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

TL;DR

This paper introduces a PDE-constrained LDDMM registration method that ensures geodesic paths by leveraging initial velocity fields and adjoint equations, achieving high accuracy and fast convergence.

Contribution

It proposes a novel approach for PDE-constrained LDDMM that guarantees geodesic paths using adjoint Jacobi equations and avoids complex dependencies in the derivations.

Findings

Method provides geodesic paths in PDE-constrained LDDMM.

Achieves performance competitive with benchmark methods.

Utilizes adjoint and incremental adjoint Jacobi equations for efficient computation.

Abstract

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 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 derivations and the computation of the adjoint equation and its incremental counterpart. The proposed method provides geodesics in the framework of PDE-constrained LDDMM, and it shows performance competitive to benchmark PDE-constrained LDDMM and EPDiff-LDDMM methods.