STOCHASTIC METAMORPHOSIS in IMAGING SCIENCE 3 Dynamics Correspond to a Certain Type of Heat Bath Used in Statistical Physics

Home , David Mumford

arXiv:1705.10149v1 [math-ph] 29 May 2017 Date References Acknowledgements. Itô representation formulation Hamiltonian 3.4. Stochastic case approach stochastic Hamilton-Pontryagin 3.3. the Stochastic for approach and 3.2. transport Notation Lie stochastic by 3.1. Metamorphosis approach Hamilton-Pontryagin 3. Deterministic formulation Hamiltonian 2.5. Deterministic Euler-Poincaré2.4. reduction Deterministic equations Euler-Lagrange determin 2.3. Deterministic the for transport Euler-Poincaré theorem Lie and 2.2. deterministic Notation by metamorphosis of 2.1. Review 2. .Introduction 1. uy1,2018. 12, July : i niaemthn otepyia ocp fodrparame order of concept physical the to matching image in sis of framework variational [ al. et Gay-Balmaz fluid [ lt acigwt yaia epae a nrdcdin introduced was templates dynamical with matching plate Abstract. xedtemto fmtmrhsst pl oiaematchin image to apply to metamorphosis of method the extend in landmarks the contrast, diffeom (In of grangian.) space space. Eulerian the in in expressed curves evolving deterministically as eedn uvso h pc fdffoopim.Teapproa uncertai for The models transport diffeomorphisms. Lie of stochastic space developing the in on curves dependent 2015 2009 o h aeo nrdcn nElra netit quantifi uncertainty Eulerian an introducing of sake the For ]; of equations metamorphosis the ] Holm rsne al. et Crisan TCATCMTMRHSSI MGN SCIENCE IMAGING IN METAMORPHOSIS STOCHASTIC [ nteptenmthn praht mgn cec,tepro the science, imaging to approach matching pattern the In 2002 NHNU FDVDMMODO I 80 HIS ON MUMFORD DAVID OF HONOUR IN .Ti eutrltdtedt tutr neligtepro the underlying structure data the related result This ]. [ 2013 [ 2017 .I atclr tcs h ocp fLgaga ah ni in paths Lagrangian of concept the cast it particular, In ]. ome al. et Holm ]. [ 1998a ru´ n Younes Trouvé and ARLD HOLM, D. DARRYL n hw ocnanteeutosfrapretcomplex perfect a for equations the contain to shown and ] Contents 1 t unicto nfli yaisin dynamics fluid in quantification nty [ rhssatn niaedt structure, data image on acting orphisms ru´ n Younes Trouvé and 2005 h tnadLDMapoc r La- are approach LDDMM standard the ainapoc niaigsine we science, imaging in approach cation eercs noteEuler-Poincaré the into recast were ] hwl egie yrcn progress recent by guided be will ch e nteter fcmlxfluids complex of theory the in ter si case istic along g TH BIRTHDAY! esof cess stochastically metamorphosis [ eso metamorpho- of cess 2005 .In ]. aigscience maging vligtime evolving ome al. et Holm ntem- in Holm 16 15 15 13 12 11 11 9 8 7 6 4 3 2 2 DARRYLD.HOLM,INHONOUROFDAVIDMUMFORDONHIS80TH BIRTHDAY!

