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

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Submitted on 23 May 2006

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

David Tschumperlé, Rachid Deriche

To cite this version:

David Tschumperlé, Rachid Deriche. Vector-Valued Image Regularization with PDE’s :A Common Framework for Different Applications. RR-4657, INRIA. 2002. �inria-00071928�

ISRN INRIA/RR--4657--FR+ENG

a p p o r t

d e r e c h e r c h 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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∂v² =c₁Iuu+c₂ Ivv

¸n

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lkzƒLhut „

c₁, c₂ w]r†µ†zv l@xkt*z|{yr„

u,v

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λ_± ^w]r† θ_± ^{]¨ G‚"zrÂ{yv †lkvÈt {

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θ₋ {yv*t*h{yƒ{yrNw]§

t {~t*hlzj€w]ƒldžz|„ x@{yrÀt*zru«zt*z|l@„…³

g%hl§zrº ™&lkt

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lk§§&ºÀr{

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rD¨¸{yvSt*hlxk§Pwy„ „ z|x…w]§"x…wy„ l‹{]¨

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zrƒ¾w]v {wyxhl@„

w]v lÈl@ÃÀ«z©Lw]§|lkrutra

(1) min

I:Ω→^z

Z

Ω

φ(k∇Ik)dΩ ^’

⇒ (2) ∂I

∂t = div φ⁰(k∇Ik) k∇Ik ∇I

!

⇒ (3) ∂I

∂t = φ⁰(k∇Ik)

k∇Ik I_ξξ+φ⁰⁰(k∇Ik)I_ηη

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η = ∇I/k∇Ik ^w]r† ξ⊥糋{yt l€t*hNw]t~t*hz|„v lkƒL«§Pw]v*z¬­w]t*z|{yrħ|l…wy†„œt { '"`4T%& Wd "&

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zt*h

†z½ÆÁlkv lkrut

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t lkr„{yv

D= ^φ

0(k∇Ik) k∇Ik Id^³

psrZt*hz|„Nw]"lkv…‚

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lǏv {y&{L„ lw

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lk§§Swy„‹v {y"{L„ lrl

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⇒^{¸š U³}

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ψ¸¨¹«rxkt*z|{yrNw]§Ba

I:Ω→min^{z n}E(I) = Z

Ω

ψ(λ+, λ−)dΩ ^¸”

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hlkv lt*hl

λ±

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G=Pn

j=1∇Ij∇I_j^T‚

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φ¸¨¹«rxkt*z|{yr–¨¸{yv*j;«§Pw]t*z|{yr ’ U³

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∂I_i

∂t = div (D∇Ii) (i= 1..n) ^¸“

®

hlkv lÈt*hl

2×2 †z½Æ&«„*z|{yrZt lkr„ {yv

D z|„†lU¼rl@†}wy„ a

D= ∂ψ

∂λ₊(λ₊, λ₋)θ₊θ₊^T + ∂ψ

∂λ₋(λ₊, λ₋)θ₋θ₋^T

p tv l@„*«§t „t*hlkrZzraw~†z©lkv*ƒlkrx@lU4™Nwy„ l@†Âl@Ãu«Nw]t*z|{yrʄ*«x hÂwy„ ¸š U‚

®

hlkv l;t*hldžz½Æ"«„ z|{yr

t lkr„ {yv

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ψ‚w]r†¿t*hlÈlkzƒlkru©l@xU

t {yv „

θ₊, θ₋ ^{]¨ G^³

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ψ

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h{y§|lÈvw]rƒl;{]¨El@xkt {yv

φ^¸¨«rxkt*z|{yrNw]§|„rV!X­•‚šLn8[Ba

ψ(λ+, λ−) =φ(p

λ++λ−)

{yvt*hl%lk§t*vw]j~zcÌN{

® ¨vw]jml

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{yv*º)V!X…š[@a

ψ(λ₊, λ₋) =p

(1 +λ₊)(1 +λ₋)

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D x…w]r¾™"l‹„ l@lkr–wy„St*hlq ^%"'Y@

p

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t {~t*hl‹¨¹«rxkt*z|{yrNw]§ ¸” U³

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zrƒ

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φ¸¨¹«rxkt*z|{yrNw]§<x…wy„l ’ U‚

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lk§§N†l@„*zƒLrl@†~t {«r†lkv „*tw]r†€ƒl@{yjmlkt*v*z|x…w]§§Œ~t*hlÅ«r†lkv*§Œdzrƒ

