Higher-order Spatial Accuracy in Diffeomorphic Image Registration

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1 Geometry, Imagng and Computng Volume, Number, 1, 214 Hger-order Spatal Accuracy n Dffeomorpc Image Regstraton Henry O. Jacobs and Stefan Sommer We dscretze a cost functonal for mage regstraton problems by dervng Taylor expansons for te matcng term. Mnma of te dscretzed cost functonals can be computed wt no spatal dscretzaton error, and te optmal solutons are equvalent to mnmal energy curves n te space of k-ets. We sow tat te solutons convergence to optmal solutons of te orgnal cost functonal as te number of partcles ncreases wt a convergence rate of O( d+k ) were s a resoluton parameter. Te effect of ts approac over tradtonal partcle metods s llustrated on syntetc examples and real mages. 1. Introducton Te goal of mage regstraton s to place dfferng mages of te same obect (e.g. MRI scans) nto a sared coordnate system so tat tey may be compared. One common means of dong ts s to deform one mage untl t matces te oter. Typcal numercal scemes for mplementng ts task are partcle metods, were partcles are used as a fnte dmensonal representaton of a dffeomorpsm. If te partcles are ntalzed on a regular grd of resoluton, ten te solutons can be O( d ) accurate at best were d s te dmenson of te mage doman. Improvng ts order of accuracy s non-trval because tradtonal ger-order numercal scemes are desgned on fxed meses (e.g. ger order fnte dfferences). In ts paper, we seek to mprove ts order of accuracy by consderng a more sopstcated class of partcles. We wll fnd tat by equppng te partcles wt et-data, one can aceve regstratons wt ger orders of accuracy. One mpact of te use of ger-order partcles s tat te mproved accuracy per partcle permts te use of fewer partcles for a desred total accuracy. For suffcently smoot ntal data, ts mples te storage requrements are mproved as well. 1

2 2 Henry O. Jacobs and Stefan Sommer 1.1. Organzaton of te paper We wll ntroduce te ger-order accurate mage regstraton framework troug te followng steps: 1. We wll ntroduce te erarcy of et-partcles. 2. We wll pose an mage regstraton problem as an optmal control problem on an nfnte dmensonal space. 3. We wll pose a sequence of deformed problems wc are easer to solve. 4. We wll reduce te deformed optmzaton problems to optmzaton problems nvolvng computaton of fnte dmensonal ODEs (.e. an nfnte dmensonal reducton). 5. We wll fnd necessary condtons for sequences of computed solutons to te deformed problems to converge to te soluton of te orgnal problem at a rate O( d+k ), were k depends on te order of te et-partcles used. Fnally, we wll dsplay te results of numercal experments comparng te use of zerot, frst, and second order et-partcles. 2. Prevous work In ts secton, we attempt a bref overvew of te large deformaton dffeomorpc metrc mappng (LDDMM) framework from ts orgns n te 199s, to ts recent marrage wt geometrc mecancs (2s-present) Matcng wt LDDMM Te noton of seekng deformatons for te sake of mage regstraton goes back a long way, see [SDP13, You1] and references teren. One of te frst attempts was to consder dffeomorpsms of te form ϕ(x) = x + f(x) for some map f : R d R d. Te map f s often denoted a dsplacement feld. Wen f s small, ϕ s a dffeomorpsm, but ts can fal wen f s large [You1, Capter 7]. Many algortms outsde te LDDMM context employ a small deformaton approac wt a dsplacement feld f. Te dsplacement can be represented for example wt B-splne bass functons. Partcle metods, as consdered n ts paper, can loosely be nterpreted as large deformaton equvalents to representng te dsplacement f wt fnte lnear combnatons of bass functons. In partcular, te kernel K can be tougt

3 Hger-order Spatal Accuracy n Dff. Image Regstraton 3 of as takng te role of e.g. B-splne bass functons n small deformaton approaces. Te breakdown for large f s a result of te fact tat te space of dffeomorpsms s a nonlnear space. One of te early obstacles n dffeomorpc mage regstraton entaled dealng wt ts nonlnearty. A key nsgt n gettng a andle on te nonlnearty of te dffeomorpsms was to consder te lnear space of vector felds. Gven a tme-dependent vector feld v(t), one can ntegrate t to obtan a dffeomorpsm ϕ t, wc s called te flow of v [CRM96]. Ts nsgt was used to obtan dffeomorpsms for magng applcatons by posng an optmal control problem on te space of vector-felds, and ten ntegratng te flow of te optmal vector feld to obtan a dffeomorpsm. Te well-posedness of ts approac was studed n [Tro95, DGM98], were te cost functonal (.e. te norm) was dentfed as a fundamental coce n ensurng well-posedness and controllng propertes of te resultng dffeomorpsms. A partcle metod based upon [DGM98] was mplemented for te purpose of medcal magng n [JM]. Te completeness of te Euler-Lagrange equatons n [DGM98] was studed torougly n [TY5], were te mage data was allowed to be of a farly general type (.e. any entty upon wc dffeomorpsms act smootly). Te analytc safe-guards provded by [DGM98] and [TY5] were ten excercsed n [BMTY5], were a number of examples were numercally nvestgated Connectons wt geometrc mecancs Followng tese early nvestgatons, connectons wt geometrc mecancs began to form. Te cost functonal cosen n [JM] was te H 1 -norm of te vector-felds. Concdentally, ts s te cost functonal of te n- dmensonal Camassa-Holm equaton (see [HM5] and references teren). In 1-dmenson, te partcle solutons n [JM] are dentcal to te peakon solutons dscovered n [CH93], and te numercal sceme reduces to tat of [HR6]. Te convergence of [JM] was proven usng geometrc tecnques n [HR6] n te one-dmensonal case. Te same proof was used n [CDTM12] for arbtrary dmensons. As mages appear as advected quanttes, te use of momentum maps became a useful conceptual tecnque for geometers to understand te numercal sceme of [BMTY5]. Te dentfcaton of numerous matematcal terms n [BMTY5] as momentum maps was performed n [BGBHR11].

4 4 Henry O. Jacobs and Stefan Sommer 2.3. Jet partcles Te partcle metod mplemented n [JM] allowed only for deformatons tat acted as local translatons (see Fgure 1(a)). Motvated by a desre to create more general deformatons [SNDP13] ntroduced a erarcy of partcles wc advect et-data. We call te partcles et-partcles n ts paper. Te frst order et-partcles modfy te Jacoban matrx at te partcle locatons and allow for locally lnear transformatons suc as local scalngs and local rotatons (see Fgure 1(b-e)). Second order et-partcles allow for deformatons wc are locally quadratc (.e. transformatons wt nontrval Hessans (see Fgure 1(f-)). Te geometrc and erarcal structure of [SNDP13] was nvestgated n [Jac13] were te Le groupod structure of et-partcles was lnked to te Le group structure of te dffeomorpsm group, tus makng te case for et-partcles as mult-scale representatons of dffeomorpsms. Independently, an ncompressble verson of ts dea was nvented for te purpose of ncompressble flud modellng n [DJR13]. Solutons to ts flud model were numercally computed n [CHJM14] based upon te regularzed Euler flud equatons developed n [MM13] and expressons for matrx-valued reproducng kernels derved n [MG14]. Te fnal secton of [CHJM14] provdes formulas wc llustrate ow et-partcles n te kt level of te erarcy yeld deformatons wc are approxmated by partcles n te (k 1)t level of te erarcy. Te approxmaton beng accurate to an order O( k ) were > s some measure of partcle spacng. Ts approxmaton s more or less equvalent to te approxmaton of a partal dfferental operator by a fnte dfferences, and t wll serve as one of te man tools used n ts paper n producng ger-order accurate numercal scemes. 3. LDDMM Let M be a manfold and let V X(M) be a subspace of te vector-felds on M equpped wt an nner-product, V : V V R. Let G V Dff(R n ) be te correspondng topologcal Le group to wc V ntegrates [You1, Capter 8]. To do mage regstraton, we try to assemble a small dffeomorpsm by mnmzng a cost functon on te space of curve n G V. Te standard cost functon takes a tme-dependent dffeomorpsm, ϕ t, and outputs a real number. Matematcally, te cost functon s often taken to be a map E GV : C 1 ([, 1] : G V ) R gven by E GV [ϕ( )] := l(v(t))dt + F (ϕ 1 ),

