Hamiltonian approach to geodesic image matching
aa r X i v : . [ m a t h . O C ] J a n HAMILTONIAN APPROACH TO GEODESIC IMAGE MATCHINGFRANÇOIS-XAVIER VIALARDAbstra t. This paper presents a generalization to image mat hing of the Hamiltonian ap-proa h for planar urve mat hing developed in the ontext of group of di(cid:27)eomorphisms. Wepropose an e(cid:30) ient framework to deal with dis ontinuous images in any dimension, for example2D or 3D. In this ontext, we give the stru ture of the initial momentum (whi h happens tobe de omposed in a smooth part and a singular part) thanks to a derivation lemma interestingin itself. The se ond part develops a Hamiltonian interpretation of the variational problem,derived from the optimal ontrol theory point of view.Contents1. Introdu tion 12. Framework and notations 32.1. The spa e of dis ontinuous images 32.2. The spa e of deformations 43. Derivation lemma 54. Minimizing the energy fun tional 64.1. Existen e and equations of geodesi s 74.2. Re onstru tion of geodesi s with the initial momentum 95. Hamiltonian generalization 125.1. Weak formulation 135.2. Uniqueness of the weak solutions 155.3. On the existen e of weak solutions 176. Con lusion 187. Proof of the lemma 188. Appendix 288.1. Central lemma of [GTL06℄ 288.2. A short lemma 28Referen es 291. Introdu tionThis paper arose from the attempt to develop the multi-modal image mat hing in the frameworkof large deformation di(cid:27)eomorphisms. Initiated by the work of Grenander, this ontext was deeplyused sin e [Tro95℄, espe ially with appli ations to omputational anatomy. The method followedis the lassi al minization of an energy on the spa e of di(cid:27)eomorphisms, whi h enables to omputegeodesi s on this spa e and to derive the evolution equations. In most of the papers, the group ofdi(cid:27)eomorphisms a ts on the support of the template; we add to this one a di(cid:27)eomorphisms groupa tion on the level set of the template. This a tion is a natural way to ope with the multi-modalDate: O tober 29, 2018.1991 Mathemati s Subje t Classi(cid:28) ation. Primary: 58b10; Se ondary: 49J45, 68T10.Key words and phrases. Variational al ulus, energy minimization, Hamiltonian System, shape representationand re ognition, geodesi , in(cid:28)nite dimensional riemannian manifolds, Lips hitz domain.1 FRANÇOIS-XAVIER VIALARDmat hing and ould be, in a ertain way, ompared to the metamorphoses approa h exposed in[TY05℄: the metamorphoses are another way to a t on the images but the goal is very di(cid:27)erent inour ase. Mat hing in our ontext, is to (cid:28)nd a ouple ( η, φ ) whi h minimizes the energy E ( η, φ ) = D ( Id, ( η, φ )) + 1 σ k η ◦ I ◦ φ − − I targ k L , (1)with I the initial fun tion (or image) and I targ the target fun tion, Id is the identity map in theprodu t of groups, σ is a alibration parameter. The distan e D is obtained through a produ t ofRiemannian metri s on the di(cid:27)eomorphisms groups.All the omplexity is then arried by the group of di(cid:27)eomorphisms and its a tion: in the parti ular ase of landmark mat hing, the geodesi s are well des ribed. The problem is redu ed in this ase to understand the geodesi (cid:29)ow on a (cid:28)nite dimensional Riemannian manifold. It should bealso emphasized that this problem an be seen as an optimal ontrol problem. In [BMTY05℄,numeri al implementation of gradient based methods are strongly developed through a semi-Lagrangian method for omputing the geodesi s. A Hamiltonian formulation an be adopted toprovide e(cid:30) ient appli ations and omputations through the use of the onservation of momenta.In [VMTY04℄, statisti s are done on the initial momenta whi h is a relative signature of thetarget fun tions. The existen e of geodesi s from an initial momentum was deeply developed in[TY05℄, but this work dealt only with smooth fun tions for I (essentially H ) however with avery large lass of momenta. An attempt to understand the stru ture of the momentum for aninitial dis ontinuous fun tion was done in the mat hing of planar urves in [GTL06℄.We propose thereafter a framework to treat dis ontinuous fun tions in any dimension: the mainpoint is to derive the energy fun tion in this ontext.Finally, we hose to give a Hamiltonian interpretation of the equations whi h is the proper wayto handle the onservation of momentum. This formulation in ludes the work done in [GTL06℄but does not apture the landmark mat hing. The formulation we adopt gives a weak sense tothe equations and we prove existen e and uniqueness for the weak Hamiltonian equations withina large set of initial data. A word on the stru ture of initial data: the arti le on planar mat hing([GTL06℄) fo uses on Jordan urves. The main result is the existen e for all time and uniqueness ofHamiltonian (cid:29)ow. The initial data are roughly a Jordan urve for the position variable and a ve tor(cid:28)eld on this urve for the momentum. In our ontext, we hoose four variables ( I , Σ , p , P ) . I is the initial fun tion with a set of di ontinuities Σ , p is the momentum on the set Σ and P isthe momentum for the smooth part of the initial fun tion. This is a natural way to understandthe problem and the hoi e to keep the set of dis ontinuity as a position variable an lead to largerappli ations than only hoose I as position variable.The paper is organized as follows. We start with a presentation of the framework underlyingequation (1). We present a key lemma on erning the data atta hment term with respe t to the η and φ variables. Its proof is postponed to the last se tion. Then we derive the geodesi s equationsand ensure the existen e of a solution for all time from an initial momentum. In the se ond part ofthis paper, we give the weak formulation of the Hamiltonian equations, and deal with the existen eand uniqueness for this Hamiltonian formulation.AMILTONIAN APPROACH TO GEODESIC IMAGE MATCHING 32. Framework and notations2.1. The spa e of dis ontinuous images. Let n ≥ and M ⊂ R n a C bounded open setdi(cid:27)eomorphi to the unit ball.We denote by BV ( M ) the set of fun tions of bounded variation. The reader is not supposed tohave a broad knowledge of BV fun tions. Below, we restri t ourselves to a subset of BV fun tionswhi h does not require the te hni al material of BV fun tions. However it is the most naturalway to introdu e our framework. Re all a de(cid:28)nition of BV fun tions:De(cid:28)nition 1. A fun tion f ∈ L ( M ) has bounded variation in M if sup { Z M f div φ dx | φ ∈ C c ( M, R n ) , | φ | ∞ ≤ } < ∞ . In this ase, Df is de(cid:28)ned by R M f div φ dx = − R M Df φ dx .De(cid:28)nition 2. We de(cid:28)ne Im ( M ) ⊂ BV ( M ) su h that for ea h fun tion f ∈ Im ( M ) , there existsa partition of M in Lips hitz domains ( U i ) i ∈ [0 ,k ] for an integer k ≥ , and the restri tion f | U i isLips hitz .Remark 1. The extension theorem of Lips hitz fun tion in R n enables us to onsider that onea h U i , f | U i is the restri tion of a Lips hitz fun tion de(cid:28)ned on R n .On the de(cid:28)nition of a Lips hitz domain U : we use here (to shorten the previous de(cid:28)nition) a largea eptation of Lips hitz domains whi h an be found in hapter 2 of [DZ01℄. Namely, U is a lipdomain if there exists a Lips hitz open set Ω su h that Ω ⊂ U ⊂ ¯Ω . In the proof of the derivationlemma 7, we give the lassi al de(cid:28)nition of Lips hitz open set that we use above. In a nutshell,an open set is Lips hitz if for every point of the boundary there exists an a(cid:30)ne basis of R n inwhi h we an des ribe the boundary of the open set as the graph of a Lips hitz fun tion on R n − .We hose to deal with Lips hitz domains be ause it makes sense in the ontext of appli ation toimages.Example 1. The most simple example is a pie ewise onstant fun tion, f = P ki =1 a i U i with a i ∈ R .Remark 2. Our framework does not allow us to treat the dis ontinuities along a usp, but we andeal with the orners respe ting the Lips hitz ondition.Let f ∈ Im ( M ) , we denote by J f the set of the jump part of f . As a BV ( M ) fun tion, we an write the distributional derivative of f : Df = ∇ f + D c f + j ( f )( x ) H n − x J f . ∇ f is theabsolutely ontinuous part of the distributional derivative with respe t to the Lebesgue measureand D c is the Cantor part of the derivative. In