„ jm{À{yt*hzrƒmv {dx@l@„ „ŏ&lkvª¨¸{yv*jml@†¿™uŒ¿t*hlLjE—° a

= 9<;C A MK7@CD7 _=h=|;C Y7:9 Y>tM MKACDMN7

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u⊥v‚&t*hz|„ȈE—° x…w]r

™&lÈl@ÃÀ«z©yw]§|lkrÀt*§Œ

®

v*zt*t lkrawy„ a

∂I_i

∂t = trace (TH_i) (i= 1..n) ^4•

®

hlkv l

H_i z|„¯t*hlr4U4"&<' <& ".

p

,+m \^< h2dm' T'Y

Ii

w]r†

T^z|„¯t*hl 2×2 ^{t lkr„ {yv}

†lU¼rl@†Âwy„Qa

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4• z|„ha

I_i(t) =I_i(t=0) ∗ G^(T,t) (i= 1..n) ^P

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G^(T,t)(x) = 1 4πt exp

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x= (x y)^T

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lk§§½4ºdr{

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{–ƒuw]«„ „ zPw]rµºlkv*rlk§|„

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G^T,t(x){…©lkvEt*hl%zj€w]ƒl

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l@lkrDwƒlkrlkv*z|x¯¨¹{yv*j {]¨"©l@xkt {yvª

©Lw]§«l@†–†z½Æ&«„*z|{yr}ˆE—°±²„ 4• w]r† ",R%D W< 0RD ",'eDt l@x hrz|ÃÀ«l@„…‚"wy„ņl@„ xkv*z™"l@†Zzr Vš‚š]’<[ ³

r{yt*hlkv„ zj~z§Pw]v‹w]v {wyx h™Nwy„ l@†Ê{yr}r{yrÈw]«„„*zPw]rÊx@{yrÀ©{y§«t*z|{yraºlkv*rlk§|„hNwy„™"l@lkrÂw]§|„ {

v {y&{L„ l@†¾¨¸{yvt*hlȄ*&l@xkz½¼NxÇx…wy„ lÈ{]¨%lk§t*vw]j~z´<§|{

® VšLš8[ ³

´<zƒL«v l;šma 9 "!#%$ + (3 /4.%/4.%/3/039( 2(*.%/ 3/4.% 1*

T

;

;%MKA@M Y> MK7@> xCL;Y7@A@M Y> 9 Y7 8=E;

—‹z½ÆÁlkv lkrx@l@„È™&lkt

®

l@lkr†z©lkv*ƒlkrx@lmt lkr„ {yv „

D ^{zr ¸š} Çw]r†Ât*vwyx@l~t lkr„ {yv „

T ^{zr 4•} ‹x…w]rµ™&l

«r†lkv „ t {À{d†Zwy„%¨¸{y§§|{

®

„…³ µl;x…w]r}†lk©lk§|{yZt*hldžz©lkv*ƒlkrx@ll@Ãu«Nw]t*z|{yr ¸š wy„ a

div (D∇I_i) = trace (DH_i) +∇I_i^T div~ (D)

® hlkv l

div~ () z|„†lU¼rl@†»wy„~w}†z©lkv*ƒlkrx@l–{y"lkvw]t {yvmwyxkt*zrƒÂ{yrej€w]t*v*z|x@l@„mw]r†ev lkt*«v*rzrƒ

©l@xkt {yv „ra

z½¨

D= (d_ij)^‚ div~ (D) =

div (d₁₁ d₁₂)^T div (d₂₁ d₂₂)^T

g%hlkrc‚Ew]r.wy††zt*z|{yrNw]§%t lkv*j

∇I_i^Tdiv~ (D) w]"l…w]v „­‚<x@{yrrl@xkt l@†t {Êt*hl 4Ddm<&"@\ "<&"' {]¨Èt*hl–t lkr„{yvm¼Nlk§|†

D^{³ p} t€j€w@Œ»"lkv*t*«v*™2t*hl}„*jm{d{yt*hzrƒ·™&lkhNw@©dz|{yv€ƒLz©lkr2™uŒ»t*hl–¼v „ t

Nw]v*t

trace (DH_i)^{‚ ®} hz|x hwyxkt*«Nw]§§Œax@{yv*v l@„*&{yr†„Èt {–w¾§|{dx…w]§E„*jm{À{yt*hzrƒZ†zv l@xkt l@†µ™uŒ}t*hl

