Recent developments in theory, methodology, and applications of statistical shape analysis

The reliable recovery and uncertainty quantification of a fixed effect function μ in a functional mixed model, for modelling population- and object-level variability in noisily observed functional data, is a notoriously challenging task: variations along the x and y axes are confounded with additive measurement error, and cannot in general be disentangled. The question then as to what properties of μ may be reliably recovered becomes important. We demonstrate that it is possible to recover the size-and-shape of a square-integrable μ under a Bayesian functional mixed model. The size-and-shape of μ is a geometric property invariant to a family of space-time unitary transformations, viewed as rotations of the Hilbert space, that jointly transform the x and y axes. A random object-level unitary transformation then captures size-and-shape preserving deviations of μ from an individual function, while a random linear term and measurement error capture size-and-shape altering deviations. The model is regularized by appropriate priors on the unitary transformations, posterior summaries of which may then be suitably interpreted as optimal data-driven rotations of a fixed orthonormal basis for the Hilbert space. Our numerical experiments demonstrate utility of the proposed model, and superiority over the current state-of-the-art. This is joint work with Fangyi Wang (Statistics, Ohio State University), Oksana Chkrebtii (Statistics, Ohio State University) and Karthik Bharath (Mathematical Sciences, University of Nottingham). 

In the 21st century, we have seen a growing availability of shape-valued and imaging data, prompting the development of new statistical methods to analyze them. Importantly, bridging the new methods and existing frameworks is advisable. In this talk, I will introduce several statistical inference methods for shapes and images based on Euler characteristics. These methods have applications in many fields, such as geometric morphometrics and radiomics. From a statistical perspective, these methods are naturally connected to functional data analysis and tensor regression. From a mathematical viewpoint, they are grounded in solid foundations, bridging various branches of mathematics: algebraic and tame topology, Euler calculus, functional analysis, and probability theory. 

In this talk, we introduce and formalize the concepts of derivative and curvature processes of Gaussian processes on compact Riemannian manifolds, both identified as Gaussian processes themselves. The derivation of covariance functions for these derivative and curvature processes forms a substantial part of our work, together with establishing the joint distribution among the original process, the derivative process, and the curvature process, which allows for principled statistical inference. Our examination finds its practicality in the case study of spheres of arbitrary dimensions. By scrutinizing the smoothness of 13 extant kernels on spheres, we derive necessary and sufficient conditions for a kernel to be continuous or differentiable on this manifold. Additionally, we determine the covariance structure of the process, its derivative, and its curvature in this setting. Empirical evidence obtained through simulations lends substantial weight to our theoretical findings. By extending the understanding of Gaussian processes on Riemannian manifolds, this study unlocks a variety of potential applications in machine learning and statistics where Gaussian processes are used.