To date, the comparison of Statistical Shape Models (SSMs) is often solely performance-based, carried out by means of simplistic metrics such as compactness, generalization, or specificity. Any similarities or differences between the actual shape spaces can neither be visualized nor quantified. In this paper, we present a new method to qualitatively compare two linear SSMs in dense correspondence by computing approximate intersection spaces and set-theoretic differences between the (hyper-ellipsoidal) allowable shape domains spanned by the models. To this end, we approximate the distribution of shapes lying in the intersection space using Markov chain Monte Carlo and subsequently apply Principal Component Analysis (PCA) to the posterior samples, eventually yielding a new SSM of the intersection space. We estimate differences between linear SSMs in a similar manner; here, however, the resulting spaces are no longer convex and we do not apply PCA but instead use the posterior samples for visualization. We showcase the proposed algorithm qualitatively by computing and analyzing intersection spaces and differences between publicly available face models, focusing on gender-specific male and female as well as identity and expression models. Our quantitative evaluation based on SSMs built from synthetic and real-world data sets provides detailed evidence that the introduced method is able to recover ground-truth intersection spaces and differences accurately.

翻译：截至目前，统计形状模型（SSM）的比较通常仅基于性能，通过紧凑性、泛化能力或特异性等简单指标进行。实际形状空间之间的任何相似性或差异既无法可视化也无法量化。本文提出了一种新方法，通过计算模型所张成的（超椭球）可行形状域之间的近似交集空间和集合论差异，对具有密集对应关系的两个线性SSM进行定性比较。为此，我们采用马尔可夫链蒙特卡洛方法近似位于交集空间中的形状分布，随后对后验样本应用主成分分析（PCA），最终生成交集空间的新SSM。我们以类似方式估计线性SSM间的差异；但在此情况下，所得空间不再具有凸性，因此我们不应用PCA，而是直接使用后验样本进行可视化。我们通过计算并分析公开人脸模型的交集空间与差异，针对性别特异性（男/女）以及身份与表情模型，定性展示了所提算法的有效性。基于合成数据集和真实世界数据集构建的SSM进行的定量评估提供了详细证据，表明本方法能够准确恢复真实交集空间与差异。