1. Introduction In recent work Arnaudon et al. [2017a,b], a new method of modelling variability of shapes has been introduced. This method uses a theory of stochastic perturbations consistent with the geometry of the diffeomorphism group corresponding to the Large Deformation Diffeomorphic Metric Mapping framework (LDDMM, see Younes [2010]). In particular, the method introduces stochastic Lie transport along stochastic curves in the diffeomorphism group of smooth invertible transformations. It models the development of variability as observed, for example, when human organs are influenced by disease processes, as analysed in computational anatomy Younes et al. [2009]. It also provides a framework including a Hamiltonian formulation for quantifying uncertainty in the development of shape atlases in computational anatomy. Hamiltonian methods for deterministic computational anatomy were recently reviewed in Miller et al. [2015]. The theory developed in Arnaudon et al. [2017a,b] treats LDDMM as a flow and uses methods based on stochastic fluid dynamics introduced in Holm [2015]. It addresses the problem of uncertainty quantification by introducing spatially correlated noise which respects the geometric structure of the data. Thus, the method provides a new way of characterising stochastic variability of shapes using spatially correlated noise in the context of the standard LDDMM framework. Numerical methods for addressing stochastic variability of shapes with landmark data structure have also been developed in Holm and Tyranowski [2016b,a]; Arnaudon et al. [2017a,b]. Although the examples were limited to landmark dynamics in the work Arnaudon et al. [2017a,b], it was clear that Lie-transport noise can be applied to any of the data structures used in the LDDMM framework, because it is compatible with the transformation theory on which LDDMM is based. The LDDMM theory was initiated by Trouvé [1995]; Christensen et al. [1996]; Dupuis et al. [1998]; Miller et al. [2002]; Beg et al. [2005] building on the pattern theory of Grenander [1994]. The LD- DMM approach models shape comparison (registration) as dynamical transformations from one shape to another whose data structure is defined as a tensor valued smooth embedding. These shape transformations are expressed in terms of the action of diffeomorphic flows, regarded as time dependent curves of smooth transformations of shape spaces. This provides a unified approach to shape modelling and shape analysis, valid for a range of structures such as landmarks, curves, surfaces, images, densities and tensor-valued images. For any such data structure, the optimal shape deformations are described via the Euler-Poincaréequation of the diffeomorphism group, usually referred to as the EPDiff equation Holm et al. [1998b]; Holm and Marsden [2005]; Younes et al. [2009]. The work Arnaudon et al. [2017a,b] showed how to obtain a stochastic EPDiff equation valid for any data structure, and in particular for the finite dimensional representation of images based on landmarks. For this purpose, the work Arnaudon et al. [2017a,b] followed the Euler-Poincaréderivation of LDDMM of Bruveris et al. [2011] based on geometric mechanics Marsden and Ratiu [1994]; Holm [2011] and the use of momentum maps to represent images and shapes. The introduction of Lie-transport noise into the EPDiff equation was implemented as cylindrical noise, obtained by pairing the deterministic momentum map the sum over eigenvectors of the spatial covariance of Stratonovich noise, each with its own Brownian motion. The work Arnaudon et al. [2017a,b] was not the first to consider stochastic evolutions in LDDMM. Indeed, Trouvéand Vialard [2012]; Vialard [2013] and more recently Marsland and Shardlow [2016] had already investigated the possibility of stochastic perturbations of landmark dynamics. In the earlier works, the noise was introduced into the landmark momentum equations, as though it were an external random force acting on each landmark independently. In Marsland and Shardlow [2016], an extra dissipative force was added to balance the energy input from the noise and to make the STOCHASTIC METAMORPHOSIS IN IMAGING SCIENCE 3 dynamics correspond to a certain type of heat bath used in statistical physics. In contrast, the work Arnaudon et al. [2017a,b] instead introduced Eulerian Stratonovich noise into the reconstruction relation used to find the deformation flows from the action of the velocity vector fields on their corresponding momenta, which are solutions of the EPDiff equation Holm and Marsden [2005]; Younes [2010]. As shown in Arnaudon et al. [2017a,b], this derivation of stochastic models is compatible with the Euler-Poincaréconstrained variational principles, it preserves the momentum map structure and yields a stochastic EPDiff equation with a novel type of multiplicative noise, depending on both the gradient and the magnitude of the solution. The model in Arnaudon et al. [2017a,b] was based on the previous works Holm [2015]; Arnaudon et al. [2016], where, respectively, stochastic perturbations of infinite and finite dimensional mechanical systems were considered. The Eulerian nature of this type of noise implies that the noise correlation depends on the image position and not, as for example in Trouvéand Vialard [2012]; Marsland and Shardlow [2016], on the landmarks themselves. This property explains why the noise is compatible with any data structure while retaining the freedom in the choice of its spatial correlation. The present work extends the Euler-Poincarévariational framework for the metamorphosis approach of Trouvéand Younes [2005]; Holm et al. [2009] from the deterministic setting to the stochastic setting. Section 2 reviews the derivation of the deterministic metamorphosis equations as cast by Holm et al. [2009] into the Euler-Poincarévariational framework of Holm et al. [1998a], as well as several other developments, including the Hamilton-Pontryagin principle and two different Hamiltonian formulations of deterministic metamorphosis. Section 3 introduces metamorphosis by stochastic Lie transport and traces out its preservation and modification of the deterministic mathematical structures reviewed in Section 2. Thus, for the sake of potential applications in uncertainty quantification, this paper extends the method of metamorphosis for image registration to enable its application to image matching along stochastically time dependent curves on the space of diffeomorphisms.

2. Review of metamorphosis by deterministic Lie transport In the pattern matching approach to imaging science, the process of “metamorphosis” in template matching with dynamical templates was introduced in Trouvéand Younes [2005]. In Holm et al. [2009] the metamorphosis equations of Trouvéand Younes [2005] were recast into the Euler-Poincarévaria- tional framework of Holm et al. [1998a] and shown to contain the equations for a perfect complex fluid Holm [2002]. This result connected the data structure underlying the process of metamorphosis in image matching to the physical concept of order parameter in the theory of complex fluids. After developing the general theory in Holm et al. [2009], various examples were reinterpreted, including point set, image and density metamorphosis. Finally, the issue was discussed of matching measures with metamorphosis, for which existence theorems for the initial and boundary value problems were provided. For more recent developments the the metamorphosis equations as well as numerical methods especially designed for metamorphosis, see Richardson and Younes [2016]. Let N be manifold, which is acted upon by a Lie group G. The manifold N contains what we can refer to as “deformable objects” and G is the group of deformations, which is the group of diffeomorphisms acting on the manifold N in our applications. Several examples for the space N were developed in the Euler-Poincarécontext in Holm et al. [2009].

Definition 1. A metamorphosis Trouvéand Younes [2005] is a pair of curves (gt, ηt) ∈ G × N parameterized by time t, with g0 = id. Its image is the curve nt ∈ N defined by the action 4 DARRYLD.HOLM,INHONOUROFDAVIDMUMFORDONHIS80TH BIRTHDAY! nt = gt.ηt, where subscript t indicates explicit dependence on time, t. The quantities gt and ηt are called, respectively, the deformation part of the metamorphosis, and its template part. When ηt is constant, the metamorphosis is said to be a pure deformation. In the general case, the image is a combination of a deformation and template variation. Before introducing stochasticity, the next section provides notation and definitions for the general problem of metamorphoses in the deterministic case. Several derivations of the fundamental metamorphosis equations are given, in order to explore the various formulations of the problem from different perspectives. 2.1. Notation and Euler-Poincarétheorem for the deterministic case. Following Trouvéand Younes [2005]; Holm et al. [2009], we will use either letters η or n to denote elements of N, the former being associated to the template part of a metamorphosis, and the latter to its image. The variational problem we shall study optimizes over metamorphoses (gt, ηt) by minimizing, for some Lagrangian L : T G × TN → R, the action integral 1 S = L(gt, g˙t, ηt, η˙t)dt , (2.1) Z0 with fixed endpoint conditions for the initial and final images n0 and n1 (with nt = gtηt) and g0 = idG. That is, the images nt are constrained at the end-points, with the initial condition g0 = id. Let g denote the Lie algebra of the Lie group G. We will consider Lagrangians defined on T G×TN, that satisfy the following invariance conditions: there exists a function ℓ defined on g × TN such that −1 L(g, Ug,η,ξη)= ℓ(Ugg ,gη,gξη ). (2.2) In other words, L is invariant under the right action of G on G × N defined by (g, η)h = (gh, h−1η). −1 For a metamorphosis (gt, ηt), we therefore have a reduced Lagrangian, upon defining ut =g ˙tgt , nt = gtηt and νt = gtη˙t, given by

