Learning Surfaces by Probing
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(2) INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE. Learning Surfaces by Probing Jean-Daniel Boissonnat — Leonidas J. Guibas — Steve Oudot. N° 5434 Décembre 2004. ISSN 0249-6399. ISRN INRIA/RR--5434--FR+ENG. Thème SYM. apport de recherche.
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(17) ´ ¶ nqcd0P!zt¢v$alcz;\oZ^P,\Uw ³N\rKhP§oP!bft~ghz(t¬Y \rj^cft~zh 0ghbft\^calz®a~]w jrP Z]\^jrcfs¦\^P7w \^a®³4oP!b oxw<s^³$cZ \^KNPZ]gh_(Z]P!\Lal*\rKhP w[cOQP!zNZ^cda0zNt~b oP!bft~ghz(t¬Y¢\^jrctlzh lgNbt~\^calz oP!xb owsBa~ w O¢t0w[PUa~\^KhPU6t0s,P,\yZLKha0Z^Pxw[gNtl b "8aljralzha0c P wh lP Z=cdz;\^P jrZ^P s,\ 8lP j^Y`v(a0cdz;\Bal$cz;\^P jrZ^P s,\^calzQa~´t "8aljralzNalchP7w[ lPcd\^K ²cfZ8\rKhP2s,P z;\^P!jBa~´t G A IDIe :9 .I oxw<s^³ C P P t®2P btlghzNt¬Y _Nt~bb8s,P z0\rP!jrP wDa0z z²vhjytls¦\rcs!Pl³4alzNbdYt Z]gh_(Z]P!\qal±2P b owss t~z²_(PQs,a0OvNg[\^P7w ³´Z]czNs!P¢ cZq&zhaz \^KNj^a0gh lKtlzaljytls!bdP \^KNt~\cfZzha~\QtlZrZ]ghOQP7w\^aw[P,\rP s¦\¢t~bb\^KhP®cz0\rP!jyZ]P7s¦\rcda0z¡v(a0cdz;\rZalq c\rK\rKhP P7w[ lP7Z§al>\^KNP "8aljralzNalc(whctl ljyt~O a~ w¢ =JLKhPqZ]gN_NZ]P!\alG2P b ows\rKNt\ w[P!\^P s,\rZ±cZzhal\^P7w2P b owstlzNw Z]\^a0j^P7w¢tlZt§Z^gh_$s!alOQvhbP,« a~2P b oxw<s^ =t0syK\rcdOQPQt v$alcz;\qcfZUtlwhw[P7w \^a w¢³ a0zhbdY®\rKhP`v(t~j^\qal8\^KhP "8aljralzNalc*whctl ljyt~O \^KNt~\qKNt0ZUsyKNt~zh 0P wDt~ \^P!j \^KNPxcdzNZ^P!j^\^calz®a~*\^KhPxv$alcz;\>KNtlZ\ra¢_(Pe;ghP j^cP w _&Y¢\^KNPxaljytls,bP > º ±KhP! Z=tlbd 0aljrc\rKhO \rtllP7Z=tlZczhvhg[\B\^KhP2Z^ghj^6tls!P>L³0tUv$a0Z^c\rcd0P¬tlbdgNtP v0³;tlZ8P!bbNt0Z8t~zQalv[\rcda0zNt~bNcdzhcd\^cft~b. v$alcz0\Z^tlOvNbdtP w¢ zhaczhcd\^cft~bhv$alcz;\±Zrt~OQvhbP>cZ= lclP!zG³~\^KNP!zQ\^KNP2t~b la0j^cd\^KhO s,a0zNZm\rj^g(s¦\rZ=alzhPczQ\^KhPoZ^tlOPLt¬Y`t0Z8algNj vhjral_hczh t~b laljrcd\^KhO