Surface comparison and matching is a challenging problem in computer vision. While elastic Riemannian metrics provide meaningful shape distances and point correspondences via the geodesic boundary value problem, solving this problem numerically tends to be difficult. Square root normal fields considerably simplify the computation of certain distances between parametrized surfaces. Yet they leave open the issue of finding optimal reparametrizations, which induce corresponding distances between unparametrized surfaces. This issue has concentrated much effort in recent years and led to the development of several numerical frameworks. In this paper, we take an alternative approach which bypasses the direct estimation of reparametrizations: we relax the geodesic boundary constraint using an auxiliary parametrization-blind varifold fidelity metric. This reformulation has several notable benefits. By avoiding altogether the need for reparametrizations, it provides the flexibility to deal with simplicial meshes of arbitrary topologies and sampling patterns. Moreover, the problem lends itself to a coarse-to-fine multi-resolution implementation, which makes the algorithm scalable to large meshes. Furthermore, this approach extends readily to higher-order feature maps such as square root curvature fields and is also able to include surface textures in the matching problem. We demonstrate these advantages on several examples, synthetic and real.

表面比较和匹配是计算机视觉中一个具有挑战性的问题。尽管弹性黎曼度量通过测地线边界值问题提供了有意义的形状距离和点对应关系，但在数值上解决该问题却趋于困难。平方根法向场大大简化了参数化曲面之间某些距离的计算。但是，它们仍然存在寻找最佳重新参数化的问题，这会导致未参数化的曲面之间产生相应的距离。近年来，这个问题投入了大量精力，并导致了几个数值框架的发展。在本文中，我们采用了一种替代方法，该方法绕过了重新参数化的直接估计：我们使用辅助参数化-盲可变可变保真度度量来放宽测地线边界约束。这种重新制定有几个显着的好处。通过完全避免重新参数化的需求，它提供了处理任意拓扑和采样模式的简单网格的灵活性。而且，该问题使其适合于从粗到细的多分辨率实现，这使得该算法可扩展到大网格。此外，这种方法很容易扩展到高阶特征图，例如平方根曲率场，并且还可以在匹配问题中包括表面纹理。我们在几个示例（合成的和真实的）上展示了这些优势。该问题使其适合于从粗到细的多分辨率实施，这使该算法可扩展到大网格。此外，这种方法很容易扩展到高阶特征图，例如平方根曲率场，并且还可以在匹配问题中包括表面纹理。我们在几个示例（合成的和真实的）上展示了这些优势。该问题使其适合于从粗到细的多分辨率实施，这使该算法可扩展到大网格。此外，这种方法很容易扩展到高阶特征图，例如平方根曲率场，并且还可以在匹配问题中包括表面纹理。我们在几个示例（合成的和真实的）上展示了这些优势。