Splines for Image Metamorphosis

This paper investigates a variational model for splines in the image metamorphosis model for the the smooth interpolation of key frames in the space of images. The original metamorphosis model is based on a simultaneous transport of image intensities and a modulation of intensities along motion trajectories and the energy functional measures the motion velocity and the material derivative of the image intensity. As in the case of cubic splines in Euclidean space where cubic splines are known to minimize the squared acceleration along the interpolation path we consider different acceleration terms to define a spline metamorphis model. In fact, the proposed spline functional combines quadratic functionals of the Eulerian motion acceleration and of the second material derivative representing an acceleration in the change of intensities along motion paths. Furthermore, a variational time discretization of this spline model is proposed and the convergence to a suitably relaxed time continuous model is discussed via $\Gamma$-convergence methodology. As a byproduct, this also allows to establish the existence of metamorphosis splines for given key frame images as minimizers of the continuous spline functional. An effective spatial discretization is proposed based on a finite difference discretization in space combined with a stable B-spline interpolation of deformed quantities. A variety of numerical examples demonstrates the robustness and versatility of the proposed method in applications using a variant of the iPALM algorithm for the minimization of the fully discrete energy functional.