SIAM Journal on Imaging Sciences, Volume 13, Issue 3, Page 1653-1691, January 2020.
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.

中文翻译：

---

用于精确图像配准的3D方向保持变化模型

SIAM影像科学杂志，第13卷，第3期，第1653-1691页，2020年1月。
最近发现，来自复杂分析的Beltrami系数为获得用于平面图像配准的方向保持和微分变换提供了鲁棒的约束。尽管存在高维拟保形图的广义理论，但在三个或更高维上没有贝尔特米系数的概念。在本文中，我们首先提出一种在三维（3D）中模仿贝尔特米（Beltrami）概念在二维（2D）中的新代数测度，然后在此基础上提出相应的配准模型。然后我们建立了所提出模型的解的存在性，并进一步提出了一种收敛的广义高斯-牛顿迭代法来解决由此产生的非线性优化问题。此外，我们还提供了另外两种3D可能的正则化器。