other words, with the lassi al notations j ( f )( x ) =( f + ( x ) − f − ( x )) ν f ( x ) , where ( f + , f − , ν f ) : J f R n × R n × S n − is a Borel fun tion. Thefun tions f + and f − are respe tively de(cid:28)ned as f + ( x ) = lim t + f ( x + tν f ( x )) and f − ( x ) =lim t − f ( x + tν f ( x )) . Naturally, j ( f ) does not depend on the hoi e of the representation of ν f ,in fa t j ( f ) is homogeneous to the gradient. See for referen e [Bra98℄ or [AFP00℄. In our ase,the Cantor part is null from the de(cid:28)nition.We then write for f ∈ Im ( M ) , FRANÇOIS-XAVIER VIALARD Df = ∇ f + j ( f )( x ) H n − x J f . (2)2.2. The spa e of deformations. We denote by V M , <, > V a Hilbert spa e of square integrableve tor (cid:28)elds on M , whi h an be ontinuously inje ted in ( χ p ( M ) , k . k p, ∞ ) , the ve tor spa e of C p with p ≥ ve tor (cid:28)elds whi h vanish on ∂M . Hen e, there exists a onstant c V su h that for all v ∈ V : k v k p, ∞ ≤ c V k v k V . Hen e this Hilbert spa e is also a RKHS (Reprodu ing Kernel Hilbert Spa e), and we denote by k V ( x, . ) α the unique element of H whi h veri(cid:28)es for all v ∈ H : < v ( x ) , α > = < k V ( x, . ) α, v > H ,where <, > is the eu lidean s alar produ t and α a ve tor in R n . This will enable an a tion onthe support M .We denote by S, <, > S a Hilbert spa e of square integrable ve tor (cid:28)elds on R , as above. We denoteby k S ( x, . ) its reprodu ing kernel. This will enable the a tion on the level set of the fun tions.Through the following paragraph, we re all the well-known properties on the (cid:29)ow of su h ve tor(cid:28)elds and its ontrol. Most of them an be found in hapter of [Gla05℄, and are elementaryappli ations of Gronwall inequalities. (See Appendix B in [Eva98℄)Let v ∈ L ([0 , , V ) , then with [Tro95℄ the (cid:29)ow is de(cid:28)ned: ∂ t φ v ,t = v t ◦ φ v ,t , (3) φ = Id. (4)For all time t ∈ [0 , , φ v ,t is a C di(cid:27)eomorphism of M and the appli ation t d x φ v ,t is ontinuousand solution of the equation: d x φ v ,t = Id + Z t d φ v ,s ( x ) v s .d x φ vs ds. (5)We dispose of the following ontrols, with respe t to the ve tor (cid:28)elds; let u and v be two ve tor(cid:28)elds in ∈ L ([0 , , V ) and T ≤ : k φ u ,t − φ v ,t k ∞ ≤ c V k v − u k L [0 ,T ] exp( c V k v k L [0 ,T ] ) , (6) k dφ u ,t − dφ v ,t k ∞ ≤ c V k v − u k L [0 ,T ] exp( c V k v k L [0 ,T ] ) . (7)And we have ontrols with respe t to the time, with [ s, t ] ⊂ [0 , T ] : k φ v ,t − φ v ,s k ∞ ≤ Z ts k v r k ∞ dr ≤ c V Z ts k v r k V dr, (8) k φ v ,t − φ v ,s k ∞ ≤ c V p | s − t |k v k L , (9) k dφ v ,t − dφ v ,s k ∞ ≤ C exp( C ′ √ T k v k L [0 ,T ] ) Z ts k v r k V dr. (10)with the onstants C and C ′ depending only on c V . Obviously these results are valid if S repla es V . In this ase, we write η ,t for the (cid:29)ow generated by s t . With the group relation for the (cid:29)ow, η t,u ◦ η s,t = η s,u .AMILTONIAN APPROACH TO GEODESIC IMAGE MATCHING 5The group we onsider is the produ t group of all the di(cid:27)eomorphisms we an obtain through the(cid:29)ow of u ∈ L ([0 , , V × S ) .We aim to minimize the following quantity, with µ the Lebesgue measure: J ( v t , s t ) = λ Z k v t k V dt + β Z k s t k S dt + Z M | η , ◦ I ◦ φ − , ( u ) − I targ ( u ) | dµ ( u ) , (11)with η ,t the (cid:29)ow asso iated to s t . Remark that the metri we pla e on the produ t groups V × S is the produ t of the metri on ea h group whi h is represented by the (cid:28)rst two terms in (11). Thefun tions I , I targ lie in Im ( M ) . In one se tion below, we prove lassi ally that there exists at leastone solution and we derive the geodesi equations whi h give the form of the initial momentum.3. Derivation lemmaThis derivation lemma may be useful in many situations where dis ontinuities arise. Considerfor exemple two Lips hitz open sets U and V . One may want to deform one of these open setswhile the se ond remains un hanged ((cid:28)gure below). The basi ase is the following: J t = Z V χ U ◦ φ − t dx = µ ( V ∩ φ t ( U )) , with µ the Lebesgue measure. We answer to the di(cid:27)erentiation of J t , we obtain a sort of Stokesformula with a perturbation term. We dis uss below a more general formula to apply in our ontext. The (cid:28)nal result is the proposition 1:Lemma 1. Let U, V two bounded Lips hitz domains of R n . Let X a Lips hitz ve tor (cid:28)eld on R n and φ t the asso iated (cid:29)ow. Finally, let g and f Lips hitz real fun tions on R n . Consider thefollowing quantity depending on t , J t = Z φ t ( U ) f ◦ φ − t g V dµ, where dµ is the Lebesgue measure, then ∂ t | t =0 + J t = Z U − < ∇ f, X > g V dµ + Z ∂U < X, n > f g ˜ V ( X ) dµ | ∂U . (12)with ˜ V ( X )( y ) = lim ǫ + ¯ V ( y + ǫX ) , if the limit exists, elsewhere. And we denote by dµ | ∂U the measure on ∂U and n the outer unit normal.As a orollary, we dedu e:Corollary 1. We have, ∂ t | t =0 + µ ( V ∩ φ t ( U )) = Z ∂U < X, n > ˜ V ( X ) dµ | ∂U , with ˜ V ( X )( y ) = lim ǫ + ¯ V ( y + ǫX ) , if the limit exists, elsewhere. And we denote by dµ | ∂U the measure on ∂U and n the outer unit normal.In this ase, the derivation formula is a Stokes' formula in whi h one takes only into a ount thedeformation viewed in V .Below is a (cid:28)gure to illustrate the lemma: FRANÇOIS-XAVIER VIALARD V U
Figure 1. Evolution of the area between two Lips hitz urves. (The arrowsrepresent X along the boundary of U )Remark 3. We ould generalize the lemma to (cid:28)nite interse tion of Lips hitz domains, with thesame s heme of the proof developed above. We gain hen e generality whi h seems to be very naturalfor on rete appli ations.This generalization for Lips hitz domains is su(cid:30) ient for the appli ation we aim, and this appli- ation is presented in the paragraph below to derive the geodesi equations. Hen e we present the orollary we use in the next paragraph.Theorem 1. Let ( f, g ) ∈ Im ( M ) , X a Lips hitz ve tor (cid:28)eld on R n and φ t the asso iated (cid:29)ow. J t = Z M f ◦ φ − t ( x ) g ( x ) dµ ( x ) , then the derivation of J t is: ∂ t | t =0+ J t = Z M − < ∇ f, X > gdx − Z ( f + − f − )˜ g < ν f , X > d H n − . (13)with ˜ g X ( x ) := lim t + g ( φ t ( x )) if the limit exists and if not, ˜ g X ( x ) = 0 .Proof: Writing f as f = P ki =1 f x ∈ U i where ( U i ) i =1 ,...,n is the partition in domains asso iatedto f , and using the same expression for g , by linearity of integration, we fall in the ase of theproposition 3. (cid:3) A last remark on the formulation of the lemma, we an rewrite the equation (13) in a more ompa tform: ∂ t | t =0+ J t = − Z M h Df, X i ˜ g, with Df the notation for the derivative for SBV fun tion and ˜ g is the fun tion de(cid:28)ned above.Remark that µ a.e. ˜ g = g , these two fun tions di(cid:27)er on J f .4. Minimizing the energy fun tionalThe existen e of geodesi s is a lassi al fa t, but in this framework the derivation of the geodesi equations did not appear to the author in the existing literature. With the metri introdu edabove, a geodesi in the produ t spa e is a produ t of geodesi s. We hose to understand theAMILTONIAN APPROACH TO GEODESIC IMAGE MATCHING 7two geodesi s separately for te hni al reasons. We ould also have des ribed the geodesi s in theprodu t spa e V × S , this point of view will be detailed in the Hamiltonian formulation of theequations.4.1. Existen e and equations of geodesi s.Theorem 2. Let ( I , I tar ) ∈ Im ( M ) , we onsider the fun tional J on H = L ([0 , , V × S ) de(cid:28)nedin (11). There exists ( v, s ) ∈ H su h that J ( v, s ) = min ( v,s ) ∈ H J ( v, s ) . For su h a minimizer, thereexists ( p a , p b , p c ) ∈ L ( M, R n ) × L ( J I , R n ) × L ( M, R ) su h that: βs t = Z M p c ( y ) d [ η t, ] I st ( y ) k S ( I st ( y ) , . ) dµ ( y ) , (14) λv t = Z M k V ( φ ,t ( x ) , . )[ dφ ,t ] − ∗ x ( p a ( x )) dµ ( x )+ Z J I k V ( φ ,t ( x ) , . )[ dφ ,t ] − ∗ x ( p b ( x )) dµ | J I ( x ) , (15)with: I st = η ,t ◦ I ◦ φ , ,I vt = η , ◦ I ◦ φ t, , and J I the jump set of I . More pre isely for the ( p a , p c ) we show, we have the equation: p a ( x ) = − Jac( φ , ( x )) ∇ | x I v p c ( x ) . (16)Proof:On the spa e H , the strong losed balls are ompa t for the weak topology. The fun tional J is lower semi- ontinuous, so we obtain the existen e of a minimizer. As referen e for