vector-valued image regularization with pde's :a common ... · vector-valued image...

Vector-Valued Image Regularization with PDE’s :A

Common Framework for Different Applications

David Tschumperle, Rachid Deriche

To cite this version:

David Tschumperle, Rachid Deriche. Vector-Valued Image Regularization with PDE’s :A Com-mon Framework for Different Applications. RR-4657, INRIA. 2002. <inria-00071928>

HAL Id: inria-00071928

https://hal.inria.fr/inria-00071928

Submitted on 23 May 2006

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https://hal.inria.fr/inria-00071928

ISS

N 0

249-

6399

ISR

N IN

RIA

/RR

--46

57--

FR

+E

NG

ap por t de r ech er ch e

THÈME 3

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Vector-Valued Image Regularization with PDE’s :A Common Framework for Different Applications

David Tschumperlé — Rachid Deriche

N° 4657

Décembre 2002

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)

⇒ (3)∂I

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© l@xkt yv raz½¨

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(

div(

(d11 d12)T)

div(

(d21 d22)T)

)

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³

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*«xhÊwy a

D = f1(λ+, λ−)θ+θT+ + f2(λ+, λ−)θ−θ

T−

Z ® zt*h

f1, f2 : R2 → R

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div (D∇Ii) =

n∑

j=1

trace(

(δijD + Qij)Hj

) WX

® hlkv lt*hlQij l@*zLrNw]t l¾w p <Q",k p n2 <&"U4

i, j = 1..nU¦lU¼rl@wyÇt*hlm*dj

jmlkt*v*z|xÇNw]v*t ]¨¦t*hl¨¹y§§| ® zrjw]t*v*z|x@l@Pij t*hlkrc

Qij = (Pij + PijT

)/2a

Pij = α ∇ITi ∇IjId

+ 2

(

∂α

∂λ+θ+θ

T+ +

∂α

∂λ−θ−θ

T−

)

∇Ij∇ITi G

+ 2

(

(α+∂β

∂λ+)θ+θ

T+ + (α+

∂β

∂λ−)θ−θ

T−

)

∇Ij∇ITi

® zt*hα = f1(λ+,λ−)−f2(λ+,λ−)

λ+−λ−

w]rβ = λ+f2(λ+,λ−)−λ−f1(λ+,λ−)

λ+−λ−

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=n∑

j=1

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(i = 1..n)WX<X

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Hil@*zLrNw]t lt*hl

l@ *zPw]rÊjw]t*v*z|x@l@]¨Ii

³ Åxkt*«Nw]§§ "t*hz|lU¶dv l@*z|yrÊxw]ra"l;l@Ãu«z©Lw]§|lkrut*§ ® v*zt*t lkr ® zt*hËw §zLhutw]« lÈ]¨¦rytw]t*z|yrNzrwS4jdD<&" Z¨¸yv*j a

∂I

∂t= ~trace (AH)

WX

® hlkv l A z|<t*hlt<&" h p " 4"&<'.D'4U4Aij ¸w]rz|<zt lk§½¨"x@yr*z|lkv l@[email protected]&"U

w]r H z|St*hlh\ ^<h p r4U4"&' %&"U4Hj

³<g%hljw]t*v*z½¶v d«xkt AH zr WX Ez|St*hlkrl@lkr)4fD<&" rdh4fD<&" "w]rt*hl;y&lkvw]t yv ~trace ()

v lkt*«v*rt*hl;© l@xkt yvzrRn

x@yv*v l@ "yrzrmt mt*hlÈt*vwyx@lÇ]¨lwyx h*«4jw]t*v*z½¶zrt*hlÈv l@ «§t*zrm© l@xkt yv]¨<jw]t*v*z|x@l@³

2CD7 C9@; L*CD=E? @; CD=E7 g%hl~S° WX<Xz|Çw«rz½¨¹dzrZl@Ãu«Nw]t*z|yrËt*hNw]tÇxw]rË&l« l@at l@ xkv*z"l~w ® z|lvw]r l~]¨© l@xkt yvª4©Lw]§«l@Zv lkL«§Pw]v*z¬w]t*z|yrga

´<zv *tNztlk© lk§|y"yt*ha©yw]v*zPw]t*z|yrNw]§w]raz© lkv* lkrx@lU4Nwy l@Ëw]v wyx hl@ t*hNw]txw]r"l ® v*zt*t lkr2wy ∂Ii

∂t = div (D∇Ii) wy~lk© lk§|y&l@»zrÄ l@xkt*z|yr ;zrut ew© lkv*§|dxw]§

¨¹yv*j«§Pw]t*z|yrc³Eg%hz| Nw]v*t*z|xk«§Pw]v*§mzrxk§«l@¯t*hl ® yv*º¯yrlzrnV YX<XLYXOXdYXZÀmX Ln" <P [Ew]jmyr¾yt*hlkv ³l@ xkv*z"l@Âw]&© lL&t*hl

2 × 2t lkr yv

Aij w]v lt*hlkrlU¼rl@¾t &l

Aij = δijD+Qij ³ yt lt*hNw]t¯t*hlQij i 6= j

x@yv*v l@*&yr%hlkv lt z½Æ"« z|yrmx@yrut*v*z«t*z|yr¯]¨"yt*hlkvEx hNw]rrlk§|

Ijzr~t*hlÅxk«v*v lkrut yrl

Ii³<g%hz|t" 4"&<'