L(gt, g˙t, ηt, η˙t)= ℓ(ut,nt,νt). (2.3)

The Lie derivative with respect to a vector field X will be denoted LX . The Lie algebra of G is identified with the set of right invariant vector fields Ug = ug, u ∈ TidG = g, g ∈ G, and we will use the notation Lu = LU . The Lie bracket [u, v] on the Lie algebra of smooth vector fields g is defined by

L[u,v] = −(LuLv −LvLu) (2.4) −1 and the associated adjoint operator is aduv = [u, v]. Letting Ig(h)= ghg and Advg = LvIg(id), we also have aduv = Lu(Adv)(id). When G is a group of diffeomorphisms, this yields aduv = du v − dv u. The pairing between a linear form µ and a vector field u will be denoted µ , u . Duality with respect to this pairing will be denoted with an asterisk. For example, N ∗ is the dual space of the manifold N with respect to this pairing. When G acts on a manifold N˜, the diamond operator (⋄) is defined on N˜ ∗ × N˜ and takes values in the dual Lie algebra g∗. That is, ⋄ : N˜ ∗ × N˜ → g∗. Forn ˜∗ ∈ N˜ ∗ andn ˜ ∈ N˜ the diamond operation is defined by ∗ ∗ n˜ ⋄ n˜ , u g := − n˜ , un˜ T N˜ , (2.5) where the action of a vector field u ∈ g onn ˜ ∈ N˜ is denoted by simple concatenation, un˜ ∈ T N˜. For example, the Lie algebra action of the vector field u ∈ g onn ˜ ∈ N˜ is denoted un˜ = Lun˜ ∈ T N˜. STOCHASTIC METAMORPHOSIS IN IMAGING SCIENCE 5

∗ R Subscripts on the pairings in the deﬁnition (2.5) indicate, as follows, · , · g : g × g → and ˜ ∗ ˜ R · , · T N˜ : N × T N → . In what follows, for brevity of notation we will often suppress these subscripts t, except where we wish to emphasise the presence or absence of explicit time dependence. Suppressing these subscripts when explicit time dependence is understood should cause no confusion.

Theorem 2 (Euler-Poincarétheorem). With the preceding notation, the following four statements are equivalent for a metamorphosis Lagrangian L that is invariant under the right action of G on G × N −1 defined by (g, η)h = (gh, h η), with fixed endpoint conditions for the initial and final images n0 and n1: i Hamilton’s variational principle 1 −1 δS = δ L(gt, g˙t, ηt, η˙t)dt = 0 for L(gt, g˙t, ηt, η˙t)= ℓ(g ˙tgt , gtηt, gtη˙t) (2.6) Z0 holds, for variations δgt of gt and δηt of ηt vanishing at the endpoints. ii gt and ηt satisfy the Euler–Lagrange equations for L on T G × TN. iii The constrained variational principle 1 δS = δ ℓ(ut,nt,νt)dt = 0 (2.7) Z0 −1 holds for Lagrangian ℓ defined on g × TN using variations of ut =g ˙tgt , nt = gtηt and νt = gtη˙t of the form ˙ δu = ξt − adut ξt, δn = ̟t + ξtnt, and δν =̟ ˙ t + ξtνt − ut̟t, (2.8) −1 where ξt = δgtgt , ̟t = gtδηt and these variations vanish at the endpoints. iv The Euler–Poincaré equations hold on g × TN

∂ δℓ ∗ δℓ δℓ δℓ + ad + ⋄ n + ⋄ ν = 0, ∂t δu ut δu δn t δν t (2.9) ∂ δℓ δℓ δℓ + u ⋆ − = 0, ∂t δν t δν δn with auxiliary equation n˙ t = νt + utnt, (2.10) −1 obtained from the deﬁnitions ut =g ˙tgt and nt = gtηt, with νt = gtη˙t, provided the endpoint condition holds, that δℓ δℓ (1) + (1) ⋄ n = 0, (2.11) δu δν 1 at time t = 1.

Corollary 3 (Coadjoint motion). Equations (2.9) and the auxiliary equation (2.10) for nt together imply the following coadjoint motion equation,

∂ δℓ δℓ ∗ δℓ δℓ + ⋄ n + ad + ⋄ n = 0. (2.12) ∂t δu δν ut δu δν 6 DARRYLD.HOLM,INHONOUROFDAVIDMUMFORDONHIS80TH BIRTHDAY!

The equivalence of statements i and ii in Theorem 2 is classical, and no other proof will be oﬀered here. The proofs of the other equivalences in Euler-Poincar´eTheorem 2 and its Corollary 3 for deterministic metamorphosis are laid out in the sections below.

