pairs. An unsupervised approach for image registration was proposed by de Vos et al. [@dlmia2017_unsupervised]; here, the loss function is the image similarity measure between images themselves and a deformation is parameterized via a spatial transformer (which essentially amounts to a parameterized model of deformation in image registration) which generates the sought-for displacement vector field. In [@hong2017fast], Hong et al. employ a low-dimensional band-limited representation of velocity fields in Fourier space [@zhang2015finite] to speed up SGR [@ref:hong] for population-based image analysis.

1ex In this work, we will build on FPIR, as it is a desirable approach for brain image registration for the following reasons: *First*, FPIR predicts the initial momentum of LDDMM and therefore inherits the theoretical properties of LDDMM. Consequently, FPIR results in diffeomorphic transformations, even though predictions are computed in a patch-by-patch manner; this can not be guaranteed by most other prediction methods. *Second*, patch-wise prediction allows for training of the prediction models based on a very small number of images, containing a large number of patches. *Third*, by using a patch-wise approach, even high-resolution image volumes can be processed without running into memory issues on a GPU. *Fourth*, none of the existing predictive methods address longitudinal data. 