the weaktopology [Bré94℄. We (cid:28)nd here [TY05℄ a proof that the (cid:29)ow is ontinuous for the weak topology, themain point to prove the semi- ontinuity: if ( u n , s n ) ⇀ ( u, v ) in H , then ( φ , , η , ) ( φ , , η , ) .We (cid:28)rst di(cid:27)erentiate w.r.t. the ve tor (cid:28)eld s ∈ L ([0 , T ] , S ) , we denote by ˜ s a perturbation of s .Using the lemma 10 in appendix, we write: ∂ ˜ s η , ( x ) = R [ dη t, ] | η ,t ( x ) ˜ s t ( η ,t ( x )) dt . We alreadyintrodu e the kernel: Z [ β < s t , ˜ s t > S + Z M I − I targ )[ dη t, ] | η ,t ( I s ( y )) < k S ( I st ( y ) , . ) , ˜ s t > S dµ ( y )] dt = 0 , it leads to: βs t + Z M I ( y ) − I targ ( y ))[ dη t, ] | I st ( y ) k S ( I st ( y ) , . ) dµ ( y ) = 0 . With the notation p c = − I − I targ ) , we have the (cid:28)rst equation announ ed.For the se ond equation, we need the derivation lemma detailed in se tion 3. In order to use thelemma, we (cid:28)rst need to develop the atta hment term: Z M | η , ◦ I ◦ φ − , ( u ) − I targ ( u ) | dµ ( u ) = Z M ( I v ) − I v I targ + I targ dµ, Now, only the (cid:28)rst two terms are involved in the derivation, and we apply the lemma to these twoterms. (A tually the lemma is ne essary only for the se ond term.) FRANÇOIS-XAVIER VIALARDAgain, we have with the lemma 10: V := ∂ ǫ φ , ( x ) = − Z d ( φ t, ) φ ,t ( x ) (˜ v [ φ ,t ( x )]) dt. We onsider the semi-derivation of (11) at the minimum with respe t to the displa ement (cid:28)eld v ,we use the notations of SBV fun tions for the derivatives: λ Z < v t , ˜ v t > dt + Z D [( I v ) ] V − Z DI v V ˜ I ˜ vtarg = 0 ,λ Z < v t , ˜ v t > dt + Z M < ∇ I v , V > ( I v − I targ ) dµ + Z J Iv (cid:0) j ([ I v ] ) − j ( I v ) (cid:1) V ˜ I targ d H n − = 0 . As ( I v ) and I v have the same dis ontinuity set, the se ond integration is only over J I v .We apply a version of the entral lemma in [GTL06℄ whi h is detailed in appendix (see lemma 9). g : L ([0 , , V ) L ([0 , , V ) × L ( M, R n ) × L ( J I v , R n )˜ v (˜ v, V , V | J I v ) , We ensure that B := { ( v, ∇ I v ( I v − I targ ) , (cid:0) j ([ I v ] ) − j ( I v ) (cid:1) ˜ I targ ) , ˜ v ∈ L ([0 , , V ) } is bounded.For ea h ˜ v , | ˜ I ˜ vtarg | ∞ ≤ | I targ | ∞ . (This assumption ould be weakened.) When e we get with thelemma 9 the existen e of ˜ I targ ∈ Conv( B ) (we observe that the Lebesgue part of ˜ I targ is equal to I targ , the modi(cid:28) ation is on the set J I v ), su h that: λ Z < v t , ˜ v t > dt + Z D [( I v ) ] V − Z DI v V ˜ I targ = 0 ,λ Z < v t , ˜ v t > dt + Z M ∇ I v ( I v − I targ ) dµ + Z J Iv j ([ I v ] ) V − Z J Iv j ( I v ) ˜ I targ V = 0 . Now, we aim to obtain the expli it geodesi equations by introdu ing the kernel, we denote by A (˜ v ) := R D [( I v ) ] V − R DI v V ˜ I targ the pseudo derivative of the atta hment term and we denotealso: ˜∆( x ) := j (( I ) ) ◦ φ , ( x ) − j ( I ) ◦ φ , ( x ) ˜ I targ ( x ) , whi h de(cid:28)nes a normal ve tor (cid:28)eld on J I v . A (˜ v ) = Z − Z M I − I targ )( y ) < k V ( φ ,t ( y ) , . ) d ( φ t, ) ∗ φ ,t ( y ) ( ∇ | φ , ( y ) I v ) , ˜ v t > V dµ ( y ) − Z φ , ( J I ) < k V ( φ ,t ( y ) , . ) d ( φ t, ) ∗ φ ,t ( y ) ( ˜∆( y )) , ˜ v t ( y ) > V dµ | φ , ( J I ) ( y ) dt. With the hange of variable x = φ , ( y ) , A (˜ v ) = Z − Z M I − I targ )( φ , ( x ))Jac( φ , ( x )) < k V ( φ ,t ( x ) , . ) d ( φ t, ) ∗ φ ,t ( x ) ( ∇ | x I v ) , ˜ v t > V dµ − Z J I Jac( φ , ( x )) | dφ , ( n x ) | < k V ( φ ,t ( x ) , . ) d ( φ t, ) ∗ φ ,t ( x ) ( ˜∆( φ , ( x )) , ˜ v t > V dµ | J I dt, (17)AMILTONIAN APPROACH TO GEODESIC IMAGE MATCHING 9with n x a normal unit ve tor to J I in x ∈ J I . Note that in the se ond term, the hange ofvariable a ts on the hypersurfa e J I . This explains the term Jac( φ , ( x )) | dφ , ( n x ) | whi h orresponds to theJa obian term for the smooth part.We are done, λv t = Z M φ , ( x ))Jac( φ , ( x )) k V ( φ ,t ( x ) , . ) d ( φ − ,t ) ∗ φ ,t ( x ) ( ∇ | x I v ) dµ + Z J I Jac( φ , ( x )) | dφ , ( n x ) | k V ( φ ,t ( x ) , . ) d ( φ − ,t ) ∗ φ ,t ( x ) ( ˜∆( φ , ( x ))) dµ | J I . With p a ( x ) = 2∆( φ , ( x ))Jac( φ , ( x )) ∇ | x ( I v ) and p b ( x ) = ˜∆( φ , ( x )) Jac( φ , ( x )) | dφ , ( n x ) | , we have thegeodesi equations. (cid:3) These geodesi equations are a ne essary ondition for optimality. In the next paragraph, we showthat if ( p a , p b , p c ) is given, we an re onstru t the geodesi s.4.2. Re onstru tion of geodesi s with the initial momentum. We (cid:28)rst demonstrate thatif a ve tor (cid:28)eld is a solution to the geodesi equations, then the norm is onstant in time.Proposition 1. Constant speed urves in ve tor (cid:28)elds spa esIf a ve tor (cid:28)eld s t is a solution of equation (14) and the kernel is di(cid:27)erentiable then k s t k is onstant.If a ve tor (cid:28)eld v t is a solution of equation (15) and the kernel is di(cid:27)erentiable then k v t k is onstant.Proof:We prove the (cid:28)rst point: k s t k S = Z M Z M p ( y ′ ) d [ η t, ] I st ( y ′ ) k S ( I st ( y ′ ) , I st ( y )) p ( y ) d [ η t, ] I st ( y ) dµ ( y ′ ) dµ ( y ) . Remark that a.e. ∂ t ( d [ η t, ] I st ( y ) ) = − d [ s t ] I st ( y ) d [ η t, ] I st ( y ) . This equation is obtained by a derivationof the group relation: d [ η , ] I s = d [ η t, ◦ η ,t ] I s , and with the derivation of the equation (14): d [ s t ] x = Z M d [ η t, ] ∗ I st ( y ) p c ( y ) ∂ k S ( x, I st ( y )) dy. As ds t ∈ L ([0 , then the equation (5) proves that d [ η t, ] I st ( . ) is absolutely ontinuous. As thespa e of absolutely ontinuous fun tions is an algebra, k s t k S is also absolutely ontinuous. Toobtain the result, it su(cid:30) es to prove that the derivate vanishes a.e. ∂ t k s t k = − Z M Z M p ( y ′ ) ds t ( I st ( y ′ )) d [ η t, ] I st ( y ′ ) k S ( I st ( y ′ )) , I st ( y ))) p ( y ) d [ η t, ] I st ( y ) dµ ( y ′ ) dµ ( y )+ Z M Z M p ( y ′ ) d [ η t, ] I st ( y ′ ) ∂ k S ( I st ( y ′ )) , I st ( y ))) s t ( I st ( y ′ )) p ( y ) d [ η t, ] I st ( y ) dµ ( y ′ ) dµ ( y ) = 0 . The se ond point is very similar. We underline that the equation (15) is a parti ular ase ofthe following, with a measure ν whi h has a Lebesgue part and a singular part on the set J I ofdis ontinuities of the fun tion I . We also de(cid:28)ne: p t ( x ) = ( d [ φ ,t ] ∗ x ) − ( p a ( x ) x/ ∈ J I + p b ( x ) x ∈ J I ) , k v t k = Z Z p t ( x ) k V ( φ ,t ( x ) , φ ,t ( y )) p t ( y ) dν ( x ) dν ( y ) . Remark that ∂ t p t ( x ) = − dv ∗ t ◦ φ ,t ( x ) p t ( x ) , and we di(cid:27)erentiate: ∂ t k v t k = − Z Z Z p t ( x ) ∂k V ( φ ,t ( x ) , φ ,t ( z )) p t ( z ) k V ( φ ,t ( x ) , φ ,t ( y )) p t ( y ) dν ( x ) dν ( y ) dν ( z )+ Z Z Z p t ( x ) ∂k V ( φ ,t ( x ) , φ ,t ( y )) p t ( y ) k V ( φ ,t ( x ) , φ ,t ( z )) p t ( z ) dν ( x ) dν ( y ) dν ( z ) = 0 . (cid:3) This proposition is ru ial to establish that the geodesi s are de(cid:28)ned for all time. Namely, weanswer to existen e and uniqueness of solutions to (the set J I but ould be mu h more generalthan the dis ontinuity set of a fun tion in Im ( M ) ): η ,t = Id + Z t s u ◦ η u du,βs t ( . ) = Z M p c ( y ) d [ η t, ] I st ( y ) k S ( I st ( y )) , . ) dµ ( y ) ,φ ,t = Id + Z t v u ◦ φ u du,λv t ( . ) = Z M k V ( ., φ ,t ( x ))[ dφ ,t ] − ∗ x ( p a ( x )) dµ ( x )+ Z J I k V ( ., φ ,t ( x ))[ dφ ,t ] − ∗ x ( p b ( x )) dµ | J I ( x ) dt. (18)On purpose, this system of equations is de oupled in v and s . The proof of the next propositiontreats both ases in the same time but it ould be separated.Proposition 2. For T su(cid:30) iently small, the system of equations (18) with ( p a , p b , p c ) ∈ L ( M, R n ) × L ( J I , R n ) × L ( M, R ) has a unique solution if the kernel is di(cid:27)erentiable and its (cid:28)rst derivative is Lips hitz.Proof:We aim to apply the (cid:28)xed point theorem on the Bana h spa e L ([0 , T ] , V × S ) . We estimate theLips hitz oe(cid:30) ient of the following appli ation: Ξ : L ([0 , T ] , V × S ) L ([0 , T ] , V × S )( v, s ) ( ξ ( v ) , ξ ( s )) , (19)with ξ ( s ) t = Z M p c ( x ) d [ η t, ] ˜ I t ( x ) k S ( ˜ I t ( x )) , . ) dµ ( x ) ,ξ ( v ) t = Z M k V ( ., φ ,t ( x ))[ dφ ,t ] − ∗ x ( p a ( x )) dµ ( x )+ Z J I k V ( ., φ ,t ( x ))[ dφ ,t ] − ∗ x ( p b ( x )) dµ | J I ( x ) dt. AMILTONIAN APPROACH TO GEODESIC IMAGE MATCHING 11For the spa e L ([0 , T ] , V ) , if we have: