Stochastic Metamorphosis in Imaging Science

TL;DR

This paper extends the metamorphosis framework in imaging science to include stochastic models, enabling uncertainty quantification in image matching by incorporating randomness into the evolution of diffeomorphic transformations.

Contribution

It introduces a stochastic extension of the metamorphosis equations, linking image matching to stochastic fluid dynamics for uncertainty quantification.

Findings

Develops a stochastic metamorphosis model for image matching

Connects metamorphosis to complex fluid theory

Provides a basis for uncertainty quantification in imaging

Abstract

In the pattern matching approach to imaging science, the process of \emph{metamorphosis} in template matching with dynamical templates was introduced in \cite{ty05b}. In \cite{HoTrYo2009} the metamorphosis equations of \cite{ty05b} were recast into the Euler-Poincar\'e variational framework of \cite{HoMaRa1998} and shown to contain the equations for a perfect complex fluid \cite{Holm2002}. This result related the data structure underlying the process of metamorphosis in image matching to the physical concept of order parameter in the theory of complex fluids \cite{GBHR2013}. In particular, it cast the concept of Lagrangian paths in imaging science as deterministically evolving curves in the space of diffeomorphisms acting on image data structure, expressed in Eulerian space. (In contrast, the landmarks in the standard LDDMM approach are Lagrangian.) For the sake of introducing an Eulerian uncertainty quantification approach in imaging science, we extend the method of metamorphosis to apply to image matching along \emph{stochastically} evolving time dependent curves on the space of diffeomorphisms. The approach will be guided by recent progress in developing stochastic Lie transport models for uncertainty quantification in fluid dynamics in \cite{holm2015variational,CrFlHo2017}.

Equations

$$S={\int }_0^{1}L(g_t,{\dot{g}}_t,{\eta }_t,{\dot{\eta }}_t)dt,$$

$$L(g,U_g,\eta ,{\xi }_{\eta })=\ell (U_g{g}^{-1},g\eta ,g{\xi }_{\eta }).$$

$$L(g_t,{\dot{g}}_t,{\eta }_t,{\dot{\eta }}_t)=\ell (u_t,n_t,{\nu }_t).$$

$${\mathcal{L}}_{[u,v]}=-({\mathcal{L}}_u{\mathcal{L}}_v-{\mathcal{L}}_v{\mathcal{L}}_u)$$

$${\big{\langle}{\tilde{n}^{*}\diamond\tilde{n}}\,,\,{u}\big{\rangle}}_{\mathfrak{g}}:=-{\big{\langle}{\tilde{n}^{*}}\,,\,{u\tilde{n}}\big{\rangle}}_{T\tilde{N}}\,,$$

$$\delta S=\delta {\int }_0^{1}L(g_t,{\dot{g}}_t,{\eta }_t,{\dot{\eta }}_t)dt=0forL(g_t,{\dot{g}}_t,{\eta }_t,{\dot{\eta }}_t)=\ell ({\dot{g}}_t{g}_t^{-1},g_t{\eta }_t,g_t{\dot{\eta }}_t)$$

$$\delta \mathcal{S}=\delta {\int }_0^1\ell (u_t,n_t,{\nu }_t)dt=0$$

$$\delta u={\dot{\xi }}_t-{\text{ad}}_{u_t}{\xi }_t,\delta n={\varpi }_t+{\xi }_t{n}_t,and\delta \nu ={\dot{\varpi }}_t+{\xi }_t{\nu }_t-u_t{\varpi }_t,$$

$$\begin{array}{rl}& \frac{\partial }{\partial t}\frac{\delta \ell }{\delta u}+{\text{ad}}_{u_t}^{\ast }\frac{\delta \ell }{\delta u}+\frac{\delta \ell }{\delta n}\diamond n_t+\frac{\delta \ell }{\delta \nu }\diamond {\nu }_t=0,\\ & \\ & \frac{\partial }{\partial t}\frac{\delta \ell }{\delta \nu }+u_t\star \frac{\delta \ell }{\delta \nu }-\frac{\delta \ell }{\delta n}=0,\end{array}$$

