´ Etat de l’art des m´ ethodes de correction des d´ eformations c´ er´ ebrales per-op´ eratoires Pierrick Coup´ e, Pierre Hellier, Christian Barillot To cite this version: Pierrick Coup´ e, Pierre Hellier, Christian Barillot. ´ Etat de l’art des m´ ethodes de correction des d´ eformations c´ er´ ebrales per-op´ eratoires. [Rapport de recherche] RR-5589, INRIA. 2005, pp.51. <inria-00070418> HAL Id: inria-00070418 https://hal.inria.fr/inria-00070418 Submitted on 19 May 2006 HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished 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´ epˆ ot 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.
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Etat de l’art des methodes de correction des
deformations cerebrales per-operatoires
Pierrick Coupe, Pierre Hellier, Christian Barillot
To cite this version:
Pierrick Coupe, Pierre Hellier, Christian Barillot. Etat de l’art des methodes de correction desdeformations cerebrales per-operatoires. [Rapport de recherche] RR-5589, INRIA. 2005, pp.51.<inria-00070418>
HAL Id: inria-00070418
https://hal.inria.fr/inria-00070418
Submitted on 19 May 2006
HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, estdestinee au depot et a la diffusion de documentsscientifiques de niveau recherche, publies ou non,emanant des etablissements d’enseignement et derecherche francais ou etrangers, des laboratoirespublics ou prives.
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¢k eJj l |$av x c l aveJ z aRjd\yjY\_cfgfcfZ_a l gfa lsx zdz e l \+Ya|Re lsl ¥;g x \_cfe ¶d¥$2dc|$e `V`Va
RC(T, f) =‖I − f(J T )‖2
V ar(I)
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f(J T, ‖∇J T‖)Z_e c\ygfa z gfj2Z zdl e[|~2a z e9Z_Z_cfbdga¯Ya
g c` x hJaKqWC¢ ÑgZ+e bd\_cfaRdda;9\yYa|$aR\]\_a` x 2c^ l agfa lsx zdz e l \Ya|Re lsl ¥;g x \_cfe bdc )xol c¥
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fZ]j l j2naRj
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IMNz eJj l g x ¢ a;Z l ¥:Z]j2g«\ x \sZ x j dc a x j Yamg x cf2Yci9jdaR9\
i9jda&gaA|~de c¡ djH| l c«\s^ l a&YaAZ]cf`Vcg xol c«\s¥ xz aRjH cf 2jda;2|$aAZ]j l g x i9j x gfc\_¥&Yj l a:| x g x hJa ¢ a$\_\_aRbMe[a l|~2e cZ]c\de 2|Ag
IMN| xol |$a;ggfa$¦Ñ|$c-daA x c\ z2x Zv [X z e \_d^:Z]aZ_j l g x |Re 2Z_a ls)x \_cfe / c9\_a;2Z_c«\s¥Vda;Z
Z]\ l j2|$\_j l a;Z¯Z]cf`Vcg x c l a:Z aR9\ l a¶daRjY¡ x |;i9jdcfZ_c\_cfe 2Z;¢ " x x gfXYZ]aYa:Z l ¥;Z_jdg\ x \sZeJbY\_a;[jda c x ga:Z¯Ya;jY¡`V¥R\_deYYa:ZYa l a;| x g x h aZ]eJjdgcfh 2aR9\iJj2a·|$aRgfgfa;Z]¦É|Rce bY\sca;ddaR9\¶Ya:Z l ¥:Z]j2g«\ x \sZZ_cf`Acfg x c l a:Z *a; c l eJ9