' k &W<'4 p Èxw]rm&lÅx@yr*z|lkv l@DwySwÈNw]v*t*z|xk«§Pw]vjdO"`' ~]¨"t*hlx@yv*v l@*"yrzr© l@xkt yvª4©yw]§«l@Êz½Æ"« z|yrE°³

v

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Édl@x@yrÁ&t*hlS° WX<Xw]§| uw]t*hlkv t*hlyv*z|lkrÀt l@dDw]§PwyxkzPw]rµ¨¸yv*j;«§Pw]t*z|yr ∂Ii∂t =

trace (THi)<u·xhÀL*zr

Aij = δijT³psr·t*hz|xwy lL¦t*hlmjw]t*v*z½¶ A z|zPw] yrNw]§

w]r¾rz½Æ&«*z|yr¿lkrlkv*Lt*vw]rª¨¸lkv¯dx@xk«v ¯"lkt ® l@lkr¿zjw] lx hNw]rrlk§|Ii

³¦g%hl© l@xkt yvx@y«§zrz| yr§~yrlt*hv y«Lh¾t*hlÅ« l]¨Át*hl t*v*«xkt*«v lt lkr yv

G¨¸yvEt*hlx@yj~«

tw]t*z|yrD]¨&t*hl§|dxw]§N*jmdyt*hzrÇ l@yjmlkt*v* ³g%hz|E«rz½¼Nl@ t*hl%¨¸yv*j;«§Pw]t*z|yr v y"Ll@zrxV!XkLLN j[ ³

7 ; E? DM ;eCM 9%C=E7

µl;v y"LlÇr ® t ml@*zLrËwrl ® © l@xkt yvª4©yw]§«l@Êv lkL«§Pw]v*z¬w]t*z|yrËE° t*hNw]t¨¸y§§| ® lU zv l@;§|dxw]§À l@yjmlkt*v*z|x%v y&lkv*t*z|l@³<g%hl@ l¯v y"lkv*t*z|l@ ® z§§rNw]t*«vw]§§lU¼rl%w*"l@xkz½¼Nx ¨¸yv*j]¨¦v lkL«§Pw]v*z¬w]t*z|yrÊS°¨¹v yj t*hlÈ© lkv*D lkrlkv*z|xÈ¨¹yv*j WX<X a

µl¿yrc± t ® w]rut~t aj~z½¶ez½Æ&«*z|yr.x@yrut*v*z«t*z|yr"lkt ® [email protected]] l¿x hNw]rrlk§|³·g%hll@*zv l@x@y«§zr¿&lkt ® l@lkrÊ© l@xkt yvx@yj~"yrlkrÀt

Ii*hy«§|yr§w]&lw]vzrÊt*hl;z½¨

¨ « z|yrE° t*hv y«Lh»t*hl¿x@yj~«tw]t*z|yr»]¨t*hlD*t*v*«xkt*«v l¾t lkr yvG

¦zr»yv lkvt x@yrut*v y§"t*hl§|dxw]§N*jmdyt*hzr;&lkhNw@©dz|yv¯]¨Át*hlÅv lkL«§Pw]v*z¬w]t*z|yr¿v dx@l@ ³¦g%hz|Ejmlw]rt*hNw]t ® lmhNw@© l~t ZlU¼rlyr§ayrlmz½Æ&«*z|yrµt lkryv

At*hlkr·x hdL l

Aij = δijA³

rl@ zv l@Zx@y«§zrmt lkv*jmw]v lÇt*hlkrÊw@© yz|l@Á³

rZhyjmy lkrl@y«v lkLz|yr ¹x@yv*v l@*&yrzrt §| ® © l@xkt yv©Lw]v*zPw]t*z|yr U ® l ® w]rutt "lkvª¨¸yv*j)w]r]"`4T%&Wd "&4<,+e"`'eat*hlkv lkzrc$z4³²lL³Dwyhlw]t;ÌN ® t*hNw]t*jmdyt*hËt*hlryz| l¯lxkz|lkrut*§ ® zt*hr]4v lU¨¸lkv*v l@zv l@xkt*z|yr a ∂Ii

∂t ' ∆Ii = trace (Hi)³pstjmlw]r

t*hNw]tt*hlÈt lkr yvA

j«*tÅ"l" 4<&WdO"&ÈzrZt*hl@ lv lkLz|yrha

lim(λ++λ−)→0

A = αId

rÄ© l@xkt yvDl@ l@ ¹x@yv*v l@*&yrzr·t ËhzLh2© l@xkt yv©Lw]v*zPw]t*z|yr U ® l ® w]rutDt µ&lkvª¨¹yv*j w]rl<'" 4<&^dO"r4<,+",'eb!'e ,+mr\ ^<4

θ−zrayv lkvt mv l@lkv*© l

t*hlkj ® hz§|lv lkjm©dzrt*hlryz| l a ∂Ii∂t = trace

(

βθ−θ−THi

) ® hlkv lβ

z|Èw¨ «rxUt*z|yrµl@xkv lwy*zrÊw]rÀ ® w@a¨¹yvÈ© lkv*ÊhzLhµ©Lw]v*zPw]t*z|yrw@© yz|zrÊ*hNw]v*µx@yv*rlkv Ç© lkvª*jmdyt*hzr³<g%hz|$jmlw]r<t*hNw]t¦t*hlSt lkr yv