$${\dot{n}}_t={\nu }_t+u_t{n}_t,$$

$$\frac{\delta \ell }{\delta u}(1)+\frac{\delta \ell }{\delta \nu }(1)\diamond n_1=0,$$

$$\displaystyle\frac{\partial}{\partial t}\Big{(}\frac{\delta\ell}{\delta u}+\frac{\delta\ell}{\delta\nu}\diamond n\Big{)}+\text{ad}_{u_{t}}^{*}\Big{(}\frac{\delta\ell}{\delta u}+\frac{\delta\ell}{\delta\nu}\diamond n\Big{)}=0.$$

$$\mathcal{S}={\int }_0^1\ell (u_t,n_t,{\nu }_t)dt$$

$$\delta\mathcal{S}=\delta\int_{0}^{1}\left({\Big{\langle}{\frac{\delta\ell}{\delta u}}\,,\,{\delta u_{t}}\Big{\rangle}}+{\Big{\langle}{\frac{\delta\ell}{\delta n}}\,,\,{\omega_{t}}\Big{\rangle}}+{\Big{\langle}{\frac{\delta\ell}{\delta\nu}}\,,\,{\dot{\omega}_{t}-u_{t}\omega_{t}-(\delta u_{t})n_{t}}\Big{\rangle}}\right)dt=0.$$

$$\frac{\delta \ell }{\delta u}+\frac{\delta \ell }{\delta \nu }\diamond n_t=0.$$

$$\frac{\partial }{\partial t}\frac{\delta \ell }{\delta \nu }+u_t\star \frac{\delta \ell }{\delta \nu }-\frac{\delta \ell }{\delta n}=0.$$

$${\Big{\langle}{\frac{\delta\ell}{\delta\nu}}\,,\,{u\omega}\Big{\rangle}}={\Big{\langle}{u\star\frac{\delta\ell}{\delta\nu}}\,,\,{\omega}\Big{\rangle}}.$$

$$\begin{array}{rl}\frac{\delta \ell }{\delta u}+\frac{\delta \ell }{\delta \nu }\diamond n_t& =0,\\ & \\ \frac{\partial }{\partial t}\frac{\delta \ell }{\delta \nu }+u_t\star \frac{\delta \ell }{\delta \nu }& =\frac{\delta \ell }{\delta n},\\ & \\ {\dot{n}}_t={\nu }_t+u_t{n}_t.& \end{array}$$

$$\int_{0}^{1}\left({\Big{\langle}{\frac{\delta\ell}{\delta u}}\,,\,{\dot{\xi}_{t}-\text{ad}_{u_{t}}\xi_{t}}\Big{\rangle}}+{\Big{\langle}{\frac{\delta\ell}{\delta n_{t}}}\,,\,{\varpi_{t}+\xi_{t}n_{t}}\Big{\rangle}}+{\Big{\langle}{\frac{\delta\ell}{\delta\nu}}\,,\,{\dot{\varpi}_{t}+\xi_{t}\nu_{t}-u_{t}\varpi_{t}}\Big{\rangle}}\right)dt=0.$$

$$-{\big{\langle}{(\delta\ell/\delta\nu)_{1}}\,,\,{\xi_{1}n_{1}}\big{\rangle}}={\big{\langle}{(\delta\ell/\delta\nu)_{1}\diamond n_{1}}\,,\,{\xi_{1}}\big{\rangle}}.$$

$$\frac{\delta \ell }{\delta u}(1)+\frac{\delta \ell }{\delta \nu }(1)\diamond n_1=0.$$

$$\frac{\partial }{\partial t}\frac{\delta \ell }{\delta u}+{\text{ad}}_{u_t}^{\ast }\frac{\delta \ell }{\delta u}+\frac{\delta \ell }{\delta n}\diamond n_t+\frac{\delta \ell }{\delta \nu }\diamond {\nu }_t=0,$$

