Cyclic Numerical Time Integration in Variational Non-Rigid Image Registration based on Quadratic Regularisation

In the present work, a novel computational framework for variational non-rigid image registration is discussed. The fundamental aim is to provide an alternative to approximate approaches based on successive convolution, which have gained great popularity in recent years, due to their linear complexity and ease of implementation. An optimise-then-discretise framework is considered. The corresponding Euler-Lagrange equations (ELEs), which arise from calculus of variation, constitute a necessary condition for a minimiser of the variational optimisation problem. The conventional, semi-implicit (SI) time integration for the solution of the ELEs is replaced by an explicit approach rendering the implementation straightforward. Since explicit methods are subject to a restrictive stability requirement on the maximal admissible time step size, they are in general inefficient and prone to get stuck in local minima. As a remedy, we take advantage of methods based on cyclic explicit numerical time integration. With this the strong stability requirement on each individual time step can be replaced by a relaxed stability requirement. This in turn results in an unconditionally stable method, which is as efficient as SI approaches. As a basis of comparison, SI methods are considered. Generalisability is demonstrated within a generic variational framework based on quadratic regularisation. Qualitative and quantitative analysis of numerical experiments based on synthetic test data demonstrates accuracy and efficiency.