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(1)Shape-from-Shading and Viscosity Solutions Emmanuel Prados, Olivier Faugeras, Elisabeth Rouy. To cite this version: Emmanuel Prados, Olivier Faugeras, Elisabeth Rouy. Shape-from-Shading and Viscosity Solutions. RR-4638, INRIA. 2002. �inria-00071947�. HAL Id: inria-00071947 https://hal.inria.fr/inria-00071947 Submitted on 23 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. Shape-from-Shading and Viscosity Solutions Emmanuel Prados — Olivier Faugeras — Elisabeth Rouy. N° 4638 Novembre 2002. ISSN 0249-6399. ISRN INRIA/RR--4638--FR+ENG. THÈME 3. apport de recherche.

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(591) acb,h8jtqz} ©xhx~b,l&}Œjnm|h(~qshx{—ˆ²^

(592) qzac}kb,§s¥ #qsj0qz}:jnr}nmWˆwy>¥LrwlŽ©

(593) h(~b,l&}„j&m|hx~ºq¬j<yBjk^

(594) bclnbmM~b,lŽ}Œ^xrw©

(595) §~h±Ò j‡hxb,b~¹m|§z§jk^

(596) b ©(}Œ©xmw§Blkb,¥Lb,lnb,hxo,b}µQ †¥@o!rw©xln}kbwy8jk^xqz}_qzh8jklnr~

(597) ©xopjnqsrMh—>b,qzh

(598) {c–wb,lnˆ ¨ b,§z§¬uvjnmwlk{Mb!jkb<~y ¨ bŽmw~–q}Œb‡qzhMjnb,lnb}Œjkb<~ lnbmM~b,l&} jkrcl&m|‹

(599) q~§zˆ{wb,j_^xrw§~rw¥’acrM}ŒjŠr|¥@°8©

(600) r|jnb~lkb,¥Lb,lnb,hxo,b}µ Ô$j_§zmM}„j<yMqzhjn^

(601) qz}$}kbopjnqsrMh ¨ b†mjkjkb,ac‹j$jkrrjkr ¨ m|lnhcjk^

(602) b:lkb<mw~bl mw—(rM©j$}krwacb†~q

(603) o!©

(604) §sjkqzb}_mwhx~ ¨ b qz§z§s©x}Œjkl&mjnb‡jk^

(605) ba ¨ qsjk^²mco!§mw}n}Œqo,mw§>b,¼

(606) m|ac‹

(607) §zbwµ fihHm )xl&}Œj jnqsacbwy ¨ b‡qsh8jklnr~©xo,b†jn^

(608) b‡o,rwh8jkqzh©

(609) rw©(}T–q}ko,rM}kq¬j„ˆ}krw§z©jkqzrwhx}$h

(610) rwjkqzrwh±µQ•†b!¼j ¨ b ¨ qs§z§>jnm|§z» mw—(rM©jŠ~q}ko,rwh8jkqzh©

(611) rw©x}_–q}ko,rM}kq¬j„ˆ}krw§z©jnqsrMhx},µ. &(%. %c++ !’" ’&%>'›’*,+%,. ]_^xb h

(612) rwjkqzrwh-r|¥(–q}ko,rM}kq¬j„ˆ0}krw§z©jkqzrwh(}’rw¥(³†mwacqs§sjkrMhui´Mmwo,rw—

(613) q8b°8©xm|jkqzrwhx}^xmM}—(bb,htqsh8jklnr~©xo,b~—ˆ  l&m|hu ~xm|§z§(m|h(~ ±qzrwhx}†Â “xy˜ y”y +WÃ>qzhjn^

