Facial expression editing plays a fundamental role in facial expression generation and has been widely applied in modern film productions and computer games. While the existing 2-D caricature facial expression editing methods are mostly realized by expression interpolation from the original image to the target image, expression extrapolation has rarely been studied before. In this article, we propose a novel expression extrapolation method for caricature facial expressions based on the Kendall shape space, in which the key idea is to introduce a representation for the 3-D expression model to remove rigid transformations, such as translation, scaling, and rotation, from the Kendall shape space. Built upon the proposed representation, the 2-D caricature expression extrapolation process can be controlled by the 3-D model reconstructed from the input 2-D caricature image and the exaggerated expressions of the caricature images generated based on the extrapolated expression of a 3-D model that is robust to facial poses in the Kendall shape space; this 3-D model can be calculated with tools such as exponential mapping in Riemannian space. The experimental results demonstrate that our method can effectively and automatically extrapolate facial expressions in caricatures with high consistency and fidelity. In addition, we derive 3-D facial models with diverse expressions and expand the scale of the original FaceWarehouse database. Furthermore, compared with the deep learning method, our approach is based on standard face datasets and avoids the construction of complicated 3-D caricature training sets.

中文翻译：

---

基于Kendall形状空间理论的漫画表达外推

面部表情编辑在面部表情生成中起着基础性作用，并已广泛应用于现代电影制作和电脑游戏。虽然现有的二维漫画人脸表情编辑方法大多是通过从原始图像到目标图像的表情插值来实现的，但之前很少有人对表情外插进行研究。在本文中，我们提出了一种基于 Kendall 形状空间的漫画面部表情的新表情外推方法，其中关键思想是引入 3-D 表情模型的表示以去除刚性变换，例如平移、缩放、和旋转，来自 Kendall 形状空间。建立在提议的代表之上，2-D 漫画表达外推过程可以通过从输入 2-D 漫画图像重建的 3-D 模型和基于 3-D 模型的外推表达生成的漫画图像的夸张表达来控制肯德尔形状空间中的面部姿势；这个 3-D 模型可以使用诸如黎曼空间中的指数映射之类的工具来计算。实验结果表明，我们的方法可以有效地自动推断漫画中的面部表情，具有高一致性和保真度。此外，我们衍生出具有多种表情的 3-D 面部模型，并扩展了原始 FaceWarehouse 数据库的规模。此外，与深度学习方法相比，