Finsler流形上的热核测地线方法研究及其在网格重构和曲面配准中的应用

测地线作为距离度量，能够反映曲面固有属性，因而被广泛应用于计算机视觉、图像图形学和离散几何等领域。目前关于测地线的研究面临两大难点问题：1）大部分的研究工作主要考虑黎曼流形上的测地线，在实际应用中较难精确地对复杂几何结构进行建模；2）随着各种几何模型的分辨率的提高和数据量的增大，传统的测地线方法无法有效处理这些大规模数据。针对上述两个问题，本项目将开展Finsler流形上热核测地线方法的研究。首先研究Finsler流形上的热核测地线方法，建立Finsler流形上热核与测地线之间的联系，然后提出一种可并行可扩展的非对称热核测地线方法，最后利用Finsler热核测地线解决网格重构和曲面配准中所面临的强各向异性和非对称长度测量问题。本项目的研究不但将为网格重构和曲面配准等应用提供快速有效的新方法，理论算法在应用数学和离散几何领域也有很强的科学意义。

As a distance metric, geodesics can reflect the intrinsic property of surfaces. Hence, it is widely applied in numerical geometry processing, computer vision, and computer graphics, etc. At present, the researches on geodesic methods face two challenges: 1)Most of the current geodesic research work mainly considers the geodesic on the Riemannian manifold, and it is difficult to accurately model complex geometric structures in practical applications; 2) As the computing and imaging equipment becomes more and more powerful, the resolution of various geometric data is getting higher and higher, and the amount of data is getting larger and larger. Current geodesic method is not able to effectively process these large-scale data. Focusing on these two challenges, this project will first generalize the geodesic by heat kernel to the Finsler manifold and then propose a parallel and scalable asymmetric heat method. Finally, the Finsler geodesic is applied to remeshing and mesh registration. The research of this project will provide fast and effective new methods for applications such as mesh reconstruction and registration. Furthermore, in terms of either applied mathematics or discrete geometry, it is of high value and importance.