(614) b:”M™w}µ’fÏj$qz}$m0–wblkˆth

(615) qo!b ¨ mWˆ-r|¥±am|»qzh

(616) {°8©xm|h8jnq¬j&mjkqz–wb:m|hx~rM‹(blŒu m|jkqzrwhxmw§

(617) jn^

(618) b†qzh8jk©

(619) qsjkqz–wb:qz~b<m0r|¥ ¨ bmw»t}krw§z©jnqsrMhx}Grw¥ )xln}ŒjŒuÏrwl&~bl†¿Rmwhx~c¥LrwlTjk^xm|j amjkjkbly}Œb<o!rMhx~uÏrwl&~b,lp¸ ‚@mwlŒjnqzmw§@À:qs¯>blkbhMjnqzmw!§ G°8©xmjnqsrMhx}¿R‚TÀ G}&¸pµfihºjn^

(620) bo!rMhMjnb!¼j0r|¥ jk^

(621) b}k^xm|‹>bt¥LlkrMa }Œ^(mw~qzh

(622) {H‹

(623) lnrw—

(624) §zb,a ¨ b0m|lnbŽrwh

(625) §zˆo!rMhxo!blkhxb~ ¨ qsjk^ )xl&}ŒjŒuÏrwl&~b,l¤‚GÀ G}µ ]_^

(626) b¥Lrw§z§sr ¨ qzh

(627) {~b )xh

(628) qsjkqzrwhx}‡m|h(~Hlkb<}Œ©

(629) §sjn}:o,mwh²—(b0¥LrM©

(630) hx~²qz+ h _m|ln§zb}Ò }5y _m|l&~q’mwhx~  m|‹x©

(631) ½,½rÀ†rM§¬u o,b!jkjnm

(632) Ò }_rw"l #qsrMhx}Ò }$—>r8rM»}ŽÂ ˜

(633) y±·|y(˜ <Ãvµ. á- /..

(634) ·<™.  MÇWÍ

(635)  |Ì WÍ :ÇÌ.     

(636)     

(637)  !  " #$&% Ÿ % %'$ ž #b!j  : .)(E0*= 0 —(bcm,+.-¥L©

(638) hxo!jkqzrwh ¿. q}Žm|hErM‹(bh}kb!jŽr|¥10*’¸:m|hx~ºo!rwh(}Œq~b,lŽm³:m|acqz§¬jnrwhu. ´Mmwo,rw—

(639) qBb<°M©(mjkqzrwhr|¥jk^xbŽ¥Lrwlna¶.  $"   # < =    #"# (5. '.B 0 / 0*Tµ  q}¤o,mw§s§zb~32&54-Ê ¬ÉÏÇ|ÈxÊ È(µ ¨ ^xb,lnb Ðq}Šmco!rwh8jnqsh©

(640) rM©x}_}kom|§m|l$¥L©

(641) hxo!jkqzrwhHrwh'.0/ 1 . ¿J˜µz·<¸. " #$&% Ÿ % %'$768’ʁÍ,ÆnÇÍpÊËÉ cÍpÌ:9pÍ!Ç ä̍ÉvÊRÇÈ';. 3<>=?+  . #A@vÍ,ÉTÇBA9nÇ̍È"   È" c̍ÈxÊ B!Ç C4  Æ&ÇÈ(ÉJÊËÈ(ÌxÇÌÍ&B&̍ÈBÆ!ÉvÊRÇÈxÍED-ÊÍ   ʁÍ,Æ&ÇWÍpÊËÉ tÍ&Ì:9pÍ,Çä̍ÉvÊRÇÈ ÇB GF!Ì |ÉJÊRÇ|ÈH@EI KJCDÊ BL;. NM O  + -.  . # P. '. sÇÆ  R4 TSMÊK4-ÌU4 ÇB  < M #    Q   Q #"W V M   Q ##X)( :Q. " #$&% Ÿ % %'$1YZ8’ʁÍ,ÆnÇÍpÊËÉ cÍpÌ\[  &Í,Ǭ̍ÉJÊRÇ|È';. 3<>=?+  . #‡ÊÍ   ʁÍ,Æ&ÇWÍpÊËÉ ÍpÌ\[&Í,Çä̍ÉvÊRÇÈ®ÇB&GF,Ì ÉvÊRÇÈ]@^I*KJGDÊ BL; M 3+ -  .. # P  Q ;. sÇÆ  R4tÊËÈxÊK4tÌU4 ÇB % < M "#     Q    Q #"W  V M   Q ## _)( " #$&% Ÿ % %'$0`a8’ʁÍ,ÆnÇÍpÊËÉ cÍ,Ǭ̍ÉJÊRÇ|È';. ­ÊÍ  ÆnÇ|ÈxÉvÊËÈxÌxÇ|ÌÍ <ʁÍ!Æ&ÇÍ&ÊËÉ cÍ,Çä̍ÉvÊRÇȹÇB GF!Ì |ÉJÊRÇ|Èb@EI KJCDcÊ B‡ÊËÉ ÊÍ.9nÇÉdc c Í&Ì:9pÍ,Çä̍ÉvÊRÇÈ  È t ÍpÌ\[Ce Í,Ç ä̍ÉJÊRÇ|ȹÇB>@^I*KJGD . fiac‹>rwlkjnmwhMj¤‹