„ "l@xkt*vw]§lk§|lkjmlkrÀt „{]¨

D^³ ‹„w~v l@„*«§t…‚Nt*hldžz©lkv*ƒlkrx@lU4™Nwy„ l@†Âl@ÃÀ«Nw]t*z|{yr ¸š %j€w…ŒZ„*jm{d{yt*h t*hlzj€w]ƒl

I ^{® zt*h ®} lkzƒLhÀt „%w]r†D†zv l@xkt*z|{yr„¯t*hNw]t¯w]v l†z½ÆÁlkv lkrut¯¨¹v {yj0t*hl„ "l@xkt*vw]§"lk§|lkjmlkrut „ {]¨

D³Gg%hz|„¾z|„¾wyxkt*«Nw]§§Œ2t*hlÊx…wy„lZ¨¸{yvDt*hla„ x…w]§Pw]v

φ^¸¨«rxkt*z|{yr ¨¸{yv*j;«§Pw]t*z|{yr ’ U‚

®

hlkv l

t*hl„*jm{d{yt*hzrƒ¿v {Àx@l@„ „‹†{dl@„*rc±t™&lkhNw@©l¼rNw]§§Œ ¸w]r†Z¨¸{yv*t*«rNw]t lk§Œ ‹wy„Èw]raz|„ {yt*v {yz|xœ{yrlL‚

†l@„ zt lÈt*hlz|„ {yt*v {yz|x¨¸{yv*j {]¨¦t*hldžz©lkv*ƒlkrx@lÇt lkr„ {yv

D= ^φ

0(k∇Ik) k∇Ik Id^³

L D= BC 7 ]9 l>BCLY; Y7BA W=|;

Åxkt*«Nw]§§Œ‚cz½¨

®

lœx@{yr„*z|†lkvt*hNw]tÈt*hl†z©lkv*ƒlkrx@l~t lkr„ {yv

D †lk&lkr†„È{yr§ŒÊ{yrÂt*hl„*"l@xkt*vw]§

lk§|lkjmlkrÀt „{]¨¦t*hlDŽ*t*v*«xkt*«v lÈt lkr„{yv

G^{‚„*«xhÊwy„ a}

D=f₁(λ₊, λ₋)θ₊θ₊^T +f₂(λ₊, λ₋)θ₋θ₋^{T Z}

®

zt*h

f₁, f₂ : R² → R^{‚ ®} hz|x hz|„Èt*hl€x…wy„lœ¨¹{yv;v {y"{L„ l@†·l@Ãu«Nw]t*z|{yr„;zr·t*hl~§zt lkvw]t*«v lU‚

t*hlkr

®

lx…w]rD†lk©lk§|{y€t*hlx@{yv*v l@„*&{yr†zrƒ†z©lkv*ƒlkrx@l‹l@Ãu«Nw]t*z|{yr

div (D∇I_i) zrut {{yv*z|lkrut l@†

{$w]§PwyxkzPw]r„…‚z4³²lL³¦t*hz|„t*vwyx@lU4™Nwy„ l@†ZˆE—° ¹†lktw]z§|„zr "lkr†z½¶ a

div (D∇Ii) =

n

X

j=1

trace (δijD+Q^ij)Hj ^WX…›

®

hlkv l€t*hl

Q^ij †l@„*zƒLrNw]t l¾w

p

<Q",k p n² <& "U4

i, j = 1..nU‚¦†lU¼rl@†wy„Çt*hlm„*Œdjœ

jmlkt*v*z|xǏNw]v*t „{]¨¦t*hl‹¨¹{y§§|{

®

zrƒ€j€w]t*v*z|x@l@„

P^{ij t*hlkrc‚} Q^ij= (P^ij+P^ijT)/2 ^a P^ij = α∇I_i^T∇I_jId

+ 2 ∂α

∂λ₊θ+θ^T₊+ ∂α

∂λ₋θ−θ^T₋

∇Ij∇I_i^TG + 2

(α+ ∂β

∂λ₊)θ₊θ₊^T + (α+ ∂β

∂λ₋)θ₋θ^T₋

∇I_j∇I_i^T

®

zt*h

α= ^f1(λ+,λ_λ⁻₊^)−f_−λ²₋^{(λ+,λ−) w]r†} β = ^λ+f2(λ+,λ_λ⁻₊^)−λ_−λ⁻₋^{f1(λ+,λ−)}

g%hz|„¯†lk©lk§|{yjmlkrut WX…› SlU¶v l@„ „ l@„%w

®

h{y§|lvw]rƒl{]¨$v lk©Àz|{y«„*§Œ~v {y&{L„ l@†D©l@xkt {yvª4©yw]§«l@†

v lkƒL«§Pw]v*z¬­w]t*z|{yr¿w]§ƒ{yv*zt*hjm„ ©Lw]v*zPw]t*z|{yrNw]§w]r†m†z©lkv*ƒlkrx@l™Nwy„ l@†~ˆE—°±²„ <zrÀt {;w]rmlU¶t lkr†l@†

