Registration maps or warps form a key element in Shape-from-Template (SfT). They relate the template with the input image, which contains the projection of the deformed surface. Recently, it was shown that isometric SfT can be solved analytically if the warp and its first-order derivatives are known. In practice, the warp is recovered by interpolating a set of discrete template-to-image point correspondences. This process relies on smoothness priors but ignores the 3D geometry. This may produce errors in the warp and poor reconstructions. In contrast, we propose to create a 3D consistent warp, which technically is a very challenging task, as the 3D shape variables must be eliminated from the isometric SfT equations to find differential constraints for the warp only. Integrating these constraints in warp estimation yields the isowarp, a warp 3D consistent with isometric SfT. Experimental results show that incorporating the isowarp in the SfT pipeline allows the analytic solution to outperform non-convex 3D shape refinement methods and the recent DNN-based SfT methods. The isowarp can be properly initialized with convex methods and its hyperparameters can be automatically obtained with cross-validation. The isowarp is resistant to 3D ambiguities and less computationally expensive than existing 3D shape refinement methods. The isowarp is thus a theoretical and practical breakthrough in SfT.

登记图或经纱在“模板形状”（SfT）中形成关键元素。它们将模板与输入图像相关联，其中包含变形表面的投影。最近，研究表明，如果已知翘曲及其一阶导数，则可以解析地求解等距SfT。实际上，通过插入一组离散的模板到图像点的对应关系来恢复变形。此过程依赖于平滑先验，但忽略了3D几何形状。这可能会在翘曲和较差的重建中产生错误。相比之下，我们建议创建3D一致的变形，从技术上讲这是一项非常具有挑战性的任务，因为必须从等距SfT方程中消除3D形状变量，才能找到仅针对变形的微分约束。将这些约束整合到翘曲估计中，得出等翘线，即与等距SfT一致的扭曲3D。实验结果表明，将等翘线纳入SfT管道可以使解析解决方案优于非凸3D形状细化方法和最新的基于DNN的SfT方法。可以使用凸方法适当地初始化isowarp，并可以通过交叉验证自动获得其超参数。与现有的3D形状细化方法相比，等翘线可抵抗3D模糊性，并且计算成本较低。因此，等翘线是SfT的理论和实践突破。