Determining the mean shape of a collection of curves is not a trivial task, in particular when curves are only irregularly/sparsely sampled at discrete points. We newly propose an elastic full Procrustes mean of shapes of (oriented) plane curves, which are considered equivalence classes of parameterized curves with respect to translation, rotation, scale, and re-parameterization (warping), based on the square-root-velocity (SRV) framework. Identifying the real plane with the complex numbers, we establish a connection to covariance estimation in irregular/sparse functional data analysis. We introduce Hermitian covariance smoothing and show how to employ this extension of existing covariance estimation methods for obtaining an estimator of the (in)elastic full Procrustes mean, also in the sparse case not yet covered by existing (intrinsic) elastic shape means. For this, we provide different groundwork results which are also of independent interest: we characterize (the decomposition of) the covariance structure of rotation-invariant bivariate stochastic processes using complex representations, and we identify sampling schemes that allow for exact observation of derivatives / SRV transforms of sparsely sampled curves. We demonstrate the performance of the approach in a phonetic study on tongue shapes and in different realistic simulation settings, inter alia based on handwriting data.

翻译：确定曲线集的平均形状不是一件微不足道的任务,特别是当曲线只在离散点不定期/不定期地抽样取样时。我们新提议一个弹性完整的平流曲线(方向)曲线形状的全彩虹平均值,这些曲线被视为根据平底速度(SRV)框架(正根速度(SRV)框架)确定的参数化曲线的等值类别。用复杂数字识别真实平面,我们在非常规/偏差功能性数据分析中建立与共变估计的连接。我们引入了Hermitian 共差平滑,并展示了如何利用现有共差估计方法的延伸,以获得(面向方向)平流曲线(方向)形状的(弹性)全彩虹曲线的估量值,这也是根据平底速度(SRV)弹性形状(SVV)框架框架尚未覆盖的稀薄情况。我们提供了不同的基础结果,这些结果也具有独立的兴趣:我们用复杂的正统(分辨)旋转易变异变量的正比差函数结构平流,我们用复杂的平流的正态模型分析方法,我们用精确的正态分析模型分析方法,在精确的平流分析中,我们用精确的平流分析方法来确定。