QZ^P!PUZ^P s¦\rcda0z h l oP!b oxw<scZ±\^KNP!z s!alOQvhg[\rP w e0gNP!jrY;czh `P!lP j^YQP7w[ lPoa~´\rKh6P "8a0j^a0zhalc4w[cft~ 0jrtlO a~ w gNZ]czh Qaljytls!bdP > \oP7tlsyKDcd\^P jrt~\^calz´³N\^KhPt~b laljrcd\^KhO cdz(Z]P j]\yZ2tzNP!v(a0cdz;\qcdz;\^R a w t~z(wDghv4wht\rP Z2oP!b ow<sr =t0syKDv(a0cdz;\qcdzNZ^P!j^\^P7w cz0\r`a w cZB\^KhPUs,P z;\^P!j±a~*t G A 0 G A IDI \9 #I oxw<s^³&\rKNt\LcZ ³&t_Ntlbdb4a~GoP!b ow<sKha0Z^Pos!P!z;\^P j >KNtlZ±_(P P!z®w[P,\rP s¦\rP w¢_&Y t~z(wDKha0Z^P§jytlwhcdgNZ2cZo ljrP t~\^P j>\^K(t~hz v; UJLKhP`tlbd 0aljrc\rKhO Z]\^a0vNZoKhP!zD\rKhP!jrP`tlj^P`zha OQa0j^P_(tlwD_NtlbdbfZoa~BoP!b oxw<s^³ KhcfsyK cbdb4P lP!z;\rgNt~bbdYQKNtlvhv$P!z cd
(18) vqcfZv(a;Z]cd\^clPUZ]czNs!PUcfZLs,a0Ov(tls¦\7 >v$alz\rP!jrOQcdzNt~\^calz´³;\rKhPqtlbd 0aljrc\rKhO j^P!\^ghjrzNZ w tlZLP!bb´tlZ 2P b ow<sr
(19) *[ $ ¶&¶ º ¶Dº
(20) ´
(21) G 9 z | ³[PqvNj^a0P wQ\rKNt6\ ±KhP ZLt~b la0j^cd\^KNO jrP,\rghjrzNZLt\rj^cft~zh 0ghbft\^P7w s!alOQvhbP,« \^K(t\>cfZt` 0a&a&w®t~vNvhj^a¬«[cO¢t\^calz a~=vNj^a&cfw[P w\^KNt~\\^KhPU¤a0bdbacdzN Qs!alzNwhc\rcda0zNZt~jrPqZrt\rcZ]°NP7ew .
(22) . . . . . . . . . . . . . !. #". !. $". &% . . '". )(. +* -,. . %. .. '". /% . . %. . ' . 0. %. ". . 132. 4. 65. 8. Q 8. 5. &% . -,. . . . C M A 3 A *6C 9# $IJ0 = CD-FL $Y -tG2 $Y *0 A &%:9 . F A M?"V. &-xCi'2#M! $* A IDIr-FLT') $*+*)'#-N)0(') $3U4 *,.*+-M8 \9. } ;"$. &%k9 A ')#-=< :9 v ?A@CB @ED6F 2'&F o s&P 8. G. -7. . C MO'.CD f'.Y 3!M.'. &CGf)0(G#% A G A IDI :9. FGFIHKJLNM ONM. 9. #I oxw<s\; :98').*+-N# . R&v$P s!c°(s t~bbdY±PvhjralP w\^KNt~\ ³´gNzNw[P!jU\rKhP Z^P¢s,a0zNw[cd\^calzNZ ³ t~zNw²\^KNP¢zhaljrO¢t~b8w[P!&cft\rcda0z²cfZU_(a0ghzNw[P7w_;Y. ov:Hcs¤j^a0O. . = G. A *0 :9 A 0$CDYLM. 0'>. A -3 VM&- v 9# . cfZxt~O`_hcP!z;\xcZ^a~\ralvhcfs\ra³Gt\?ot~gNZrw[alj w[cfZm\yt~zNs!P cfZUcdzNs!bdg(w[P wcz²\rKhP¢ghzhcalza~\^KNP. ov s^ TDaljrP!alP j ³G.