5 Hger-order Spatal Accuracy n Dff. Image Regstraton (a) translaton (b) expanson (c) rotaton (d) stretc (e) sear (f) 2nd order (g) 2nd order () 2nd order Fgure 1: Deformatons of ntally square grds. (a) zerot order, (b-e) frst order, (f-) second order. A sngle et-partcle s located at te blue dots before movng wt te flows to te red crosses. Grds are colored by log- Jacoban determnant. were v(t) V s te Euleran velocty feld v(t, x) = t ϕ t (ϕ 1 t (x)) and l s a control-cost. Explctly, ϕ t G V s obtaned from v(t) V va te ntal value problem (1) { d dt ϕ t = v(t) ϕ t ϕ = d. One ten obtans extremzers of E GV by solvng te Euler-Lagrange equatons on G V. However, G V s a non-commutatve group, and can be very dffcult to work wt. It s typcal to express E GV as a cost functon on te vector-space V and ncorporate (1) as a constrant. Ts means optmzng a cost functon E : C 1 ([, 1], V ) R wt respect to constraned varatons. Any extremzer, v( ), of E must necessarly satsfy a symmetry reduced form of te Euler-Lagrange equatons, known as te Euler-Poncaré equaton. In essence, te Euler-Poncaré equatons are notng but te Euler-Lagrange equatons pulled to te space V. For a generc l, te Euler-Poncaré equatons take te form (2) d dt ( ) l + ad v v ( ) l =. v

6 6 Henry O. Jacobs and Stefan Sommer We suggest [MR99] for furter nformaton on te Euler-Poncaré equatons. Equaton (2) s an evoluton equaton, wc allows us to searc over te space of ntal condtons (.e. V ) n place of optmzng over space of curves (.e. C 1 ([, 1]; V )). Explctly, ts s done by consderng te map evol EP : V C 1 ([, 1]; V ) wc sends eac v V to te curve v( ) C 1 ([, 1]; V ) obtaned by ntegratng (2) wt ntal condton v. We can ten pre-compose E wt evol EP to produce te functon (3) e := E evol EP : V R. Te ntal condton v V mnmzes e f and only te soluton v( ) = evol EP (v ) of (2) mnmzes E. Generally, solutons to (2) are extremzers of E, and one must appeal to ger-order varatons n order to obtan suffcent condtons for an extremzer to be a mnmzer. However, we wll not pursue tese matters n ts artcle Overvew of te problem and our soluton Partcle metods are typcally used to approxmate a dffeomorpsm n te followng way. We usually compute all quanttes wt respect to an ntal condton were all te partcles le on a grd/mes and prove convergence as te mes wdt,, tends to. However, t would be nce to ave an order of accuracy as well. PROBLEM: Can we solve for a mnmzer of E wt a convergence rate of O( p ) for some p N? Our strategy for tacklng ts problem s to approxmate E wt a sequence of O( p )-accurate curve energes E for wc we can compute te mnmzers exactly up to tme dscretzaton (.e. te computed solutons ave no spatal dscretzaton error). More specfcally, te mes sze wll determne a contnuous sequence of subgroups G G. We wll approxmate te matcng functonal F, wt a G -nvarant functonal F : G V R suc tat for a fxed ϕ G V F (ϕ) F (ϕ) = O( p ) for some p N. We fnd te curve energes to be O( p ) accurate as well, and ts accuracy wll transfer to te solutons for a suffcently wde range of scenaros.

7 Hger-order Spatal Accuracy n Dff. Image Regstraton 7 4. Reducton teory In ts secton, we revew subgroup reducton of a class of optmzaton problems usng Clebsc varables. In te Hamltonan context, Clebsc varables are also called symplectc varables, and consttute a Posson map ψ : T R n P. Ts s useful wen 2n < dm(p ), snce solutons to certan Hamltonan equatons on P can be derved by solvng Hamltonan equatons on T R n frst [MW83, We83]. Te Lagrangan verson of ts dea was furter developed n te context of equatons wt ydrodynamc background n [HM5]. It s ts later perspectve wc we sall take n ts paper, snce problem setup s stated n Lagrangan form. A more toroug overvew of te role of reducton by symmetry n LDDMM can be found n [SJ15]. Let G be a Le group and G s G be a Le subgroup wt Le algebras g and g s respectvely. We wll denote te omogenous space of rgt cosets by Q = G/G s, and we wll denote te correspondng prncpal bundle proecton by π : G Q. Note tat G naturally acts on Q troug te formula g π( g) = π(g g). Gven ts acton, te correspondng (left) momentum map, J : T Q g, s defned by te condton J(q, p), ξ = p, ξ q, ξ g. Let L : T G R be te Lagrangan and let F : G R. We ws to mnmze te curve energy or acton (4) E[g( )] = 1 L(g(t), ġ(t))dt + F (g(1)) over te space of curves g(t) G on te nterval [, 1] wt g() = d. Tat s to say E : Cd 1 ([, 1]; G) R were C1 d ([, 1]; G) denotes te space of C1 curves n G orgnatng from te dentty. Extremzaton of E means takng a varaton n Cd 1 ([, 1]; G), wc s a varaton of a curve wt a fxed endpont at t = but not at t = 1. It s smple to sow tat any soluton must satsfy te boundary value problem ( ) d L dt ġ L g (5) = L g() = d, + df (g(1)) =. t=1 ġ If te dmenson of G s large, ntegratng ts equaton can be troublesome. However, n te presence of a G s -symmetry a reducton can be appled to reduce te problem to a boundary value problem on Q.

