État de l'art des méthodes de correction des déformations ... · ISSN 0249-6399 ISRN INRIA/RR--5589--FR+ENG apport de recherche INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET

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

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INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

État de l’art des méthodes de correction desdéformations cérébrales per-opératoires

Pierrick Coupé , Pierre Hellier and Christian Barillot

N˚5589

Juin 2005

Systèmes biologiques

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P (vi|J) ∝ P (J |vi)P (vi)eJ¨P (vi|J)

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P (J |vi)a$\Yj\sa l Àa

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j + δ)3/2 a zxol~x `V^$\ l aδa;Z]\jd\_cfgcZ]¥ z e j l |Re 9\ l gfa l ga:Z z2l e zdl cf¥$\s¥;Z©geY| x gfa;ZYag x ¬e 2|$\_cfe Yaqb x Z_a ¢Yrucf2Z_c7§

z gj2Zδx j2h `VaR9\_aJ§ z gfj2Z¯g x Re 2a c 2jdaR|$ada¶g x ¬e |3\_cfe da¶b x Z]aYc`Vcf[jda ¢ a:Zh x j2Z_ca;dda;Z

z eJZsZ_^;Ya;J\g xS)x 9\ x h auYavYe 2da l jddavc` z e l \ x 2|$ayZ_cfh dc| x \_c a z2l cf2|$c z2x ga;Àa;9\ x j e cZ_c x h aydjz e cf9\Ya·|$eJJ\ l JgaJ¢*r g c2Z]\ x l Ya

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tkYa+g x \ l~x 2Z]¬e l ` x \_cfe

Tl a:|~da l |~d¥;a "$ g jY\_cfgfcfZ x \sceJda k WVZ]eJ9\-¥;h x ga;Àa;9\*d¥$2dcfa;Z"|$e `V`Vajd|$eJ`&bdcf x cfZ_e Agcfd¥ x c l aYa+¬eJ2|3\sceJ2Z l~x Yc x gfa;Z-Yaub x Z_a

θ(rj)` x cfZ x jdh `Va;J\s¥Ra:Z jd2a z2xol \_cfa x da

t(x, y, z) = a1 + a2x + a3y + a4z +

n∑

j=1

bjθ(rj)

Kyda\ l~x Z¬e l ` x \sceJT j2 z e cf9\uaR9\ l a YaRjd¡cf` x h a:Z 3 a;Z]\ x gfe l Z+Y¥$22ca|Re `V`Va

T (x, y, z) = (t1(x, y, z), t2(x, y, z), t3(x, y, z))T

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θ(rj) =

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r2j log(r2

j )Z]c

d = 2rj

Z]cd = 3

áá æ_^-`àb

HH /'!2 "$ "22

r a:| ga:Z¶|Re 2Yc\_cfe Z¶ c9\sa lsz eJg x \_cfe T (φi) = φ′

i

l~x `Va;d¥Ra:ZZ]eJj2ZV¬e l `Va·` x \ l c|$cfaRgfgfae z aRjY\¥:| l c l a

(

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ba

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bjdda` x \ l c|$a

n × 3|Re 9\_a; x 9\

gfa;Z©|$e[a |Rca;J\~Z"2e x V2aua$\aRY2Θjdday` x \ l cf|Ra

n×ne ¨

Θij = θ(‖φi −φj‖)" xS)x 9\ x h auda;Z

Z z gcfda:Z Ya©\X z a +dcfY¦ z g x \_a a;Z]\-Ya z eJj eJc l |~deJcfZ_c l!xol bdc\ l~x c l a;Àa;9\g x YcZ\ l cfbdjY\sceJ¯Z z2x \sc x gaYa;Zz e cf9\Ya|$eJ9\ l gfa § ` x gda;j l aRj2Z_aR`VaR9\+|$aRjd¡9¦Ñ|$cMe 9\jddavc jdaR2|RayhJgeJb x gfa x ge l Z©iJj2aygfa;Z"¬e 2|$\_cfe 2Zl~x dc x ga:Z¯Yab x Z]ajY\_cfgfcfZ_¥Ra:ZAZ_e 9\¸Z_j zdz e l \A2dc=¢ a;Z k W Z_e 9\AhJ¥Rd¥ lsx ga;`VaR9\¯jY\scgfcZ]¥:Z¯gfe l ZsiJj2ai9jda *g x Yc»C¥ l aR2|RahJ¥ReJÀ¥R\ l cfi9jdaaR9\ l a ga:ZYaRjd¡ c` x h a;ZA a;Z]\ z2x ZA\ l ^:ZAc` z e l \ x 9\sa §*gfa;Z z e cfJ\daH|$eJJ\ l Jga·Z_e 9\¶a:Z zx |$¥:Z xz aRj zdl ^;ZYa·` x 2c^ l a·j2dc«¬e l `Va·aR\i9jda·gfaHdeJ`&b l a·Ya z e cf9\Ya|Re lsl a:Z z e x 2|Ray a;Z]\ z2x Z\ l ^;Z+cf` z e l \ x 9\;¢#&#JÀ ½ S #EDÉ½) d¿-:Ã½ 9 a;Z ZZ_e 9\ Y¥$dca:Z Z]j l jddaVh l cgfgfa

