We introduce an implicit representation of continuous, bijective, orientation-preserving maps between genus zero surfaces with or without boundary. The distortion of these maps can easily be minimized by optimizing the Ginzburg-Landau functional - a ubiquitous model in physics and differential geometry - leading to a simple algorithm for computing bijective correspondences using only standard tools of the tangent vector field toolbox. The method avoids combinatorial mesh modifications and does not require barrier functions to enforce bijectivity making it more robust to noise and simpler to implement. Moreover, the algorithm does not assume a bijective initialization and can untangle non-bijective correspondences generated by computationally cheaper methods such as functional maps. It supports the use of both landmark points and landmark curves to guide the correspondence. The key idea is that a bijection between surfaces defines a two-dimensional mapping surface sitting inside the four-dimensional product space of the two inputs, and this mapping surface can be stored implicitly as the zero set of a complex section - essentially a complex function defined on the product space. Now the distortion of the map can be optimized by minimizing the area of this mapping surface, which amounts to minimizing the Ginzburg-Landau functional of the complex section. We demonstrate the practical benefits of our method by comparing to state-of-the-art correspondence algorithms and show that our implicit representation offers improved stability and naturally supports constraints that are difficult to enforce with explicit map representations.

翻译：我们提出一种连续、双射、保定向映射的隐式表示，适用于带边界或不带边界的亏格零曲面。通过优化Ginzburg-Landau泛函（物理学和微分几何中的经典模型），可轻松最小化这些映射的畸变，从而得到一种仅使用切向量场标准工具即可计算双射对应的简单算法。该方法避免了组合网格修改，无需使用障碍函数强制双射性，因此对噪声更具鲁棒性且更易实现。此外，算法不假设双射初始化，可解缠由计算成本较低的方法（如函数映射）生成的非双射对应。它还支持使用标志点和标志曲线引导对应关系。核心思想在于：曲面间的双射定义了嵌入两个输入四维乘积空间中的二维映射曲面，该映射曲面可隐式存储为复截面的零集——本质上是在乘积空间上定义的复函数。通过最小化该映射曲面的面积（等价于最小化复截面的Ginzburg-Landau泛函）即可优化映射畸变。通过与最先进的对应算法比较，我们展示了该方法的实际优势，表明隐式表示具有更优的稳定性，并能自然支持显式映射表示难以实施的约束条件。