k ξ ( v ) t − ξ ( u ) t k ≤ M k v − u k L [0 ,T ] , the result is then proven with Cau hy-S hwarz inequality: k ξ ( v ) − ξ ( u ) k L [0 ,T ] ≤ √ M T k v − u k L [0 ,T ] . This an be obtained with: k ξ ( v ) t − ξ ( u ) t k = < ξ ( v ) t , ξ ( v ) t − ξ ( u ) t > − < ξ ( u ) t , ξ ( v ) t − ξ ( u ) t >, k ξ ( v ) t − ξ ( u ) t k ≤ | < ξ ( v ) t , ξ ( v ) t − ξ ( u ) t > | , | < ξ ( u ) t , ξ ( v ) t − ξ ( u ) t > | ) . For one of the two terms in the equation above: < ξ ( v ) t , ξ ( v ) t − ξ ( u ) t > = Z Z [ dφ v ,t ] − ∗ x ( p a ( x ))[ k ( φ v ,t ( x ) , φ v ,t ( y ))[ dφ v ,t ] − ∗ y ( p a ( y )) − k ( φ v ,t ( x ) , φ u ,t ( y ))[ dφ u ,t ] − ∗ y ( p a ( y ))] dµ ( y ) dµ ( x ) . On the unit ball of L ([0 , T ] , V × S ) denoted by B , and with the inequality (6), we ontrol thedi(cid:27)eomorphisms: k φ u ,t − φ v ,t k ∞ ≤ c V k v − u k L [0 ,T ] exp( c V ) , (20) k dφ u ,t − dφ v ,t k ∞ ≤ c V k v − u k L [0 ,T ] exp( c V ) . With the triangle inequality, we get: | < ξ ( v ) t , ξ ( v ) t − ξ ( u ) t > | ≤ Z Z [ dφ v ,t ] − ∗ x ( p a ( x )) | [ k ( φ v ,t ( x ) , φ v ,t ( y ))[ dφ v ,t ] − ∗ y ( p a ( y )) − k ( φ v ,t ( x ) , φ v ,t ( y ))[ dφ u ,t ] − ∗ y ( p a ( y ))] | + | [ k ( φ v ,t ( x ) , φ v ,t ( y ))[ dφ u ,t ] − ∗ y ( p a ( y )) − k ( φ v ,t ( x ) , φ u ,t ( y ))[ dφ u ,t ] − ∗ y ( p a ( y ))] | dµ ( x ) dµ ( y ) . On the unit ball B , we have: k dφ u ,t k ∞ ≤ c V exp( c V ) , k φ u ,t − Id k ∞ ≤ c V . Let M k ∈ R a bound for the kernel and its (cid:28)rst derivative on the unit ball B . Su h a onstantexists thanks to the hypothesis on the kernel and its (cid:28)rst derivative.A bound for the (cid:28)rst term an be found with the se ond inequality of (20): c V k v − u k L [0 ,T ] exp( c V ) M k (1 + 2 c V exp( c V )) k p a kk p b k , the se ond term is ontrolled with the (cid:28)rst inequality of (20) with the Lips hitz hypothesis on thekernel: c V k v − u k L [0 ,T ] exp( c V ) M k (1 + 2 c V exp( c V )) k p a kk p b k . Finally we get, k ξ ( v ) t − ξ ( u ) t k ≤ c V k v − u k L [0 ,T ] exp( c V ) M k (1 + 2 c V exp( c V )) k p a kk p b k . We have now on luded for the (cid:28)rst omponent of the appli ation Ξ . For the se ond term, theproof is essentially the same, we do not give the details. (cid:3) T > su h that we have existen e and uniqueness to the system(18), we prove now that the solutions are non-exploding i.e. we an hoose T = + ∞ in the lastproposition. This property shows that the asso iated riemannian manifold of in(cid:28)nite dimension is omplete, sin e the exponential map is de(cid:28)ned for all time. Without the hypothesis on the kernel,we an (cid:28)nd simple ounter-examples to this fa t.Proposition 3. The solution proposition 2 is de(cid:28)ned for all time.Proof:Thanks to propostion 1, we know that the norm of the solution u t is onstant in time, whi h willenable the extension for all time. Consider a maximal solution with interval of de(cid:28)nition [0 , T ] with T < + ∞ , then with the inequalities from (8) and after, we de(cid:28)ne the limit lim t T φ ,t := φ ,T ,sin e for all x , φ ,t ( x ) is a Cau hy sequen e. This is the same for lim t T dφ ,t ( x ) . This limit isalso a di(cid:27)eomorphism, sin e we an de(cid:28)ne the limit of the inverse as well. The proof is the sameto extend η t for all time.We an then apply the proposition (existen e for small time) to the urrent image I t instead of I , we obtain di(cid:27)eomorphisms ˜ φ ,s and ˜ η ,s in a neighborhood of , [0 , ǫ ] . Composing with φ ,T and η ,T , we extend the maximal solution on [0 , T + ǫ ] . This is a ontradi tion. (cid:3) We de oupled the equations in s and v to give a simple proof of the existen e in all time of the (cid:29)ow.The formulation of (18) implies the following formulation, whi h is the (cid:28)rst step to understandthe weak Hamiltonian formulation. If we have the system (18) and the relation (16), through the hange of variable u = φ ,t ( x ) , we get easily: η ,t = Id + Z t s u ◦ η u du,βs t ( . ) = − Z M P t ( x ) k S ( I t ( x )) , . ) dµ ( x ) ,φ ,t = Id + Z t v u ◦ φ u du,λv t ( . ) = Z M k V ( ., u ) P t ( u ) ∇ | u I t dµ ( x ) , + Z J I k V ( ., φ ,t ( x ))[ dφ ,t ] − ∗ x ( p b ( x )) dµ | J I ( x ) dt, (21)with, P t ( x ) = Jac( φ t, ) d [ η t, ] I st ( x ) P ◦ φ t, ,P ( x ) = − p c ( φ , ( x )) d [ η , ] I ( x ) Jac( φ , ( x )) .
5. Hamiltonian generalizationIn numerous papers on large deformation di(cid:27)eomorphisms, the Hamiltonian framework arises.The simplest example is probably the Landmark mat hing problem for whi h the geodesi equa-tions and the Hamiltonian version of the evolution are well known ([VMTY04℄,[ATY05℄). Ourgoal is to provide an Hamiltonian interpretation of the initial variational problem. The maindi(cid:27)eren e is that we want to write Hamiltonian equations in an in(cid:28)nite dimensional spa e, whi hAMILTONIAN APPROACH TO GEODESIC IMAGE MATCHING 13is roughly the spa e of images. The (cid:28)rst step was done in [GTL06℄ where Hamiltonian equationswere written on the representations of losed urves. Our work generalizes this approa h to thespa e of images. We use the point of view of the optimal ontrol theory (as it is developed in[ATY05℄) to formally introdu e the Hamiltonian. Then, we state a weak Hamiltonian formulationof the equations obtained by the variational approa h. We prove uniqueness for the solutions tothese equations. At the end of this se tion, we dis uss the existen e of the solutions with the helpof the existen e of the solutions for the variational problem.In the whole se tion, we maintain our previous assumptions on the kernel for the existen e ofsolutions for all time.5.1. Weak formulation. In this paragraph, we slightly modify the approa h in order to developthe idea of de omposing an image in "more simple parts". Let introdu e the position variables.We onsider in the following that the dis ontinuity boundary is a position variable. Instead of onsidering the fun tion I t as the se ond position variable, we introdu e a produ t spa e whi h an be proje ted on the spa e Im ( M ) . Let ( U , . . . , U n ) be a partition in Lips hitz domain of M .We denote by Σ = ∪ ni =1 ∂U i the union of the boundaries of the Lips hitz domains. We onsiderthe proje tion: p : n Y i =1 W , ∞ ( M ) Im ( M ) (22) ( I i ) i =1 ,...,n I = n X i =1 I i U i . (23)Dis ontinuities give derivatives with a singular part. Maybe we ould have treated this aseadopting only the variable I t , but we (cid:28)nd the idea of de omposing an image into more simpleparts ri h enough to study the ase. Observe that Σ is endowed with an important role in thede(cid:28)nition of the proje tion: to write down a Hamiltonian system on the large spa e, we need tointrodu e the deformation of Σ and the deformation of ea h fun tion in the produ t spa e. Wewill derive the Hamiltonian equations from this optimal ontrol problem: (the position variable is Q and the ontrol variable is U , c ( U ) is the instantaneous ost fun tion) Q = ( Q i ) ≤ i ≤ r = (Σ , ( I i ) ≤ i ≤ r ) ∈ L (Σ , M ) × W , ∞ ( M ) r ,U = ( v, s ) ∈ V × S, ˙ Q = f ( Q, U ) = ( v ◦ Q , ( −h∇ Q i , v i + s ( Q i )) ≤ i ≤ r ) ,c ( U ) = λ | v | V + β | s | S . The otangent spa e D of the position variable ontains F = L ∞ (Σ , R n ) × L ( M, R ) r ⊂ D . Wewrite the formal minimized Hamiltonian of the ontrol system on the subspa e F , with P ∈ F : H ( P, Q ) = min U Z Σ h P ( x ) , ˙ Q ( x ) i dµ | Σ ( x ) + r X i =1 Z M P i ˙ Q i dµ ( x ) − c ( U ) . (24)Minimizing in U , we obtain optimality onditions in ( u, v ) a minimizer su h that for any pertur-bation ( δv, δs ) :4 FRANÇOIS-XAVIER VIALARD λ h v, δv i = Z Σ h P , δv ◦ Q i dµ | Σ ( x ) − r X i =1 Z M P i h∇ Q i , δv i dµ ( x ) ,β h s, δs i = r X i =1 Z M P i δs ( I i ) dµ ( x ) . Using the kernel, it an be rewritten, λv = Z Σ k ( Q ( x ) , . ) P dµ | Σ ( x ) − r X i =1 Z M k V ( x, . ) P i ∇ Q i dµ ( x ) , (25) βs = r X i =1 Z M k S ( Q i , . ) P i dµ ( x ) . (26)We dedu e the expression of the Hamiltonian, H ( P, Q ) = 12 λ [ Z Σ Z Σ P ( x ) k V ( Q ( x ) , Q ( y )) P ( y ) dµ | Σ ( x ) dµ | Σ ( y )+ Z M Z M P j ( y ) ∇ Q j ( y ) k S ( y, x ) ∇ Q i ( x ) P i ( x ) dµ ( x ) dµ ( y ) − X ≤ i ≤ r Z M Z Σ P ( y ) k V ( Q ( y ) , x ) P i ( x ) ∇ Q i ( x ) dµ ( x ) dµ | Σ ( y )]+ 12 β X ≤ i,j ≤ r Z M Z