Log-Euclidean free-form deformation

The Free-Form Deformation (FFD) algorithm is a widely used method for non-rigid registration. Modifications have previously been proposed to ensure topology preservation and invertibility within this framework. However, in practice, none of these yield the inverse transformation itself, and one loses the parsimonious B-spline parametrisation. We present a novel log-Euclidean FFD approach in which a spline model of a stationary velocity field is exponentiated to yield a diffeomorphism, using an efficient scaling-and-squaring algorithm. The log-Euclidean framework allows easy computation of a consistent inverse transformation, and offers advantages in group-wise atlas building and statistical analysis. We optimise the Normalised Mutual Information plus a regularisation term based on the Jacobian determinant of the transformation, and we present a novel analytical gradient of the latter. The proposed method has been assessed against a fast FFD implementation (F3D) using simulated T1- and T2-weighted magnetic resonance brain images. The overlap measures between propagated grey matter tissue probability maps used in the simulations show similar results for both approaches; however, our new method obtains more reasonable Jacobian values, and yields inverse transformations.