We address the problem of registering two surfaces, of which a natural bijection between them does not exist. More precisely, only a partial subset of the source surface is assumed to be in correspondence with a subset of the target surface. We call such a problem an inconsistent surface registration (ISR) problem. This problem is challenging as the corresponding regions on each surface and a meaningful bijection between them have to be simultaneously determined. In this paper, we propose a variational model to solve the ISR problem by minimizing mapping distortions. Mapping distortions are described by the Beltrami coefficient as well as the differential of the mapping. Registration is then guided by feature landmarks and/or intensities, such as curvatures, defined on each surface. The key idea of the approach is to control angle and scale distortions via quasiconformal theory as well as minimizing landmark and/or intensity mismatch. A splitting method is proposed to iteratively search for the optimal corresponding regions as well as the optimal bijection between them. Bijectivity of the mapping is easily enforced by a thresholding of the Beltrami coefficient. We test the proposed method on both synthetic and real examples. Experimental results demonstrate the efficacy of our proposed model.

我们解决了配准两个曲面的问题，两个曲面之间不存在自然双射。更精确地，仅假设源表面的一部分子集与目标表面的子集相对应。我们称此问题为不一致的表面配准（ISR）问题。该问题具有挑战性，因为必须同时确定每个表面上的相应区域以及它们之间有意义的双射。在本文中，我们提出了一种变分模型，通过最小化映射失真来解决ISR问题。映射失真由Beltrami系数以及映射的微分来描述。然后通过每个表面上定义的特征界标和/或强度（例如曲率）引导配准。该方法的关键思想是通过准保形理论控制角度和比例失真，并最大程度地减少界标和/或强度不匹配。提出了一种分割方法来迭代搜索最优的对应区域以及它们之间的最优双射。可以通过对Beltrami系数进行阈值化来轻松实现映射的双射性。我们在综合实例和实际实例上测试了所提出的方法。实验结果证明了我们提出的模型的有效性。我们在综合实例和实际实例上都对提出的方法进行了测试。实验结果证明了我们提出的模型的有效性。我们在综合实例和实际实例上都对提出的方法进行了测试。实验结果证明了我们提出的模型的有效性。