Aj;« t<&l2<'" 4<&^dO"Szrt*hl@ l v lkLz|yr a

lim(λ++λ−)→0

A = βθ−θT−

g%hlE¨¹y§§| ® zrj«§t*z©yw]§«l@v lkL«§Pw]v*z¬w]t*z|yr~E°Êv l@*"l@xkt ¦w]§§ut*hl@lE§|dxw]§u l@yjmlkt*v*z|x v ylkv*t*z|l@ha

∂Ii∂t

= trace (THi) (i = 1..n)WXn

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'O "$#j 'Ni4k³³p tz|jmyv lv l@xkz| lL&*zrx@lt*hl;x@yj~«t l@a§|dxw]§$º lkv*rlk§¦x@yv*v l@*&yrlU¶wyxkt*§Zt t*hl

*jmdyt*hzrt ;"lkvª¨¸yv*jZ³ zj~v l@xkz| l l@x@yrlkv*z©Lw]t*z© l@¯hNw@© lt Ç&lx@yj~«t l@¹´<z³ n U³

Ê¨¹yvÂ*hyv*t x@yj~zr µ]¨¾t*hz|Â x hlkjmlL ® l»hNw@© lt jmlkrut*z|yr0t*hNw]tÂztÂz|Âjmyv lt*zjmlUx@yr «j~zr*zrx@l ® lhNw© lt Êx@yj~«t l¾wz½Æ&lkv lkrÀtuw]« zPw]reº lkv*rlk§ z4³²lL³ZlU¶d&yrlkrut*zPw]§¨¹«rxkt*z|yr ¦w]tElwyxh~zjw] lÅ"yzrÀtuw]r¨¸yvlwyx hmzt lkvw]t*z|yrc³´yvEy«vElU¶&lkv*zjmlkrut ® lÅx hLl5 × 5

x@yrÀ© y§«t*z|yrÊº lkv*rlk§|³

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¡ Ò ~ Î Î u" Ò Ò Ï ¤ Ò ¡ Î Î Ó w ·¢ Î aÅzLztw]§¦zjw] l@«lt Dt*hlkzvz~jmlkjmyv*¾ z¬lLw]v lÈ]¨¹t lkr*t yv l@¿zrZwjmyv lÈx@yj~Nwyxkt%¨¹yv*j ytw]zrl@ ® zt*h)! 44k

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X E4T+ee2d /Q"+m

zt Èlkv*z©Lw]t*z© l@ ® zt*hËv l@*&l@xktt Iix

w]rIiyU&zrÂt lkv*jmÇ]¨¯lkv*z©Lw]t*z© l@ ® zt*hËv l@*&l@xktÈt

t*hlgkl

a∂λ±∂Iix

=∑

k,l

∂λ±∂gkl

∂gkl∂Iix

and∂λ±∂Iiy

=∑

k,l

∂λ±∂gkl

∂gkl∂Iiy

WX

g%hl;lU¶dv l@ z|yr ∂gkl

∂Iix

w]r ∂gkl

∂Iiy

w]v lÇNw]v*t*z|xk«§Pw]v*§¿ zj~§|la

∂g11∂Iix

= 2Iix

∂g11∂Iiy

= 0

and

∂g12∂Iix

= Iiy

∂g12∂Iiy

= Iix

and

∂g22∂Iix

= 0

∂g22∂Iiy

= 2Iiy

z4³²l WX Åxw]rZ"l ® v*zt*t lkrawy a

∂λ±∂Iix

∂λ±∂Iiy

=

2∂λ±∂g11

∂λ±∂g12

∂λ±∂g12

2∂λ±∂g22

∇IiWXP

g%hÀ«yrlÈ§Pwy*tÅy*twyxk§|lÇv lkjw]zrÅt "lxkv L l@Át*hNw]tz|¯¼rzrmt*hl¨¸yv*jw]§clU¶dv l@ z|yr]¨ ∂λ±

∂gkl

³lkj~zr·t*hNw]tt*hl

λ±w]r

θ±w]v lDt*hlDlkz lkrÀ©yw]§«l@mw]rlkz lkru© l@xkt yv ]¨t*hlD t*v*«xkt*«v l

t lkryvG

aG = λ+ θ+θ

T+ + λ− θ−θ

T−g%hl;lkv*z©yw]t*z|yr]¨¦t*hz|Åt lkr yv ® zt*hZv l@*&l@xktÅt myrlÇ]¨¦zt x@Àlxkz|lkrÀt

gklz| a

∂G

∂gkl=

∂λ+

∂gklθ+θ

T+ +

∂λ−∂gkl

θ−θT−

WXZ

+ λ+∂θ+∂gkl

θT+ + λ−∂θ−∂gkl

θT−

+ λ+ θ+∂θT+∂gkl

+ λ− θ−∂θT−∂gkl

ayv l@© lkv"wyÅt*hlθ±

w]v lÇ«rztw]v*w]ryv*t*hy yrNw]§$lkz lkru© l@xkt yv ® lÇhNw@© la

θT+θ+ = θT−θ− = 1

θT+θ− = θT−θ+ = 0

and

∂θT+∂gkl

θ+ = θT+∂θ+∂gkl

= 0

∂θT−∂gkl

θ− = θT−∂θ−∂gkl

= 0

¸L

v

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µl¼v tÈj;«§t*z§Ât*hlml@ÃÀ«Nw]t*z|yr WXZ uθT±

