Efficient momentum conservation constrained PDE-LDDMM with Gauss–Newton–Krylov optimization, Semi-Lagrangian Runge–Kutta solvers, and the band-limited parameterization

Abstract

This paper proposes three efficient variants of Jacobi EPDiff PDE-LDDMM, where efficiency is achieved through Semi-Lagrangian Runge–Kutta integration and the band-limited parameterization. During Gauss–Newton–Krylov optimization, the method computes the gradient and the Hessian-vector product on the final time sample, and transports these magnitudes towards the initial time using the adjoint Jacobi equations and their incremental counterparts. Then, the optimization is performed on the initial time sample. The proposed methods have effectively achieved a considerable reduction of the computational complexity at a competitive accuracy. These variants constitute a contribution to the efficient computation of diffeomorphisms belonging to geodesics, suitable for statistical shape analysis or the construction of transversal and longitudinal models of shape variability using Principal Geodesic Analysis and Geodesic Regression in spaces of diffeomorphisms.