ΦYa

nx × ny × nzz e cfJ\~Z Ya|$eJJ\ l JgaAa;Z z2x |$¥;ZYa

δx × δy × δz¢ a;ZyY¥$¬e l ` x \sceJ2ZYag c` x hJa Z]eJJ\ l ¥ x gcZ_¥Ra;Za;·Y¥ z g x ;x 9\u|$a:Z z eJc9\~ZuYa|$eJ9\ l gfa

da¯gfaRj lz eJZ_c«\sceJHe l cfh cf x gfa¯| l ¥ x 9\ x cZ]c-jdda z2x lsx `V¥$\ l cZ x \_cfe ·|$eJ` z2x |$\_a¯daAg x \ l~x 2Z]¬e l ` x \_cfe ¢¹©a;ggfa$¦Ñ|$c z eJj )x 9\©Z ¥:| l c l auZ_e j2Z*g x ¬e l `Vay jd z2l eYYjdc\©Yau\_aRZ]a;j l 3 Yav®"¦ÑZ z gcfda:Z x `Vcgfcfa l ZR¢

T (x, y, z) =

3∑

l=0

3∑

m=0

3∑

n=0

Bl(u)Bm(v)Bn(w)φi+l,j+m,k+n

eJ¨φl a zdl ¥;Z_aR9\_ag x h l cgfgaYa:Z z e cfJ\~ZmYau|Re 9\ l gfai9jdc z2xol~x `V¥$\ l cZ_a©g x §

i, j, kcf2Yci9jdaug cf2Yc|$a

da;Z z eJc9\~ZAda |Re 9\ l gfa¶aR\u, v, w

|$e l_l a;Z z eJ2¸g x¸z eJZ_c«\sceJ l aRg x \sc aYa(x, y, z)

x j Z_aRcf Ya:Z|Re[e l Ye d2¥Ra;Z+Yag x h l cgfgaJ¢ a:Z ®©¦ÑZ z gfcfda;Z jY\_cfgfcfZ_¥;Zu|Re `V`Vav¬e 2|$\_cfe 2ZYa b x Z_a;Z+¥$\ x 9\

B0(s) = (1 − s)3 B2(s) = (−3s3 + 3s2 + 3s + 1)/6B1(s) = (3s3 − 6s2 + 4)/6 B3(s) = s3/6

k eJj l \ l eJj a l g x \ lsx 2Z¬e l ` x \_cfe e z \sc` x gacg[Z_j V\- cfJ\ l eYYj2c l a"jd2a"¬eJ2|3\sceJ&|$eY\CZ]j l g x i9jdaRgfga

j2da zdl e[|R¥;Yj l a e z \sc`VcZ x \_cfe Z]cf` z gaZ_a l~x¯l ¥ x gfcfZ_¥RaJ¢

C = −Csimilarite + λCregularite

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g x \ lsx 2Z]¬e l ` x \_cfe 1¢ x |$eJJ\ lsx c9\sayYa l ¥;h jdg xol c«\s¥yjY\scgfcZ]¥;a z aRjY\+ª$\ l avg x `Vcfdc`VcZ x \_cfe ¶daqg x d¥ l c ¥z2l a;Àcf^ l a

∫ ∫ ∫(

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à¬í!á1à°ã

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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

Jac(T ) < 0µ$¢ÑgYdeJj2Z l a:Z\sav|Re 2|Rgj l aaRZ_e jdgfchJ x 9\*i9jda+ga:Z Z z aRj aR9\

x |$cfga;`VaR9\¯ªR\ l ac` z gf¥R`Va;J\s¥Ra:Z¯Ya` x dcf^ l adcf¥ l~xol |~dciJj2aa$\¯2e jddc¬e l `Va 7 8+aR Y¥$22cfZsZ x 9\da;Z+h l cfggfa;Z+Ya z e cf9\sZ|$eJ9\ l gfa x j` x cfgfg x h aYa z gjZaR z gfj2ZZ]a l_l ¥ve ¨ |$a l \ x c Ya |$a:Z z eJc9\~ZuZ]eJ9\x |3\sc«ÃZ aR\& x jY\ l a z2x Z_Z_c«ÃZ³7|3 !ch¢m!H[µ3¢1ryc2Z_c=§1cg"a;Z]\ z e9Z_Z_cb2gaYa l ¥;djdc l aVg x Y¥R¬e l ` x \_cfe jdda;e da cfJ\s¥ l ª$\` x cZ¥Rh x gfaR`VaR9\+ga\saR` z Zuda| x g|$jdg=¢

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\se `VciJj2a § z jdcZg cf9\_a lsz e g x \_cfe /YaVg x Y¥R¬e l ` x \_cfe /\se jY\gfa e gfjd`Va £cfgfZaR¡YcfZ]\_a;J\gfa;Z`V¥$\sdeYYa;Zb x Z_¥Ra;ZVYc l a;|3\saR`VaR9\Z_j l g cfJ\saR2Z_c\_¥Ya;ZVcf` x hJa;ZAe ¨ºga:ZA\ l~x 2Z]¬e l ` x \sceJ2Z¯Z_e 9\Ya\X z a|~ x ` zdaR2Z_a YaY¥$¬e l ` x \_cfe e j |$e `¯bdcf x cZ]eJYa\ lsx 2Z]¬e l ` x \_cfe 2Z©gfcd¥ x c l a:ZR¢Â7½2¿V½ O¿:$U£#x 2eo\ye z \_ci9jdaa:Z\ujd2a \_a:|~ddciJj2ab x Z_¥RaZ_j l g 9X z eo\sd^;Z_a&Ya g x |$eJ2Z_a lsSx \_cfe Yag cf9\_a;2Z]c\_¥

davg c` x hJa x j|$eJj l Z+Yj\saR` z Z 7Ý 8É¢

I(p, t) = I(p + ∆p, t + ∆t)

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∂z

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sεS

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sa$\

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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)

ISSN 0249-6399

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État de l'art des méthodes de correction des déformations ... · ISSN 0249-6399 ISRN INRIA/RR--5589--FR+ENG apport de recherche INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET