4D Deformable Models Using Corresponding Control Points and Spatio-Temporal Radial Functions

4D imaging techniques make it possible to investigate the dynamic process of 3D objects. This paper aims to provide a 4D deformable model for 4D registration using corresponding control points and spatio-temporal radial basis functions (RBFs), which takes advantage of the temporal information contained in control point sequences, and exploits the temporal coherence of the control point sequences. It preserves the displacements of control points at each time point exactly, and calculates the displacement fields of other voxels at any time point by spatio-temporal interpolation. The solvability, spatial smoothness, temporal smoothness, and separability of our deformation model are then discussed in theory. Evaluation of the 4D deformable model is performed on given motion models and shows that the target registration error and spatial smoothness of our deformation model are dependent on the spatial RBFs and temporal RBFs used in the spatio-temporal RBFs. Different RBFs can be combined to construct the spatio-temporal RBFs with better performance in respect of registration accuracy and temporal smoothness. Most importantly, the temporal smoothness of our spatio-temporal transformation is superior to other transformation