w]tÈt*hl§|lU¨¹tcÀθ±

w]tÈt*hlmv*zLhutct*hlkr·« lt*hlv y&lkv*t*z|l@ ¸L U³pst¯w]§§| ® %hzLh¾ zj~§z½¼Nxw]t*z|yrw]r§|lwy¯t ;t*hl@ lt ® v lk§Pw]t*z|yr a

∂λ+

∂gkl= θT+

∂G

∂gklθ+

w]r ∂λ−∂gkl

= θT−∂G

∂gklθ−

¸mX

°SÃu«Nw]t*z|yr ¸mXÈ¨¹yv*jw]§§et lk§§%«;h ® lkz lkru©Lw]§«l@~]¨wz½Æ&«*z|yr»t lkr yvG

©yw]v* ® zt*hv l@ "l@xktEt wÈNw]v*t*z|xk«§Pw]v%x@Àlxkz|lkrÀt

gkl]¨

G³ Åxkt*«Nw]§§ dt*hz|SzrÀt lkv l@*t*zrv y"lkv*tsmxw]r&l

v © l@Â¨¸yv;w]ruË*dj~jmlkt*v*z|x~jw]t*v*z½¶&³~´NyvÇzr*tw]rx@lL$w]«t*hyv ]¨qV!X8[Ev y&L l@·w¿ zj~z§Pw]vlkjmyr t*vw]t*z|yrÊzrÊwm«v lk§¿jw]t*v*z½¶¨¹yv*jZN§|lwyzrt mt*hl;w]jmlÇv l@*«§t³¯g%hlk« l@ztt lw]§ ® zt*hZ lkrlkvw]§x@©Lw]v*zPw]rx@lÇjw]t*v*z|x@l@³

ayv l@© lkvzry«vxwy lLt*hlÈjw]t*v*z|x@l@ ∂G∂gkl

w]v l© lkv**zj~§|l.a

∂G

∂g11=

(

1 00 0

)

,∂G

∂g12=

(

0 11 0

)

and∂G

∂g22=

(

0 00 1

)

zt*hw]§§ct*hl@ lÈlk§|lkjmlkrÀt ® lÇxw]rlU¶v l@ WXP Åwyha

∂λ+

∂Iix

∂λ+

∂Iiy

= 2 θ+θT+∇Ii and

∂λ−∂Iix

∂λ−∂Iiy

= 2 θ−θT−∇Ii

¸L

´<zrNw]§§ v lk§Pwyxkzr ¸L ¾zr t*hlµ°«§|lkvªDw]Lvw]r ll@ÃÀ«Nw]t*z|yr WX w]r WX ULz© l@Zt*hl© l@xkt yvª4©Lw]§«l@ZLvwyz|lkrÀtl@ x@lkrÀt]¨¦t*hl¨ «rxkt*z|yrNw]§ ¸ qa

minI:Ω→ z n

∫

Ωψ(λ+, λ−) dΩ =⇒ ∂Ii

∂t= 2 div

([

∂ψ

∂λ+θ+θ

T+ +

∂ψ

∂λ−θ−θ

T−

]

∇Ii)

¸Ln

f¨¸yvi = 1..n

yt lt*hNw]t ¸Ln ¯z|wz© lkv* lkrx@lU4Nwy l@Êl@ÃÀ«Nw]t*z|yr*«xhZt*hNw]tha

∂Ii∂t

= div (D∇Ii) where D = 2∂ψ

∂λ+θ+θ

T+ + 2

∂ψ

∂λ−θ−θ

T−

D ∈ P(2)z|Åt*hlkrÊw

2 × 2z½Æ&«*z|yrt lkr yv ® hLlÇlkz lkru©Lw]§«l@w]v la

λ1 = 2∂ψ

∂λ+and λ2 = 2

∂ψ

∂λ−

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wy ÀxkzPw]t l@¿t mt*hl@ lÈx@yv*v l@*&yrzrDyv*t*hyryv*jw]§$lkz lkrÀ© l@xkt yv ra

u1 = θ+ and u2 = θ−pst~z|Dw]§| ® yv*t*h2t Ëjmlkrut*z|yr t*hNw]tx@yj~«t*zrt*hz|mLvwyz|lkrut¾l@ x@lkrut¾z|myrlÊlU¶wyxkt*§zrt*hlw]jml ® w@ ® hlkrZlw]§zr ® zt*hZzjw] lÈyjw]zr