(23) _NtlbdbfZ¢a~oP!b oxw<s§\rKNt\ s!cdjys,gNOQZrs,jrc_(P \rKhPD6tls!P,\yZa~ JLKhcfZ¢cdOQvhbcP Z¢\rKNt\Rw cZ t v Zrt~OQvhbP a~`³\^K(t\ cfZ.
(24) q o qcv s (w
(25) 0 >P zNs,P0³ vhjra;czh ¢\rKhP¢s,a0j^jrP s,\^zhP7Z^Z2a~ ±KhP ZUtlbd 0aljrc\rKhO ³$KNP!zj^ghz²c\rKZ^alOQP`aljytls,bP >³ jrP whgNs,P7Z>\ra®Z^Khaczh \^K(t\2tlZrZ^P!j^\^calzNZi§,³Giq<Lt~z(w 2 Ltlj^PxZrt\rcZ]°NP w cd\^K 2P b owsBghv$alz®\^P j^OQczNt\rcda0z´ zDtlwhwhc\rcda0z´³[±PxvNj^a0P wc8 z | ´\^K(t\\^KNPxalg[\rvhg[\>v$alcz0\2Zrt~OQvhbP?w a~\^KhPtlbd 0aljrc\rKhO cfZ M/4 A )M# ³Nc Pl 8\rKhP!jrPUP,«[cZ]\rZ tQs,a0zNZm\yt~z;\ \^KNt~\2w[a&P ZLzhal\2w[P!v$P!z(w a0z¡zha0jalz v;³hZ]gNsyK®\rKNt\@
(26) q (o q v s 1wL R&czNs,8P w cfZt¢Z]vNtljrZ^P v Zrt~OQvhbPqa~8L³h±PUKNt¬0P ¶&º ¶ Q r FL) $ f.3 :9 w "!$# %' &(*) ,+.- f'3A0( / 21 4 ;5= FL. f1-iF (') $*+M&- A *+-5CD* -iF 0 u#M *, -0 N4#*01 * * $ ! $* v P jrP t o s6$5 v 2tlzNw¢KhP zNs,P7 w 0tlj^P2vhj^a0v(a0j]\rcda0zNt~bN\^a§\^KhOP v98 #*6- f &46% al³&c. Pl \rKhPoOQczhcO§ghO z&ghO`_(P j±al*Z]vNKhP!jrP ZBa~ jytlw[cgNHZ vU\rKNt\2s t~z s,alP j2 7 . . >. F. ". . 2. /% . >. ,. . B. 2. G. 2. B. G. G. :H. : ; !
(27) ,
(28) § h a0jx\^KNP Z^tllPQa~>s!btlj^cd\mYl³*±Pt0Z^Z^ghOQP¢cdzZ^P s,\^calzNZ§h³G}²t~zNw¡| \^K(t\Q cfZ§s,a0zhzhP7s¦\^P7w Pw[P!¤P!j§\^KhP¢\rj^P7t\rOP z;\a~ Z^P!lP jrtlb s,alzNzhP s,\^P w®s,a0OQv(a0zhP!z;\rZ\raZ]P7s¦\rcda0Bz DN s s,a0jrw[czh `\^a l³[Pq&zha tv(a0cdz;\ E L±PQP,«hs!P!v[\U\^KNP¢Oa0P ZUalB\^KNP¢vhj^a0_hczh Dw[P!&cfs,Pl³Galghjtlbd 0aljrc\rKhOWcfZUlP!jrYZ^cdOQcbtljq\^a ±KhP Zxtlbd 0aljrc\rKhO QJLKNP O¢t~czDw[c 4P!jrP!z(s,PUcZ\^KhPxa0jrt0s,bP2\rKNt\cfZgNZ^P w \^a¢w[cfZ^s!alP j±\rKhPZ^ghj]6t0s,P' z alghj>s tlZ^Pl³&\raQsyKNP syKhP!\^KhP j>t"8aljralzha0c P wh lP<xcz0\rP!jyZ]P7s¦\yZ2¡a0jzha~\7³[±PUO§g(Zm\°NjyZ]\LOQa0Pqa0ghjLvhjral_hczh ¢w[P!&cfs,Pq\^aQa0zhPUa~c\yZP!zNwhv(a0cdz;\rZ 8JLKhcfZj^P7e;ghcdjrP Z±\ma \^KNcdzh ;.Z ±°NjyZm\7³N\rKNt\qt\obP t0Zm\>a0zhPP zNw[v$alcz0\>= al?<§_$P§ba[s!t\rP w czO l>