2.2. Deterministic Euler-Lagrange equations. We compute the Euler-Lagrange equations associated with the minimization of the symmetry reduced action

1 S = ℓ(ut,nt,νt)dt Z0 with ﬁxed boundary conditions n0 and n1. We therefore consider variations δu and ω = δn. The variation δν can be obtained from n = gη and ν = gη˙ yieldingn ˙ = ν + un andω ˙ = δν + uω + (δu)n. Here and in the following of this paper, we assume that computations are performed in a local chart on TN with respect to which we take partial derivatives. We therefore have 1 δℓ δℓ δℓ δS = δ , δut + , ωt + , ω˙ t − utωt − (δut)nt dt = 0. 0 δu δn δν Z D E D E D E The δu term yields the equation δℓ δℓ + ⋄ n = 0. δu δν t where, in a slight abuse of notation, δℓ/δν ∈ T (TN)∗ has been considered as a linear form on TN by δℓ/δν , z := δℓ/δν , (0, z) . From the terms involving ω, we ﬁnd, after an integration by parts

∂ δℓ δℓ δℓ + u ⋆ − = 0 . (2.13) ∂t δν t δν δn Here, we have introduced notation for the star (⋆) operation, δℓ δℓ , uω = u ⋆ , ω . (2.14) δν δν D E D E ∗ T ∗ For uω = Luω, the star (⋆) operation denotes the dual of the Lie derivative, u⋆ν = Lu ν . We therefore obtain the system of equations δℓ δℓ + ⋄ n = 0 , δu δν t

∂ δℓ δℓ δℓ (2.15) + u ⋆ = , ∂t δν t δν δn

n˙ t = νt + utnt .

δℓ δℓ Note that the sum δu + δν ⋄n is the momentum map arising from Noether’s theorem for the considered invariance of the Lagrangian. The special form of the boundary conditions (ﬁxed n0 and n1) ensures that this momentum map vanishes. STOCHASTIC METAMORPHOSIS IN IMAGING SCIENCE 7

2.3. Deterministic Euler-Poincaréreduction. As explained in in Holm et al. [2009], a system equivalent to that in (2.15) can be obtained via Euler-Poincaréreduction Holm et al. [1998a]. In this setting, we make the variation in the group element and in the template instead of the velocity and −1 the image. We denote ξt = δgtgt and ̟t = gtδηt. From these definitions, we obtain the expressions of the variations, δu, δn and δν. We first have δut = ξ˙t + [ξt, ut], which arises from the standard Euler-Poincaréreduction theorem, as provided in Holm et al. [1998a]; Marsden and Ratiu [1994]. We also have δnt = δ(gtηt)= ̟t +ξtnt. From νt = gtη˙t, we get δνt = gtδη˙t + ξtνt and from ̟t = gtδηt we also have̟ ˙ t = ut̟t + gtη˙t. This yields δνt =̟ ˙ t + ξtνt − ut̟t. We also compute the boundary conditions for ξ and ̟. At t = 0, we have g0 = id and n0 = g0η0 = cst which implies ξ0 = 0 and ̟0 = 0. At t = 1, the relation g1η1 = cst yields ξ1n1 + ω1 = 0. Now, the first variation of is 1 δℓ ˙ δℓ δℓ , ξt − adut ξt + , ̟t + ξtnt + , ̟˙ t + ξtνt − ut̟t dt = 0. 0 δu δnt δν Z D E D E D E In the integration by parts to eliminate ξ˙t and̟ ˙ t, the boundary term is (δℓ/δu)1 , ξ1 + (δℓ/δν)1 , ω1 . Using the boundary condition, the last term can be re-written

− (δℓ/δν)1 , ξ1n1 = (δℓ/δν)1 ⋄ n1 , ξ1 . We therefore obtain the endpoint equation δℓ δℓ (1) + (1) ⋄ n = 0. δu δν 1 The evolution equation for δℓ/δu is

∂ δℓ ∗ δℓ δℓ δℓ + ad + ⋄ n + ⋄ ν = 0, ∂t δu ut δu δn t δν t and δℓ/δν evolves by ∂ δℓ δℓ δℓ + u ⋆ − = 0. ∂t δν t δν δn Consequently, we obtain the following system of equations,

∂ δℓ ∗ δℓ δℓ δℓ + ad + ⋄ n + ⋄ ν = 0, ∂t δu ut δu δn t δν t (2.16) ∂ δℓ δℓ δℓ + u ⋆ − = 0, ∂t δν t δν δn as well as the auxiliary equation ∂ n = ν + u n , (2.17) ∂t t t t t −1 obtained from the deﬁnitions ut =g ˙tgt and nt = gtηt. Moreover, the endpoint condition holds, that δℓ δℓ (1) + (1) ⋄ n = 0, (2.18) δu δν 1 at time t = 1. ✷ 8 DARRYLD.HOLM,INHONOUROFDAVIDMUMFORDONHIS80TH BIRTHDAY!