ΩlU¼rl@zrhzLhlkvzjmlkr*z|yrNw]§

Nwyx@l@ Ω ⊂ R

p ® hlkv lp > 2

Êyv l%Nw]v*t*z|xk«§Pw]v*§ Àt*hlxwy lÅ]¨cny © y§«jmlÅv lkL«§Pw]v*z¬w]t*z|yrp = 3

Åxw]r"l ® v*zt*t lkrÂwy a

minI:Ω→ z n

∫

Ωψ(λ1, λ2, λ3) dΩ =⇒ ∂Ii

∂t= 2 div

([

∂ψ

∂λ1θ1θ

T1 +

∂ψ

∂λ2θ2θ

T2 +

∂ψ

∂λ3θ3θ

T3

]

∇Ii)

psr¾t*hz|Åxwy lLt*hlλ1,2,3

w]v lt*hlt*hv l@lÈlkz lkru©Lw]§«l@]¨$t*hl3× 3

*t*v*«xkt*«v lt lkr yvG

w]rθ1,2,3

w]v lt*hlÇx@yv*v l@*&yrzrDyv*t*hyryv*jw]§<lkz lkru© l@xkt yv ³

6GFF E';!

aL*tz© lkv* lkrx@lU4Nwyl@v lkL«§Pw]v*z¬w]t*z|yrËE°±²wyxkt*zrDyrÊj;«§t*z©Lw]§«l@zjw] l@hNw@© l;t*hl¨¸y§§| ® zrm¨¹yv*j a

∂Ii∂t

= div (D∇Ii) (i = 1..n)¸]

® hlkv lD

z|%wÇz½Æ&«*z|yr¾t lkr yv¯Nwy l@n<' k ' @U4T <WT;y"lkvw]t yv ³g%hl¨4wyxkt z| t*hNw]tDz|]¨¹t lkrx@yj~«t l@¨¹v yj t*hl*t*v*«xkt*«v lt lkr yv

G =∑n

j=1 ∇Ij∇ITjw]rmlk&lkrjw]zr§

yrt*hl*Nw]t*zPw]§Nlkv*z©Lw]t*z© l@Iix

w]rIiy

³<psrut*«zt*z© lk§ wySt*hlz© lkv* lkrx@ldiv () = ∂

∂x + ∂∂yz|zt lk§½¨w%¼v t$yv lkv¦lkv*z©yw]t*z© l¯y"lkvw]t yv ® l hy«§|Ç&l w]§|lEt ® v*zt l ¸] yr§ ® zt*h¼v *t

w]r l@x@yrm*Nw]t*zPw]§"lkv*z©yw]t*z© l@Iix

Iiy

Iixx

Iixy

w]rIiyy

³$g%hÀ«ÀztSx@y«§|~&llU¶dv l@ l@® zt*h¿yv*z|lkrut l@ w]§PwyxkzPw]r¯zr¿lwyx h¾zjw] lxhNw]rrlk§

Iiwy ® lk§§4Àz4³²lw]rDlU¶v l@ *z|yr¾Nwy l@Dyr

t*hlt*vwyx@l;y"lkvw]t yv ∂Ii∂t = trace (DHi)

³

µl ® w]rutt jw]º lmt*hlm§zrºa&lkt ® l@lkr·t*hl~t ® Êz½Æ&lkv lkrÀtz½Æ&«*z|yrµt lkr yv D

w]rT

zrt*hlz© lkv* lkrx@lU4Nwy l@w]r·t*vwyx@lU4Nwy l@µv lkL«§Pw]v*z¬w]t*z|yr.E°±²zrµt*hlxwy l ® hlkr

Dz|

' <B<'4'Oa∂Ii∂t

= div (D∇Ii)w]r ∂Ii

∂t= trace (THi)

® lryt*z|x@l@¿zr¿t*hlv lk©Àz|y« l@xkt*z|yrct*hl@ lt ® ¨¹yv*j«§Pw]t*z|yrw]v lÈw]§jmL*tÅl@Ãu«z©Lw]§|lkrut«t Dw]rwyzt*z|yrNw]§t lkv*j lk&lkrzrDyrt*hl \ "<&"') p ,+mrD'4 !

Da

div (D∇Ii) = trace (DHIi) + ∇ITi ~div (D)¸L

v

!"$#%&"&')(*"&,+.-0/21.3 4 XZ

® hlkv l ~div ()z|Åt*hl <&" ",\ ' Uk³

ArNw]t*«vw]§z|lwz|Åt*hlkrt ~l@x@yj~"L lÇt*hlÇwyzt*z|yrNw]§t lkv*j ∇ITi ~div (D)zrÀt "'OD^

EdO!%T"'4klU¶dv l@ l@ ® zt*hÊwyzt*z|yrNw]§$z½Æ&«*z|yrZt lkr yv Q

zrt*hlÈt*vwyx@l;y"lkvw]t yv³

´Nyv;t*hz|Ç«v*&L lL ® l ® z§§Sx@yr*z|lkvt*hNw]t;t*hlmz© lkv* lkrx@lDt lkr yvD

z|;lU¼rl@w]tlwyx h&yzrut

x ∈ Ωu

D = f1(λ+, λ−) θ+θT+ + f2(λ+, λ−) θ−θ

T− with f1/2 : R

2 → R¸L

pstjmlw]rt*hNw]tD

z|yr§¿lU¶v l@ l@¨ v yj t*hl;lkz lkru©Lw]§«l@λ±

w]rt*hl;lkz lkrÀ© l@xkt yv θ±]¨¦t*hlÈ*t*v*«xkt*«v l;t lkr yv

Ga

G = λ+ θ+θT+ + λ− θ−θ

T−

g%hz|z|zrl@l@Âw~© lkv* lkrlkv*z|xhud&yt*hl@*z|t*hNw]tz|© lkv*z½¼Nl@Àt*hlÇjw£*yv*zt ]¨Et*hl;v ]&L l@e© l@xkt yvª4©yw]§«l@Äv lkL«§Pw]v*z¬w]t*z|yrÄjmlkt*hÀ¦¨¹yvzr*tw]rx@l¿t*hl¿yrl¾v y"Ll@ezr &lkrz½¶ a