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|Re ` z2x lsx J\-gfaSSD
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TaR z2x Z_Z x 9\ z2xol jdA¬e l `¯jdg x \_cfe zdl e b x bdcgfcZ\sa ¢ k*xol aR¡[a;` z ga
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IaR bdgfe[| §dgfa bdgeY| |Re lsl a:Z z e 2 x 9\+ x 2Z
Ja;Z]\ l a;|~da l |~d¥ x j·Z]a;2ZuYj` x ¡Ycf`&jd` Ya [l~x cZ]a;`&bdg x |$aZ]eJj2Zg [X z e \_d^:Z]a¯ j2da&Y¥;h l~x x \_cfe
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vMLaR¶` x ¡[cf`VcfZ x 9\ z eJj l |~ x i9jdavbdgfe[|
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aR»Ma:|3\_j2a l jd l a:| x g x hJadeJ l chJcfda §Cde z2x Zva;¸Z_a¯b x Z x 9\Z_j l g cfJ\saR2Z_c\_¥Ya;Z eS¡[a;gfZ;§M` x cfZva;`VaR\]\ x 9\aR |$e lsl a;Z z eJ2 x |$aVYa;Z z e cf9\sZ;§|$eJj l bMa;ZveJj Z_j l x |Ra;Zvde `VeJgeJh jda:Zqa;9\ l aAgfa;Z&YaRjY¡ eo¦gfjd`Va;Z¯cf` x h a:ZR¢ k eJj l |$a\X z aYa Y¥;` x l |~2aga:Z¯| l c\_^ l a:Z&YaZ]cf`Vcg xol c\_¥da Z_e 9\ z2x Z&gfa;Z¯`VªR`Va;Zi9jda zdl ¥:|$¥:YaR`V`VaR9\q| xol c|$ccg x jd\ xS eJc l j2da xoz2zdl eY|~da hJ¥ReJ`A¥R\ l cfi9jdadj zdl e bdgf^R`Va ¢ a&|Re 2|Ra z \
gfaR`Va;J\¯ª$\ l a l~x `VaRd¥ x 2Z&jd| x l ab x X ¥;Z_cfaR1¢ a;Z\ lsxS)x jY¡/Ya cdh !4 7´ 3 §- 8 z e9Z]a;9\&gaz2l eJbdg^;`VavYj l a;| x g x h av|$eJ`A`Vavg xAl aR`VcZ]a yne j l Yag xAz e9Z]c\_cfe Ya;Z z e cf9\sZYa |Re 9\_eJj l
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Z_j l x |RavZ_aRhJ`Aa;9\_¥Ra x 2Z+g w F zdl ¥Re z ¥ lsx \_eJc l a c x da;Z+cf` x h a:Z KqW[c=¢ x ¬e l `¯jdg x \_cfe b x X ¥:Z]cfaRd2agfaRj l*z a l `Va$\" x ne jY\sa l Ya+g x |$e 2 x cZ_Z x |$a /Y"$" x jVdc a x jAYa+g x Yc l a:|3\_cfe VYag x Y¥$¬e l ` x \sceJ` x cfZ"¥Rh x ga;`VaR9\"YaqZ_e x ` z ga;j l h ls² |$au g x |$eJd x cfZsZ x 2|$audau`Va;Z_j l a:Zma;` z c l cfi9jda:ZR¢ a z2l eJbdg^;`VaZ_a lsx `V^Rda YeJ2|
P (vi|J) ∝ P (J |vi)P (vi)eJ¨P (vi|J)
a:Z\yg xVzdl e b x bdcfgc\_¥ z e j l iJj2a g x 2e j aRgfga z eJZ_c«\sceJ·YaBiZ]eJc«\
¢ a·\sa l `Aa Ya [lsx cfZ_aR`¯bdg x 2|$aa:Z\ g x zdl e b x bdcfgc\_¥/Ya¸\ l e j a l j2da¸cf9\_a;2Z]c\_¥zxol \_c|$jdgfc^ l am x 2Zg c` x h a KqWYc9Z x |~ x 9\1ga"d¥ z g x |RaR`VaR9\
vi¢Ñg z a;jY\!YeJ2|*ªR\ l ama$¡ zdl c`V¥"|Re `V`Vamjdda
¬eJ2|3\sceJAdag c` x hJaKvW[c=§fUSi
l a;gc x 9\-g xvz eJZ_c«\sceJ l ¥Ra;ggfaYa;Z z e cfJ\~Z*Ya|$eJJ\se j l*x jY¡c9\saR2Z_c«\s¥;Z*Ya
áá æ_^-``ab