As discussed in in Holm et al. [2009], the system (2.9) is equivalent to (2.15), since they characterize the same critical points. Direct evidence of this fact may be obtained from the proof of Corollary 3, that ∂ δℓ δℓ ∗ δℓ δℓ + ⋄ n + ad + ⋄ n = 0. (2.19) ∂t δu δν ut δu δν Proof of Corollary 3. A solution of (2.9) satisﬁes the coadjoint motion equation, ∂ δℓ δℓ ∂ δℓ ∂ δℓ δℓ + ⋄ nt = + ⋄ nt + ⋄ n˙ t ∂t δut δν ∂t δu ∂t δν δν ∂ δℓ δℓ δℓ δℓ = + − u ⋆ ⋄ n + ⋄ (ν + u n ) ∂t δu δn t δν t δν t t t ∂ δℓ δℓ δℓ δℓ δℓ = + ⋄ n + ⋄ ν − u ⋆ ⋄ n + ⋄ (u n ) ∂t δu δn t δν t t δν t δν t t ∗ δℓ ∗ δℓ = −ad − ad ( ⋄ n ). ut δu ut δν t In the last equation, we have used the fact that, for any α ∈ g, δℓ δℓ δℓ ⋄ (un) − u ⋆ ⋄ n , α = , α(un) − u(αn) δν δν δν D E D δℓ E = − , [u, α]n δν D δℓ E = − ⋄ n , [u, α] δν D ∗ δℓ E = − ad ( ⋄ n ) , α . ut δν t D E

Remark 4. Corollary 3 combined with the relation (δℓ/δu)1 + (δℓ/δν)1 ⋄ u1 = 0 implies the first equation in (2.15). Namely, the zero level set of the momentum map is preserved by coadjoint motion. 2.4. Deterministic Hamiltonian formulation. The Euler-Poincaréformulation of metamorphosis in (2.9) and (2.10) in Theorem 2 allows passage to its Hamiltonian formulation via the following Legendre transformation of the reduced Lagrangian ℓ in the velocities u and ν, in the Eulerian fluid description, δℓ δℓ µ = , σ = , h(µ,σ,n)= µ , u + σ, ν − ℓ(u, ν, n). (2.20) δu δν Accordingly, one computes the variational derivatives of h as δh δh δh δℓ = u , = ν , = − . (2.21) δµ δσ δn δn Consequently, the Euler-Poincaréequations (2.9) and the auxiliary kinematic equation (2.10) for metamorphosis imply the following equations, for the Legendre-transformed variables, (µ,σ,n),

∗ δh δh ∂ µ + ad µ + σ ⋄ − ⋄ n = 0, t δh/δµ δσ δn (2.22) δh ∂ σ + LT σ + = 0, t δh/δµ δn STOCHASTIC METAMORPHOSIS IN IMAGING SCIENCE 9 as well as the auxiliary equation δh ∂ n = L n + . (2.23) t δh/δµ δσ These equations are Hamiltonian. That is, they may be expressed in the form ∂z δh = {z, h} = b · , (2.24) ∂t δz where z ∈ (µ,σ,n) and the Hamiltonian matrix b defines the Poisson bracket δf δh {f, h} = d n x · b · , (2.25) δz δz Z which is bilinear, skew symmetric and satisfies the Jacobi identity, {f, {g, h}} + {g, {h, f}} + {h, {f, g}} = 0. Assembling the metamorphosis equations (2.22) - (2.23) into the Hamiltonian form (2.24) gives, ∗ ∂tµ ad✷µ − σ ⋄ ✷ ✷ ⋄ n δh/δµ (= u) T ∂ σ = − L✷σ 0 −1 δh/δσ (= ν) (2.26)  t      ∂tn −L✷n 1 0 δh/δn In this expression, the operators  act to the right on all terms in a product by the chain rule. Remarks about the Hamiltonian matrix. The Hamiltonian matrix in equation (2.26) was discovered some time ago in the context of complex fluids in Holm and Kupershmidt [1988]. There, it was proven to be a valid Hamiltonian matrix by associating its Poisson bracket as defined in equation (2.25) with the dual space of a certain Lie algebra of semidirect-product type which has a canonical two-cocycle on it. The mathematical discussion of Lie algebras with two-cocycles is given in Holm and Kupershmidt [1988]. See also Holm [2002]; Gay-Balmaz and Ratiu [2009]; Gay-Balmaz and Tronci [2010]; Gay-Balmaz et al. [2013] for further discussions of semidirect-product Poisson brackets with cocycles for complex fluids. Being dual to the semidirect-product Lie algebra g s T ∗N, our Hamiltonian matrix in equation (2.26) is in fact a Lie-Poisson Hamiltonian matrix. See, e.g., Marsden and Ratiu [1994] and references therein for more discussions of such Hamiltonian matrices. For our present purposes, its rediscovery in the context of metamorphosis links the physical and mathematical interpretations of the variables in the theory of imaging science with earlier work in complex fluid dynamics and with the gauge theory approach to condensed matter, see, e.g., Kleinert [1989]. 2.5. Deterministic Hamilton-Pontryagin approach. An alternative formulation to either the Euler-Lagrange equations, or the Euler-Poincaréapproach is obtained in the Hamilton–Pontryagin principle. In this approach, the diffeomorphic paths appear explicitly. Theorem 5 (Hamilton–Pontryagin principle). The Euler–Poincaréequations in Corollary 3 for coadjoint motion given by ∂ δℓ δℓ ∗ δℓ δℓ + ⋄ n + ad + ⋄ n = 0, ∂t δu δν u δu δν (2.27) ∂ δℓ δℓ δℓ + u ⋆ − = 0, ∂t δν δν δn as well as the auxiliary equation n˙ = ν + un, (2.28) 10 DARRYLD.HOLM,INHONOUROFDAVIDMUMFORDONHIS80TH BIRTHDAY! on the space g∗ × T ∗N × TN are equivalent to the following implicit variational principle, δS(u, n, n,ν,g,˙ g˙) = 0, (2.29) for a constrained action 1 − S(u, n, n,ν,g,˙ g˙)= ℓ(u, n, ν)+ h M , (gg ˙ 1 − u) i + h σ , (n ˙ − ν − un) i dt . (2.30) 0 Z h i Proof. The variations of S in formula (3.7) are given by 1 δℓ δℓ 0= δS = − M + σ ⋄ n , δu − σ˙ + u⋆σ − , δn 0 δu δn Z D E D E (2.31) δℓ − + − σ, δν + M , δ(g 1g˙) dt . δν D E D E −1 −1 After a side calculation, one finds δ(gg ˙ )= ξ˙−aduξ, with ξ = δgg for the last term in (3.7). Then, integrating by parts yields the familiar relation 1 1 −1 M , δ(gg ˙ ) dt = M , (ξ˙ − aduξ) dt Z0 Z0 D E 1 D E 1 ˙ ∗ = − M − adu M , η dt + M,ξ , 0 0 Z D E D E −1 where ξ = δgg vanishes at the endpoints in time. Thus, stationarity δS = 0 of the Hamilton–Pontryagin variational principle with constrained action integral (3.7) yields the following set of equations:

δℓ δℓ ∂σ δℓ ∂M ∗ M = + σ ⋄ n, σ = , + u⋆σ − = 0 , + ad M = 0 , (2.32) δu δν ∂t δn ∂t u as well as the constraint equations − gg˙ 1 − u = 0 andn ˙ − ν − un = 0 . (2.33) This ﬁnishes the proof of the Hamilton–Pontryagin principle in Theorem 5. Proposition 6 (Untangling the Lie-Poisson structure (2.26)). By the change of variables h(µ,σ,n)= H(M,σ,n) , (2.34) the Lie-Poisson structure (2.26) transforms into ∗ ∂tM ad✷M 0 0 δH/δM (= u) ∂tσ = − 0 0 1 δH/δσ + LδH/δM n (= ν + Lun) , (2.35)      T T  ∂tn 0 −1 0 −δH/δn −LδH/δM σ (= −δH/δn −Lu σ)       and thereby recovers equations (2.27) and (2.28) in Hamiltonian form. Proof. The proof follows from the expanding out the change of variables formula for variational derivatives, δh(µ,σ,n)= δH(M,σ,n) , STOCHASTIC METAMORPHOSIS IN IMAGING SCIENCE 11 where (M,σ,n) := (µ + σ ⋄ n,σ,n). Namely, one substitutes the corresponding terms, δh δh δh δh(µ,σ,n)= , δµ + , δσ + , δn , δµ δσ δn D E D E D E δH δH δH T δH(M,σ,n)= , δµ + + L δH n, δσ − + L δH σ, δn , δM δσ δM δn δM D E D E D E into the transformed Hamiltonian structure.

Remark 7. The Lie-Poisson Hamiltonian structure (2.35) is the variable transformation (2.34) of the corresponding structure (2.26). The corresponding Lie-Poisson bracket is defined on the dual Lie algebra of the vector fields over the domain, D; namely, g = X(D) with canonical 2-cocycle X(D)∗ × T ∗F(D,N), where F(D,N) denotes smooth functions from the domain, D, to the data structure manifold, N. For more details about how the untangling of Lie-Poisson structures is applied in geometric mechanics in the theory of complex fluids and for further citations in this literature to earlier work, see Gay-Balmaz and Tronci [2010]; Gay-Balmaz et al. [2013].

3. Metamorphosis by stochastic Lie transport 3.1. Notation and approach for the stochastic case. To derive the stochastic partial differential equations (SPDEs) for uncertainty quantification in the metamorphosis approach to imaging science, we combine the recent developments for uncertainty quantification in fluid dynamics in Holm [2015] with the Hamilton-Pontryagin principle for metamorphosis discussed in the previous section. The idea is to replace the deterministic evolutionary constraints in equation (2.33) by the following stochastic processes,

−1 i i dgg − ut(x) dt + ξi(x) ◦ dWt =0 and dnt − νt dt − ut dt + ξi ◦ dWt n = 0 , (3.1) ! ! Xi Xi where d is brief notation for the stochastic evolution operator, which strictly speaking is an integral operator. The ﬁrst of these stochastic processes may be written equivalently as a stochastic version of ∗ the Lagrange-to-Euler map by using the notation gt for pullback by the stochastic diﬀeomorphism gt,

∗ i dgt − gt ut(X) dt + ξi(X) ◦ dWt = 0 . (3.2) ! Xi In this form, one sees that gt is a stochastic process with time dependent drift term given by the ∗ pullback operation, ut(x) dt = gt ut(X) dt = ut(gtX) dt, in which subscript t on gt and ut(X) indicates that both ut and gt depend explicitly on time, t. Thus, the dynamical drift velocity ut(x) depends on time explicitly and also through the Lagrange-to-Euler map x = gtX governed by (3.2) with initial value g0 = Id. The Lagrange-to-Euler map in (3.2) also contains a Stratonovich stochastic term, comprising a finite sum over time independent spatial functions ξi, i = 1, 2,...,N, each composed in a i Stratonovich sense (denoted by the symbol ◦) with its own Brownian motion in time, dWt . This type of Stratonovich stochasticity, called “cylindrical noise”, was introduced in Schaumlöffel [1988]. In the cylindrical noise term, the ξi(x), i = 1, 2,...,N, are interpreted as describing the spatial correlations of the noise in fixed Eulerian space, e.g., as eigenvectors of the correlation tensor, or covariance, for a process which is assumed to be statistically stationary. 12 DARRYLD.HOLM,INHONOUROFDAVIDMUMFORDONHIS80TH BIRTHDAY!