t*vwyx@lU4™Nwy„l@†Dl@ÃÀ«Nw]t*z|{yrc‚x@{yj~"{L„ l@†¾{]¨*4T\ ^|" e4"' U'Y& "fm&"'4t*hNw]t¯hNw@©lwDŽ zj~§|l

ƒl@{yjmlkt*v*z|xµzrut lkv*v lktw]t*z|{yrAzr t lkv*jq{]¨§|{dx…w]§¼§t lkv*zrƒ³ g%hlÂzrut lkv l@„*t*zrƒÄ&{yzrut}z|„¿t*hNw]t

%"&&"&' e" 4"&<'D'4T< U4

Q^ijw]v lSw]&l…w]v*zrƒ‹w]r†;x@{yrut*v*z™«t l t { "

p

k,+mB4%,+"`'

fU+ \"

®

hz|xhZz|„%¼rNw]§§Œ 'O<v"`\'xfk,+m ",'",&"E",\ ' U D'4

D^³

7BC Y> B; C=E7

´v {yj t*hl@„ lv lk©dz|{y«„†lk©lk§|{yjmlkrÀt „…‚

®

lx…w]rÊr{

®

†lU¼rlœwS' " \^< \8^

! "!#j<&"' -0/1„a

∂I_i

∂t =

n

X

j=1

trace A^ijH_i

(i= 1..n) ^WX<X

®

hlkv l‹t*hl

A^ij ¨¸{yv*jm„ÅwȨ4w]j~z§ŒD{]¨

2×2 „ ŒÀj~jmlkt*v*z|x‹j€w]t*v*z|x@l@„…‚w]r†Dt*hl

H_i †l@„*zƒLrNw]t l‹t*hl

‹l@„ „*zPw]rÊj€w]t*v*z|x@l@„{]¨

I_i^³ Åxkt*«Nw]§§Œ‚"t*hz|„lU¶dv l@„„*z|{yrÊx…w]ra™"l;l@Ãu«z©Lw]§|lkrut*§Œ

®

v*zt*t lkr

®

zt*hËw

„ §zƒLhutw]™«„ lÈ{]¨¦r{ytw]t*z|{yr„…‚Nzr}wS4jd D<& "Z¨¸{yv*j a

∂I

∂t =trace~ (AH) ^WX…š

®

hlkv l

Az|„<t*hlt<& "h

p

" 4"&<'.D'4 U4

A^ij ¸w]r†z|„<zt „ lk§½¨"x@{yr„*z|†lkv l@†~wy„@4kQ.T& "U‚

w]r† Hz|„St*hlh\^< h

p

r4U4"&' %& "U4

H_j³<g%hlj€w]t*v*z½¶€v {d†«xkt

AH ^{zr WX…š} Ez|„St*hlkr

„l@lkr)4fD<& "rd h4fD<& "‚"w]r†}t*hl;{y&lkvw]t {yv

trace~ () v lkt*«v*r„t*hl;©l@xkt {yvzr

R^n‚

x@{yv*v l@„ "{yr†zrƒmt {mt*hlÈt*vwyx@lÇ{]¨l…wyx h}„*«™4j€w]t*v*z½¶–zr–t*hlÈv l@„ «§t*zrƒm©l@xkt {yv{]¨<j€w]t*v*z|x@l@„…³

2CD7 C9@; L*CD=E? @; CD=E7

g%hl~ˆS—‹° WX<Xz|„Çw€«rz½¨¹ŒdzrƒZl@Ãu«Nw]t*z|{yrËt*hNw]tÇx…w]r˙&l«„ l@†at {–†l@„ xkv*z™"l~w

®

z|†lœvw]rƒl~{]¨

©l@xkt {yvª4©Lw]§«l@†Zv lkƒL«§Pw]v*z¬­w]t*z|{yrga

´<zv „*t…‚Nzt‹†lk©lk§|{y„‹™"{yt*ha©yw]v*zPw]t*z|{yrNw]§w]r†a†z©lkv*ƒlkrx@lU4™Nwy„ l@†Ëw]v {wyx hl@„ t*hNw]tx…w]r