(29) Z]P7s,a0zNw ³h\rKNt\q±Px&zha t 9) 2254 A -F ¤jralO l Z ¤KhP j^P2\^KhPqvhjral_hczh w[P!&cfs,Pos t~z Oa0P¤jrP!P!bY 8\ra=$³ C P PBt`s!alz;\^cz&ghalg(ZBs,gNj^0P2czNs,bgNw[P7wcz lB \rKNt\L la&P7Z ¤jralO l Z \^a7=$ JLKhPUO¢t~cz cw[P7ta~G\rKhPxvNtlv(P jcZL\ra¢s,alz(Zm\rj^gNs,\vhcdP7s,P cZ^PUbdczhP tljvNt\rKNZO¢tlwhPqal= "8a0j^a0zhalc4P7w[ lP7Z! ®¶2@ º } BkA CD"V.* A 4 $CD*+-M#.- w<; -FL "8a0j^a0zhalc$ ljyt~vhK¢al w<;DC A ow<sf; C MO-FLtKc A 46F(3 A 0c! :9-iF O"$# &-xCi')&M A *,0 )0.Kg&M! \98-iF C $ f *, $C]0 C A Kc A 3 \9 w(P ¯oghj`_NtlZ^cs cdz;\rghc\rcda0zcfZ\ra²s,a0zNZ]\^jyt~cz\^KhP®vhjral_hczh w[P &cs!P\^a²OQalP tlbda0zh \^KhP®P wh lP Z§a~0 "2Tn oxw<s lx³8KhcfsyKt~jrP s!tlbdbP w\^KhP 9) f) )0.Kg#M3E w[c ¢s!ghbd\mYt~jrcZ^P Z>¤j^a0O \^KhP`6tls,\U\^KNt~\ ³ KhP z²t zhP!©v$alcz;k\ j²cfZoczNZ^P!j^\^P7wcnz w¢³GZ^alOQP`a~ \^KNPs,ghjrj^P z;\ "8a0j^a0zhalc 0P!j^\^cfs,P7Zt~zNw®P wh lP ZLO¢t¬Y whcZrt~vhv$P tlj ' \¤a0bdba>Z\^K(t\>v$alj^\^calzNZal= "2Tn oxw<sl \^KNt~\2s,algNbw®_$P jrP tlsyKNP w®_&Y\^KNPvhjral_hczh w[P ;cfs,P¤j^a0O l Z _$P,¤a0j^Px\rKhPczNZ^P!j^\^calz a~r jDO¢t¬Y zNa¢bda0zh lP j>_(P§j^P7tlsyKNtl_hbdPxt~ \^P!jrLt~jywhZ V Z^P!Pq°N 0ghj^PQq¤alj>tlz cbbdgNZ]\^jyt\rcda0z´ . . >. . 5=*. *. 5. . this point becomes unreachable. FHGJI2KMLNOQPRTSQUMU*NWVYXGJZ
(30) GJX6[\V]*^_UMI2NW`aGJUbX ]MNdceS2L S2UMSQGgfhGi^_IQL^_jlkDmn]MNdL Nof\pSQGqU2XDGi`aGqU*` N0L X NofrGqUtsu^_U*frX ]MNvL NofrNofhIQNo` j7^_Lw7X ]*NxyS2KMU*fM^QL [7SQzHGJX`{ceS2L S2UMSQG.V0N0|q|}k ~3DY}eY6 6} 0QYo66JQY36Y{603 6 6}Y }H6Yo} ¡$¢£Q¥¤$¦}§$§3 ¨©Hª¨ «.