$$\frac{\partial }{\partial t}\frac{\delta \ell }{\delta \nu }+u_t\star \frac{\delta \ell }{\delta \nu }-\frac{\delta \ell }{\delta n}=0.$$

$$\begin{array}{rl}& \frac{\partial }{\partial t}\frac{\delta \ell }{\delta u}+{\text{ad}}_{u_t}^{\ast }\frac{\delta \ell }{\delta u}+\frac{\delta \ell }{\delta n}\diamond n_t+\frac{\delta \ell }{\delta \nu }\diamond {\nu }_t=0,\\ & \\ & \frac{\partial }{\partial t}\frac{\delta \ell }{\delta \nu }+u_t\star \frac{\delta \ell }{\delta \nu }-\frac{\delta \ell }{\delta n}=0,\end{array}$$

$$\frac{\partial }{\partial t}{n}_t={\nu }_t+u_t{n}_t,$$

$$\frac{\delta \ell }{\delta u}(1)+\frac{\delta \ell }{\delta \nu }(1)\diamond n_1=0,$$

$$\displaystyle\frac{\partial}{\partial t}\Big{(}\frac{\delta\ell}{\delta u}+\frac{\delta\ell}{\delta\nu}\diamond n\Big{)}+\text{ad}_{u_{t}}^{*}\Big{(}\frac{\delta\ell}{\delta u}+\frac{\delta\ell}{\delta\nu}\diamond n\Big{)}=0.$$

$$\displaystyle\frac{\partial}{\partial t}\Big{(}\frac{\delta\ell}{\delta u_{t}}+\frac{\delta\ell}{\delta\nu}\diamond n_{t}\Big{)}$$

$$\displaystyle{\Big{\langle}{\frac{\delta\ell}{\delta\nu}\diamond(un)-\Big{(}u\star\frac{\delta\ell}{\delta\nu}\Big{)}\diamond n}\,,\,{\alpha}\Big{\rangle}}$$

$$\mu=\frac{\delta\ell}{\delta u}\,,\ \sigma=\frac{\delta\ell}{\delta\nu}\,,\quad h(\mu,\sigma,n)={\big{\langle}{\mu}\,,\,{u}\big{\rangle}}+{\big{\langle}{\sigma}\,,\,{\nu}\big{\rangle}}-\ell(u,\nu,n).$$

Full text

For the sake of introducing an Eulerian uncertainty quantification approach in imaging science, we extend the method of metamorphosis to apply to image matching along stochastically evolving time dependent curves on the space of diffeomorphisms. The approach will be guided by recent progress in developing stochastic Lie transport models for uncertainty quantification in fluid dynamics in Holm [2015]; Crisan et al. [2017].

1. 1 Introduction

2. 2 Review of metamorphosis by deterministic Lie transport

3. 2.1 Notation and Euler-Poincaré theorem for the deterministic case

4. 2.2 Deterministic Euler-Lagrange equations

5. 2.3 Deterministic Euler-Poincaré reduction

6. 2.4 Deterministic Hamiltonian formulation

7. 2.5 Deterministic Hamilton-Pontryagin approach

8. 3 Metamorphosis by stochastic Lie transport

9. 3.1 Notation and approach for the stochastic case

10. 3.2 Stochastic Hamilton-Pontryagin approach

11. 3.3 Stochastic Hamiltonian formulation

12. 3.4 Itô representation

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 LDDMM 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-Poincaré equation 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-Poincaré derivation 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, Trouvé and 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 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-Poincaré constrained 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 Trouvé and 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-Poincaré variational framework for the metamorphosis approach of Trouvé and 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-Poincaré variational 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 Trouvé and Younes [2005]. In Holm et al. [2009] the metamorphosis equations of Trouvé and Younes [2005] were recast into the Euler-Poincaré variational 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-Poincaré context in Holm et al. [2009].