8 8 Henry O. Jacobs and Stefan Sommer Trougout ts secton we wll assume tat F s G s nvarant. As a result tere exsts a functon f : Q R defned by te condton f(q) = F (g) q Q, g G suc tat q = π(g). More succnctly, f = F π. We wll also assume tat L(g, ġ) s G-nvarant, and comes from a reduced Lagrangan functon l : g R. Fnally, we wll l assume tat te Legendre transformaton, ξ : g g, s nvertble. Te reduced Hamltonan : g R s ten gven by (µ) = µ, l ξ 1 (µ) l ( l 1 ) (µ). ξ Teorem 4.1 (c.f. [BGBHR11]). Let H := J : T Q R. If te curve (q, p)(t) T Q satsfes (6) { q = H p, ṗ = H q q() = π(d), p(1) + df(q(1)) = ten te curve g(t) obtaned by ntegratng te ntal value problem ġ(t) = ξ(t) g(t), ξ = (l/ξ) 1 (J(q, p)), g() = d satsfes (5). Moreover, all mnmzers of (4) must be of ts form. Proof. We can replace E wt te (equvalent) curve energy E 2 : C 1 ([, 1]; g) R gven by (7) E 2 [ξ] = 1 l(ξ(t))dt + F (g(1)) were g(1) G s mplctly obtaned troug te reconstructon equaton dg dt = ξ g wc we vew as a constrant. Mnmzers of E 2 are related to mnmzers of E troug te reconstructon equaton as well. We are now gong to use te G s symmetry of (7) to reduce te dmensonalty of te problem. Te G s nvarance of F mples te exstence of a functon f : Q R suc tat F = f π. Terefore, we may equvalently express E 2 as te energy functonal (8) E 2 [ξ] = 1 l(ξ(t))dt + f(q(1))

9 Hger-order Spatal Accuracy n Dff. Image Regstraton 9 were q(1) s obtaned troug te reconstructon equaton q(t) = ξ(t) q(t) wt te ntal condton q() = π s (d). Agan, te dynamc constrant q = ξ q makes ts a constraned optmzaton problem. We may take te dual of ts constraned optmzaton problem by usng Lagrange multplers to get an equvalent unconstraned optmzaton problem [BV4]. In our case, te dual problem s tat of extremzng te (unconstraned) curve energy E 3 : C 1 ([, 1]; g T Q) R gven by E 3 [ξ, q, p] = 1 l(ξ(t)) + p(t), q(t) ξ(t) q(t) dt + f(q(1)). Usng te defnton of J we can re-wrte ts as E 3 [ξ, q, p] = 1 l(ξ) + p, q J(q, p), ξ dt + f(q(1)). We fnd tat statonarty wt respect to arbtrary varatons of ξ mples (9) l ξ = J(q(t), p(t)). We may vew (9) as a constrant wc defnes ξ n terms of te q s and p s. Explctly, (9) tell us ξ = l 1 (J(q, p)). ξ We can substtute ts nto te prevous curve energy to elmnate te varable ξ and express E 3 solely n terms of p and q. We tus obtan te curve energy E 4 [q, p] = By observng 1 l + f(q(1)). H(q, p) = (J(q, p)) = ( l 1 ) (J(q, p)) + p, q ξ J(q, p), l ξ J(q, p), l ξ 1 (J(q, p)) l 1 (J(q, p)) dt ( l 1 ) (J(q, p)), ξ

10 1 Henry O. Jacobs and Stefan Sommer we can wrte E 4 as (1) E 4 [q, p] = 1 p, q H(q, p)dt + f(q(1)). By takng arbtrary varatons of q and p, we fnd tat extremzaton of E 4 yelds te desred result. As a corollary, we fnd tat extremzers of E n (4) can be derved by fndng te extremzers of E 4 n (1). As E s a curve energy over T G and E 4 s a curve energy over T Q = T (G/G s ), we can see te computatonal sgnfcance of ts result most clearly wen te dmenson of Q s small compared to G (e.g. fnte compared to nfnte). More specfcally, Teorem 4.1 allows us to mnmze curve energes usng te followng gradent descent algortm. Algortm for general Le groups 1. Solve for (q(t), p(t)) T Q n (6). 2. Set ξ(t) = (l/ξ) 1 J(q(t), p(t)) 3. If necessary, obtan g(t) G as a soluton to te ntal value problem, ġ = ξ g, g() = d. 4. Evaluate cost functon E 4, and backward compute te adont equatons [Son98] to compute te gradent of te cost functon wt respect to a new ntal condton. 5. If te gradent s below some tolerance, ɛ, ten stop. Oterwse use te gradent to create a new ntal condton and return to step 1. We say f neccessary n step 3 because computaton of g(t) s not needed n te context of mage regstraton. We wll fnd tat only q(t) s needed. In any case, f g(t) were to be computed, te resultng curve would mnmze te orgnal curve energy E gven n equaton (4), and all te mnmzers of te orgnal problem are obtaned n ts way. Agan, te advantage of ts metod s tat te bulk of te computaton s performed on te lower dmensonal space T Q rater tan T G. In te next sectons we wll consder te case were G s a dffeomorpsm group, and G s s a subgroup suc tat Q = G/G s s te (fnte-dmensonal) space of et-partcles.

11 Hger-order Spatal Accuracy n Dff. Image Regstraton Jets as Homogenous spaces In order to nvoke te fndngs of te prevous secton, we must fnd a way to caracterze te space of et-partcles as a omogenous space (.e. a group modulo a subgroup). Ts s te content of Proposton 5.1, te man result n ts secton. Let M be a fnte set of dstnct ponts n M. If f s any k- dfferentable map from a negborood of, te k-et of f s denoted J (k) (k) (f). In coordnates, J (f) s represented by te coeffcents of te kt order Taylor expansons of f about eac of te ponts n. We call J (k) te kt order et functor about. Ts s ndeed a functor, and can be appled to any k-dfferentable map from subsets of M wc contan, ncludng real valued functons, dffeomorpsms, and curves supported on [KMS99, Capter IV]. Let G = Dff(M) and let e G denote te dentty transformaton on M. We can consder te subgroup G () and te normal subgroups G (k) := {ψ G ψ(x) = x x } := {ψ G() J (k) ψ = J (k) e} Moreover, te Le algebra of G (k) s g (k) = {η(k) X(M) J (k) η(k) = J (k) ()}. In oter words, g (k) s te sub-algebra of X(M) consstng of vector felds wt vansng partal dervatves up to order k at te ponts of. Proposton 5.1. Te functor, J (k), s te prncpal bundle proecton from G to Q (k) = G/G (k). Proof. Ts s merely te defnton of J (k), and a more toroug descrpton of ts statement can be found n [KMS93]. Noneteless, we wll attempt a skeletal proof ere. If ϕ 2 = ϕ 1 ψ for some ψ G (k) ten J k(ϕ 2) = J k(ϕ 1 ψ). However, ψ as absolutely no mpact on te kt order Taylor expanson because te Taylor expanson of ψ s trval to kt order. Tus J (k) (ϕ 1) = J (k) (ϕ 2) and so J (k) s a well defned map on te coset space Q (k). Conversely, for

12 12 Henry O. Jacobs and Stefan Sommer eac element q Q (k) one can sow tat te nverse mage (J (k) ) 1 (q) s composed of a sngle G (k) orbt and no more. For example f conssts of only two dstnct ponts ten Q () = {(y 1, y 2 ) M 2 y 1 y 2 }, Q (1) = {(f 1, f 2 ) Fr(M) 2 π Fr (f 1 ) π Fr (f 2 )}. were π Fr : Fr(M) M s te frame bundle of M. Proposton 5.2. J (k) functor J (k) (G() ) s a (fnte dmensonal) Le group, and te (k) restrcted to G() s a group omomorpsm. Moreover J (G() s a normal subgroup of J (k) (G(l) ) for all l N. Corollary 5.3. Te space Q (k) s a (fnte-dmensonal) prncpal bundle wt structure group J (k) (G() ). For k = ts structure group s trval. At k = 1 ts structure group s dentfable wt GL(d) were d = dm(m). 6. An O( p ) accurate algortm In ts secton, we descrbe te basc strategy for usng et-partcles to get g order accuracy n solutons to LDDMM problems posed on M = R d. Te algortm uses an O( p ) approxmaton to te matcng term wc s G (k) nvarant. We ten nvoke Teorem 4.1 to reduce te problem to a fnte dmensonal boundary problem on te space of kt order et-partcles Q (k). We solve ts problem to obtan an approxmaton of te soluton to te orgnal problem. As dscussed n Secton 2.3, a fnte number of et-partcles n Q (k 1) can approxmate et-partcles n Q (k). In partcular, zerot order et-partcles can approxmate any of te ger-order et-partcles n te erarcy. Wle usng low order partcles may perform well n practcal applcatons, tey do not represent exact solutons to ger-order dscretzatons of te matcng term. Terefore, tey cannot represent te exact solutons to te approxmated problem tat we seek ere. Furtermore, usng lower-order partcles to represent ger-order ets can be numercally unstable as te approxmaton s n essence a fnte dfference approxmaton: te partcles need to be very close and te momentum can be of very large magntude. We wll assume tat te problem s defned on a reproducng kernel Hlbert space (RKHS), wc we denote by V X(R n ) were X(R) wll )