∂Ii∂t

= div (D∇Ii) ® zt*h(26)

w]r

f1(λ+, λ−) = 2∂ψ

∂λ+

f2(λ+, λ−) = 2∂ψ

∂λ−psrÊyv lkvt lk© lk§|yÂt*hlwyzt*z|yrNw]§¦z½Æ&«*z|yrat lkv*j ∇ITi ~div (D)

zrt*hll@Ãu«Nw]t*z|yr ¸L U® lÈv y&L lt ® v*zt l

Dwyw§zrlw]vx@yj;zrNw]t*z|yrÊ]¨

Gw]r

Ida

D = α(λ+, λ−)G + β(λ+, λ−)Id¸

z4³²l ® lÇ lkNw]vw]t l;t*hl " 4<&WdO"&w]rx'"`4T%&Wd "&Nw]v*t ]¨D

® zt*h

α =f1(λ+, λ−) − f2(λ+, λ−)

λ+ − λ−and β =

λ+f2(λ+, λ−) − λ−f1(λ+, λ−)

λ+ − λ−

¸<P psrl@l@Á ® lÈhNw@© l

αG + βId =f1 − f2

λ+ − λ−(λ+ θ+θ

T+ + λ− θ−θ

T−) +

λ+f2 − λ−f1

λ+ − λ−(θ+θ

T+ + θ−θ

T−)

=1

λ+ − λ−

[

θ+θT+ (λ+f1 − λ−f1) + θ−θ

T− (λ+f2 − λ−f2)

]

= f1 θ+θT+ + f2 θ−θ

T−

= D

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L E4T+ee2d /Q"+m

lkv l ® lÇwy «jml@Dt*hNw]tλ+ 6= λ−

z4³²lt*hl*t*v*«xkt*«v lt lkr yvG

z|Åw]rz| yt*v yz|xU³ rÀ ® w@ z½¨

Gz|Çz|yt*v yz|xLyrl lkrlkvw]§§·xhÀLl@w]r_" 4<&WdO"&¾z½Æ&«*z|yr·t lkryv

Dt À<zr·t*hl

z© lkv* lkrx@ly"lkvw]t yvm]¨ ¸L UEz4³²lf1(λ+, λ−) = f2(λ+, λ−)

³psret*hz|mxwylL ® lxhÀL lα = 0

w]rβ = f1(λ+, λ−)

³

g%hz|l@x@yj~"L*zt*z|yrÊz|Å« lU¨¹«§Át v l ® v*zt lÇt*hljw]t*v*z½¶Zz© lkv* lkrx@l ~div (D)zrut a

~div (D) = α ~div (G) + G∇α+ ∇β ¸<Z w]rZt*hlÇwyzt*z|yrNw]§t lkv*j ]¨¦t*hlÇl@ÃÀ«Nw]t*z|yr ¸L ® y«§|Z"lÈx@yj~«t l@awy a

∇IT ~div (D) = trace(

~div (D)∇ITi)

= αtrace(

~div (G)∇ITi) ¸nL

+ trace(

G∇α∇ITi) ¸nmX

+ trace(

∇β∇ITi) ¸nL

psrt*hl¨¸y§§| ® zr ® lÈv y&L lÈt ¼r¿¨¸yv*jw]§lU¶v l@ *z|yr]¨ ¸nL U ¸nmXÅw]r ¸nL U³

• ´<zv *tv lkjmlkj"lkvt*hNw]tt*hlÈ*t*v*«xkt*«v lÈt lkryvG

z|ÅlU¼rl@awy a

G =

n∑

j=1

∇Ij∇IjT

µlhNw© lÇt*hlkrxa

~div (G) =n∑

j=1

~div

(

I2jx IjxIjy

IjxIjy I2jy

)

=

n∑

j=1

(

2 IjxIjxx + IjxIjyy + IjyIjxy

IjxIjxy + IjyIjxx + 2 IjyIjyy

)

=n∑

j=1

(

Ijx(Ijxx + Ijyy)Ijy(Ijxx + Ijyy)

)

+

(

IjxIjxx + IjyIjxy

IjxIjxy + IjyIjyy

)

=

n∑

j=1

∆Ij∇Ij + Hj∇Ij

® hlkv l∆Ij

w]rHj

w]v lav l@*"l@xkt*z© lk§2t*hl w]§PwyxkzPw]rw]r t*hl l@*zPw]r ]¨t*hlazjw] lx@yj~&yrlkrut

Ij³

v

!"$#%&"&')(*"&,+.-0/21.3 4 mX

g%hlkrc ® lÇxw]r ® v*zt lÇt*hlÈlU¶dv l@ z|yrÊnL~wy a

αtrace(

~div (G)∇IiT)

