With the advancement of computer technology, there is a surge of interest in effective mapping methods for objects in higher-dimensional spaces. To establish a one-to-one correspondence between objects, higher-dimensional quasi-conformal theory can be utilized for ensuring the bijectivity of the mappings. In addition, it is often desirable for the mappings to satisfy certain prescribed geometric constraints and possess low distortion in conformality or volume. In this work, we develop a unifying framework for computing $n$-dimensional quasi-conformal mappings. More specifically, we propose a variational model that integrates quasi-conformal distortion, volumetric distortion, landmark correspondence, intensity mismatch and volume prior information to handle a large variety of deformation problems. We further prove the existence of a minimizer for the proposed model and devise efficient numerical methods to solve the optimization problem. We demonstrate the effectiveness of the proposed framework using various experiments in two- and three-dimensions, with applications to medical image registration, adaptive remeshing and shape modeling.
翻译:随着计算机技术的进步,对高维空间物体的有效绘图方法的兴趣激增。为了在天体之间建立一对一的对应关系,可以使用高维准正态理论来确保绘图的双偏向性。此外,绘图往往有必要满足某些规定的几何限制,在一致性或体积方面进行低度扭曲。在这项工作中,我们为计算美元维度准正态绘图开发了一个统一框架。更具体地说,我们提出了一个变式模型,将准正态扭曲、体积扭曲、标志性对应、强度不匹配和量前信息结合起来,以处理大量变形问题。我们进一步证明,拟议的模型存在一个最小化器,并设计有效的数字方法来解决优化问题。我们用二维和三维的实验方法展示了拟议框架的有效性,并应用医学图像登记、适应性重视和成型模型。