(31) J*aa0P!jys,a0OP2\^KhcfZw[c s,ghbd\mYl³ha0zNs,Pxt`¤jrP!PxvNt~\^K o=6s=¤j^a0OUl Z \^a¢Z]a0OQP "8a0j^a0zhalc40P!j^\^P,« = KNtlZL_$P!P z ¤a0ghzNw ³ P Zm\raljrP o=6sqczOQP!OQa0j^YZ]aD\rKNt\v=cbdbjrP!O¢t~cz¡j^P7tlsyKNtl_hbP_&Y²\^KhP vhjral_hczh w[P ;cfs,P v(P j^O¢t~zNP!z;\^bYl 2P!zNs!P algNj vNt~\^KNZUt~jrP§O¢t0w[PQa~=\ma\mY&v$P ZUa~±P w[ 0P Zu2P wh lP Z2\^K(t\x_(P bda0zh \^a \^KhP¢s,gNj^jrP!z;\6"8a0j^a0zhalc ljyt~vNK´³ t~zNwP w[ 0P Zo\^KNt~\Uwha zhal\>_hg[\>P!jrPoP7w[ lP7ZLcdzDZ^alOQPq¤aljrOP j "8aljralzha0c w[ctl ljyt~O Y²OQa&czh \^KhP®vhjral_hczh w[P &cs!P t~balzN Dalghj§¤j^P PvNt\rKNZ ³=t~zNw_&Yvhj^a0_hczh D¤jralOP tlsyK¡&cfZ]cd\^P7w "8aljralzNalc=0P!j^\^P,« \^aLt~jywhZcd\rZBzhP!c lK&_$alj=0P!j^\^cfs,P Z=cd$z "8a0uj oxw<s^³;±Pos!t~z¢w[P!\^P7s¦\txZ^gh_NZ^P,\ ¡a~4\^KNP>v(a0cdz;\yZ=a~ "2Tn oxw<s_ §t~zNw¢s!alzNZ]\^jrgNs,\Bt Z^gh_$s!alOQvhbP,«¢a~2P b ow<s8s!tlbdbP w\^KhP "uC M&CG#IJ! f&M&-x &Ci'#-N)0 .I A Y * A %T-x &C A *gKcY I A -Ci *n :9 w¢³&aljLZ]cOQvhbY2P b ow<sr 8lP j^Y v$alcz0\>al ¨cfZL\^KhPs!P!z;\^P ja~toP!bft~gNzNt¬Y_Ntlbdb ³hs!tlbdbP w G A IDI \9 #I ows^³[\rKNt\2s!cdjys,ghO¢Zrs,jrcd_$P Zt`6t0s,P,\aloP!b oxw<s^ . . . . . . . W
(32) 5 *-3 ,3 *,. &(%. PqvNj^a[s,P P w tlZcdz ±KhP Ztlbd 0aljrc\rKhO ³;_&YZ]\^a0j^czh Q2P b ow<s=t0ZtQZ^gh_4s,alOQvhbP,«aloP!bxow<sr @ zNZ^cwhPoP lP!jrYoP!bft~ghzNt¬Y \^P!\^jyt~KhP7w[jralz´³¬P=O¢t~jrqP7tlsyKxal[\^KhP±¤alghj*6tls!P,\yZ*t0ZG_$P!czh 2a0jGzhal\_$P!czh ovNt~j^\*alN2P b ow<sr JLKNcZ*t¬Y0³¬P!lP j^YUoP!bft~ghzNt¬Y 6tls!P,\>cfZLO¢t~jrlP w\mcs!PZ]czNs!Pqcd\>_$P!balzh ;Z\^aQ\maoP!bft~ghz(t¬Y¢\^P!\^jyt~KhP7w[jrtN z®aljyw[P!j\^a¢Z]\^a0j^Po\^KhPxv(t\^K(Z¤aljL\^KNPqvNj^a0_hcdzN QwhP!&cs!Pl³[P lP!jr$ Y "8aljralzNalc$lP!j^\^P! «