Definition 1**.**

A metamorphosis Trouvé and Younes [2005] is a pair of curves $(g_{t},\,\eta_{t})\in G\times N$ parameterized by time $t$, with $g_{0}={\mathrm{id}}$. Its image is the curve $n_{t}\in N$ defined by the action $n_{t}=g_{t}.\eta_{t}$, where subscript $t$ indicates explicit dependence on time, $t$. The quantities $g_{t}$ and $\eta_{t}$ are called, respectively, the deformation part of the metamorphosis, and its template part. When $\eta_{t}$ 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-Poincaré theorem for the deterministic case

Following Trouvé and Younes [2005]; Holm et al. [2009], we will use either letters $\eta$ 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 $(g_{t},\eta_{t})$ by minimizing, for some Lagrangian $L:TG\times TN\to\mathbb{R}$, the action integral

[TABLE]

with fixed endpoint conditions for the initial and final images $n_{0}$ and $n_{1}$ (with $n_{t}=g_{t}\eta_{t}$) and $g_{0}={\mathrm{id}}_{G}$. That is, the images $n_{t}$ are constrained at the end-points, with the initial condition $g_{0}={\mathrm{id}}$.

Let $\mathfrak{g}$ denote the Lie algebra of the Lie group $G$. We will consider Lagrangians defined on $TG\times TN$, that satisfy the following invariance conditions: there exists a function $\ell$ defined on $\mathfrak{g}\times TN$ such that

[TABLE]

In other words, $L$ is invariant under the right action of $G$ on $G\times N$ defined by $(g,\eta)h=(gh,h^{-1}\eta)$.

For a metamorphosis $(g_{t},\eta_{t})$, we therefore have a reduced Lagrangian, upon defining $u_{t}=\dot{g}_{t}g_{t}^{-1}$, $n_{t}=g_{t}\eta_{t}$ and $\nu_{t}=g_{t}\dot{\eta}_{t}$, given by

[TABLE]

The Lie derivative with respect to a vector field $X$ will be denoted $\mathcal{L}_{X}$. The Lie algebra of $G$ is identified with the set of right invariant vector fields $U_{g}=ug$, $u\in T_{\mathrm{id}}G=\mathfrak{g}$, $g\in G$, and we will use the notation $\mathcal{L}_{u}=\mathcal{L}_{U}$.

The Lie bracket $[u,v]$ on the Lie algebra of smooth vector fields $\mathfrak{g}$ is defined by

[TABLE]

and the associated adjoint operator is $\text{ad}_{u}v=[u,v]$. Letting $I_{g}(h)=ghg^{-1}$ and $Ad_{v}g=\mathcal{L}_{v}I_{g}({\mathrm{id}})$, we also have $\text{ad}_{u}v=\mathcal{L}_{u}(\text{Ad}_{v})({\mathrm{id}})$. When $G$ is a group of diffeomorphisms, this yields $ad_{u}v=du\,v-dv\,u$.

The pairing between a linear form $\mu$ and a vector field $u$ will be denoted {\big{\langle}{\mu}\,,\,{u}\big{\rangle}}. 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 $\tilde{N}$, the diamond operator $(\diamond)$ is defined on $\tilde{N}^{*}\times\tilde{N}$ and takes values in the dual Lie algebra $\mathfrak{g}^{*}$. That is, $\diamond:\tilde{N}^{*}\times\tilde{N}\to\mathfrak{g}^{*}$. For $\tilde{n}^{*}\in\tilde{N}^{*}$ and $\tilde{n}\in\tilde{N}$ the diamond operation is defined by

[TABLE]

where the action of a vector field $u\in\mathfrak{g}$ on $\tilde{n}\in\tilde{N}$ is denoted by simple concatenation, $u\tilde{n}\in T\tilde{N}$. For example, the Lie algebra action of the vector field $u\in\mathfrak{g}$ on $\tilde{n}\in\tilde{N}$ is denoted $u\tilde{n}=\mathcal{L}_{u}\tilde{n}\in T\tilde{N}$. Subscripts on the pairings in the definition (2.5) indicate, as follows, {\big{\langle}{\,\cdot}\,,\,{\cdot\,}\big{\rangle}}_{\mathfrak{g}}:{\mathfrak{g}}^{*}\times{\mathfrak{g}}\to\mathbb{R} and {\big{\langle}{\,\cdot}\,,\,{\cdot\,}\big{\rangle}}_{T\tilde{N}}:{\tilde{N}}^{*}\times{T\tilde{N}}\to\mathbb{R}. 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-Poincaré theorem).**

