# A Lagrangian Gauss-Newton-Krylov Solver for Mass- and Intensity-Preserving Diffeomorphic Image Registration

## TL;DR

This paper introduces an efficient Lagrangian Gauss-Newton-Krylov solver for diffeomorphic image registration that preserves mass and intensity, supporting both stationary and non-stationary velocity fields with scalable optimization techniques.

## Contribution

It develops a novel PDE constraint elimination method using Lagrangian hyperbolic PDE solvers and integrates it into a scalable, efficient registration framework supporting various models.

## Key findings

- Supports large-scale problems with up to 14.7 million degrees of freedom.
- Achieves fast convergence with spectral preconditioners and Gauss-Newton methods.
- Demonstrates effectiveness on synthetic and real-world datasets.

## Abstract

We present an efficient solver for diffeomorphic image registration problems in the framework of Large Deformations Diffeomorphic Metric Mappings (LDDMM). We use an optimal control formulation, in which the velocity field of a hyperbolic PDE needs to be found such that the distance between the final state of the system (the transformed/transported template image) and the observation (the reference image) is minimized. Our solver supports both stationary and non-stationary (i.e., transient or time-dependent) velocity fields. As transformation models, we consider both the transport equation (assuming intensities are preserved during the deformation) and the continuity equation (assuming mass-preservation). We consider the reduced form of the optimal control problem and solve the resulting unconstrained optimization problem using a discretize-then-optimize approach. A key contribution is the elimination of the PDE constraint using a Lagrangian hyperbolic PDE solver. Lagrangian methods rely on the concept of characteristic curves that we approximate here using a fourth-order Runge-Kutta method. We also present an efficient algorithm for computing the derivatives of final state of the system with respect to the velocity field. This allows us to use fast Gauss-Newton based methods. We present quickly converging iterative linear solvers using spectral preconditioners that render the overall optimization efficient and scalable. Our method is embedded into the image registration framework FAIR and, thus, supports the most commonly used similarity measures and regularization functionals. We demonstrate the potential of our new approach using several synthetic and real world test problems with up to 14.7 million degrees of freedom.