3D orientation-preserving variational models for accurate image registration

The Beltrami coefficient from complex analysis has recently been found to provide a robust constraint for obtaining orientation-preserving and diffeomorphic transformations for registration of planar images. There exists no such concept of the Beltrami coefficient in three or higher dimensions, although a generalized theory of quasi-conformal maps in high dimensions exists. In this paper, we first propose a new algebraic measure in three dimensions (3D) that mimics the Beltrami concept in two dimensions (2D) and then propose a corresponding registration model based on it. We then establish the existence of solutions for the proposed model and further propose a converging generalized Gauss--Newton iterative method to solve the resulting nonlinear optimization problem. In addition, we also provide another two possible regularizers in 3D. Numerical experiments show that the new model can produce more accurate orientation-preserving transformations than competing state-of-the-art registration models.