  Anatomically plausible image registration often requires volumetric
preservation. Previous approaches to incompressible image registration have
exploited relaxed constraints, ad hoc optimisation methods or practically
intractable computational schemes. Divergence-free velocity fields have been
used to achieve incompressibility in the continuous domain, although, after
discretisation, no guarantees have been provided. In this paper, we introduce
stationary velocity fields (SVFs) parameterised by divergence-conforming
B-splines in the context of image registration. We demonstrate that sparse
linear constraints on the parameters of such divergence-conforming B-Splines
SVFs lead to being exactly divergence-free at any point of the continuous
spatial domain. In contrast to previous approaches, our framework can easily
take advantage of modern solvers for constrained optimisation, symmetric
registration approaches, arbitrary image similarity and additional
regularisation terms. We study the numerical incompressibility error for the
transformation in the case of an Euler integration, which gives theoretical
insights on the improved accuracy error over previous methods. We evaluate the
proposed framework using synthetically deformed multimodal brain images, and
the STACOM11 myocardial tracking challenge. Accuracy measurements demonstrate
that our method compares favourably with state-of-the-art methods whilst
achieving volume preservation.