(642) l&mwo!jkqo,mw§   

(643) ®¡\  f±ž ¶ ·wµ_fihÁ‹

(644) l&mwopjnqzo,bwy#jkrEo&^

(645) b<o&»²jk^

(646) b^8ˆ‹>r|jn^

(647) b}kb}Žrw¥ jn^

(648) b}kb~b )(h

(649) q¬jnqsrMhx}y’qsj-o,mwh®—(b©x}kb!¥L©x§Gjkr )(ln}Œj )

(650) ¼ jk^

(651) bc‹>rwqzh8j  Q y±mwhx~²jn^

(652) b,hjkr–wblkqs¥Lˆ²q¬¥„yB¥LrMl0m|§z§’¥L©xhxopjnqsrMhx} M qsg h +.-  . #Š}Œ©xo&^¹mw}  Q q}Žm§zro,m|§ am¼qsa-©

(653) a¿Rlkb<}Œ‹#µacqsh

(654) qza©xa¸0r|¥ %;< M #ky ¨ b^xmW–wbtjk^xbm|‹

(655) ‹

(656) lnrw‹xlkqmjnb-qzh

(657) b,{8m|§zq¬j„ˆMµ]_^©x}y#qzh ‹

(658) l&mwopjnqzo,bwyWjnr:‹

(659) lnr–wbGjk^xm|j©tqz}@m†–8q}no!rM}kqsj„ˆŽ}krw§z©jnqsrMh±y ¨ b$o,mwhrwhx§sˆ0qzh8jkb,lnb}Œj’©(}#jnr†jn^

(660) b$‹xm|lkjkqo!©

(661) §m|l ‹(rMqsh8j&} ¨ ^

(662) b,lnb‡jk^

(663) b0‹xlkrM‹(blŒjnqsb<}$mw—(r–Mb‡mwlkbŽh

(664) rwj¤jklnqs–qm|§vµ ˜µ_fÏj0qz}Žb<mw}kˆHjkr‹

(665) lkr–Mbjn^xmj0qzhºjn^

(666) bc‹

(667) lkb–qsrM©x}:~

(668) b )xh

(669) qsjkqzrwh(},y ¨ bco,m|h®m|§}krqzat‹>rM}kbtjk^xm|jŽjk^

(670) btjkb<}„j ¥L©

(671) hxo!jkqzrwhx} M –wblkqs¥Lˆ M.  Q #     Q # . ]_^©x}_¥Lrwl¤b,¼

(672) m|ac‹

(673) §zbwyjk^xb~b )xhxq¬jnqsrMhrw¥}k©

(674) —x}krw§z©jkqzrwhHo,mwh—>b0lnb ¨ lnq¬jkjkb,hHmw}¶. ô õ ô ø.

