Acceleration controlled diffeomorphisms for nonparametric image regression

The analysis of medical image time-series is becoming increasingly important as longitudinal imaging studies are maturing and large scale open imaging databases are becoming available. Image regression is widely used for several purposes: as a statistical representation for hypothesis testing, to bring clinical scores and images not acquired at the same time into temporal correspondence, or as a consistency filter to enforce temporal correlation. Geodesic image regression is the most prominent method, but the geodesic constraint limits the flexibility and therefore the application of the model, particularly when the observation time window is large or the anatomical changes are non-monotonic. In this paper, we propose to parameterize diffeomorphic flow by acceleration rather than velocity, as in the geodesic model. This results in a nonparametric image regression model which is completely flexible to capture complex change trajectories, while still constrained to be diffeomorphic and with a guarantee of temporal smoothness. We demonstrate the application of our model on synthetic 2D images as well as real 3D images of the cardiac cycle.