  Diffeomorphic deformable image registration is one of the crucial tasks in
medical image analysis, which aims to find a unique transformation while
preserving the topology and invertibility of the transformation. Deep
convolutional neural networks (CNNs) have yielded well-suited approaches for
image registration by learning the transformation priors from a large dataset.
The improvement in the performance of these methods is related to their ability
to learn information from several sample medical images that are difficult to
obtain and bias the framework to the specific domain of data. In this paper, we
propose a novel diffeomorphic training-free approach; this is built upon the
principle of an ordinary differential equation.
  Our formulation yields an Euler integration type recursive scheme to estimate
the changes of spatial transformations between the fixed and the moving image
pyramids at different resolutions. The proposed architecture is simple in
design. The moving image is warped successively at each resolution and finally
aligned to the fixed image; this procedure is recursive in a way that at each
resolution, a fully convolutional network (FCN) models a progressive change of
deformation for the current warped image. The entire system is end-to-end and
optimized for each pair of images from scratch. In comparison to learning-based
methods, the proposed method neither requires a dedicated training set nor
suffers from any training bias. We evaluate our method on three cardiac image
datasets. The evaluation results demonstrate that the proposed method achieves
state-of-the-art registration accuracy while maintaining desirable
diffeomorphic properties.