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 $G\times N$ defined by $(g,\eta)h=(gh,h^{-1}\eta)$, with fixed endpoint conditions for the initial and final images $n_{0}$ and $n_{1}$:

1. i

Hamilton’s variational principle

[TABLE]

holds, for variations $\delta g_{t}$ of $g_{t}$ and $\delta\eta_{t}$ of $\eta_{t}$ vanishing at the endpoints. 2. ii

$g_{t}$* and $\eta_{t}$ satisfy the Euler–Lagrange equations for $L$ on $TG\times TN$.* 3. iii

The constrained variational principle

[TABLE]

holds for Lagrangian $\ell$ defined on $\mathfrak{g}\times TN$ using variations of $u_{t}=\dot{g}_{t}g_{t}^{-1}$, $n_{t}=g_{t}\eta_{t}$ and $\nu_{t}=g_{t}\dot{\eta}_{t}$ of the form

[TABLE]

where $\xi_{t}=\delta g_{t}g_{t}^{-1}$, $\varpi_{t}=g_{t}\delta\eta_{t}$ and these variations vanish at the endpoints. 4. iv

*The Euler–Poincaré equations hold on *$\mathfrak{g}\times TN$

[TABLE]

with auxiliary equation

[TABLE]

obtained from the definitions $u_{t}=\dot{g}_{t}g_{t}^{-1}$ and $n_{t}=g_{t}\eta_{t}$, with $\nu_{t}=g_{t}\dot{\eta}_{t}$, provided the endpoint condition holds, that

[TABLE]

at time $t=1$.

Corollary 3** (Coadjoint motion).**

Equations (2.9) and the auxiliary equation (2.10) for $n_{t}$ together imply the following coadjoint motion equation,

[TABLE]

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-Poincaré Theorem 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

[TABLE]

with fixed boundary conditions $n_{0}$ and $n_{1}$. We therefore consider variations $\delta u$ and $\omega=\delta n$. The variation $\delta\nu$ can be obtained from $n=g\eta$ and $\nu=g\dot{\eta}$ yielding $\dot{n}=\nu+un$ and $\dot{\omega}=\delta\nu+u\omega+(\delta u)n$. 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

[TABLE]

The $\delta u$ term yields the equation

[TABLE]

where, in a slight abuse of notation, $\delta\ell/\delta\nu\in T(TN)^{*}$ has been considered as a linear form on $TN$ by {\big{\langle}{\delta\ell/\delta\nu}\,,\,{z}\big{\rangle}}:={\big{\langle}{\delta\ell/\delta\nu}\,,\,{(0,z)}\big{\rangle}}. From the terms involving $\omega$, we find, after an integration by parts

[TABLE]

Here, we have introduced notation for the star $(\star)$ operation,

[TABLE]

For $u\omega=\mathcal{L}_{u}\omega$, the star $(\star)$ operation denotes the dual of the Lie derivative, $u\star\nu^{*}=\mathcal{L}_{u}^{T}\nu^{*}$.

We therefore obtain the system of equations

[TABLE]

Note that the sum $\frac{\delta\ell}{\delta u}+\frac{\delta\ell}{\delta\nu}\diamond n$ is the momentum map arising from Noether’s theorem for the considered invariance of the Lagrangian. The special form of the boundary conditions (fixed $n_{0}$ and $n_{1}$) ensures that this momentum map vanishes.

2.3. Deterministic Euler-Poincaré reduction

As explained in in Holm et al. [2009], a system equivalent to that in (2.15) can be obtained via Euler-Poincaré reduction Holm et al. [1998a]. In this setting, we make the variation in the group element and in the template instead of the velocity and the image. We denote $\xi_{t}=\delta g_{t}g_{t}^{-1}$ and $\varpi_{t}=g_{t}\delta\eta_{t}$. From these definitions, we obtain the expressions of the variations, $\delta u$, $\delta n$ and $\delta\nu$.

