti*g<efj0z uvz>rt{veGzdc<i*j €0c<bdbdj8štgqpdrtc<{­ovg<zdj0~gqštefc<bžrœp vi*z8wrt{v~vj0}j0{v~vj0{<pdšœGcxŸgf{<cqp‚vj0b -S{c0› štj0~ve<j gfocfuSpp‚vj ›cfb>št~W¥ -ž{ p vj ª4bhzžp}vg<bžpMcqŸp‚†rtz«}vg<}Wj0b›jG~vj0zh€0bžrtojPzhcfij!i*j8p vc†~vzp c%€0cfi}vu†p‚j«p vj ŸKu{v~vg<i*j0{<p‚gqš igxp bžr¤£GŸ=bdc<i g*}vgqrtbcqŸ.rti*g<e<j0z gfzhzduvirt{vep‚vgqpzdc<i*j)rti*g<e<jM}cxrt{<p€0c<bdbhj0zd}cf{v~j0{v€8j0z g<bd.j -_{vc0›{W¥ ™ vjij8p‚vc†~vz gfbhj¡{cxp.~vj8p gxrœštj0~icfbhj.p‚gf{{vj0€0j0zhzdg<bžŠoj0€0gfuzdjJp vj8MvgyfjoWj0j0{«gqštbdj0g<~†M}vbhj0zdj0{<p‚j0~ rt{­p vjšœrœp j0bdgqp‚uvbh0j /!$% Jµ & 91 +32 A & 94165 &=7“ & 91 +%–†,• & S(98 ¥ -ž{ p‚jzhj0€0cf{~P}gfb>p.p¢›c«~†rœ©j0bdj0{<p i*j8p vc†~vz^g<bdj}vbdj0zhj0{fp j0~«p‚cŠ€0c<i*}u†p‚j6p‚vj€0c0yxg<bžrtg<{v€0ji*gqp‚b>r¤£ cqŸJp‚vjŸKuv{~vgfij0{<p‚gqšvi*gqp‚bžr¤£¥S™ vjMz>p‚g<o†rœšœrœp¢ZcqŸJp‚jŸKu{v~vg<i*j0{<p‚gqšvi*gqp‚b>r¤£Gvg<zgqštbdj0g<~†*oj0j0{Gzžp‚u#~  rtj0~@o<; :P¥ ™Š¥ Juvc<{v<e /=!*% Jµ & 9>( w !*%#& L?(98 w6ovu†p p‚j*€0c<bdbdj0zh}Wc<{v~†rt{eZ€0c0yqgfb>rtgf{v€0ji*gqp‚b>r¤£ ›g<z!{vcqp g<€0€0uvbdgqp‚j8šœj0zžphrti*gqp‚j0~Ww rt{G}vg<bžpdrt€0uSštgfb6p‚vjyxg<bžrtg<{v€0j0z6›j0bhj p‚c†c«i uv€ *c0y<j0bdj0zžphrti*gqp‚j0~p cPoj uvzhj0~!rt{ Ÿ=uvbžp vj0b€0cfi}vu†p‚gqpdrtc<{vz8¥ ¦j«zdvgqšœšJzdvc› rt{Gp‚jŠštg<zžp }vg<bžp cqŸJp‚†rtz bhj0}Wc<bžpp‚vgqp uvzžrt{e p‚vjp¢›ci*j8p‚c†~vz ›j«}vbdc<}Wc<zdj›j c<o†p‚gqrt{Gg!i uv€ Eoj8pdp j0b¡j0z>pdrtigxphrtcf{cxŸJp vjŠuv{€0j0bžp‚gqrt{<p¢*cqŸJp‚vjŸKuv{v~gfij0{fp gxš i*gqp‚bžr¤£¥ @ A 