(33) a=¢cfZ 0cd0P!z tv$alcz;\^P!j 4, f." \^a\^KNP vhjrP!&calgNZ0P!j^\^P!« alz®\rKhP§vNt~\^K ¤jralO 8 l Z \^a7=$ Y s!alz&lP z0\rcda0z´³M= 4, f." ²OP7t~zNZ>\^KNt~\2P&zha zNaQ¤j^P P vNt~\^K®¤j^a0O l Z \^a7=$' z®Z^gNsyKDtQs!tlZ^Pl³h=¢cfZ>Zrt~cfw\raQ_(P CD* A '.-CD"$ ¯2\^KhP j^cfZ^Pl³
(34) =cfZ>s!tlbdbP w A '.-CD"$ 8tQzhP!bY s!j^P7t\^P7w "8a0j^a0zhalc 0P!j^\^P!«= _$P!balzh ;Z\^a O l Z ³h\^KNP!z ±PZ^P,\¢= 46 f." = _&Y whP,6t~gNb\2Z^cdz(s,P= s t~z _$P jrP tlsyKNP w®_&Y \rKhPvhjral_Ncdzh w[P!&cfs,Pl zDvNt~j^\^cfs,ghbft~j7³(tlzDcdzh°Nzhcd\^P "8a0j^a0zhalc*lP j]\rP,9« C P Po\rKhP§P zNw[v$alcz;\qt\ocdz[°(zhc\mY al8tlz ghz&_$alghzNwhP >w "8aljralzNalc P wh lP >t~bt¬Y[ZbdcP Z2alg[\yZ]cfw[P Z ³(KhcsyKDcfZ>s!alOQvNtls,\ LJLK&gNZ!³(c\yZ 4, f#" °NP!bfw cfZzhP!0P!j g= _$P!balzh ;Z\^a Z ³[\^KNP!z ±PUczhcd\^cft~bc P= 46 f." _;Y w[P,6tlghb\7 J*a§s,a0zNZm\rj^g(s¦\tlzNwQ\rKhP!z ghv$wNt\^Pq2P b ows^³[P2gNZ^Pot§j^a0g[\^czhP>zNtlOQP w !" #%$&&& (')'7³lcz0\rj^a[w[g(s,P wcdz¢°( lghjrPo[ R;\yt~j^\^czh ¢¤jralO t~zt0s¦\rcd0PlP j]\rP,«l+= *-,/.102, 3³ !" #%$&&& (')'v$P!j^¤aljrOQZqtw[P v[\^K .°NjyZ]\2\^jyt¬lP jrZrt~b al@ "2Tn ow<sl> \^a Z]P P KhcfsyKvhjrP!&calgNZ^bdYczNtls,\^clP>0P!j^\^cfs,P Z±s!tlz¢_(Pqj^P7tlsyKhP7w_;Y\rKhP2vhjral_Ncdzh `w[P!&cfs,P>¤jralO = *4,5.10-, \rKhj^a0gh lKQ¤jrP!PoP w[ 0P ZBa~ \^KNP "8aljralzNalc4 ljyt~vhKG 6 ! !" #%$&&& !')'= *-,/.102, & 7 º ¶ h zhP cd 0K&_(a0j£= a~g= *-,/.102, » º 8(9(:; P w[ 0=P < = *4,5.10-, q != >
(35) -7 ?< += *4,5.102,$q != >
(36) A C @ B< ¶ tlwNw \rKhPw[gNtlb a~ < += *4,5.10-,uq != >´\ra2P b ows
(37) ¶ ¤ ¶ 27 = 4, f." C ¶ D(D =_(P7s,a0OP7Ztls,\^clP 27 = Z ¶ = 46 f." += *-,/.102,
(38) ¶ ¤ ¶ = 46 f." =
(39) ¶ G» -7 EF:"G vhj^a0_hczh ¢w[P!&cfs,Pq¤j^a0O = *4,5.102, \^ad =
(40) 6 ! !" #%$&&& !')'=
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