基于拟共形映射Teichmüller理论的曲面配准研究

Constrained surface registration with large non-rigid deformation and anisotropic deformation is the one of the most challenging and important. Teichmüller quasi-conformal mapping is a diffeomorphism minimizing the maximum angle distortion. This kind of mapping, which can not only satisfy the constrains between surfaces, but also describe the anisotropic non-rigid surface deformation, is an effective method to solve the surface registration problem. We plan to research on the Teichmüller mapping between surfaces with constrained conditions and the applications of surface registration based on Teichmüller theory. Our researches include the following aspects. We will analyze the properties of non-rigid and anisotropic deformations on the surface with constrained conditions by Teichmüller theory and model the surface registration problems. Then by researching on the properties of the metrics on the original surface and target surface, we will develop the algorithm to search the Teichmüller mapping between general surfaces. By combining with the method in Riemannian geometry, we will develop the algorithm to tackle the incomplete surface registration problem. Research on using the method of surface registration based on Teichmüller mapping to the application fields, which include face matching, expression extraction, expression analysis and medical morphologic diagnosis with the brain surface.

带约束大的非刚性和各向异性形变曲面配准是3D曲面配准的难点。Teichmüller拟共形映射是最小化最大角度扭曲的微分同胚映射，该映射既可满足曲面约束条件的精确配准，又能描绘非刚性各向异性的曲面变形得到曲面间1-1映射，有望成为解决曲面配准问题的有效方法。本课题将研究基于Teichmüller映射带约束任意拓扑曲面的配准问题及应用。内容包括：通过Teichmüller理论分析带约束各向异性形变曲面的特有变形性质对该类配准问题进行统一建模；研究任意拓扑曲面间Teichmüller拟共形映射的计算方法；将通过Teichmüller映射配准任意黎曼曲面到特定的二维平面区域从而计算任意曲面的全局等温坐标；与黎曼几何相结合解决非完整曲面配准问题；将基于Teichmüller映射的曲面配准方法应用于人脸曲面匹配、表情提取及分析、大脑医学形态学诊断等具体领域，给出更高效准确的方法。

曲面配准是研究曲面间的1-1映射，是计算机图形学、计算机视觉以及医学影像等领域中的重要问题。Teichmüller映射具有共性映射的保形性又具有微分同胚映射的灵活性，是非常先进的曲面配准理论方法。我们对Teichmüller理论进行了深入的学习理解，实现了计算两个曲面间极值映射的算法，以及证明了算法的正确性、唯一性。到目前为止该算法是计算任意曲面间Teichmüller映射唯一可行算法。我们将极值映射用于人脸序列曲面配准表情追踪和曲面全局参数化及四边形化研究中，取得了较好的实验结果。在项目期间，我们开发了一套完整的算法软件，对于曲面的配准和曲面全局四边形化有着较高的效率。人脸序列的追踪算法使用极值映射配准相邻两帧的三维人脸曲面，由于极值映射的唯一性和各向异性，使得大形变的人脸表情配准精度大幅提高。从我们的追踪结果视频中可以看出，算法精度和效率都达到较高水平。通过极值映射，我们得到一种人脸全局四边形化算法，从而将人脸重新拓扑化。算法可以用于网格迁移，这在电影工业有着广泛的应用。在四边形化算法上，我们通过共性映射将曲面共形变化到参数区域，然后通过整数优化和极值映射，我们可以得到最终的四边形化曲面。曲面的四边形化是各种工业流程的开始，有着非常广泛的应用。结合拟共形映射和最优传输理论，我们得到了一种曲面参数化方法，称为极性分解的参数化方法。