=

n∑

j=1

αtrace(

Hj

[

∇IiT∇IjId + ∇Ij∇IiT]) ¸nLn

• µl~¼rNw]§§ÊhNw© lmt Zx@yj~«t l ∇α w]r ∇β Ázrµt*hlmlU¶v l@ *z|yr ¸nmXÈw]r ¸nL U³g%hz|xw]r"lÈyrlÀ¾t*hl;l@x@yj~"L*zt*z|yrca

∇α =∂α

∂λ+∇λ+ +

∂α

∂λ−∇λ− and ∇β =

∂β

∂λ+∇λ+ +

∂β

∂λ−∇λ−

¸n]

w]rÊwyÅt*hlλ±

lkz lkrÀ©yw]§«l@]¨<t*hl;*t*v*«xkt*«v lÈt lkr yvG

lk"lkryrt*hlIjx

w]rIjy

a

∇λ± =

(

λ±x

λ±y

)

=

n∑

j=1

( ∂λ±∂Ijx

Ijxx + ∂λ±∂Ijy

Ijxy

∂λ±∂Ijx

Ijxy + ∂λ±∂Ijy

Ijyy

)

=

n∑

j=1

HIj

∂λ±∂Ixj

∂λ±∂Iyj

psr "lkrz½¶ ® llkv*z©Lw]t l@mlkz lkrÀ©yw]§«l@¯]¨ÁwÇ*t*v*«xkt*«v lt lkr yvG

® zt*hv l@ "l@xktt Èt*hl Nw]t*zPw]§czjw] lÇlkv*z©Lw]t*z© l@³ µlÇlkrl@« ® zt*ht*hl¨¹y§§| ® zrv lk§Pw]t*z|yrxa

∂λ±∂Ixj

∂λ±∂Iyj

= 2θ±θT±∇Ij

g%hlkrc

∇λ± =

n∑

j=1

2Hjθ±θT±∇Ij

¸nL

µlÅxw]r~v lk§Pwyx@l ¸nL <zrut t*hllU¶v l@ *z|yr]¨¸n] U zrmyv lkvt ¼rmt*hlÅ*Nw]t*zPw]§Lvwyz|lkrut ]¨α

w]rβa

∇α =

n∑

j=1

2Hj

(

∂α

∂λ+θ+θ

T+ +

∂α

∂λ−θ+θ

T+

)

∇Ij

∇β =

n∑

j=1

2Hj

(

∂β

∂λ+θ+θ

T+ +

∂β

∂λ−θ+θ

T+

)

∇Ij

¸nL

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*zr ¸nL U ® l¼rNw]§§x@yj~«t lt*hlt ® j~z|*zrNw]v*t ¸nmX w]r ¸nL ]¨t*hlwyzt*z|yrNw]§t lkv*j ∇ITi ~div (D)

a

trace(

G∇α∇ITi)

=

n∑

j=1

trace

(

2 GHj

(

∂α

∂λ+θ+θ

T+ +

∂α

∂λ−θ−θ

T−

)

∇Ij∇ITi)

trace(

∇β∇ITi)

=n∑

j=1

trace

(

2 Hj

(

∂β

∂λ+θ+θ

T+ +

∂β

∂λ−θ−θ

T−

)

∇Ij∇ITi)

¸n

• g%hlÇ¼rNw]§$*t lkax@yr*z|*t zrÊ«t*t*zrDt y lkt*hlkvt*hl;l@ÃÀ«Nw]t*z|yr ¸nLn w]r ¸n UzrÊyv lkvt ~lU¶v l@ Åt*hlwyzt*z|yrNw]§t lkv*j ∇ITi ~div (D)

zrt*hlÇS° ¸L U³

∇ITi ~div (D) =n∑

j=1

trace(

HjPij) ¸n<P

® hlkv lÇt*hlPij w]v lt*hl¨¸y§§| ® zr

2 × 2jw]t*v*z|x@l@ha

Pij = α ∇ITi ∇IjId

+ 2

(

∂α

∂λ+θ+θ

T+ +

∂α

∂λ−θ−θ

T−

)

∇Ij∇ITi G

+ 2

(

(α+∂β

∂λ+)θ+θ

T+ + (α+

∂β

∂λ−)θ−θ

T−

)

∇Ij∇ITi¸n<Z

yt lt*hNw]tÈt*hlzrz|x@l@i, j

zrËt*hlrytw]t*z|yrPij %b' <t4"R<'O<Dt*hl~x@Àlxkz|lkrÀt ;]¨w

jw]t*v*z½¶P

$«t;t*hl~Nw]vw]jmlkt lkv ]¨%t*hl~¨4w]j~z§Ëx@yr*z|*t*zr]¨n2 jw]t*v*z|x@l@

Pij ¹lwyxhe]¨t*hlkj z|w

2 × 2jw]t*v*z½¶ U³

g%hl;jw]t*v*z|x@l@Pii w]v l Àj~jmlkt*v*z|xL«t lkrlkvw]§§rytt*hl

Pij ® hlkv li 6= j

U&*zrx@lÇt*hlLvwyz|lkrÀt ∇Ii

w]r ∇Ijw]v lrytw]§zLrl@zrZt*hlÇ lkrlkvw]§xwy lL³

Slkt ® l ® w]rÀt;t lU¶v l@ Èt*hlDl@Ãu«Nw]t*z|yr ¸n<P Èyr§ ® zt*h·*dj~jmlkt*v*z|xmjw]t*v*z|x@l@$zreyv lkvt ¿zrÀt lkv*v lktÇzt;wyÇw¿*«j?]¨¯§|Àxw]§E*jmdyt*hzrv dx@l@ l@Èyv*z|lkrut l@·u " 4"&'gD'4U4k³´Nyv*t*«rNw]t lk§ t*hlÈt*vwyx@lÇy&lkvw]t yvhNwyÅt*hz|*zj~§|lv y&lkv*t )a