(35) <$& 'n

(36) Š  ,@'( .0/ ™ vj€0gfij0bdg!ic†~vj8šJuvzdj0~%rtz)p‚j €8štgfzhzžrt€0gqCš B %+

(37) ')]&

(38) + ¥ -¢Ÿ¡p vj cfo†|>j0€8pŠzd}vg<€0jrtz«€0cf{vz>rt~vj0bdj0~­p c oj«p‚vj E :~†rtij0{vzžrtc<{vgqšC·uv€8šœrt~vj0g<{­zd}vg<€0 j DFEj0i oj0~v~vj0~rt{%p‚vjuvzhuvgqšC›g0%rt{%p vj E Â~†rtij0{vzžrtc<{vgqš }vbhcx|žj0€8phrœyfjzd}gf€0Hj G E g<{v~%p vj)rti*g<efjzd}gf€0j«p‚j Â~†rti*j0{vz>rtcf{gxš^·¡u€8šœrt~vj0gf{%zh}vg<€0j DJI j0i oWj0~v~j0~ rt{p‚vj uvzhuvgqšv›g!rt{!p vj Â~†rtij0{vzžrtc<{vgqš}bdcx|>j0€8pdrœyfj zh}vg<€0j G I w_p vj €0g<i*j0bdgrtz p vj0{!~vj0zh€0bžrtoj0~ gfz+g + /

(39) ( B LK

(40)  

(41)  B>B + * *( ŸKbhcfM i G.E p c G I /Âzhj0Oj N ¹vg<uvŽ EQP 8 ¥ -¢ŸW›j~vj0{vcqp‚jo<S TXR p‚j EU k Âi*gqp‚b>r¤£ bhj0}vbdj0zhj0{fphrt{ve«p‚†rtz g<}v}†šœrt€0gqpdrtc<{Wwf›jvg0y<j V B]\ % W YX[Z T /dŒ 8 ^ _a` b . Tc. ›vj0bhj B ~vj0{vcqp‚j0zJp vjštgfz>p¡€0cqštuvi*{ cqŸ TR w%dXeN fhgjikg=lg0Œ Pnm bdj0}bdj0zdj0{<p zJgf{!c<o†|žj0€8p }cqrt{fpo w pX N q

(42) gsr4g0Œ Pnm w†rœp z }vbdcq|žj0€8pdrtc<{ut c<{<p‚cp vj rti*g<e<jgf{~vW­g {vc<{v®0j0bhcMzh€0gxštg<b¥f¦j«gfzhzduvij6w>x,yCz/{T 8 X E ¥ -¢Ÿ.›jMvgyfj«p¢›c€0gfij0bdg<zw_bhj0}vbdj0zhj0{<p‚j0~*o<eT R g<{v~eT.R | w4o<­j8šœrti!rt{vgqpdrt{e % ŸKbdc<ij0s_uvgxphrtcf{/dŒ w ›jcfo†p gxrt{p‚jŠbdj8štgqpdrtc<{GoWj8p¢›j0j0{*p vjŠ}bdcx|>j0€8pdrtc<{vz¡cqŸ o cf{<p‚cPp‚vjª4bdz>pg<{v~­zdj0€0c<{v~rti*g<efj V 8 W | | X}W?+ ~ ± | /¢ 8 €{`€ƒ‚.

(43)

Références

Documents relatifs

To test whether the vesicular pool of Atat1 promotes the acetyl- ation of -tubulin in MTs, we isolated subcellular fractions from newborn mouse cortices and then assessed

Néanmoins, la dualité des acides (Lewis et Bronsted) est un système dispendieux, dont le recyclage est une opération complexe et par conséquent difficilement applicable à

Cette mutation familiale du gène MME est une substitution d’une base guanine par une base adenine sur le chromosome 3q25.2, ce qui induit un remplacement d’un acide aminé cystéine

En ouvrant cette page avec Netscape composer, vous verrez que le cadre prévu pour accueillir le panoramique a une taille déterminée, choisie par les concepteurs des hyperpaysages

Chaque séance durera deux heures, mais dans la seconde, seule la première heure sera consacrée à l'expérimentation décrite ici ; durant la seconde, les élèves travailleront sur

A time-varying respiratory elastance model is developed with a negative elastic component (E demand ), to describe the driving pressure generated during a patient initiated

The aim of this study was to assess, in three experimental fields representative of the various topoclimatological zones of Luxembourg, the impact of timing of fungicide

Attention to a relation ontology [...] refocuses security discourses to better reflect and appreciate three forms of interconnection that are not sufficiently attended to