™"l

®

v*zt*t lkr2wy„

∂Ii

∂t = div (D∇Ii)^‚ wy„~†lk©lk§|{y&l@†»zrĄ l@xkt*z|{yr š ;zrut {ew}©lkv*Œ§|{dx…w]§

¨¹{yv*j«§Pw]t*z|{yrc³Eg%hz|„ Nw]v*t*z|xk«§Pw]v*§Œmzrxk§«†l@„¯t*hl

®

{yv*º„¯†{yrlzrnV”‚YX<XL‚YX…š‚OX­•d‚YXZ‚ÀšmX‚

šLn‚"š•‚š<P[Ew]jm{yrƒ¾{yt*hlkv „­³ ‹„‹†l@„ xkv*z™"l@†Âw]™&{…©lL‚&t*hl

2×2 t lkr„ {yv „

A^ij w]v lt*hlkr

†lU¼rl@†¾t {™&l

A^ij=δ_ijD+Q^ij³ ‹{yt l‹t*hNw]t¯t*hl

Q^ij i6=j x@{yv*v l@„*&{yr†„%hlkv lt {

†z½Æ"«„ z|{yrmx@{yrut*v*z™«t*z|{yr„¯{]¨"{yt*hlkvEx hNw]rrlk§|„

I_j zr~t*hlÅxk«v*v lkrut {yrl

I_i³<g%hz|„t" 4"&<' ' k & W<'4

p

Èx…w]rm™&lÅx@{yr„*z|†lkv l@†Dwy„SwȏNw]v*t*z|xk«§Pw]vjdO"`' ~{]¨"t*hlx@{yv*v l@„*"{yr†zrƒ

©l@xkt {yvª4©yw]§«l@†Ê†z½Æ"«„ z|{yr}ˆE—°³

Édl@x@{yr†Á‚&t*hlˆS—‹° WX<X‹w]§|„ {€ƒuw]t*hlkv „t*hl{yv*z|lkrÀt l@†dD{w]§PwyxkzPw]rµ¨¸{yv*j;«§Pw]t*z|{yr„

∂Ii

∂t = trace (TH_i)‚<™uŒ·xh{À{L„*zrƒ

A^ij=δ_ijT³}psr·t*hz|„x…wy„ lL‚¦t*hlmj€w]t*v*z½¶

A z|„†zPw]ƒ{yrNw]§

w]r†¾r{†z½Æ&«„*z|{yr¿lkrlkv*ƒLŒ€t*vw]r„ª¨¸lkv¯{dx@xk«v „¯™"lkt

®

l@lkr¿zj€w]ƒlx hNw]rrlk§|„

I_i³¦g%hl‹©l@xkt {yv x@{y«§zrƒz|„ {yr§Œ~†{yrlt*hv {y«ƒLh¾t*hlÅ«„ l{]¨Át*hl„ t*v*«xkt*«v lt lkr„ {yv

G¨¸{yvEt*hlx@{yj~«

tw]t*z|{yrD{]¨&t*hl§|{dx…w]§N„*jm{d{yt*hzrƒÇƒl@{yjmlkt*v*Œ³g%hz|„E«rz½¼Nl@„ t*hl%¨¸{yv*j;«§Pw]t*z|{yr„ v {y"{L„l@†

