Studying the changes of shape is a common concern in many scientific fields. We address here two problems: (1) quantifying the deformation between two given shapes and (2) transporting this deformation to morph a third shape. These operations can be done with or without point correspondence, depending on the availability of a surface matching algorithm, and on the type of mathematical procedure adopted. In computer vision, the re-targeting of emotions mapped on faces is a common application. We contrast here four different methods used for transporting the deformation toward a target once it was estimated upon the matching of two shapes. These methods come from very different fields such as computational anatomy, computer vision and biology. We used the large diffeomorphic deformation metric mapping and thin plate spline, in order to estimate deformations in a deformational trajectory of a human face experiencing different emotions. Then we use naive transport (NT), linear shift (LS), direct transport (DT) and fanning scheme (FS) to transport the estimated deformations toward four alien faces constituted by 240 homologous points and identifying a triangulation structure of 416 triangles. We used both local and global criteria for evaluating the performance of the 4 methods, e.g., the maintenance of the original deformation. We found DT, LS and FS very effective in recovering the original deformation while NT fails under several aspects in transporting the shape change. As the best method may differ depending on the application, we recommend carefully testing different methods in order to choose the best one for any specific application.

研究形状的变化是许多科学领域中普遍关注的问题。我们在此解决两个问题：（1）量化两个给定形状之间的变形，以及（2）传输此变形以变形第三个形状。这些操作可以在有或没有点对应的情况下完成，具体取决于表面匹配算法的可用性以及所采用的数学过程的类型。在计算机视觉中，重新定位映射到面部的情感是很常见的应用。我们在这里对比了四种不同的方法，一旦根据两个形状的匹配估计了变形，就可以将变形用于向目标传输。这些方法来自非常不同的领域，例如计算解剖学，计算机视觉和生物学。我们使用了大的亚变形变形度量映射和薄板样条，为了估计经历不同情绪的人脸的变形轨迹中的变形。然后，我们使用朴素传输（NT），线性移位（LS），直接传输（DT）和扇形方案（FS）将估计的变形向由240个同源点组成的四个外来面传输，并识别416个三角形的三角剖分结构。我们使用本地和全局标准来评估这4种方法的性能，例如，保持原始变形。我们发现DT，LS和FS在恢复原始变形方面非常有效，而NT在传输形状变化的某些方面却失败了。由于最佳方法可能会因应用程序而异，因此我们建议仔细测试不同的方法，以便为任何特定应用程序选择最佳方法。