We propose regression models for curve-valued responses in two or more dimensions, where only the image but not the parametrization of the curves is of interest. Examples of such data are handwritten letters, movement paths or outlines of objects. In the square-root-velocity framework, a parametrization invariant distance for curves is obtained as the quotient space metric with respect to the action of re-parametrization, which is by isometries. With this special case in mind, we discuss the generalization of 'linear' regression to quotient metric spaces more generally, before illustrating the usefulness of our approach for curves modulo re-parametrization. We address the issue of sparsely or irregularly sampled curves by using splines for modeling smooth conditional mean curves. We test this model in simulations and apply it to human hippocampal outlines, obtained from Magnetic Resonance Imaging scans. Here we model how the shape of the irregularly sampled hippocampus is related to age, Alzheimer's disease and sex.

翻译：我们提出针对二维及以上维度曲线值响应的回归模型，其中仅关注曲线的图像而非参数化。此类数据包括手写字母、运动轨迹或物体轮廓等实例。在平方根速度框架下，曲线参数化不变距离可表示为重参数化作用下的商空间度量（该作用通过等距变换实现）。基于这一特例，我们首先讨论将"线性"回归推广至一般商度量空间的问题，随后阐明该方法对模重参数化曲线的实用性。针对稀疏或非规则采样曲线，我们采用样条函数建模光滑条件均值曲线。通过仿真实验验证模型后，我们将该方法应用于磁共振成像扫描获取的人类海马体轮廓数据，建模非规则采样海马体形态与年龄、阿尔茨海默病及性别的关联。