13 Hger-order Spatal Accuracy n Dff. Image Regstraton 13 denote te space of C k vector felds. We wll denote te kernel of V by K : R d R d R [You1, Capter 9], and we wll assume V satsfes te admssblty condton (11) v V v k, for a constant C > a postve nteger k N and all v V. We wll denote te topologcal group wc ntegrates V by G V. To make precse wat we mean by an O( p ) approxmaton to a matcng term, we wll recall te bg O notaton. Defnton 6.1. Let F : G V R, and let F : G V R depend on a parameter >. We say tat F s an O( p )-approxmaton to F f ( ) F (x) F (x) < lm p for all x R d. Moreover, O( p ) wll serve as a place-older for an arbtrary functon wtn te equvalence class of all functons of wc vans at a rate of p or faster as. Under ts notaton, F s an O( p )- approxmaton of F f F = F + O( p ). To llustrate ow we may produce O( p )-approxmatons to matcng functons we wll consder te followng example. Example Let I, I 1 C k (R d ; [, 1]) be two greyscale mages wt compact support. We can consder te matcng functonal F : Dff(R d ) R gven by F (ϕ) = 1 σ I (I 1 ϕ) 2 L 2 = 1 I (x) I 1 (ϕ(x)) 2 dx. σ R d As I and I 1 eac ave compact support, te ntegral term can be restrcted to a compact doman. We wll contnue to wrte our ntegrals as ntegratons over R d, but we wll explot ts compactfcaton wen we need to. Consder te regular lattce = Z d wereupon, for suffcently small >, te L 2 -ntegral can be approxmated to order O( d ) wt a Remann sum F () (ϕ) = d (I (x) I 1 (ϕ(x))) 2 x Wle te order of te set s nfnte, te sum over used to compute F as only fntely many non-zero terms to consder because I and I 1 ave

14 14 Henry O. Jacobs and Stefan Sommer compact support. Moreover, F s G () nvarant because t only depends on ϕ(x) for x. An O( d+2 ) approxmaton s gven by F (2) (q) = d (I (x) I 1 (ϕ(x))) 2 x + d d+2 [( I (x) I 1 (ϕ(x)) ϕ (x)) 2] [ (I (x) I 1 (ϕ(x))) ( 2 12 I (x) γ I 1 (ϕ(x)) ϕ (x) ϕ γ (x) γ I 1 (ϕ(x)) ϕ γ (x)) ], and we can observe tat F (2) s G (2) nvarant because F (2) (ϕ) only depends on te 2nd order Taylor expanson of ϕ centered at eac x. Gven a G (k) -nvarant O( p )-approxmaton F : Dff(R d ) R to te matcng term F, we may consder te alternatve curve energy E [ϕ] = v(t) 2 V + F (ϕ 1 ), were v(t) V s te Euleran velocty feld v(t, x) = t ϕ t (ϕ 1 t (x)). For a fxed curve ϕ t, we observe tat E s an O( p )-approxmaton to E. One mgt surmse tat te extremzers of E provde good approxmatons of te extremzers of E. Ts s mportant because E s G (k) -nvarant, and we can nvoke Teorem 4.1 to solve for extremzers of E, but we can not do ts for E. Fortunately, for many coces of F, tere wll be mnmzers of E tat converge to tose of E as wt a known convergence rate. Teorem 6.2. Let F : G V R be C 2 wt respect to te topology nduced by V. 1 Let F : G V R be C 2 and an O( p )-approxmaton for F wt G (k) nvarance, constructed as above. Consder te curve energes E, E : C([, 1], V ) R E[v( )] = 1 2 E [v( )] = v(t) V dt + F (ϕ 1 ) v(t) V dt + F (ϕ 1 ). 1 We wll assume tat G V s a smoot manfold and a topologcal Le group. For example te space of H s dffeomorpsms wt s suffcently large.

15 Hger-order Spatal Accuracy n Dff. Image Regstraton 15 were ϕ 1 G V s te Le ntegraton of v(t). Let v mnmze e = E evol EP : V R If te Hessan at v V s bounded, postve defnte, and non-degenerate at v V, and evol EP exbts C 2 dependency upon te ntal velocty feld, ten, for suffcently small, tere exst a mnmzer v of e = E evol EP : V R wc s an O( p )-approxmatons of v n te V -norm. We wll employ te followng well-known result to approxmate vectorfelds n V wt fnte lnear combnatons of te RKHS kernel K. Lemma 6.3. Assume V satsfes te admssblty assumpton (11). Consder te subspace of vector-felds V (k) = {v X(R d ) v = y K(x y)}, y, k for k < ( k 1)/2. Te set W = > V () s dense n V wt respect to, V. Proof. Let { > } be a sequence suc tat lm ( ) =. Let v V be ortogonal to W. Tus v, w V = for all w W. Tat s to say v(x) = for all x and all N. However, any pont y R n s te lmt of a sequence {x }. Snce all members of V are contnuous, t must be te case tat v =. A drect corollary s tat W (k) = > V (k) s dense n V snce V () V (k) for any k N. Lemma 6.3 wll allow us to approxmate our cost functonal on V. Proof of Teorem 6.2. By te Morse Lemma (sutably generalzed to Hlbert Manfolds [Tro83, GM83]), tere exsts a smoot coordnate cart around v, Φ : U V, suc tat Φ(v ) = v and ẽ(v + w) = ẽ(v ) + Dv 2 ẽ(w, w), were ẽ := e Φ. Defne also ẽ := e Φ Note tat e(v ) e (v ) = F (ϕ ) F (ϕ ) for ϕ := evol EP (v ). By te assumpton on evol EP, and snce F, F C 2 (G), we observe tat e e s C 2 at v V. Moreover, we know tat e e = (F F ) evol EP = O( p ). We can dscard te bg O notaton and wrte e(v) e (v) = A(v) p + B(v, )