M P j ( y ) k S ( Q j ( y ) , Q i ( x )) P i ( x ) dµ ( x ) dµ ( y ) . Now, we want to give a sense to the Hamiltonian equations, ∀ i ∈ [1 , r ] : ˙ Q t = ∂ P H ( P t , Q t )( . ) , (27) ˙ Q it = ∂ P i H ( P t , Q t ) ∀ i ∈ [1 , r ] , ˙ P t = − ∂ Q H ( P t , Q t ) , ˙ P it = − ∂ Q i H ( P t , Q t ) ∀ i ∈ [1 , r ] . These derivatives should be understood as distributions, for Ψ ∈ C + ∞ ( M, R ) and u ∈ C + ∞ ( M, R n ) and with the notation introdu ed in (25), ∀ i ∈ [1 , r ] : ∂ P H ( P, Q )( u ) = Z Σ h v ◦ Q ( y ) , u ( y ) i dµ | Σ ( y ) , (28) ∂ P i H ( P, Q )(Ψ) = Z M Ψ( y ) (cid:0) s ( Q i ( y )) − h v ( y ) , ∇ Q i ( y ) i (cid:1) dµ ( y ) ,∂ Q H ( P, Q )( u ) = Z Σ h [ dv ] | Q ( y ) ( u ( y )) , P ( y ) i dµ | Σ ( y ) ,∂ Q i H ( P, Q )(Ψ) = Z M Ψ( y )[ ds ] | Q i ( y ) P i ( y ) − h v ( y ) , P i ( y ) ∇ Ψ( y ) i dµ ( y ) . Remark that only the last equation really needs to be de(cid:28)ned as a distribution and not as afun tion. Now we an give a sense to the Hamiltonian equations but only in a weak sense:AMILTONIAN APPROACH TO GEODESIC IMAGE MATCHING 15De(cid:28)nition 3. An appli ation χ ∈ C ([0 , T ] , L (Σ , M ) × W , ∞ ( M ) r × L (Σ , R n ) × L ( M, R ) r ) issaid to be a weak solution if it veri(cid:28)es for Ψ ∈ C + ∞ ([0 , T ] × M, R ) and u ∈ C + ∞ ([0 , T ] × M, R n ) :(we denote χ ( t ) = ( Q t , P t ) .) Z T Z M − ∂ t Ψ Q it dµdt = Z T ∂ P i H ( P t , Q t )(Ψ) dt ∀ i ∈ [1 , r ] , (29) Z T Z M − ∂ t Ψ P it dµdt = − Z T ∂ Q i H ( P t , Q t )(Ψ) dt ∀ i ∈ [1 , r ] , (30) Z T Z Σ − ∂ t u Q t dµ | Σ dt = Z T ∂ P H ( P t , Q t )( u ) dt, (31) Z T Z Σ − ∂ t u P t dµ | Σ dt = − Z T ∂ Q H ( P t , Q t )( u ) dt. (32)5.2. Uniqueness of the weak solutions. In this paragraph, the uniqueness to the weak Hamil-tonian equations is proven, and the proof gives also the general form of the solutions. This formis losely related to the solution of the variational problem of the previous se tion.Theorem 3. Every weak solution is unique and there exists an element of L ([0 , T ] , V × S ) whi hgenerates the (cid:29)ow ( φ ,t , η ,t ) su h that: Q t ( x ) = φ ,t ( x ) , x ∈ Σ , (33) Q it ( u ) = η ,t ◦ Q i ◦ φ t, ( u ) , u / ∈ φ ,t (Σ ) , i ∈ [0 , n ] ., (34)and for the momentum variables: P t ( x ) = d [ φ ,t ] − ∗ x ( P ( x )) , x ∈ Σ , (35) P it ( u ) = P i ◦ φ t, Jac( φ t, ) d [ η t, ] Q it ( u ) , u / ∈ φ ,t (Σ ) . (36)Proof: Let χ a weak solution on [0 , T ] , we introdu e t v t ( . ) = 1 λ Z Σ k ( Q t ( x ) , . ) P t dµ | Σ ( x ) − r X i =1 Z M k V ( x, . ) P it ∇ Q it dµ ( x ) , whi h lies in L ([0 , T ] , V ) . This ve tor (cid:28)eld is uniquely determined by the weak solution. Fromthe preliminaries, we dedu e that φ ,t ( x ) = R t v s ( φ s, ( x )) ds is well de(cid:28)ned. We introdu e also, t s t ( . ) = r X i =1 Z M k S ( Q it , . ) P it dµ ( x ) . For the same reasons, we an integrate the (cid:29)ow: η ,t ( x ) = R t s r ( η r, ( x )) dr is well de(cid:28)ned. In-trodu ing ˜ Q it ( x ) = η t, ◦ Q it ◦ φ ,t for i ∈ [1 , r ] , we obtain, with S t ◦ η t, ( x ) = ∂ t η t, ( x ) and V t ◦ φ ,t ( x ) = ∂ t φ ,t ( x ) :6 FRANÇOIS-XAVIER VIALARD Z T Z M − ∂ t Ψ ˜ Q it dµ = Z Z M − ∂ t Ψ η t, ◦ Q it ◦ φ ,t dµdt = Z T Z M − η t, ◦ Q it [ ∂ t Ψ] ◦ φ t, Jac( φ t, ) dµdt = Z T Z M − η t, ◦ Q it ( ∂ t [Ψ ◦ φ t, ] − < ∇ Ψ ◦ φ t, , v t ◦ φ t, > )Jac( φ t, ) dµdt = Z T Z M ( S t ( ˜ Q it ) − < ∇ ˜ Q it , V t > + dη t, ( ˙ Q it ◦ φ ,t ))Ψ dµdt = Z T Z M ( S t ( ˜ Q it ) − < ∇ ˜ Q it , V t > + dη t, ( − < ∇ Q i ◦ φ ,t , v ◦ φ ,t > + s ( Q i ◦ φ ,t )))Ψ dµdt. The an elation of the equation above relies on the group relation of (cid:29)ows of ve tor (cid:28)elds. Wehave the equality: S t + dη t, ( s t ◦ η ,t ) = 0 , then the (cid:28)rst and last terms an el. The remaining terms an el too be ause of the relations: ∇ ˜ Q it = dφ ∗ ,t (cid:0) dη t, ( ∇ Q it ◦ φ t, ) (cid:1) ,v t + dφ ,t ( V t ◦ φ t, ) = 0 . Then, we on lude: Z T Z M − ∂ t Ψ ˜ Q it dµdt = 0 . Introdu ing Ψ( t, x ) = λ ( t ) γ ( x ) , with λ ∈ C + ∞ ([0 , T ]) and γ ∈ C + ∞ ( M ) , we have: R T − λ ′ ( t )( R M γ ( x ) ˜ Q it dµ ) dt = 0 , hen e: R M γ ( x ) ˜ Q it dµ = R M γ ( x ) ˜ Q it dµ , i.e. ˜ Q it = ˜ Q i = Q i , and: Q it = η ,t ◦ Q i ◦ φ t, . Now, we introdu e for i ∈ [1 , r ] , ˜ P it ( . ) = P it ◦ φ ,t Jac( φ ,t ( . )) d [ η t, ] η ,t ◦ Qi . ) , this quantity is well de(cid:28)ned be auseof the inversibility of the (cid:29)ow of s t and v t . Remark that Jac( φ ,t ( . )) d [ η t, ] η ,t ◦ Qi . ) is di(cid:27)erentiable almosteverywhere be ause Q i is Lips hitz on M . We want to prove that R R M − ∂ t Ψ ˜ P it dµdt = 0 with Ψ ∈ C + ∞ ( M ) , whi h leads to: ˜ P it = ˜ P i = P i , and then we are done.To prove the result, we (cid:28)rst use the hange of variable y = φ ,t ( x ) , this is a straightforward al ulation, we will also use the equality: d [ η t, ] η ,t ( . ) dη ,t ( . ) = 1 . Z T Z M Ψ ∂ t ˜ P it dµdt = − Z T Z M ˜ P it ∂ t Ψ ◦ φ t, Jac( φ t, ) dµdt, Z T Z M Ψ ∂ t ˜ P it dµdt = − Z T Z M P it d [ η t, ] Q it ( . ) ∂ t Ψ ◦ φ t, dµdt, Z T Z M Ψ ∂ t ˜ P it dµdt = Z T Z M − P it ∂ t ( Ψ ◦ φ t, d [ η t, ] Q it ( . ) ) + P it d [ η t, ] Q it ( . ) < ∇ Ψ | φ t, , − dφ t, ( v t ) > + P it Ψ ◦ φ t, ∂ t ( d [ η ,t ] Q i ◦ φ ,t ) dµdt, (37)AMILTONIAN APPROACH TO GEODESIC IMAGE MATCHING 17The third term of the last equation an be rewritten: Z T Z M P it Ψ ◦ φ t, ∂ t ( d [ η ,t ] Q i ◦ φ ,t ) dµdt = Z T Z M P it Ψ ◦ φ t, ( < ∇ ( d [ η ,t ] Q i ) , − dφ t, ( v t ) > + d [ s t ] Q it d [ η ,t ] Q i ◦ φ ,t ) dµdt. (38)Now, we an apply the hypothesis on ∂P it , the (cid:28)rst term of the expression is equal to: Z T Z M − P it ∂ t ( Ψ ◦ φ t, d [ η t, ] Q it ( . ) ) dµdt = Z T Z M P it < ∇ (Ψ ◦ φ t, d [ η ,t ] Q i ◦ φ ,t ) , v t > − Ψ ◦ φ t, d [ η ,t ] Q i ◦ φ ,t d [ s t ] Q it P it dµdt, (39) Z Z M P it < ∇ (Ψ ◦ φ t, d [ η ,t ] Q i ◦ φ ,t ) , v t > dµdt = Z Z M P it < dφ ∗ t, ( ∇ Ψ | φ t, ) , v t > d [ η ,t ] Q i ◦ φ ,t + P it < dφ ∗ ,t ∇ ( d [ η ,t ] Q i ) , v t > dµdt. (40)All the terms of the equation an el together, so we obtain the result.With ˜ P t ( x ) = d [ φ ,t ] ∗ x P t ( x ) and ˜ Q t = φ t, ◦ Q , we get the result for the last two terms of thesystem in the same way than the pre eding equations, but it is even easier. (cid:3) Remark that we only have to suppose the weak solution is L to obtain the result. More thanuniqueness, we know that the weak solution "looks like" a variational solution of our initial prob-lem.5.3. On the existen e of weak solutions. We onsider in this se tion a Hamiltonian equationwhi h in ludes our initial ase, for whi h we have proved existen e results. Namely, we an rewriteour result on the last se tion in terms of existen e of weak solution of the system (27).Proposition 4. Let ( U , . . . , U n ) be a partition in Lips hitz domain of M , Σ = ∪ ni =1 ∂U i . Forany intial data, I ∈ W , ∞ ( M ) r , Q = (Σ , I ) and P ∈ L ∞ (Σ , R n ) × L ( M, R ) r su h thatSupp ( P i ) ⊂ U i , then there exists a solution to the Hamiltonian equations.Remark that this solution has the same stru ture than a variational solution of our initial problem.In this ase, all the momenta ( i ∈ [1 , r ] ) an be viewed as one momentum: with P t = P ri =1 P it ,we have all the information for the evolution of the system.Now, we an say a little bit more on the general Hamiltonian equations. We will not give a proofhere of the existen e if we relax the ondition on the support of P i , but the reader an onvin ehimself that the existen e is somehow a by-produ t of the last se tion.Summing up our work at this point, from a pre ise variational problem we obtain generalizedHamiltonian equations, for whi h we an prove results on existen e and uniqueness. A naturalquestion arises then, from what type of variational problems ould appear these solutions? Theanswer ould be based on the remark: be ause the de omposition we hoose is the dire t produ tof spa es, we an put a sort of produ t metri on it. A simple generalized minization is obtainedby modifying the equation (1): E ( η, φ ) = D ( Id, ( η, φ )) + r X i =1 σ i k η ◦ ( I i U i ) ◦ φ − − I itarg k L , (41)8 FRANÇOIS-XAVIER VIALARDwith for i ∈ [1 , r ] I i ∈ W , ∞ ( M ) , and I ∈ Im ( M ) .6. Con lusionThe main point of this paper is the derivation lemma whi h may be of useful appli ations.This te hni al lemma gives a larger framework to develop the large deformation di(cid:27)eomorphismstheory. The a tion on the level lines is far to be none of interest but we aim to obtain numeri alimplementations of the ontrast term applied to smooth images. Finally, the interpretation as aHamiltonian system through optimal ontrol theory ends up with giving a proper understandingof the momentum map. To go further, the te hni al lemma seems to be easily enlarged to re ti(cid:28)-ables domains, and there may be a useful generalization to SBV fun tions. This would enable ageneralization of a part of this work to