zrxV!Xk’‚šL›‚šL“‚Nš•j[ ³

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

µl;v {y"{L„lÇr{

®

t {m†l@„*zƒLrËwœrl

®

©l@xkt {yvª4©yw]§«l@†Êv lkƒL«§Pw]v*z¬­w]t*z|{yrˈE—° t*hNw]t¨¸{y§§|{

®

„‹†lU

„ zv l@†;§|{dx…w]§Àƒl@{yjmlkt*v*z|x%v {y&lkv*t*z|l@„…³<g%hl@„ l¯v {y"lkv*t*z|l@„

®

z§§rNw]t*«vw]§§Œ†lU¼rl%w„*"l@xkz½¼Nx ¨¸{yv*j

{]¨¦v lkƒL«§Pw]v*z¬­w]t*z|{yrʈS—‹°‚¨¹v {yj t*hlÈ©lkv*ŒDƒlkrlkv*z|xȨ¹{yv*j WX<X a

µl¿†{yrc±t ®

w]rut~t {aj~z½¶e†z½Æ&«„*z|{yr.x@{yrut*v*z™«t*z|{yr„€™"lkt

®

l@lkr.zj€w]ƒl¿x hNw]rrlk§|„…³·g%hl

†l@„*zv l@†}x@{y«§zrƒ¿™&lkt

®

l@lkrÊ©l@xkt {yvx@{yj~"{yrlkrÀt „

I_i „*h{y«§|†}{yr§Œ}w]&l…w]vzrÊt*hl;†z½¨

¨«„ z|{yrˆE—° t*hv {y«ƒLh»t*hl¿x@{yj~«tw]t*z|{yr»{]¨t*hlD„*t*v*«xkt*«v l¾t lkr„ {yv

G‚¦zr»{yv †lkvœt { x@{yrut*v {y§"t*hl§|{dx…w]§N„*jm{d{yt*hzrƒ;™&lkhNw@©dz|{yv¯{]¨Át*hlÅv lkƒL«§Pw]v*z¬­w]t*z|{yr¿v {dx@l@„ „…³¦g%hz|„Ejml…w]r„

t*hNw]t

®

lmhNw@©l~t {Z†lU¼rl€{yr§Œa{yrlm†z½Æ&«„*z|{yrµt lkr„{yv

A‚t*hlkr·x h{d{L„ l

A^ij = δ_ijA^³

r†l@„ zv l@†Zx@{y«§zrƒmt lkv*jm„w]v lÇt*hlkrÊw@©{yz|†l@†Á³

ŠrZh{yjm{yƒlkrl@{y«„v lkƒLz|{yr„ ¹x@{yv*v l@„*&{yr†zrƒ€t {œ§|{

®

©l@xkt {yv©Lw]v*zPw]t*z|{yr„ U‚

® l ® w]rutt {

"lkvª¨¸{yv*j)w]r]"`4T%& Wd "&4<,+e"`'eat*hlkv lkzrc‚$z4³²lL³Dw}šy—hl…w]t;ÌN{

®

t*hNw]t„*jm{d{yt*hËt*hl

r{yz|„ l¯l€xkz|lkrut*§Œ

®

zt*hr{]4v lU¨¸lkv*v l@††zv l@xkt*z|{yr„ a

∂Ii

∂t '∆I_i= trace (H_i)³pstjml…w]r„

t*hNw]tt*hlÈt lkr„ {yv

A j«„*tř"l"4<& WdO"&ÈzrZt*hl@„ lv lkƒLz|{yr„ha

(λ++λlim−)→0A=αId

ŠrÄ©l@xkt {yvDl@†ƒl@„ ¹x@{yv*v l@„*&{yr†zrƒ·t {ËhzƒLh2©l@xkt {yv€©Lw]v*zPw]t*z|{yr„ U‚

® l ®

w]rutDt {µ&lkvª

¨¹{yv*j w]rl<'"4<& ^dO"r4<,+",'eb!'e ,+mr\^< 4

θ₋‚zra{yv †lkv‹t {mv l@„lkv*©l t*hlkj

®

hz§|lœv lkjm{…©dzrƒ–t*hlœr{yz|„ l a

∂Ii

∂t = trace βθ−θ−TH_i^‚ ® hlkv l

β ^{z|„Èw€¨ «rxU}

t*z|{yrµ†l@xkv l…wy„*zrƒÊw]rÀŒ

®

w@Œa¨¹{yvÈ©lkv*ŒÊhzƒLhµ©Lw]v*zPw]t*z|{yr„…‚w@©{yz|†zrƒÊ„*hNw]v*µx@{yv*rlkv „Ç{…©lkvª

„*jm{d{yt*hzrƒ³<g%hz|„$jml…w]r„<t*hNw]t¦t*hlSt lkr„ {yv

Aj;«„ t<™&l2<'"4<& ^dO"Szrt*hl@„ l v lkƒLz|{yr„ a

(λ++λlim−)→0A=βθ−θ₋^T

g%hlE¨¹{y§§|{

®

zrƒj«§t*z©yw]§«l@†v lkƒL«§Pw]v*z¬­w]t*z|{yr~ˆE—°Êv l@„*"l@xkt „¦w]§§ut*hl@„lE§|{dx…w]§uƒl@{yjmlkt*v*z|x v {y

lkv*t*z|l@„ha

∂Ii

∂t = trace (TH_i) (i= 1..n) ^WX…n