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g c` x hJa ¢ a:Z x jY\_a;j l ZY¥R2dcZ_Z_aR9\|Ra$\_\_aq¬e 2|$\_cfe &g x cYavYaqg c` x hJa KvW[c 3 a$\+YavZ_e ¶h l~x YcfaR9\
P (J |vi) = kfUSi + (1 − k)‖∇fUSi‖
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dcrz a$\_c\~µ*ga`VeYY^Rgfa x jY\se l cfZ_a l~x j2¯¬e l \
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M <.M N ½ 9 Á ÂÃ!:ý 9vx 2Z+|Ra$\]\sa z2x l \scaJ§J2e j2Z xS eJ2ZmaR¡ z eJZ_¥ygfa;ZYc»M¥ l a;9\_a;Z"\_a:|~ddci9jda;ZYa l a:| x g x hJa xS a:|c` x h a l cfa
jY\_cfgfcfZ x \sceJ jd x J\ `Va xS a:|yeJjZ x 2Zd¥$¬e l ` x \sceJ2Z 7:Y§ L)8=§ jY\_cfgfcfZ x \sceJV cf` x h a:Z!c9\ l~x e z ¥ lsx \_eJc l a;Z w F ¹ )pycY¥Re |Re `V`Va ¥ l c«\s¥"\sa lslsx c 7´Hd§JJY§JJ 8=§ jY\_cfgfcfZ x \sceJ&Yj z e cf9\_a;j l dj&Z]XYZ]\_^;`AaYa"2aRj l e xS cfh x \_cfe z eJj l d¥$\_a l `Vcfda l gfa;Z!|$e[e l Ye 2d¥Ra:Zl ¥;aRgfga:ZYa ` xol i9jdaRj l Z z2xol \_c|$j2gcfa l ZR¢
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e jd\_a;Zy|$a;Zu`Aa:Z]j l a:Z cfZ_aR9\yi9j x 9\sc«2a l g a l_l aRj ll ¥ x gcZ_¥RaaR·|~ x i9jda z eJc9\r§2a;9\ l a g x \ lsx 2Z]¬e l ¦
` x \_cfe &| x g|$jdgf¥RaTa$\-g x \ l~x 2Z]¬e l ` x \sceJ l ¥;aRgfga
a l_l aRj l ³ w µ 7 3 8 TRE(r) = |T (r) − Tv(r)|x |Re d x cZ_Z x 2|RaqYag x \ l~x 2Z]¬e l ` x \_cfe l ¥;aRgfgavd¥;|Ra;ZsZ]c\ x 9\jd2ay ¥ l c«\s¥u\_a l_l~x cf[± l~xol aR`Va;J\+YcZ z eo¦
aR\yYa g x cYj2|Rc x g eY| x gfc x \_cfe lsl a;j l ³ µi9jdc1\ lsx Yjdc\ug cf` z2l ¥:|$cZ]cfe Z]j l g x gfeY| x gfcZ x \_cfe Ya;Z` x l i9jda;j l Z x jZ_aRcfYag c` x h a ¢ xAl a;g x \_cfe gc x 9\u|$a:ZYaRjY¡\X z a:Z a l_l aRj l a;Z]\d¥$2dcfa z2x l
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l a zdl ¥:Z]a;9\_a g a;Z z ¥ l~x 2|Ra` x \_d¥;` x \_ci9jdaHY¥R2dcfaH|$eJ`V`Aa< FLE2 >=< FLE2
1 >+ < FLE2
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rYa¶g c` x hJa ¢ qx 2Zgfa | x l a¶dj
l a;| x g x hJa l chJcfYa x `Va l Z+Yav\X z a z e cfJ\:§dgfa| x g|$jdg£Ya g x w x ¥R\_¥ Y¥R2dc z2x l !c\ z2x \ l c| ! 7 3 d§o 8|Re `V`Va ¥$\ x 9\¥Rh x ga"
< TRE2(r) >=< FLE2 >
n
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; $6 < N ½MPO :Ã:½ 9 #[Á 9 dÂ$VM½ :ÿ-B! # SEDÉ&# 9 Á#vx 2Z+gfa | x Z jd l a;| x g x h a l cfh cYade b x Z_¥ Z]j l g x `VcfZ_aaR |$e lsl a;Z z eJ2 x |$ayYa z e cf9\sZ;§dYe 2|
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aR¡YaR` z gfa §*Z]c\ l eJcfZ¯c` x h a;Zt1§t2a$\
