EPDiff Equation Research Articles

Moment Evolution Equations and Moment Matching for Stochastic Image EPDiff

Models of stochastic image deformation allow study of time-continuous stochastic effects transforming images by deforming the image domain. Applications include longitudinal medical image analysis with both population trends and random subject-specific variation. Focusing on a stochastic extension of the LDDMM models with evolutions governed by a stochastic EPDiff equation, we use moment approximations of the corresponding Itô diffusion to construct estimators for statistical inference in the full stochastic model. We show that this approach, when efficiently implemented with automatic differentiation tools, can successfully estimate parameters encoding the spatial correlation of the noise fields on the image.

Well-posedness of the EPDiff equation with a pseudo-differential inertia operator

In this article we study the class of right-invariant, fractional order Sobolev-type metrics on groups of diffeomorphisms of a compact manifold M. Our main result concerns well-posedness properties for the corresponding Euler-Arnold equations, also called the EPDiff equations, which are of importance in mathematical physics and in the field of shape analysis and template registration. Depending on the order of the metric, we will prove both local and global well-posedness results for these equations. As a result of our analysis we will also obtain new commutator estimates for elliptic pseudo-differential operators.

PDE-constrained LDDMM via geodesic shooting and inexact Gauss–Newton–Krylov optimization using the incremental adjoint Jacobi equations

The class of non-rigid registration methods proposed in the framework of PDE-constrained large deformation diffeomorphic metric mapping is a particularly interesting family of physically meaningful diffeomorphic registration methods. Inexact Gauss–Newton–Krylov optimization has shown an excellent numerical accuracy and an extraordinarily fast convergence rate in this framework. However, the Galerkin representation of the non-stationary velocity fields does not provide proper geodesic paths. In this work, we propose a method for PDE-constrained LDDMM parameterized in the space of initial velocity fields under the EPDiff equation. The derivation of the gradient and the Hessian-vector products are performed on the final velocity field and transported backward using the adjoint and the incremental adjoint Jacobi equations. This way, we avoid the complex dependence on the initial velocity field in the computations. We also avoid the computation of the adjoint equation and its incremental counterpart that has been recently identified as a subtle problem in PDE-constrained LDDMM. The proposed method provides geodesics in the framework of PDE-constrained LDDMM, and it shows performance competing with benchmark PDE-constrained LDDMM and EPDiff-LDDMM methods.

An explicit structure‐preserving numerical scheme for EPDiff

AbstractWe present a new structure‐preserving numerical scheme for solving the Euler‐Poincaré Differential (EPDiff) equation on arbitrary triangle meshes. Unlike existing techniques, our method solves the difficult non‐linear EPDiff equation by constructing energy preserving, yet fully explicit, update rules. Our approach uses standard differential operators on triangle meshes, allowing for a simple and efficient implementation. Key to the structure‐preserving features that our method exhibits is a novel numerical splitting scheme. Namely, we break the integration into three steps which rely on linear solves with a fixed sparse matrix that is independent of the simulation and thus can be pre‐factored. We test our method in the context of simulating concentrated reconnecting wavefronts on flat and curved domains. In particular, EPDiff is known to generate geometrical fronts which exhibit wave‐like behavior when they interact with each other. In addition, we also show that at a small additional cost, we can produce globally‐supported periodic waves by using our simulated fronts with wavefronts tracking techniques. We provide quantitative graphs showing that our method exactly preserves the energy in practice. In addition, we demonstrate various interesting results including annihilation and recreation of a circular front, a wave splitting and merging when hitting an obstacle and two separate fronts propagating and bending due to the curvature of the domain.

Local well-posedness of the EPDiff equation: A survey

This article is a survey on the local well-posedness problem for the general EPDiff equation. The main contribution concerns recent results on local existence of the geodesics on $\mathrm{Diff}(\mathbb{T}^{d})$ and $\mathrm{Diff}(\mathbb{R}^{d})$ when the inertia operator is a non-local Fourier multiplier.

Local and global well-posedness of the fractional order EPDiff equation on [formula omitted

Of concern is the study of fractional order Sobolev-type metrics on the group of H∞-diffeomorphism of Rd and on its Sobolev completions Dq(Rd). It is shown that the Hs-Sobolev metric induces a strong and smooth Riemannian metric on the Banach manifolds Ds(Rd) for s>1+d2. As a consequence a global well-posedness result of the corresponding geodesic equations, both on the Banach manifold Ds(Rd) and on the smooth regular Fréchet–Lie group of all H∞-diffeomorphisms is obtained. In addition a local existence result for the geodesic equation for metrics of order 12≤s<1+d/2 is derived.

Integration of the EPDiff equation by particle methods

The purpose of this paper is to apply particle methods to the numerical solution of the EPDiff equation. The weak solutions of EPDiff are contact discontinuities that carry momentum so that wavefront interactions represent collisions in which momentum is exchanged. This behavior allows for the description of many rich physical applications, but also introduces difficult numerical challenges. We present a particle method for the EPDiff equation that is well-suited for this class of solutions and for simulating collisions between wavefronts. Discretization by means of the particle method is shown to preserve the basic Hamiltonian, the weak and variational structure of the original problem, and to respect the conservation laws associated with symmetry under the Euclidean group. Numerical results illustrate that the particle method has superior features in both one and two dimensions, and can also be effectively implemented when the initial data of interest lies on a submanifold.

Geodesic flows on semidirect-product Lie groups: geometry of singular measure-valued solutions

The EPDiff equation (or the dispersionless Camassa–Holm equation in one dimension) is a well-known example of geodesic motion on the Diff group of smooth invertible maps (diffeomorphisms). Its recent two-component extension governs geodesic motion on the semidirect product DiffⓈ , where denotes the space of scalar functions. This paper generalizes the second construction to consider geodesic motion on DiffⓈ , where denotes the space of scalar functions that take values on a certain Lie algebra (e.g. = ⊗ (3)). Measure-valued delta-like solutions are shown to be momentum maps possessing a dual pair structure, thereby extending previous results for the EPDiff equation. The collective Hamiltonians are shown to fit into the Kaluza–Klein theory of particles in a Yang–Mills field and these formulations are shown to apply also at the continuum partial differential equation level. In the continuum description, the Kaluza–Klein approach produces the Kelvin circulation theorem.

Soliton dynamics in computational anatomy

Computational anatomy (CA) has introduced the idea of anatomical structures being transformed by geodesic deformations on groups of diffeomorphisms. Among these geometric structures, landmarks and image outlines in CA are shown to be singular solutions of a partial differential equation that is called the geodesic EPDiff equation. A recently discovered momentum map for singular solutions of EPDiff yields their canonical Hamiltonian formulation, which in turn provides a complete parameterization of the landmarks by their canonical positions and momenta. The momentum map provides an isomorphism between landmarks (and outlines) for images and singular soliton solutions of the EPDiff equation. This isomorphism suggests a new dynamical paradigm for CA, as well as new data representation.