SBV fun tions. But to understand the weak Hamiltonianformulation would have been mu h more di(cid:30) ult within the
SBV framework. From the numeri alpoint of view, some algorithms that are urrently developed to treat the evolution of urves ouldbe used e(cid:30) iently but they need strong developments.7. Proof of the lemmaAfter re alling some lassi al fa ts about Lips hitz fun tions, we prove the derivation lemma:Lemma 1. Let
U, V two bounded Lips hitz domains of R n . Let X a Lips hitz ve tor (cid:28)eld on R n and φ t the asso iated (cid:29)ow. Finally, let g and f Lips hitz real fun tions on R n . Consider thefollowing quantity depending on t , J t = Z φ t ( U ) f ◦ φ − t g V dµ, where dµ is the Lebesgue measure, then ∂ t | t =0 + J t = Z U − < ∇ f, X > g V dµ + Z ∂U < X, n > f g ˜ V ( X ) dµ | ∂U . (42)with ˜ V ( X )( y ) = lim ǫ + ¯ V ( y + ǫX ) , if the limit exists, elsewhere. And we denote by dµ | ∂U the measure on ∂U and n the outer unit normal of ∂U .We will use,Theorem 4. Radema her's theoremLet f : U R n a Lips hitz fun tion de(cid:28)ned on an open set U ⊂ R n , then f is di(cid:27)erentiable µ a.e.Theorem 5. Let F : R d × R n R d , Lips hitz ontinuous on a neighborhood of ( v , y ) and F ( v , y ) = 0 . Suppose that ∂ F ( v , y ) exists and is invertible. Then there exists a neighborhood W of ( v , y ) , on whi h there exists a fun tion g : R n R d su h that in W : • g ( y ) = v . • F ( g ( y ) , y ) = 0 f or ( g ( y ) , y ) ∈ W . • | g ( y ) − g ( y ) | ≤ c | y − y | ,with c = 1 + ( Lip ( F ) + 1) k ∂ v F ( v , y ) − k .AMILTONIAN APPROACH TO GEODESIC IMAGE MATCHING 19This theorem an be found in [PS03℄ and in a more general exposition than we will use hereafter.Now an obvious lemma of derivation under the integral,Lemma 2. Let f a Lips hitz fun tion de(cid:28)ned on an open set U ⊂ R n . Let X a Lips hitz ve tor(cid:28)eld on R n of ompa t support and φ t the asso iated (cid:29)ow. Consider J t = R U f ◦ φ t dx , then: ∂ t | t =0 J t = Z U < ∇ f, X > dx. (43)Proof: Using Radema her's theorem, this is a staightforward appli ation of dominated onvergen etheorem. Note that under the ondition that f is Lips hitz, if both f and ∇ f are integrable and X is a bounded ve tor (cid:28)eld, we an relax the hypothesis of a ompa t support for the ve tor (cid:28)eld,whi h is repla ed here by the integrability ondition. (cid:3) We will need the following hara terization of derivation for real fun tions to prove the lemma 4 .Lemma 3. Let w : R n R a fun tion, then w is di(cid:27)erentiable in x ∈ R n if and only if thereexist f and g two C fun tions and a neighborhood V of x , su h that f ( x ) = g ( x ) and if y ∈ V , g ( y ) ≤ w ( y ) ≤ f ( y ) . (44)Proof: Suppose w di(cid:27)erentiable in x , it su(cid:30) es to prove that there exists f C su h that w ≤ f in a neighborhood of x . (To obtain g , onsider then − w .) we an suppose x = 0 , w (0) = 0 and w ′ (0) = 0 then lim y x w ( y ) | y | = 0 . Hen e there exists a ontinuous fun tion v de(cid:28)ned on B (0 , r > su h that: w ( y ) | y | ≤ v ( y ) and v (0) = 0 . With the notation | x | for the eu lidean norm in R n , let m ( | x | ) = sup | y |≤| x | v ( y ) , we have m ( | x | ) ≥ v ( x ) for x ∈ B (0 , r ) . The fun tion m is non de reasingand ontinuous with m (0) = 0 . At last, let f ( x ) = R | x || x | m ( t ) dt , then w ( x ) ≤ | x | m ( | x | ) ≤ f ( x ) .Moreover, the fa t f is C is straightforward to verify.Suppose f and g are C , and denote by f ′ and g ′ their derivative in x , then f ′ ( x ) = g ′ ( x ) sin e f − g ≥ and has a minimum in x . On V we have: g ( x + h ) − g ′ ( x ) .h ≤ w ( x + h ) − g ′ ( x ) .h ≤ f ( x + h ) − f ′ ( x ) .h. Hen e, w ( x + h ) − g ′ ( x ) .h = o ( h ) and w is di(cid:27)erentiable. (Remark that we only use the fa t that f and g are di(cid:27)erentiable in x .) (cid:3) Using the lemma above, we study the deformation of an epigraph of a Lips hitz fun tion underthe a tion of a ve tor (cid:28)eld, whi h leads to study the deformation of the graph of the fun tion:Lemma 4. Let φ t the (cid:29)ow of the ve tor (cid:28)eld Lips hitz X on R n (with k X k bounded on R n ). Let V = { ( x, z ) ∈ R n − × R | z > w ( x ) } with w a Lips hitz fun tion, and w t ( x ) = inf { z | ( x, z ) ∈ φ t ( V ) } .Then, a.e. ∂ t | t =0 w t ( x ) = − < ∇ w ( x ) , p ( X ( x, w ( x ))) > + p ( X ( x, w ( x ))) , with p and p orthogonal proje tions respe tively on R n − × and n − × R .Proof: Remark that w t is well de(cid:28)ned for all t by onnexity reason, but w t might be dis ontinuousfor t large enough. However it is Lips hitz ontinuous for t in a neighborhood of : we (cid:28)rst applythe impli it fun tion theorem for Lips hitz maps to the fun tion: F ( x, t ) = p ( φ t ( x, w ( x ))) − x , ∂ x F ( x ,
0) = Id | R n − , so we obtain for ea h x ∈ R n − a fun tion x : t x ( t ) su hthat x is Lips hitz and the equation F ( x, t ) = 0 ⇔ x = x ( t ) on a neighborhood of ( x , . Notethat the impli it fun tion theorem in [PS03℄ gives only existen e but not uniqueness. We developnow the uniqueness. The Lips hitz ondition on w an be written with the one property. Let ( A, B ) two points on the graph of w , then | y A − y B | < M | x A − x B | , for a Lips hitz onstant. Thisopen ondition is then veri(cid:28)ed in a neighborhood of { t = 0 } . We see that, F ( x , t ) = F ( x , t ) implies φ t ( x , w ( x )) = φ t ( x , w ( x )) , hen e x = x .If w is C , we get by impli it fun tion theorem the (cid:28)rst derivative of x ( t ) : ∂ t | t =0 x t = − p ( X ( x t , w ( x t ))) . We dedu e, w t ( x ) = p ( φ t ( x t , w ( x t ))) , (45)and that t w t ( x ) is a Lips hitz fun tion for ea h x . In the C ase, we get by di(cid:27)erentiationof the equation (45), ∂ t | t =0 w t ( x ) = − < ∇ w ( x ) , p ( X ( x, w ( x ))) > + p ( X ( x, w ( x ))) . We now observe that there is an obvious monotoni ity in w of w t . Indeed, if w ≤ v then w t ≤ v t .We then use the lemma 3 to prove the result in the ase w is Lips hitz . Let x su h that w isupper and lower approximated by C fun tions: let u and v su h that u ( x ) = w ( x ) = v ( x ) , and u ≤ w ≤ v . We obtain: t ( u t ( x ) − u ( x )) ≤ t ( w t ( x ) − w ( x )) ≤ t ( v t ( x ) − v ( x )) , (46)We dedu e the result: lim t t ( w t ( x ) − w ( x )) = − < ∇ w ( x ) , p ( X ( x, w ( x ))) > + p ( X ( x, w ( x ))) , for all the points of derivability of w , i.e. almost everywhere sin e w is Lips hitz. (cid:3) Now, we prove the following lemma, whi h an be seen as a onsequen e of the oarea formula. Itwill be used in the proposition 5.Lemma 5. Let w : R n R , an a.e. di(cid:27)erentiable fun tion, A = w − ( { } ) and B = { x |∇ w ( x ) =0 } then µ ( A ∩ B ) = 0 .Proof: For n = 1 , the lemma is obvious be ause the point of A ∩ B are isolated with Taylorformula. For n > , we generalize with Fubini's theorem: A ∩ B = n [ i =1 A ∩ B i , with B i = { x ∈ R n | < ∇ f ( x ) , e i > = 0 } . To prove the result, it su(cid:30) es to see that µ ( A ∩ B n ) = 0 .Consider ( x, t ) ∈ R n − × R , f x ( t ) = f ( x + t ) and D x = x × R . Then we apply the ase n = 1 tothe fun tion f x : µ ( A ∩ B n ∩ D x ) = 0 , and with Fubini's theorem, µ ( A ∩ B n ) = 0 . (cid:3) Remark that this lemma an be applied to a Lips hitz fun tion. Below lies the fundamental stepto prove the derivation lemma.AMILTONIAN APPROACH TO GEODESIC IMAGE MATCHING 21Proposition 5. Let w : R n − R a Lips hitz fun tion. Let V := { ( x, y ) | y > w ( x ) } and U := { ( x, y ) | y > } . Let X a Lips hitz ve tor (cid:28)eld on R n and φ t the asso iated (cid:29)ow. Finally,let g and f Lips hitz real fun tions on R n of ompa t support. Consider the following quantitydepending on t , J t = Z φ t ( U ) f ◦ φ − t g V dµ, where dµ is the Lebesgue measure also denoted by dx , then ∂ t | t =0 + J t = Z U − < ∇ f, X > g V dx + Z ∂U < X, n > f g ˜ V ( X ) dµ | ∂U . (47)with ˜ V ( X )( y ) = lim ǫ + ¯ V ( y + ǫX ) , if the limit exists, elsewhere. And we denote by dµ | ∂U the measure on ∂U and n the outer unit normal. U w V