16 16 Henry O. Jacobs and Stefan Sommer were A C 2 (X(M)) s ndependent of and k =B = for k p. Snce D 2 e(v ) s nondegenerate, tere exsts κ > suc tat D 2 ẽ(w) κ 2 w. Terefore ẽ(v + w) ẽ(v ) + κ w 2. Tus, for suffcently small r >, ẽ(v +w) ẽ(v )+ r wt r = κ 2 r 2 wen w = r. Gven suc r >, coose suffcently small so tat e(v) e (v) < r /3 n U and so tat tere exsts v Φ 1 (V (k) U) wt ẽ(v ) ẽ(v ) < r /3 (we know suc a v exsts by Lemma 6.3). Ten ẽ (v ) < ẽ(v ) + 2 r /3 and ẽ (v +w) > ẽ(v +w) r /3 ẽ(v )+2 r /3 wen w = r. Tus tere exsts a pont nsde te ntersecton of te r-ball of v and Φ 1 (V (k) U) were ẽ s strctly smaller tan on te boundary of ts ntersecton. Snce V (k) s fnte dmensonal, ts mples te exstence of a local mnmzer ṽ Φ 1 (V (k) U) of ẽ Φ 1 (U V (k) ). By Teorem 4.1, ṽ s also a local mnmzer of ẽ on U. As, we can let r, and te local mnma ṽ wll approac v. In addton, v = Φ 1 (ṽ ) s a local mnmum for e wc must also approac v. Ts proves convergence as. We now address te order of accuracy. As v s a crtcal pont of e, we ave tat De (v ) =, were De s te Frecét dervatve of e. If we defne w = ṽ ṽ ten we observe = Dẽ (ṽ ) = Dẽ(ṽ ) DÃ(ṽ )p D B(ṽ, ) = Dẽ(v ) + D 2 ẽ(v )( w, ) DÃ(ṽ )p D B(ṽ, ). Moreover Dẽ(v ) = because v s a crtcal pont of ẽ. Tus we observe D 2 ẽ(v )( w, ) = DÃ(ṽ )p + D B(ṽ, ) We can observe tat te Hessan D 2 ẽ(v ) s related to te Hessan D 2 e(v ) va pre-composton by te bounded lnear operator DΦ(v ). Tus D 2 ẽ(v ) s a bounded operator from U nto V. By assumpton, ts Hessan s non-degenerate, and tus nvertble. Tus we observe w = [D 2 ẽ(v )] 1 (DÃ(ṽ )p + D B(ṽ, ) ). In oter words, v = ṽ + O(p ). So tere exsts an O(1) functon C(v) and a functon F (v, ) suc tat v = v + C(v)p + F (v, ) were d k F/d k = for k p. Tus we fnd v = Φ 1 (v ) = Φ 1 (ṽ + C(v)p + F (v, )) = Φ 1 (ṽ ) + DΦ 1 (ṽ ) (C(v)p + F (v, )) + O( 2p ) = v + O(p ).

17 Hger-order Spatal Accuracy n Dff. Image Regstraton 17 Te assumpton tat te Hessan of te curve energy be non-degenerate s generally dffcult to ceck n practce. We can stll nvoke ts teorem n specfc examples because te mnmzer of E(v) = v(t) 2 V dt, v(t) = evol t EP (v) s v =, and te Hessan s dentcal to te nner product on V. We can vew all relevant examples as perturbatons of ts curve energy, and use te contnuty of te Hessan operator to nvoke Teorem 6.2. Settng G = G V, Q = Q (k) = G V /G (k) n Algortm 1, we obtan te specal case of Algortm 1 gven by Algortm 2: 1. Solve for (q(t), p(t)) T Q (k) n (6). 2. If necessary, set u(t) = K J(q(t), p(t)) and obtan ϕ t G V troug te reconstructon formula ϕ t (x) = u(ϕ(x)) for all x M. 3. Evaluate te cost functon, and backward compute te adont equatons to compute te gradent of te cost functon wt respect to a new ntal condton. 4. If te gradent s below some tolerance, ɛ, ten stop. Oterwse use te gradent to create a new ntal condton and return to step 1. Step (1) concerns solvng a system of Hamltonan equatons on te space of kt order et-partcles. Here te confguraton s gven by numbers [q ] were {1,..., d}, s a mult-ndex on R d of degree less tan or equal to k, and {1,..., N} were N s te number of et-partcles. Te Hamltonan can usually be computed n closed form f te Green s kernel, K, of te RKHS V s known n closed form. For example f k =, we obtan tradtonal partcles (.e. te mult-ndex s of degree ) and te momentum map s gven by J(q, p) = N p q =1 were (q, p ) T R d = R d R d, and were p q s a one-form densty representaton of te element of V gven by p q, w := p w(q ) for all

18 18 Henry O. Jacobs and Stefan Sommer w V. Te Hamltonan s ten gven by H(q, p) = 1 2 and Hamlton s equatons are q = =1 N (p p )K(q q ),,=1 N N p K(q q ), ṗ = (p p )DK(q q ). Te velocty feld n step (2) s u(x) = =1 N p K(x q ). =1 Steps (3) and (4) are obtaned troug formulas of comparable complexty. For arbtrary k, te relatonsp between te momenta (q, p) T Q (k) and te velocty feld u nvokes te multvarate Faà d Bruno formula, and so te algebra rses n complexty very quckly [CS96]. In te case of k = 2, te Hamltonan s substantally more complex but stll tractable (see (12) n Appendx A). However, beyond k = 2, a symbolc algebra package s advsed. Te examples we wll be consderng n ts paper concern te case were d = 2, and k = or 2. By Teorem 6.2, we sould be able to approxmate mnmzers of E wt O( 2 ) and O( 4 ) accuracy respectvely n te V -norm. 7. Numercal Results In ts secton we, wll llustrate te deformatons encoded by et-partcles of varous orders and numercally verfy te O( d+k ) convergence rate of te matcng functonal approxmaton for k =, 2. In addton, we wll sow tat te second order approxmaton F (k), k = 2 allows matcng of second order mage features. We use smple examples to descrbe te dfferent capabltes of ger order et-partcles over lower order et-partcles. We do ts by llustratng structures tat cannot be matced wt low numbers of regular zerot order landmarks, but can stll be matced successfully wt frst and second order et-partcles. Tese effects mply more precse matcng of small scale features on larger mages were more spatal dervatves can be leveraged. In all examples, te et-partcles wll be postoned on regular grds n te mage domans.

19 Hger-order Spatal Accuracy n Dff. Image Regstraton 19 We do not pursue approxmatons wt k > 2 due to te dffcultes of takng very g order dervatves of mages and kernel functons. In addton, code complexty rses rapdly as te order ncreases beyond Implementaton Te results are obtaned usng te etflows code avalable at ttp://www. gtub.com/stefansommer/etflows. Te package nclude scrpts for producng te fgures dsplayed n ts secton. Te mplementaton follows Algortm 2. Te flow equatons tat are gven n explct form n Appendx A and te adont equatons tat are gven n explct form n Appendx C are numercally ntegrated usng ScPy s odent solver (ttp://scpy.org). Bot equaton systems requre a seres of tensor multplcatons. Te optmzaton s performed wt a quas-newton BFGS optmzer. Te algortm uses sotropc Gaussan kernels. Te mages to be matced are pre-smooted wt a Gaussan flter, and mage dervatves are computed as analytc gradents of B-splne nterpolatons of te smooted mages Jet Deformatons Fgure 1 (page 5) sows te deformatons encoded by zerot, frst and second order et-partcles on ntally square grds. Note te locally affne deformatons arsng from te zerot and frst order et-partcles. Up to rotaton of te axes, te tree frst order examples n te fgure consttute a bass of te 4 dmensonal space of frst order et-partcles wt fxed lower-order components. Lkewse, up to rotaton, te tree second order examples consttute a bass for te 6 dmensonal space of second order et-partcles Matcng Functonal Approxmaton We ere llustrate and test te convergence rate of te matcng functonal approxmatons. In Fgure 2 (page 2), te approxmatons F (p) are compared for p =, 2 and varyng grd szes on tree syntetc mages supported on te unt square. Te frst two mages (a,b) are generated by frst and second order polynomals, respectvely, wle te last mage (c) s generated by a trgonometrc functon. A truncated Taylor expanson can terefore only approxmate te mage (c). Te second order approxmaton F (2) models F locally wt a second order polynomal, and t s tus expected tat te error sould vans on te mages (a,b). As te mes wdt decreases, we expect to observe O( 2 ) convergence rate for te zerot order approxmaton

