We develop a framework for polynomial regression on Riemannian manifolds. Unlike recently developed spline models on Riemannian manifolds, Riemannian polynomials offer the ability to model parametric polynomials of all integer orders, odd and even. An intrinsic adjoint method is employed to compute variations of the matching functional, and polynomial regression is accomplished using a gradient-based optimization scheme. We apply our polynomial regression framework in the context of shape analysis in Kendall shape space as well as in diffeomorphic landmark space. Our algorithm is shown to be particularly convenient in Riemannian manifolds with additional symmetry, such as Lie groups and homogeneous spaces with right or left invariant metrics. As a particularly important example, we also apply polynomial regression to time-series imaging data using a right invariant Sobolev metric on the diffeomorphism group. The results show that Riemannian polynomials provide a practical model for parametric curve regression, while offering increased flexibility over geodesics.

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This paper presents a novel approach for diffeomorphic image regression and atlas estimation that results in improved convergence and numerical stability. We use a vector momenta representation of a diffeomorphism's initial conditions instead of the standard scalar momentum that is typically used. The corresponding variational problem results in a closed-form update for template estimation in both the geodesic regression and atlas estimation problems. While we show that the theoretical optimal solution is equivalent to the scalar momenta case, the simplification of the optimization problem leads to more stable and efficient estimation in practice. We demonstrate the effectiveness of our method for atlas estimation and geodesic regression using synthetically generated shapes and 3D MRI brain scans.

Machine learning and artificial intelligence (ML/AI) are rapidly becoming an indispensable part of physics research, with domain applications ranging from theory and materials prediction to high-throughput data analysis. In parallel, the recent successes in applying ML/AI methods for autonomous systems from robotics through self-driving cars to organic and inorganic synthesis are generating enthusiasm for the potential of these techniques to enable automated and autonomous experiment (AE) in imaging. Here, we aim to analyze the major pathways towards AE in imaging methods with sequential image formation mechanisms, focusing on scanning probe microscopy (SPM) and (scanning) transmission electron microscopy ((S)TEM). We argue that automated experiments should necessarily be discussed in a broader context of the general domain knowledge that both informs the experiment and is increased as the result of the experiment. As such, this analysis should explore the human and ML/AI roles prior to and during the experiment, and consider the latencies, biases, and knowledge priors of the decision-making process. Similarly, such discussion should include the limitations of the existing imaging systems, including intrinsic latencies, non-idealities and drifts comprising both correctable and stochastic components. We further pose that the role of the AE in microscopy is not the exclusion of human operators (as is the case for autonomous driving), but rather automation of routine operations such as microscope tuning, etc., prior to the experiment, and conversion of low latency decision making processes on the time scale spanning from image acquisition to human-level high-order experiment planning. Overall, we argue that ML/AI can dramatically alter the (S)TEM and SPM fields; however, this process is likely to be highly nontrivial, be initiated by combined human-ML workflows, and will bring new challenges both from the microscope and ML/AI sides. At the same time, these methods will enable fundamentally new opportunities and paradigms for scientific discovery and nanostructure fabrication.