(675) È" <ʁÍ!Æ&ÇÍ&ÊËÉ Í,Ǭ̍ÉJÊRÇ|È

(676) Í . " #$R% Ÿ % % $   Q ;..  . < =?+. ·M·.  . #‡ÊÍ&cÍpÌ:9pÍ,Çä̍ÉvÊRÇÈÇB@EI KJCDÊ BA; NM 3+ -  . † # ÍpÌxÆLcÉdc*|É.  Q #     Q #P M _ ª  ÇÈ È ,Ê  c 9nÇÌGcÇ<Ç ÇB :Q P  ?c(   'Q "  :Q #  V M :Q ##!X)(  M. A‡lnmw‹

(677) ^

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(679) b )xh

(680) qsjkqzrwhq}¤a©xo&^acrwlnbŽqsh8jk©xq¬jnqs–MbŽjk^xmwhjn^

(681) b0‹

(682) lnb,–qsrM©x}_rwhxbwµ. d

(683) µ¤]_^

(684) b}kbŽ~b )xhxq¬jnqsrMhx}T—>b,qzh

(685) {t—xmw}kb~rwh§zro,m|§>‹

(686) lnrw‹>b,lkjkqzb}yMb–wbhq¬¥’q¬j_acbm|h(}Tlnb~©(o!qzh

(687) {jn^

(688) b‡}kqz½,b†rw¥ jk^

(689) b0hxb,qz{w^—(rM©

(690) ln^

(691) rr~±y  C[ c8ÊR$Æ     ¨ bo,m|hHo!rMhx}kqz~bl$jk^xm|j¤jk^

(692) bŽ¥L©

(693) h(opjkqzrwh(} M mwlkb0 m

(694) h

(695) bMµ   U$ ¢ % % ž¢ % ž % Ÿ  ž % $ #Ÿ % % $ ž. :qz}no!r8}Œqsj„ˆ®}ŒrM§s©jnqsrMhx}mwlkb ¨ b<m|»®}ŒrM§s©

(696) jkqzrwhx}µ]_^

(697) bˆÁm|lnbh

(698) r|jc~qs¯>blkbhMjnqzmw—

(699) §zb

(700) •†b,–Mb,lkjk^

(701) b§sb<}k}y ¨ bm|lnb {Mrwqzh

(702) {-jkr}kb,b‡jk^xm|j¤jk^

(703) q}_h

(704) rwjkqzrwhHqz}¤o,rwhx}kq}„jnb,h8j ¨ q¬jn^jn^

(705) b0h

(706) rwjkqzrwhr|¥Qo!§mw}n}Œqo,mw§B}krw§z©jnqsrMhx},µ   % % ž % Ÿ % % $ 6,É  |Ê  !È(ÉJÊ 9/ EÇÈ . P&®Æ  Í&Í&ÊRÆ  _Í!Ç ä̍ÉvÊRÇÈ\Ç B @EI KJGD  B  1<>=?+  . # P dÉ c*,È  ʁ& Í <ʁÍ!Æ&ÇÍ&ÊËÉ Í,Ǭ̍ÉJÊRÇ|È   ‚Qlkrrw¥„¶ fihc¥RmMopj<yMqs¥ qz} ~qs¯>blkbhMjnqzmw—

(707) §zbwy|jn^

(708) b,h¥LrwlTmw§s§¥L©xhxopjnqsrMhx} M  +.-  . #’q¬¥ 'Q¤qz} mŽ§srom|§xam|¼qza©xa¦¿Llnb}k‹±µ acqzh

(709) qza©

(710) a¸$r|,¥ % < M #kyxjk^

(711) bh   % . mwhx~jk^

(712) bh . /< =  'Q #   V M :Q #.   Q "   Q #"W V M  Q #"# (. fihHrwjk^

(713) bl$lnb}k‹>bopj&},y ¨ bŽ^xmW–Mb‡mw§z}krtjk^

(714) bŽ¥LrM§s§zr ¨ qshx{-lnb}k©

(715) §sj¶  *%   

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