20 log() log() log() 2 Henry O. Jacobs and Stefan Sommer Matc Term Approxmaton.13 Matc Term Approxmaton 2 Matc Term Approxmaton measured L 2 dssmlarty order order 2 measured L 2 dssmlarty order order 2 measured L 2 dssmlarty order order =1,,6 ( =cel(2 )) =1,,6 ( =cel(2 )) =1,,6 ( =cel(2 )) log L 2 dssmlarty error Matc Term Approxmaton, Log-log-scale 3 4 order order (a) f(x, y) = x + y convergence rate (neg. slope of log-error) log L 2 dssmlarty error Matc Term Approxmaton, Log-log-scale 2 3 order order (b) f(x, y) = (x + y) convergence rate (neg. slope of log-error) log L 2 dssmlarty error Matc Term Approxmaton, Log-log-scale 4 order 2 order (c) f(x, y) = sn(6πx)+x convergence rate (neg. slope of log-error) Fgure 2: Convergence of matcng functonal F (k), k =, 2. Top row: (a) lnear, (b) quadratc, and (c) non-polynomal mages. Lower rows, orz. axs: decreasng (ncreasng nr. of sample ponts); vert. axs: F (k) (sold, left axs) and convergence rate (dased, rgt axs). Wt lnear and quadratc mages, te error s vansng wt k = 2 and usng only one sample pont. Average convergence rates, k = : quadratc; k = 2: quartc as expected. (c, top row) sample ponts for = 2 3 (2 3 sample ponts per axs). F () on all tree mages. Lkewse, we expect a convergence rate of O( 4 ) for F (2) on mage (c). In accordance wt tese expectatons, we see te vansng error for F (2) on (a,b) and decreasng error on (c) (lower row, sold green lnes). Te non-monotonc convergence seen on (c) s a result of te polynomal approxmaton beng ntegrated over a compact doman. Te zerot order approxmaton F () lkewse decreases wt 2 convergence rate (lower row, dased blue lnes). Te convergence rate of F (2) on mage (c) stablzes at approxmately 4 untl t decreases due to numercal errors ntroduced wen te error approaces te macne precson.

21 Hger-order Spatal 25 Accuracy n Dff. 25Image Regstraton (a) (b) (c) (d) (e) (f) (g) () () () (k) (l) Fgure 3: Matcng movng mages (b-d) to fxed mage (a) usng four etpartcles (blue ponts). Enlarged fxed mage and movng mages after warpng (e-). Correspondng deformatons of an ntally square grd (-l). (b/f/) Order ; (c/g/k) order 1; (d//l) order 2. Red crosses mark locaton of etpartcles n movng mages after matcng, green boxes deformed by te warp Jacoban at te partcle postons. Movng mages at te red crosses sould matc fxed mage at blue dots; second row mages sould matc te fxed mage (a/e) Matcng Smple Structures Wt te followng set of examples, we ws to llustrate te effects of ncludng second order nformaton n te matcng term approxmaton. We vsualze ts usng smple test mages. In all examples, we wll employ te approxmatons F (k) for k =, 2. In addton, we wll matc usng only zerot and frst order nformaton wt a matcng term tat results from droppng te second order terms from F (2). Wle ts approxmaton does not arse naturally from a Taylor expanson of F, t allows vsualzaton of

22 Henry O. 25Jacobs and Stefan 25 Sommer (a) (b) (e) () (f) (c) 5 (g) (d) () Fgure 4: Frst order (lnear/affne) deformatons can be matced wt multple zerot order et-partcles (a,b) or one frst order et-partcle (c,d). A rotated bar (b/d) s matced to a bar (a/c). Te warps tat transform te movng mages (b/d) to te fxed mages (a/c) are appled to ntally square grds (/) (rotated 9 ). Red crcles are deformed wt warp dervatve at te partcle postons. (e-) sows enlarged fxed and warped movng mages. Te amount of bendng at te edges (f/) s a functon of te kernel sze () te dfferences between ncludng frst and second order mage nformaton n te matc. In Fgure 3, a bar (movng mage) s matced to a square (fxed mage). Te fgure sows ow four et-partcles move from ter postons on a grd n te fxed mage (a) to postons n te movng mage tat contan features matcng te fxed mage up to te order of te approxmaton. For zerot order (b), only pontwse ntensty s matced and te et-partcles move vertcally (red crosses) resultng n only a slgt deformaton. Wt frst order matcng (c), te et-partcles locally rotate te doman (warp Jacoban matrces sown wt green boxes) to account for te mage gradent at te corners of te square. Ts produces a damond-lke sape. Wt second

23 Hger-order Spatal Accuracy n Dff. Image Regstraton (a) (b) (c) 5 Fgure 5: Wtout ger order features, 2nd order et-partcles do not cange te matc: A blob (a) s translated and matced n movng mages (b,d) wt red crosses markng postons of et-partcles after matc. Grds (c,e) llustrate te deformatons tat are equvalent for t order (b,c) and 2nd order (d,e) (d) 1 8 (e) order (d), te corners are matced and te et-partcles move towards te corners of te movng mage bar. Te mddle row sows te warped movng mages enlarged. Te second order matc () s close to te fxed mage (a) wle bot frst and zerot order fal to produce satsfyng matces. Fgure 4 sows te result of matcng mages dfferng by an affne transformaton wt eter one frst order et or multple zerot order et-partcles. Wle tree zerot order et-partcles can approxmate a frst order deformaton n 2D, four partcles are used to produce a symmetrc pcture. Te warp Jacobans deform te ntally square green boxes dsplayed at te et postons. Te resultng warps n bot cases approxmate an affne transformaton. Wt translaton only, ncludng second order nformaton n te matc does not cange te result as llustrated n Fgure 5 were te matc s performed on an mage and a translated verson of te mage Real mage data We llustrate te effect of te ncreased order on real mages by matcng two md-sagttal slces of 3D MRI from te MGH1 dataset [KAA + 9]. In Fgure 6, red boxes mark te ventrcle area of te bran on wc te matcng s performed. We perform te matc wt 9 et-partcles (3 per axs), 16 et-partcles (4 per axs) and 64 et-partcles (8 per axs) and k =, 2. Wt 9 second order et-partcles (e), te movng mage (d) approaces te fxed (b). A vsually good matc s obtaned wt 16 or more et-partcles. 9 and 16 zerot order et-partcles are not suffcent to correctly encode te expanson of te ventrcle. Wt 64 zerot order et-partcles, te transformed mage s close to te results of te second order matces.