Figure 2. The main aseProof: The (cid:28)rst ase to treat is when w > , we an then integrate on V instead of φ t ( U ) . J t = Z V f ◦ φ − t g V dµ, we di(cid:27)erentiate under the integral, we get: ∂ t | t =0 J t = Z U − < ∇ f, X > g V dµ, this is the formula be ause the se ond term is null. In the following, we have to use this ase.To treat the general ase, we (cid:28)rst do a hange of variable: J t = Z R n − Z + ∞ w t ( x ) + f g ◦ φ t Jac( φ t ) dµ. We introdu e some notations: x + = max(0 , x ) = H ( x ) , ∇ H ( x )( v ) = 0 , x < , or x = 0 et v < , ∇ H ( x )( v ) = v , elsewhere,w t ( x ) = inf { z | ( x, z ) ∈ φ − t ( V ) } . Let p and p the orthogonal proje tions respe tively on R n − × { } and n − × R . With thelemma 4, ∂ t | t =0 + w t ( x ) = < ∇ w ( x ) , p ( X ( x, w ( x ))) > − p ( X ( x, w ( x ))) . ∂ t | t =0 + w t ( x ) + = ∇ H ( w ( x ))( < ∇ w ( x ) , p ( X ( x, w ( x ))) > − p ( X ( x, w ( x )))) . Using f ( y )( < ∇ g, X > +div( X ) g ) = div( f gX ) − g < ∇ f, X > , we get: ∂ t | t =0 + J t = Z R n − − ∂ t | t =0 + w t ( x ) + f ( x, w ( x ) + ) g ( x, w ( x ) + ) dx − Z R n − Z + ∞ w t ( x ) + < ∇ f ( x, z ) , X ( x, z ) > g ( x, z ) dxdz + Z ∂ ( U ∩ V ) f g < X, n > dµ | ∂ ( U ∩ V ) . Here n is the outer unit normal of ∂ ( U ∩ V ) . Rewrite the last term: Z ∂ ( U ∩ V ) f g < X, n > dµ | ∂ ( U ∩ V ) = Z ∂V ∩ U f g < X, n > dµ | ∂V + Z w − (] −∞ , f ( x, g ( x, < X, n > dx. In a neighborhood C of x su h that w ( x ) > , we have demonstrated that the formula holds, sothe (cid:28)rst term in the equation above is equal to: − Z ∂V ∩ U f g < X, n > dµ | ∂V − Z w − (] −∞ , ∂ t | t =0 w t ( x ) + f ( x, g ( x, dx. Moreover, on the set F = { x : w ( x ) = 0 } , we have, with the lemma 5, a.e. ∇ w = 0 . Then, we have: ∂ t | t =0 + w t ( x ) + = < X, n > + , a.e. On the set G = { x : w ( x ) < } , we have: ∂ t | t =0 + w t ( x ) + = 0 .We now get the result with: Z w − (] −∞ , ( < X, n > − ∂ t | t =0 w t ( x ) + ) f ( x, g ( x, dx = Z w − (] −∞ , ˜ V ( X )( x, < X, n > f ( x, g ( x, dx. Indeed, if < X, n > = 0 the result is straightforward be ause the limit exists in the de(cid:28)nition of ˜ V . If < X, n > = 0 , the ontribution is null. (cid:3) Our goal is to prove the formula for Lips hitz open sets, we present some de(cid:28)nitions.De(cid:28)nition 4.An open set U = ∅ of R n is said to be lo ally Lips hitz if for ea h x ∈ ∂U , there exist: • an a(cid:30)ne isometry I , of R n , • an open neighborhood V ( x ) of x , • a Lips hitz fun tion w de(cid:28)ned on R n − with a Lips hitz onstant K ( x ) su h that, I ( V ( x ) ∩ U ) = I ( V ( x )) ∩ { ( x, y ) ∈ R n − × R | y > w ( x ) } If the onstant K ( x ) an be hosen independent of x , U is said to be Lips hitz.Remark 4.(1) An open bounded set of R n whi h is lo ally Lips hitz is also Lips hitz .(2) By Radema her's theorem, the outer unit normal n ( x ) exists for H n − a.e. x ∈ ∂U .AMILTONIAN APPROACH TO GEODESIC IMAGE MATCHING 23(3) We will say that ( V ( x ) , I ) trivializes the Lips hitz domain in x .The three lemmas below prove that one an des ribe a Lips hitz domain in many systems of oordinates. This is a key point to understand the two boundaries at a point of interse tion andenables to use the proposition 5.Lemma 6. Let ψ a C di(cid:27)eomorphism of R n and V a Lips hitz domain, then ψ ( V ) is a Lips hitzdomain.Proof: The proof is straightforward with the hara terization of Lips hitz domains with the uni-form one property, whi h an be found in [DZ01℄. (cid:3) Lemma 7. Let ( e , . . . , e n ) an orthonormal basis of R n , and w a Lips hitz fun tion de(cid:28)ned on R n − of Lips hitz onstant M . Let U the Lips hitz open set whi h is above the graph of w : U := { ( x, y ) ∈ R n − × R | y > w ( x ) } , then for ea h n in the open one C := { n = ( x, y ) ∈ R n − × R + | y > M | x |} one an trivializethe boundary of U through the graph of a fun tion de(cid:28)ned on n ⊥ (with an orthonormal basis).Moreover, this fun tion is Lips hitz . wCn Figure 3. Trivializing with respe t to the orthogonal hyperplan to n .Proof: Note that, n an be represented with the angle between the hyperplan and the ve tororthogonal to this hyperplan, and also the Lips hitz onstant an be represented as the tangenteof su h an angle. Let two points a, b ∈ R n whi h belong to the ∂U , then b − a and n are not olinear, be ause of the Lips hitz property. As a onsequen e, ∂U is de(cid:28)ned as the graph of afun tion ˜ w on n ⊥ . And, one an verify that, if n is normalized, a Lips hitz onstant for ˜ w is equalto: tan ( | θ − θ | ) , if θ is the angle of n and θ asso iated to the Lips hitz onstant. (cid:3) Lemma 8. Let U ⊂ R n a Lips hitz domain with ∈ ∂U , then there exist a neighborhood V of , w a Lips hitz fun tion de(cid:28)ned on R n − and a linear transformation A su h that: ( e , . . . , e n − ) ⊂ Ker ( A − Id ) and V ∩ A ( U ) = V ∩ { ( x, y ) ∈ R n × R | y > w ( x ) } . Here is the illustration of the idea driving the proof.Proof: In some oordinates, we write U as the epigraph G + of a Lips hitz fun tion v on a hyperplan H in a neighborhood W of . We note by n a normal ve tor to H . We fa e two ases:4 FRANÇOIS-XAVIER VIALARD M Figure 4. The linear transformation A • If < n, e n > = 0 , let A ∈ L ( R n ) de(cid:28)ned by: A ( n ) = e n and ( e , . . . , e n − ) = Ker ( A − Id ) . Denoteby p the orthogonal proje tion on ( e , . . . , e n − ) . Let M ∈ G de(cid:28)ned by its proje tions z on H : M = z + w ( z ) n , by de(cid:28)nition of A , A ( M ) = ( p ( z ) , w ( z )) + A ( z − p ( z )) . But z − p ( z ) = λ ( z ) n with λ ∈ ( R n ) ′ so we obtain: A ( M ) = ( p ( z ) , w ( z ) + λ ) . Also p | H is a linear isomorphism, wenote the inverse p − , then with the hange of variable x = p ( z ) , we get G = { ( x, λ ( p − ( x )) + w ◦ p − ( x )) | x ∈ p ( H ∩ W ) } .w ◦ p − is learly Lips hitz and we obtain the lemma in this ase. • If < n, e n > = 0 , we an hoose by lemma 7 another system of oordinates for whi h we fall inthe (cid:28)rst ase, and the lemma is demonstrated. (cid:3) We present a smooth ( C ) version of the derivation lemma.Proposition 6. Let U a bounded C domain of R n and V a bounded Lips hitz domain. Let X aLips hitz ve tor (cid:28)eld on R n and φ t the asso iated (cid:29)ow. Finally, let g and f Lips hitz real fun tionson R n . Consider the following quantity depending on t , J t = Z φ t ( U ) f ◦ φ − t g V dµ, where dµ is the Lebesgue measure also denoted by dx , then ∂ t | t =0 + J t = Z U − < ∇ f, X > g V dx + Z ∂U < X, n > f g ˜ V ( X ) dµ | ∂U . (48)with ˜ V ( X )( y ) = lim ǫ ¯ V ( y + ǫX ) , if the limit exists, elsewhere. And we denote by dµ | ∂U themeasure on ∂U and n the outer unit normal.Proof: Let K a Lips hitz onstant of V . Applying the de(cid:28)nition of a C domain with the ompa tboundary of U , there exist a (cid:28)nite overing W , . . . , W k of ∂U with open balls and (Ψ , . . . , Ψ k ) C di(cid:27)eomorphisms su h that for ea h i ∈ [1 , k ] , Ψ i ( W i ∩ U ) = { ( x, y ) ∈ R n − × R | y > } ∩ Ψ i ( W i ) , also one has: Ψ i ( W i ∩ ∂U ) = R n − × { } ∩ Ψ i ( W i ) . Let ( θ , . . . , θ n , θ k +1 ) a partition of unity asso iated to the family ( W = R n \ ¯ U , W , . . . , W k , W k +1 = U ) . It means:AMILTONIAN APPROACH TO GEODESIC IMAGE MATCHING 25 • ≤ θ i ≤ , ∀ i ∈ [0 , k + 1] and P i ∈ [0 ,k +1] θ i = 1 on R n . • Supp θ i ⊂ W i for i ∈ [1 , k + 1] . • Supp θ ⊂ R n \ ¯ U .Through the hange of variable y = φ t ( x ) , the quantity is: J t = k +1 X i =1 Z U f θ i g ◦ φ t V ◦ φ t Jac( φ t ) dµ. Four ases appear: • ¯ U ∩ W i ⊂ V • W i ∩ U ⊂ R n \ V • ¯ V ∩ W i ⊂ U • ∂V ∩ ∂U ∩ W i = ∅ .In the (cid:28)rst ase, the formula is the result of the derivation under the integral, whi h is allowedbe ause g is Lips hitz. ∂ t | t =0 + J t = Z W i f ( y )( < ∇ g, X > +div( X ) g ) dµ, with, f ( y )( < ∇ g, X > +div( X ) g ) = div( f gX ) − g < ∇ f, X > , and applying Stokes theorem truefor a re ti(cid:28)able open set and Lips hitz fun tions, we obtain the result.In the se ond