3.2. Stochastic Hamilton-Pontryagin approach. Theorem 8 (Stochastic Hamilton–Pontryagin principle). Stochastic metamorphosis is governed by coadjoint motion represented as SPDE given by

δℓ δℓ ∗ δℓ δℓ d + ⋄ n + ad ◦ i + ⋄ n = 0, δu δν (ut(x) dt+Pi ξi(x) dWt ) δu δν (3.3)

δℓ i δℓ δℓ d + ut(x) dt + ξi(x) ◦ dWt ⋆ − = 0, δν ! δν δn Xi as well as the auxiliary equation

i dn = ν + ut(x) dt + ξi(x) ◦ dWt n, (3.4) ! Xi on the space g∗ × T ∗N × TN are equivalent to the following implicit variational principle, δS(u, n, dn,ν,g, dg) = 0, (3.5) for a stochastically constrained action

1 −1 i S(u, n, dn,ν,g, dg)= ℓ(ut,nt,νt) dt + M , dgtgt − ut(x) dt − ξi(x) ◦ dWt 0 Z h D Xi E (3.6) i + σ , (dnt − νt dt − ut(x) dt + ξi(x) ◦ dWt nt) . D Xi Ei Remark 9 (Stratonovich versus Itôrepresentations). In dealing with the stochastic variational principle, we will work in the Stratonovich representation, because it admits ordinary variational calculus. However, later, when we consider expected values for the solutions, we will transform to the equivalent Itôrepresentation. In transforming to the Itôrepresentation, we will discover that the effective diffusion from the Itôcontraction term is by no means a Laplacian. Instead, the Itôcontraction term turns out to produce a double Lie derivative with respect to the sum of vector fields ξi(x). After this remark, we return to the proof of Theorem 8 for the Stochastic Hamilton–Pontryagin principle.

Proof. The variations of S in formula (3.6) are given by

1 δℓ i δℓ 0= δS = − M + σ ⋄ n , δu dt − dσ + ut(x) dt + ξi(x) ◦ dWt ⋆σ − dt, δn 0 δu δn Z D E D Xi E δℓ − + − σ, δν dt + M , δ(dgg 1) dt . δν D E D E (3.7) In a side calculation, one ﬁnds

−1 −1 d d i δ( gg )= ξ − ad(ut(x) dt+Pi ξi(x)◦dWt )ξ , with ξ = δgg , STOCHASTIC METAMORPHOSIS IN IMAGING SCIENCE 13 for substitution into the last term. Then, integrating by parts yields the relation 1 1 −1 d d i M , δ( gg ) dt = M , ξ − ad(ut(x) dt+Pi ξi(x)◦dWt )ξ dt 0 0 Z D E Z 1 D E ∗ 1 = − dM − ad i M , η dt + M,ξ , (ut(x) dt+Pi ξi(x)◦dWt ) 0 0 Z D E D E −1 where ξ = δgg vanishes at the endpoints in time. Thus, stationarity δS = 0 of the Hamilton–Pontryagin variational principle with stochastically constrained action integral (3.7) yields the following set of SPDEs: ∗ dM + ad i M = 0 , (ut(x) dt+Pi ξi(x)◦dWt ) δℓ (3.8) dσ + u (x) dt + ξ (x) ◦ dW i ⋆σ − dt = 0 , t i t δn Xi for the quantities δℓ δℓ M = + σ ⋄ n , and σ = , (3.9) δu δν as well as the stochastic constraint equations −1 i dgg − ut(x) dt + ξi(x) ◦ dWt = 0 , Xi (3.10) i dn − ν − ut(x) dt + ξi(x) ◦ dWt n = 0 . Xi This ﬁnishes the proof of the Hamilton–Pontryagin principle for stochastic metamorphosis formulated in Theorem 8. 3.3. Stochastic Hamiltonian formulation. By Corollary 3, the stochastic equations (3.8) through (3.10) above imply the corresponding stochastic versions of in (2.9) and (2.10) in Theorem 2. Namely,

δℓ ∗ δℓ δℓ δℓ d + ad + ⋄ n dt + ⋄ ν dt = 0, δu u˜ δu δn δν

δℓ δℓ δℓ (3.11) d + LT − dt = 0, δν u˜ δν δn

dn = Lu˜nt + ν dt with Stratonovich stochastic transport velocityu ˜ given by i u˜ := ut(x) dt + ξi(x) ◦ dWt . (3.12) Xi At this point the Hamiltonian structure of the deterministic metamorphosis equations reveals how we can write the stochastic metamorphosis equations in Hamiltonian form. Namely, we deform the deterministic Hamiltonian by adding the stochastic part to it as being linear in the momentum µ and paired with the Stratonovich noise perturbation, ˜ i h := h(µ.ν,n) dt + µ , ξi(x) ◦ dWt . (3.13) D Xi E 14 DARRYLD.HOLM,INHONOUROFDAVIDMUMFORDONHIS80TH BIRTHDAY!