24 24 Henry O. Jacobs and Stefan Sommer (a) fxed, full mage (b) fxed, regon (c) movng, full mage (d) mov., regon (e) 2nd order, 9 (f) 2nd order, 16 (g) 2nd order, 64 et-partcles et-partcles et-partcles () t order, 9 () t order, 16 () t order, 64 et-partcles et-partcles et-partcles Fgure 6: 2D regstraton of MRI slces, (a-b) fxed mage, (c-d) movng mage, red boxes: regons to be matced. Lower rows: matcng results usng 2nd order et-partcles (e-g), t order et-partcles (-). Images n lower rows sould be close to (b). Wt 9 2nd order et-partcles (3 per axs), te movng mage approaces te fxed. Te matc s vsually good wt 16 etpartcles (4 per axs). Te ventrcle regon can equvalently be nflated wt 64 t order et-partcles. 8. Concluson and Future Work A pror, te LDDMM framework of mage regstraton poses an optmzaton problem on te space of dffeomorpsm. Here, we ntroduced a famly of dscretzed cost functons on a fnte dmensonal pase space tat can be mnmzed numercally. Te solutons of te dscretzed problem can be related to solutons of te full nfnte-dmensonal problem wt O( d+k )

25 Hger-order Spatal Accuracy n Dff. Image Regstraton 25 accuracy, were s a grd spacng and k s te order of approxmaton. We provded numercal examples of deformatons parametrzed by zerot, frst, and second order et-partcles, and we sowed examples of te ger order convergence of te smlarty measure. Te ger-order smlarty measure allows matcng of ger order features, and we use ts fact to regster varous sapes and mages wt low numbers of et-partcles. Representng a C k mage requres muc less nformaton tan representng a C mage. Heurstcally, te mpact of ts for computaton s tat we may use dfferent tecnques to approxmate and advect smoot mages wt a sparse set of parameters. Te ger-order accuracy scemes ere consttutes a partcular example of usng reducton by symmetry to remove redundant nformaton, and specalze advecton to te data at and. In ts case, we reduce te dmensonalty from nfnte to fnte for a gven dscretzaton, and we specalze te dscretzaton to C 2 mages. Wle te applcablty of ts specalzaton s lmted to mages of suffcent regularty, te bgger pont of ts artcle s te noton of talorng dscretzatons to data. Ts approac s applcable for reducng te dmensonalty of data beyond mages. For example, accurate dscretzatons of curves wt tangents, surfaces wt tangent planes, and ger-order tensors can be derved wt correspondng reducton n dmensonalty. Te present framework tus ponts to a general approac for ger-order accurate dscretzatons of general classes of matcng problems. Future work wll consttute testng tese areas of wder applcablty. Acknowledgments Henry O. Jacobs s supported by te European Researc Councl Advanced Grant FCCA. Stefan Sommer s supported by te Dans Councl for Independent Researc wt te proect Image Based Quantfcaton of Anatomcal Cange. We would lke to tank Klas Modn for summarzng te lterature on analyss of te EPDff equaton, especally n regards to smootness of ntal condtons. Appendx A. Equatons of moton Te equatons of moton are expressble as Hamltonan equatons wt respect to a non-canoncal Posson bracket. If we denote q () smply by q and

26 26 Henry O. Jacobs and Stefan Sommer p () smply by p ten te Hamltonan s (12) H(q, p, µ (1), µ (2) ) = 1 2 p p K (q q ) p [µ (1) γk (q q ) (13) (14) (15) + p [µ (2) γ K (q q ) 1 2 [µ(1) + [µ (1) ] ɛ [µ (2) [µ(2) [µ (2) ɛγ K (q q ) γɛφ K (q q ) ] [µ (1) γk (q q ) Were K (x) = e x 2 /2σ 2. Hamton s equatons are ten gven n sort by (16) (17) (18) (19) q = H p ṗ = H q ξ = H µ µ = ad ξ (µ). More explctly, equaton (16) s gven by q = p K (q q ) [µ (1) equaton (17) s gven by te sum γk (q q ) + [µ (2) ṗ = T + T 1 + T 2 + T 12 + T 11 + T 22 Were we defne te sx terms n ts sum as T = p γ p K γ (q q ) T 1 =(p [µ (1) p [µ (1) ) γk (q q ) T 2 = (p ɛ [µ (2) + p ɛ [µ (2) = ([µ (1) ] φ ɛ [µ(2) [µ (1) T 12 T 11 =[µ (1) ] ɛ [µ (1) T 22 = [µ (2) ζ [µ (2) ) γ K ɛ (q q ) ] φ ɛ [µ(2) γk ɛ (q q ) ɛγφ K ζ (q q ) γ K (q q ) ) ɛγ K φ (q q )

27 Hger-order Spatal Accuracy n Dff. Image Regstraton 27 Next, we calculate te quanttes ξ () = H/ µ () for = 1, 2 of equaton (18) to be [ξ (1) ] =p,γ K γ (q q ) [µ (1) γk (q q ) + [µ (2) ] γ =p γ K (q q ) [µ (1) ] ɛ ɛγk (q q ) + [µ (2) [ξ (2) wc allows us to compute µ () n equaton (19) as [ µ (1) ] [ µ (2) ] γ =[µ (1) =[µ (2) [ξ (1) ] γ [µ (1) + [µ (2) ] γ [ξ (2) ] γ [ξ (1) ] γ [µ(2) ] + [µ(2) [ξ (1) ] γ [ξ (2) ] [ξ (1) ] γ [µ (2) [µ(2) A.1. Computng q as a functon of ξ ɛ γ K ɛ (q q ) ] φ ɛ γφ K ɛ (q q ) [ξ (2) ] γ ] γ [ξ (1) ] Te acton of ξ on q s gven by ξ q. We set q = ξ q. We ve already calculated q (). We need only calculate q (1) and q (2). Componentwse we calculate tese to be [ q (1) ] =[ξ(1) ] γ[q (1) [ q (2) ] γ =[ξ(2) ] ɛ [q(1) ] [q(1) ] ɛ γ + [ξ (1) ] [q(2) ] γ Appendx B. Frst varaton equatons Te frst varaton equatons are equvalent to applyng te tangent functor to our evolutons. We fnd te veloctes: d dt q =p K (q q ) + p (q γ qγ ) γk (q q ) [µ (1) + [µ (2) γk (q q ) [µ (1) γ K (q q ) + [µ (2) γk (q q )(q q ) γɛ K (q q )(q ɛ q ɛ ) [ξ (1) ] =p γ K γ (q q ) + p γ (q q ) K γ (q q ) [µ (1) + [µ (2) γk (q q ) [µ (1) φ γk φ (q q ) + [µ (2) (qɛ q ɛ ) γɛ K (q q ) φ (qɛ q ɛ ) γɛ K φ (q q )

28 28 Henry O. Jacobs and Stefan Sommer [ξ (2) ] γ =p γ K () + p ɛ (q q ) γ K ɛ () [µ (1) ] ɛ γ K ɛ () [µ (1) ] φ (qɛ q) ɛ γɛ K φ () + [µ (2) ] ɛ φ γɛk φ () + [µ (2) ] λ ɛ (qφ qφ ) γɛφk λ () and te momenta: d dt p =T + T 1 + T 2 + T 12 + T 11 + T 22 Te frst-varaton equaton for µ (1) s d dt [µ(1) ] =[µ (1) ] γ [ξ (1) ] γ + [µ (1) ] γ [ξ (1) ] γ [µ (1) ] γ + [µ (2) ] γ [µ (2) ] γ [µ (2) [ξ (1) [µ (1) [ξ (1) [ξ (2) ] γ + [µ(2) ] γ [ξ (2) ] γ [ξ (2) ] γ [µ (2) ] γ [ξ (2) ] γ [ξ (2) ] γ [µ (2) [ξ (2) ] γ and fnally d dt [µ(2) ] γ =[µ (2) ] γ + [µ (2) ] [µ (2) ] γ [ξ (1) ] + [µ(2) ] γ [ξ (1) ] [ξ (1) + [µ(2) ] [ξ (1) [ξ (1) ] [µ (2) ] γ [ξ (1) ] were te T s are gven by T = p γ p K γ (q q ) p γ p K γ (q q ) p γ p K γ (q q )(q q )