We first have $\delta u_{t}=\dot{\xi}_{t}+[\xi_{t},u_{t}]$, which arises from the standard Euler-Poincaré reduction theorem, as provided in Holm et al. [1998a]; Marsden and Ratiu [1994]. We also have $\delta n_{t}=\delta(g_{t}\eta_{t})=\varpi_{t}+\xi_{t}n_{t}$. From $\nu_{t}=g_{t}\dot{\eta}_{t}$, we get $\delta\nu_{t}=g_{t}\delta\dot{\eta}_{t}+\xi_{t}\nu_{t}$ and from $\varpi_{t}=g_{t}\delta\eta_{t}$ we also have $\dot{\varpi}_{t}=u_{t}\varpi_{t}+g_{t}\dot{\eta}_{t}$. This yields $\delta\nu_{t}=\dot{\varpi}_{t}+\xi_{t}\nu_{t}-u_{t}\varpi_{t}.$

We also compute the boundary conditions for $\xi$ and $\varpi$. At $t=0$, we have $g_{0}={\mathrm{id}}$ and $n_{0}=g_{0}\eta_{0}=\text{cst}$ which implies $\xi_{0}=0$ and $\varpi_{0}=0$. At $t=1$, the relation $g_{1}\eta_{1}=\text{cst}$ yields $\xi_{1}n_{1}+\omega_{1}=0$.

Now, the first variation of is

[TABLE]

In the integration by parts to eliminate $\dot{\xi}_{t}$ and $\dot{\varpi}_{t}$, the boundary term is {\big{\langle}{(\delta\ell/\delta u)_{1}}\,,\,{\xi_{1}}\big{\rangle}}+{\big{\langle}{(\delta\ell/\delta\nu)_{1}}\,,\,{\omega_{1}}\big{\rangle}}. Using the boundary condition, the last term can be re-written

[TABLE]

We therefore obtain the endpoint equation

[TABLE]

The evolution equation for ${\delta\ell}/{\delta u}$ is

[TABLE]

and ${\delta\ell}/{\delta\nu}$ evolves by

[TABLE]

Consequently, we obtain the following system of equations,

[TABLE]

as well as the auxiliary equation

[TABLE]

obtained from the definitions $u_{t}=\dot{g}_{t}g_{t}^{-1}$ and $n_{t}=g_{t}\eta_{t}$. Moreover, the endpoint condition holds, that

[TABLE]

at time $t=1$. $\Box$

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

[TABLE]

Proof of Corollary 3.

A solution of (2.9) satisfies the coadjoint motion equation,

[TABLE]

In the last equation, we have used the fact that, for any $\alpha\in\mathfrak{g}$,

[TABLE]

∎

Remark 4**.**

Corollary 3 combined with the relation $(\delta\ell/\delta u)_{1}+(\delta\ell/\delta\nu)_{1}\diamond u_{1}=0$ 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-Poincaré formulation 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 $\ell$ in the velocities ${u}$ and $\nu$, in the Eulerian fluid description,

[TABLE]

Accordingly, one computes the variational derivatives of $h$ as

[TABLE]

Consequently, the Euler-Poincaré equations (2.9) and the auxiliary kinematic equation (2.10) for metamorphosis imply the following equations, for the Legendre-transformed variables, $(\mu,\sigma,n)$,

[TABLE]

as well as the auxiliary equation

[TABLE]

These equations are Hamiltonian. That is, they may be expressed in the form

[TABLE]

where $\mathbf{z}\in(\mu,\sigma,n)$ and the Hamiltonian matrix b defines the Poisson bracket

[TABLE]

which is bilinear, skew symmetric and satisfies the Jacobi identity,

[TABLE]

Assembling the metamorphosis equations (2.22) - (2.23) into the Hamiltonian form (2.24) gives,

[TABLE]

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 $\mathfrak{g}\,\circledS\,T^{*}N$, 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-Poincaré approach is obtained in the Hamilton–Pontryagin principle. In this approach, the diffeomorphic paths appear explicitly.