ase, the quantity is null for t su(cid:30) iently small. So the formula is obvious.In the third ase, we an integrate on V instead of φ t ( U ∩ W i ) : J t = Z V f ◦ φ − t g V dµ, we di(cid:27)erentiate under the integral, we get: ∂ t | t =0 + J t = Z U − < ∇ f, X > g V dµ, be ause the se ond term of the formula is null.We deal hereafter with the last ase: as Ψ i is a C di(cid:27)eomorphism, Ψ i ( V ) is also Lips hitz.Consequently, we an (cid:28)nd a (cid:28)nite overing B , . . . , B m of W i , for whi h one of the following onditions holds: • B i ⊂ V • B i ⊂ R n \ ¯ V • B i ∩ ∂V = ∅ and there exists I su h as (Ψ i ( B i ) , I ) trivializes the Lips hitz domain Ψ i ( V ) .In the (cid:28)rst two ases, we have already demonstrated that the formula is true.With the lemma 8, we know that after a linear transformation A whi h is the identity on R n − ×{ } ,the Lips hitz domain an be represented as the epigraph of a Lips hitz fun tion de(cid:28)ned on R n − .We repla e Ψ i by A ◦ Ψ i = Ψ . We then have the following situations: Ψ( W i ∩ V ) = { ( z, t ) | t > w ( z ) } , or Ψ( W i ∩ V ) = { ( z, t ) | t < w ( z ) } , w : R n − ∩ B (0 , ρ ) R a Lips hitz fun tion. This situation (or the symetri situation whi his essentially the same) is treated in the proposition 5. (cid:3) We generalize the proposition 6 to the ase of Lips hitz domains, we need some additional resultsof approximation:Theorem 6. C approximationLet f : R n R a Lips hitz fun tion. Then for ea h ǫ > , there exists a C fun tion ¯ f : R n R su h that: µ ( { x | ¯ f ( x ) = f ( x ) orD ¯ f ( x ) = Df ( x ) } ) ≤ ǫ. In addition, sup R n | D ¯ f | ≤ C Lip ( f ) , for some onstant C depending only on n .See the proof of [EG92℄.Remark 5. A dire t onsequen e of the theorem is that we have, k f − ¯ f k ∞ ≤ C, Lip ( f ) √ n − ǫ n − . On ea h ube of volume ǫ there exists a point where the two fun tions are equal, then we dedu eeasily the laimed bound. Thus, we get also µ ( { ( x, y ) | ¯ f ( x ) < y < f ( x ) or f ( x ) < y < ¯ f ( x ) } ) ≤ C, Lip ( f ) √ n − ǫ n − . We dedu e a orollary:Corollary 2. Let U a bounded Lips hitz domain, for ea h ǫ > there exists V a C domain su hthat, S = U \ ¯ V ∪ V \ ¯ U , is a re ti(cid:28)able open set verifying: µ ( S ) < ǫ (49) H n − ( ∂S ) < ǫ. (50)Proof: We just present the main points for the proof of the orollary.By ompa ity of ∂U , there exists a (cid:28)nite open overing ( V , . . . , V k ) of ∂U , su h that for ea h openset we an trivialize the boundary. On ea h open set V i , we have by previous theorem a Lips hitzappli ation g i : ∂U ∩ V i R n whi h gives a C hypersurfa e. We have H n − ( { x ∈ ∂U ∩ V i | x = g ( x ) } ) ≤ ǫ. Moreover we an assume that this overing satis(cid:28)es the following property. Let < η < ǫ and Z := ∪ i = j V i ∩ V j , H n − ( ∂U ∩ Z ) ≤ η . We thus obtain an appli ation g : ∂U R n whi h isLips hitz ( ∂U is endowed with the indu ed metri by the eu lidean metri on R n ) and is theboundary of a C domain V , for whi h we have: H n − ( { x ∈ ∂U | x = g ( x ) } ) ≤ ( k + 1) ǫ. Then, H n − ( ∂S ) < Lip ( g ) n − ( k + 1) ǫ . And also with the same argument given in the pre- eding remark, there exists a onstant K su h that, µ ( S ) ≤ K Lip ( g ) (( k + 1) ǫ ) n − , with K = √ n − Lip ( ∂U ) . (cid:3) AMILTONIAN APPROACH TO GEODESIC IMAGE MATCHING 27We now turn to the proof of the lemma 1.Proof: We use the orollary 2, let U ǫ a C domain for ǫ as in the orollary. Let M a onstantsu h that in a ompa t neighborhood of U , | f ◦ φ t − f | ≤ M t , g ≤ M , | f | ≤ M and | X | ≤ K .We have, with S ǫ = ∆( U ǫ , U ) = U ǫ \ ¯ U ∪ U \ ¯ U ǫ , We denote by θ = U − U ǫ , so we have (triangular inequality for the se ond inequation): | ( J t ( U ) − J ( U )) − ( J t ( U ǫ ) − J ( U ǫ )) | ≤ Z V | θ ◦ φ − t f ◦ φ − t g − θf g | dµ, | ( J t ( U ) − J ( U )) − ( J t ( U ǫ ) − J ( U ǫ )) | ≤ Z V | θ ◦ φ − t ( f ◦ φ − t g − f g ) | dµ + Z V | ( θ ◦ φ − t − θ ) f g | dµ, | ( J t ( U ) − J ( U )) − ( J t ( U ǫ ) − J ( U ǫ )) | ≤ Z V ∩ φ t ( S ǫ ) | f − f ◦ φ − t || g | V dµ + Z V ∩ ∆( φ t ( S ǫ ) ,S ǫ ) | f g | V dµ, | ( J t ( U ) − J ( U )) − ( J t ( U ǫ ) − J ( U ǫ )) | ≤ tM M µ ( φ t ( S ǫ ) ∩ V ) + M M µ ( V ∩ ∆( φ t ( S ǫ ) , S ǫ )) We (cid:28)rst treat the last term. We laim that, for s > su h that Lip ( φ t ) ≤ , we have, for t ∈ [ − s , s ] , µ (∆( φ t ( S ǫ ) , S ǫ )) ≤ t max(2 , M ) n H n − ( ∂S ǫ )) . Introdu e
Ψ : ( t, x ) ∈ [ − s , s ] × R n φ t ( x ) ∈ R n .We have Lip (Ψ) ≤ max(2 , M ) , and H n ([0 , t ] × ∂S ǫ )) = t H n − ( ∂S ǫ ) . Hen e, H n (Ψ([ − s , s ] × ∂S ǫ )) ≤ t max(2 , M ) n H n − ( ∂S ǫ )) . To (cid:28)nish, we prove that: ∆( φ t ( S ǫ ) , S ǫ ) ⊂ [ s ≤ t φ t ( ∂S ǫ ) . Let z ∈ ∆( φ t ( S ǫ ) , S ǫ ) , • Suppose z / ∈ S ǫ , there exists x ∈ S ǫ su h that φ t ( x ) = z . The map c : s ∈ [0 , t ] φ s ( x ) veri(cid:28)es c (0) = x ∈ S and c ( t ) = z / ∈ S . By onnexity, there exists u ∈ [0 , t ] , su h that c ( u ) ∈ ∂S ǫ . By omposition of (cid:29)ow, φ t − u ( c ( u )) = z . • Suppose z / ∈ φ t ( S ǫ ) , there exists x ∈ φ t ( S ǫ ) su h that φ − t ( x ) = z . The map c : s ∈ [0 , t ] φ − s ( x ) veri(cid:28)es c (0) ∈ φ t ( S ǫ ) and c ( t ) = z / ∈ φ t ( S ǫ ) . By onnexity, there exists u ∈ [0 , t ] ,su h that c ( u ) ∈ ∂φ t ( S ǫ ) . By omposition of (cid:29)ow, m = φ − u ( z ) ∈ ∂S ǫ and obviously, φ u ( m ) = z .We give a bound for the (cid:28)rst term in the same neighborhood for t ∈ [ − s , s ] , µ ( φ t ( S ǫ ) ∩ V ) ≤ µ ( φ t ( S ǫ )) ≤ n tǫµ ( S ǫ ) . Consequently, lim sup t + | t [( J t ( U ) − J ( U ) − ( J t ( U ǫ ) − J ( U ǫ ))] | ≤ M M µ ( S ǫ ) + M H n − ( ∂S ǫ )) . We an now obtain the on lusion. Let ǫ > ,8 FRANÇOIS-XAVIER VIALARD lim sup t + | t [( J t ( U ) − J ( U ) − ( J t ( U ǫ ) − J ( U ǫ ))] | ≤ ( M M + M ) ǫ. We use now the formula already demonstrated for C domains, lim sup t + | t [( J t ( U ) − J ( U ) − Z U ǫ − < ∇ f, X > g V dµ + Z ∂U ǫ < X, n > f g ˜ V ( X ) dµ | ∂U ǫ ] | ≤ ( M M + M ) ǫ, and the result is proven. (cid:3)
8. Appendix8.1. Central lemma of [GTL06℄. We present here a di(cid:27)erent version of the lemma, whi h isessentially the same, but from another point of view.Lemma 9. Let H a Hilbert spa e and B a non-empty bounded subset of E a Hilbert spa e su h thatthere exists a ontinuous linear appli ation g : H E . Assume that for any a ∈ H , there exists b a ∈ B su h that h b a , g ( a ) i ≥ . Then, there exists b ∈ Conv( B ) su h that h b, g ( a ) i = 0 , ∀ a ∈ H .Proof: We denote by H = g ( H ) . Let p the orthogonal proje tion on H and Z = Conv( B ) is anon-empty losed bounded onvex subset of H , when e weakly ompa t. As p is weakly ontinuousand linear, p ( Z ) is a weakly ompa t onvex subset and thus strongly losed. From the proje tiontheorem on losed onvex subset, there exists b ∈ p ( Z ) su h that: | b | = inf c ∈ Z | c | and h b, c − b i ≥ for c ∈ p ( Z ) . As a dire t onsequen e, we have also: h b, u − b i ≥ ∀ u ∈ Z. (51)The element b lies in the adheren e of g ( H ) then there exists a sequen e b n ∈ g ( H ) su h that lim b n = b . From the hypothesis, there exists u n ∈ B su h that: h u n , − b n i ≥ . As B is bounded, lim h u n , b − b n i = 0 , hen e lim sup h u n , b n i ≤ . By (51), we get lim sup h u n , b i ≥ h b, b i . As a result, h b, b i ≤ and b = 0 . By de(cid:28)nition, there exists v ∈ Z , p ( v ) = b = 0 . Now, v ∈ H ⊥ and φ ( . ) = h v, . i gives the result. (cid:3) C but it an be proven with weaker assumptions on the regularity ofve tor (cid:28)elds. (See [Gla05℄, for a detailed proof following another method.)Lemma 10. Let u t and v t be two time dependent C ve tor (cid:28)elds on R n , and denote by φ ǫ ,t the(cid:29)ow generated by the ve tor (cid:28)eld u t + ǫv t , then we have: ∂ ǫ φ , ( x ) = Z [ dφ t, ] φ ,t ( x ) v ( φ ,t ( x )) dt. Proof: Introdu e the notation A t ∈ R n de(cid:28)ned by: A t ( φ ,t ( x )) = ∂ ǫ φ ǫ ,t ( x ) . Deriving this expres-sion with respe t to the time variable: ddt A t ( φ ,t ( x )) = du t ( A t ( φ ,t ( x ))) + v t ( φ ,t ( x )) , AMILTONIAN APPROACH TO GEODESIC IMAGE MATCHING 29Remark that the expression above an be written as (with L the Lie derivative): L u t A t = ddu | u =0 [ dφ ,t + u ] − ( A t + u ( φ t + u ( x ))) = [ dφ ,t ] − ( v t ( φ ,t ( x ))) . By integration in time, we obtain the result. (cid:3)(cid:3)