The field of computational anatomy has developed rigorous frameworks for analyzing anatomical shape, based on diffeomorphic transformations of a template. However, differences in algorithms used for template warping, in regularization parameters, and in the template itself, lead to different representations of the same anatomy. Variations of these parameters are considered as confounding factors as they give rise to non-unique representation. Recently, extensions of the conventional computational anatomy framework to account for such confounding variations have shown that learning the equivalence class derived from the multitude of representations can lead to improved and more stable morphological descriptors. Herein, we follow that approach, estimating the morphological appearance manifold obtained by varying parameters of the template warping procedure. Our approach parallels work in the computer vision field, in which variations lighting, pose and other parameters leads to image appearance manifolds representing the exact same figure in different ways. The proposed framework is then used for groupwise registration and statistical analysis of biomedical images, by employing a minimum variance criterion on selected complete morphological descriptor to perform manifoldconstrained optimization, i.e. to traverse each individual's morphological appearance manifold until group variance is minimal. Effectively, this process removes the aforementioned confounding effects and potentially leads to morphological representations reflecting purely biological variations, instead of variations introduced by modeling assumptions and parameter settings. The nonlinearity of a morphological appearance manifold is treated via local approximations of the manifold via PCA.