trace (AH) = trace

(

A + AT

2H

)

v

!"$#%&"&')(*"&,+.-0/21.3 4 Ln

® hlkv l(A + AT )/2

z|w2 × 2

4k .T&"jw]t*v*z½¶ t*hlÇ*dj~jmlkt*v*z|xÈNw]v*t]ÄU³

g%hÀ« ® llU¼rlÇt*hl*Àj~jmlkt*v*z|x;jw]t*v*z|x@l@Qij x@yv*v l@*&yrzr¾t t*hl;*dj~jmlkt*v*z|xÇNw]v*t

]¨¦t*hlPij a

Qij =Pij + PijT

2

u

w]r ® lÈhNw© la∇ITi ~div (D) =

n∑

j=1

trace(

HjQij)

´<zrNw]§§ t*hl;z© lkv* lkrx@lU4Nwy l@E° ¸L %xw]rZ&l ® v*zt*t lkrÊwy a

div (D∇Ii) =

n∑

j=1

trace(

(δijD + Qij)Hj

) X

® hlkv lδij

z|Åt*hl v yrl@xº lkv±²*dj;&y§0a

δij =

0z½ï 6= j

1z½ï = j

g%hlv lkL«§Pw]v*z¬w]t*z|yr.E° XÈz|;l@ÃÀ«z©yw]§|lkrÀtt t*hlDz© lkv* lkrx@lU4Nwy l@»l@Ãu«Nw]t*z|yr ∂Ii∂t =

div (D∇Ii)«t ® zt*hawt*vwyx@lU4Nwy l@¨¸yv*j;«§Pw]t*z|yrc³

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V L8[³]gx hu«j~&lkv*§|w]r³£lkv*z|x hlL³ yr*t*vw]zrl@mw]r«rx@yr*t*vw]zrl@"lL±²$¨¸yv¦© l@xkt yvzjw] l%v l@*t yvw]t*z|yrc³psrps©Lw]v «*t*© y§§4Ll@zt yvm- WjU^",'e 4 p ,+m ,+ 'O<"`'O<\"&<'*' p ^T'OU ' 'Ok 4"`4UdNw] l@ XLnOXL %lkv* lkrc yv ® w L«rlLLmXL³

V j[³Sg xhu«j~&lkv*§|Âw]r2;³Elkv*z|xhlL³ z½Æ&«*z|yr E°±²Dyr l@xkt yvª Èw]§«l@ zjw] l@³^10101 Y"R<'OO- Û44",'e nj%j#T",' kKXZ ¸ a!XdL&LLL³

V <P8[ ³ µlkz|x º lkv*t³ 'Y" 4<&WdO"&]/ " 4"&' ",' - Û44",'eL³ glk«rlkvª Elkv*§Pw]ÉÀt*«t*t*uw]v*tEXZ<Z<P³

V <Z8[ ³µlkz|x º lkv*tw]r ³NÉdx hryv*v³ At*hl@yv lkt*z|xw]§¨ vw]jml ® yv*ºD¨¸yvx@yru© lU¶v lkL«§Pw]v*z¬lkv zr"lU4Nwy l@x@yj~«tw]t*z|yr.]¨zjw] lDjmyt*z|yrc³ E+m 'OD' <&"&<'O e'O p *<QdO DT K"`4"'Ndu ¸n a ]udL]"l@x@lkj"lkvLLmXL³

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v

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v

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!0$ (%(*/ 1 .%1*.% ( % !64 × 64

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Unité de recherche INRIA Sophia Antipolis2004, route des Lucioles - BP 93 - 06902 Sophia Antipolis Cedex (France)

Unité de recherche INRIA Lorraine : LORIA, Technopôle de Nancy-Brabois - Campus scientifique615, rue du Jardin Botanique - BP 101 - 54602 Villers-lès-Nancy Cedex (France)

Unité de recherche INRIA Rennes : IRISA, Campus universitaire de Beaulieu - 35042 Rennes Cedex (France)Unité de recherche INRIA Rhône-Alpes : 655, avenue de l’Europe - 38330 Montbonnot-St-Martin (France)

Unité de recherche INRIA Rocquencourt : Domaine de Voluceau - Rocquencourt - BP 105 - 78153 Le Chesnay Cedex (France)

ÉditeurINRIA - Domaine de Voluceau - Rocquencourt, BP 105 - 78153 Le Chesnay Cedex (France)

ISSN 0249-6399

vector-valued image regularization with pde's :a common ... · vector-valued image...

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