Theorem 5** (Hamilton–Pontryagin principle).**

The Euler–Poincaré equations in Corollary 3 for coadjoint motion given by

[TABLE]

as well as the auxiliary equation

[TABLE]

on the space $\mathfrak{g}^{*}\times T^{*}N\times TN$ are equivalent to the following implicit variational principle,

[TABLE]

for a constrained action

[TABLE]

Proof.

The variations of $S$ in formula (3.7) are given by

[TABLE]

After a side calculation, one finds $\delta(\dot{g}g^{-1})=\dot{\xi}-{\rm ad}_{u}\xi$, with $\xi=\delta gg^{-1}$ for the last term in (3.7). Then, integrating by parts yields the familiar relation

[TABLE]

where $\xi=\delta gg^{-1}$ vanishes at the endpoints in time.

Thus, stationarity $\delta S=0$ of the Hamilton–Pontryagin variational principle with constrained action integral (3.7) yields the following set of equations:

[TABLE]

as well as the constraint equations

[TABLE]

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

[TABLE]

the Lie-Poisson structure (2.26) transforms into

[TABLE]

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,

[TABLE]

where $(M,\sigma,n):=(\mu+\sigma\diamond n,\sigma,n)$. Namely, one substitutes the corresponding terms,

[TABLE]

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, $\mathcal{D}$; namely, $\mathfrak{g}=\mathfrak{X}(\mathcal{D})$ with canonical 2-cocycle $\mathfrak{X}(\mathcal{D})^{*}\times T^{*}\mathcal{F}(\mathcal{D},N)$, where $\mathcal{F}(\mathcal{D},N)$ denotes smooth functions from the domain, $\mathcal{D}$, 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,

[TABLE]

where ${\boldsymbol{d}}$ 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 $g_{t}^{*}$ for pullback by the stochastic diffeomorphism $g_{t}$,

[TABLE]

In this form, one sees that $g_{t}$ is a stochastic process with time dependent drift term given by the pullback operation, $u_{t}(x)\,dt=g_{t}^{*}u_{t}(X)\,dt=u_{t}(g_{t}X)\,dt$, in which subscript $t$ on $g_{t}$ and $u_{t}(X)$ indicates that both $u_{t}$ and $g_{t}$ depend explicitly on time, $t$. Thus, the dynamical drift velocity $u_{t}(x)$ depends on time explicitly and also through the Lagrange-to-Euler map $x=g_{t}X$ governed by (3.2) with initial value $g_{0}=Id$. The Lagrange-to-Euler map in (3.2) also contains a Stratonovich stochastic term, comprising a finite sum over time independent spatial functions $\xi_{i}$, $i=1,2,\dots,N$, each composed in a Stratonovich sense (denoted by the symbol $\circ$) with its own Brownian motion in time, $dW^{i}_{t}$. This type of Stratonovich stochasticity, called “cylindrical noise”, was introduced in Schaumlöffel [1988]. In the cylindrical noise term, the $\xi_{i}(x)$, $i=1,2,\dots,N$, 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.

3.2. Stochastic Hamilton-Pontryagin approach

Theorem 8** (Stochastic Hamilton–Pontryagin principle).**

Stochastic metamorphosis is governed by coadjoint motion represented as SPDE given by

[TABLE]

as well as the auxiliary equation

[TABLE]

on the space $\mathfrak{g}^{*}\times T^{*}N\times TN$ are equivalent to the following implicit variational principle,

[TABLE]

for a stochastically constrained action

[TABLE]

Remark 9** (Stratonovich versus Itô representations).**

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 Itô representation. In transforming to the Itô representation, we will discover that the effective diffusion from the Itô contraction term is by no means a Laplacian. Instead, the Itô contraction term turns out to produce a double Lie derivative with respect to the sum of vector fields $\xi_{i}(x)$.