29 Hger-order Spatal Accuracy n Dff. Image Regstraton 29 T 1 = p [µ (1) γk () p [µ (1) γk () p [µ (1) (qɛ q) ɛ ɛγ K () + p [µ (1) γk () + p [µ (1) + p [µ (1) (qɛ q) ɛ ɛγ K () γk () T 2 = p ɛ [µ (2) γ K ɛ () p ɛ [µ (2) γ K ɛ () p ɛ [µ (2) p ɛ [µ (2) p ɛ [µ (2) (q φ qφ ) γφk ɛ () γ K ɛ () p ɛ [µ (2) (q φ qφ ) γφk ɛ () γ K ɛ () T 12 = [µ (1) [µ (1) ] ɛ [µ (1) ] ɛ ] φ ɛ [µ(2) φ [µ(2) φ [µ(2) + [µ (1) ] ɛ + [µ (1) ] ɛ + [µ (1) ] ɛ φ [µ(2) φ [µ(2) φ [µ(2) ɛγ K φ () ɛγ K φ () (q ζ qζ ) ζɛγk φ () ɛγ K φ () ɛγ K φ () (q ζ qζ ) ζɛγk φ () T 11 =([µ (1) ] ɛ [µ (1) + [µ(1) ] ɛ [µ (1) + [µ (1) ] ɛ [µ (1) ) γk ɛ () (qφ qφ ) φγk ɛ ()

30 3 Henry O. Jacobs and Stefan Sommer T 22 = ([µ (2) [µ (2) ζ [µ (2) ζ [µ (2) + [µ (2) ζ [µ (2) (q λ q λ ) λɛγφ K ζ () ) ɛγφ K ζ () Fnally, we compute te varaton equatons for q (1) and q (2) to be [ q (1) ] =[ξ(1) ] γ[q (1) + [ξ(1) ] γ[q (1) [ q (2) ] γ =[ξ(2) ] ɛ [q(1) ] [q(1) ] ɛ γ + [ξ (2) ] ɛ [q(1) ] [q(1) ] ɛ γ + [ξ (2) ] ɛ [q(1) ] [q(1) ] ɛ γ + [ξ (1) ] [q(2) ] γ + [ξ(1) ] [q(2) ] γ Appendx C. Computaton of te adont equatons Gven any ODE ẋ = f(x) on M, we may consder te equatons of moton for varatons d dt x = T xf x. In partcular, T x f s a lnear operator over te pont x wc as a dual operator. Te adont equatons are and ODE on T M gven by dλ dt = T x f λ. Ts s useful for us n te followng way. Gven an ntegral curve, x(t), and a varaton n te ntal condton, x, we see tat te quantty λ(t), x(t) s constant wen x(t) satsfes te frst varaton equaton wt ntal condton x and λ(t) satsfes te adont equaton. In our case we are able to compute te gradent of te energy wt respect to varyng an ntal condton n ts way. More explctly, we sould be able to express T x f as a matrx M(x) B A so tat te frst varaton equatons are d dt xa = M(x) A Bx B and te adont equatons can be wrtten as λ A = λ B M(x) B A were λ A s te covector assocated to te A-t coordnate and M(x) B A s te coeffcent for x A n te equaton for d dt B. More specfcally, te elements of MA B s te partal dervatve of Ḃ wt respect to A. So we compute all tese (36) quanttes below.

31 Hger-order Spatal Accuracy n Dff. Image Regstraton 31 [ q () ] ( [q () ] = p kγ K γ (k) [µ (1) k [ q () ] [q (1) ] γ p γ K γ () + [µ (1) [ q () ] =, [q (2) ] =, γ [ q () ] ] [p () ] =K (), [ q () ] [µ (1) γk (k) + [µ (2) k γk () [µ (2) = γ K (), ) ] ɛ γ γ K ɛ (k) ] ɛ γ γ K ɛ (), [ q () ] [µ (2) = γ K (), [ q (1) ] [q () (1) ] ] = [ξ γ [q () [ q (1) ] [p () (1) ] [q(1) ], = [ξ [p () [q (1) ], [ q (1) ] [µ (2) ɛ = [ξ(1) ] φ [µ (2) ] ɛ γ [q (1) ] φ, [ q (1) ] [q (1) [ q (1) ] [µ (1) = [ξ (1) ] γ,, [ q (1) = [ξ(1) [µ (1) ] ɛ ] γ [q (1) ] ɛ, ] [q (2) ɛ =, [ q (2) ] γ [q () = ] [ q (2) ] γ [q (1) ] ɛ [ q (2) ] γ [q (2) ] ɛφ [ξ (2) ] φɛ [q () ] [q(1) =([ξ (2) ] φ [q(1) =[ξ (1) ] ɛ φ γ, ] φ [q(1) ] ɛ γ + [ξ(1) ] ɛ [q () ] [q(2) ] φ γ ɛ + [ξ(2) ] φ [q(1) ] φ ɛ γ), ] ɛ γ,

32 32 Henry O. Jacobs and Stefan Sommer [ q (2) ] γ [p () = ] [ q (2) ] γ [µ (1) ] ɛ [ q (2) ] γ [µ (2) [ξ (2) ] φɛ [p (1) [q (1) ] = [ξ(2) [µ (1) = [ξ(2) [µ (2) ] φζ ] ɛ ] ζλ [q (1) [q (1) ] φ [q(1) ] φ [q(1) ] ζ [q(1) ] ɛ γ + [ξ(1) ] ɛ [p () [q (2) ] ɛ γ ], ] ζ γ + [ξ(1) [µ (1) ] φ ] ɛ ] λ γ + [ξ(1) ] ζ [µ (2) [q (2) ] φ γ, [q (2) ] ζ γ, [ṗ () ] [q () ] = [T ] [q () ] + [T 1 ] [q () ] + [T 11 ] [q () ] + [T 12 ] [q () ] + [T 2 ] [q () ] + [T 22 ] [q () ], [ṗ () ] [q (1) ] γ =, [ṗ () ] [q (2) ] γ =, [ṗ () ] [p () = [T ] ] [p () + [T 1 ] ] [p () + [T 2 ] ] [p (), ] [ṗ () ] [µ (1) [ṗ () ] [µ (2) = = [T 1 ] [µ (1) [T 2 ] [µ (2) + + [T 11 ] [µ (1) + [T 12 ] [µ (2) [T 12 ] [µ (1) +, [T 22 ] [µ (2), [ µ (1) ] [q () =[µ (1) ] + [µ (2) [ µ (1) ] [q (1) =,, [ξ (1) ] [q () ] γ [µ(1) ] ɛ ] [ξ (2) ] ɛ [q () ] γ [µ(2) [ µ (1) ] [q (2) =, [ξ (1) ] [q () ] ɛ [ξ (2) ] ɛ [q () ] γ [µ(2) ] ɛ [ξ (2) ] ɛ [q () ], γ