Parametric non-rigid registration using a stationary velocity field

The Free-Form Deformation (FFD) algorithm is a widely-used approach 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 parameterisation. 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 velocity field. The proposed method has been assessed against the conventional FFD using T1-weighted magnetic resonance brain images, following a published protocol with an openly available data-set (MGH10) to enable comparison with many other algorithms. The proposed method performed similarly to the state of the art.