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

[TABLE]

In a side calculation, one finds

[TABLE]

for substitution into the last term. Then, integrating by parts yields the relation

[TABLE]

where $\xi=\delta gg^{-1}$ vanishes at the endpoints in time.

Thus, stationarity $\delta S=0$ of the Hamilton–Pontryagin variational principle with stochastically constrained action integral (3.7) yields the following set of SPDEs:

[TABLE]

for the quantities

[TABLE]

as well as the stochastic constraint equations

[TABLE]

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,

[TABLE]

with Stratonovich stochastic transport velocity $\tilde{u}$ given by

[TABLE]

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 $\mu$ and paired with the Stratonovich noise perturbation,

[TABLE]

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,

[TABLE]

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,

[TABLE]

one finds that the Lie-Poisson structure in (3.14) transforms into

[TABLE]

and thereby recovers equations (3.8) - (3.10) in Hamiltonian form.

Remark 10**.**

The advantage of writing equations (3.16) in terms of the total momentum 1-form density $M:=\mathbf{M}\cdot d\mathbf{x}\otimes d^{3}x$ may be seen by recalling that ${\rm ad}^{*}_{\widetilde{u}}M=\mathcal{L}_{\widetilde{u}}M$ for 1-form densities. Consequently, the first equation in (3.16) implies $({\boldsymbol{d}}+\mathcal{L}_{\widetilde{u}})M=0$, which in turn implies that

[TABLE]

This means the total momentum 1-form density $M:=\mathbf{M}\cdot d\mathbf{x}\otimes d^{3}x$ is preserved by the stochastic flow given by the pullback of the stochastic diffeomorphism $g_{t}$ in (3.2), which is the flow of the stochastic vector field $\widetilde{u}$. That is, the stochastic Lagrange-to-Euler flow $g_{t}$, which is the solution of ${\boldsymbol{d}}gg^{-1}=\widetilde{u}$ in (3.1),

[TABLE]

preserves the quantity $M$ along its flow. Hence, we say that the total momentum 1-form density $M:=\mathbf{M}\cdot d\mathbf{x}\otimes d^{3}x$ 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 of the flow. The Hamiltonian flow of the momentum $\langle\mu\,,\,\sum_{i}\xi_{i}(x)\circ dW^{i}_{t}\rangle$ produces stochastic translation in Eulerian space with velocity $\sum_{i}\xi_{i}(x)\circ dW^{i}_{t}$. Thus, adding the stochastic part, linear in the momentum, to the metamorphosis Hamiltonian $h(\mu,\nu,n)$ adds a stochastic transport to the deterministic flow. This is consistent with our 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.

3.4. Itô representation

In preparation for writing the Itô representation, we first substitute ${\rm ad}_{u}^{*}\mu=\mathcal{L}_{u}\mu$ to show in the more familiar Lie derivative notation the action of the vector field $u$ on its dual momentum, the 1-form density $\mu={\delta\ell}/{\delta u}$. The equivalent Itô representations of equations (3.19) are then given by

[TABLE]

with stochastic transport velocity $\widehat{u}$ given in Itô form by

[TABLE]

plus the Itô contraction drift terms. Likewise, the stochastic advection of the total momentum 1-form density $M=\mu+\sigma\diamond n=\mathbf{M}\cdot d\mathbf{x}\otimes d^{3}x$, expressed in Stratonovich form as $({\boldsymbol{d}}+\mathcal{L}_{\widetilde{u}})M=0$, is expressed in Itô form as

[TABLE]

In which the last sum is the Itô contraction term.

In Itô form, the expectation of the noise terms vanish. The noise interacts multiplicatively with both the solution $\mathbf{M}$ and the gradient of the solution $\nabla\mathbf{M}$, through the Lie derivative, as

[TABLE]

Likewise, the Itô contraction 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 $\xi_{i}(x)$ 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. Trouvé and L. Younes for their collaboration in developing the Euler-Poincaré description 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. During the present work the author was partially supported by the European Research Council Advanced Grant 267382 FCCA and the EPSRC Grant EP/N023781/1.