Space-time metamorphosis

TL;DR

This paper introduces a novel space-time metamorphosis framework for image registration, formulating it as an infinite-dimensional optimization problem and solving it analytically with finite element methods and gradient descent.

Contribution

It develops a variational Eulerian space-time approach for image registration, including analytical solutions, finite element analysis, and a gradient descent algorithm for the control variable.

Findings

Analytical solutions to geodesic equations for registration.

Finite element method ensures well-posedness and convergence.

Numerical results demonstrate effectiveness of the proposed approach.

Abstract

We study the problem of registering images. The framework we use is metamorphosis and we construct a variational Eulerian space-time setting and pose the registration problem as an infinite-dimensional optimisation problem. The geodesic equations correspond to a system of advection and continuity equations and are solved analytically. Well-posedness of a primal conforming finite element method is established and its convergence is investigated numerically. This provides a discrete forward operator for the matching parameterized by a space-time velocity field. We propose a gradient descent method on this control variable and show several promising numerical results for this approach.