We then use the same Lie-Poisson Hamiltonian structure as in the deterministic case. Assembling the metamorphosis equations (3.19) into the Hamiltonian form (2.24) gives,

∗ ✷ ✷ ˜ i dµ ad✷µ − σ ⋄ ⋄ n δh/δµ (=u ˜ := ut(x) dt + i ξi(x) ◦ dWt ) T dσ = − L✷σ 0 −1 δh/δσ˜ = δh/δσ (= ν) . (3.14)      P  dn −L✷n 1 0 δh/δσ˜ = δh/δn       As before, the operators in the Hamiltonian matrix act to the right on all terms in a product by the chain rule. By the change of variables corresponding to (2.34) for this stochastic case,

˜ i h(µ,σ,n)= H(M,σ,n)= H(M,σ,n) dt ++ M , ξi(x) ◦ dWt , (3.15) D Xi E e one ﬁnds that the Lie-Poisson structure in (3.14) transforms into

∗ dM ad✷M 0 0 δH/δM (= u) dσ = − 0 0 1 δH/δσ + L n (= ν + Lun) , (3.16)      δH/δMe e  dn 0 −1 0 −δH/δn −LT e σ (= −eδH/δn −LT σ) δH/δM u  e e e        and thereby recovers equations (3.8) - (3.10) in Hamiltoniane form. e

Remark 10. The advantage of writing equations (3.16) in terms of the total momentum 1-form den- 3 ∗ sity M := M·dx⊗d x may be seen by recalling that aduM = LuM for 1-form densities. Consequently, e e the first equation in (3.16) implies (d + Lu)M = 0, which in turn implies that e ∗ ∗ d(g M)= g (d + Lu)M = 0 . (3.17) e This means the total momentum 1-form density M := M · dx ⊗ d3x is preserved by the stochastic flow given by the pullback of the stochastic diffeomorphism gt in (3.2), which is the flow of the stochastic −1 vector field u. That is, the stochastic Lagrange-to-Euler flow gt, which is the solution of dgg = u in (3.1), e e ∗ ∗ i dgt − gt u = dgt − gt ut(X) dt + ξi(X) ◦ dWt = 0 , (3.18) ! Xi preserves the quantity M alonge its flow. Hence, we say that the total momentum 1-form density M := M · dx ⊗ d3x is stochastically advected.

Summary. The preservation of the Hamiltonian structure achieved in (3.14) for the present formulation of the stochastic metamorphosis equations provides the interpretation of the stochastic part i of the flow. The Hamiltonian flow of the momentum hµ , i ξi(x) ◦ dWt i produces stochastic trans- i lation in Eulerian space with velocity i ξi(x) ◦ dWt . Thus, adding the stochastic part, linear in the momentum, to the metamorphosis Hamiltonian h(µ,ν,n)P adds a stochastic transport to the deterministic flow. This is consistent with ourP intention of modelling stochastic metamorphosis as motion generated by a temporally stochastic flow on the diffeomorphisms, with spatial correlations given by the prescribed, time-independent correlation eigenvectors determined from data assimilation. STOCHASTIC METAMORPHOSIS IN IMAGING SCIENCE 15

∗ 3.4. Itôrepresentation. In preparation for writing the Itôrepresentation, we first substitute aduµ = Luµ to show in the more familiar Lie derivative notation the action of the vector field u on its dual momentum, the 1-form density µ = δℓ/δu. The equivalent Itôrepresentations of equations (3.19) are then given by δℓ δℓ δℓ 1 δℓ δℓ δℓ d + L dt + L dW i − L L dt + ⋄ n dt + ⋄ ν dt = 0, δu u δu ξi(x) δu t 2 ξi(x) ξi(x) δu δn δν Xi Xi

δℓ δℓ δℓ 1 δℓ δℓ d + LT dt + LT dW i − LT LT dt − dt = 0, δν u δν ξi(x) δν t 2 ξi(x) ξi(x) δν δn (3.19) Xi Xi

1 dn = L n dt + L n dW i − L L n dt + ν dt, u t ξi(x) t t 2 ξi(x) ξi(x) t Xi Xi with stochastic transport velocity u given in Itˆoform by

i bu := ut(x) dt + ξi(x)dWt , (3.20) Xi plus the Itôcontraction drift terms.b Likewise, the stochastic advection of the total momentum 1-form 3 density M = µ + σ ⋄ n = M · dx ⊗ d x, expressed in Stratonovich form as (d + Lu)M = 0, is expressed in Itôform as e 1 dM + L M dt + L MdW i − L L M dt = 0 , (3.21) u ξi(x) t 2 ξi(x) ξi(x) Xi Xi In which the last sum is the Itôcontraction term. In Itôform, the expectation of the noise terms vanish. The noise interacts multiplicatively with both the solution M and the gradient of the solution ∇M, through the Lie derivative, as

i T i 3 Lξi(x)MdWt = ξi(x) ·∇ M + ∇ξi(x) · M + M divξi(x) dWt · dx ⊗ d x. (3.22) i i X X nh i o Likewise, the Itˆocontraction drift terms are not Laplacians, as would have been the case for additive noise with constant amplitude. Instead, in (3.21) they are sums over double Lie derivatives with respect to the vector ﬁelds ξi(x) associated with the spatial correlations of the stochastic perturbation. This double Lie derivative combination was called the Lie Laplacian in Holm [2015] has many properties of potential interest in the mathematical analysis of these equations Crisan et al. [2017].

Acknowledgements. I am grateful to A. Trouvéand L. Younes for their collaboration in developing the Euler-Poincarédescription of metamorphosis. I am also grateful to F. Gay-Balmaz, T. S. Ratiu and C. Tronci for many illuminating collaborations in complex fluids and other topics in geometric mechanics during the course of our long friendship. Finally, I am also grateful to M. I. Miller and D. Mumford for their encouragement over the years to pursue the role of EPDiff in imaging science. Dur- ing the present work the author was partially supported by the European Research Council Advanced Grant 267382 FCCA and the EPSRC Grant EP/N023781/1. 16 DARRYLD.HOLM,INHONOUROFDAVIDMUMFORDONHIS80TH BIRTHDAY!

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