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da;Zv\ l~x Z¬e l ` x \sceJ2ZvaR bMe j2|RgaJ§1Z]eJj2Z[X z eo\sd^;Z_a jd x gfh e l c\_d`Va z2xol x c\;§1de c\&|$e Yjdc l ag x\ lsx 2Z]¬e l ` x \_cfe cfYa;9\_c\_¥
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g ¥ )x gfj x \sceJ&Yj \ x jY¡Ya l a;|$eJj [l a;`VaR9\Ya+Z\ l j2|$\_j l a;Z x x \_eJ`Aci9jda;Ze j zx \_2e gfe h ci9jda;Z;¢ a$\_\_aRbMe[a l
à¬í!á1à°ã
!#"$%&'#()"$%*+-,! ./*0 /! 1"2+ 3
! 4 7´oY§ 8jd\_cfgcZ]a;9\uga:Z+\_jd`Va;j l Zy|$¥ l ¥Rb lsx ga:Z|Re `V`VaZ\ l j|3\_j l aYa l ¥$¬¥ l a;2|$aJ¢dr zdl ^;ZyZ]a;h `VaRY¦\ x \sceJ` x [jdaRgfgaAYa&g x \sjd`VaRj l x Zugfa;Zqc` x hJa;Z zdl ¥&a$\ z a l e z ¥ l~x \se c l a:ZR§2ga:Z x jd\_aRj l Zq|$e ` z2x l a;9\gfa+\ x jY¡Ya l a:|$eJj [l a;`Aa;9\*day|$aR\]\_aqZ\ l j2|$\_j l a xS)x 9\*a$\ x zdl ^;Z l a;| x g x h aJ¢ a\ x jY¡VYa l a;|$eJj [l a;`VaR9\¥R\ x J\uY¥R2dc|Re `V`Va
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2 + (z − zj)2]1/2
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θ(rj) = rj|$jdbdci9jda;Zθ(rj) = ‖rj‖
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j + δ)1/2`&jdg\_ci9j x l ciJj2a;Zc a l Z]a:Zθ(rj) = (r2
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t(x, y, z) = a1 + a2x + a3y + a4z +
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T (x, y, z) = (t1(x, y, z), t2(x, y, z), t3(x, y, z))T
a;Z k W Z_e 9\g jdda+Ya;Z lsx l a:Z1|$g x ZsZ_a;Zda w ® jY\_cfgfcfZ x 9\*Ya;Z¬e 2|$\_cfe 2ZYab x Z_a"Yc»M¥ l a;9\_a;Z!Z_jdc )x J\g x Ycf`Aa;2Z_ceJYj zdl e b2g^;`Aa
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C = −Csimilarite + λCregularite
λ¥$\ x 9\+gfav x |$\_aRj l Ya z eJcfdZ z2l c cgf¥RhJc x J\+g x Z_j z a lsz e9Z]c\_cfe Ya:ZYaRjY¡c` x hJa;Ze jg x¯l ¥RhJjdg xol c«\s¥vYa
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Jac(T ) ≈ 1µ x j Y¥$\ l cf`VaR9\ Ya;Z|$cZ x cfggfaR`VaR9\~ZA³
Jac(T ) =∞µ!a$\ l a z gfcfaR`VaR9\sZ³
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Unité de recherche INRIA Lorraine, Technopôle de Nancy-Brabois, Campus scientifique,615 rue du Jardin Botanique, BP 101, 54600 VILLERS LÈS NANCY
Unité de recherche INRIA Rennes, Irisa, Campus universitaire de Beaulieu, 35042 RENNES CedexUnité de recherche INRIA Rhône-Alpes, 655, avenue de l’Europe, 38330 MONTBONNOT ST MARTIN
Unité de recherche INRIA Rocquencourt, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY CedexUnité de recherche INRIA Sophia-Antipolis, 2004 route des Lucioles, BP 93, 06902 SOPHIA-ANTIPOLIS Cedex
ÉditeurINRIA, Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France)