Structure-exploiting uncertainty quantification and control : Bayesian inference and Itô SDE methods from MRI segmentation to autonomous sensing

Uncertainty is a defining feature of both medical imaging and autonomous decision-making. In clinical diagnostics, inaccurate predictions can lead to harmful outcomes, while in autonomous sensing, uncertainty in partial observations can hinder safe and efficient control. This thesis develops methods that both quantify uncertainty and exploit problem-specific structure, demonstrating their effectiveness across two distinct but connected domains. The first part addresses challenges in cardiac magnetic resonance imaging, where automated segmentation methods often produce overconfident estimates of ventricular volumes. A post-hoc uncertainty quantification method based on Itô stochastic differential equations is introduced to model prediction error dynamics; we establish the existence of a unique strong solution that is non-negative almost surely and prove that the procedure is bias-free with respect to the underlying automatic segmentation. Moving from classical cine MRI to real-time MRI, where thousands of frames must be processed to capture both cardiac and respiratory motion, the main challenge shifts from overconfidence to the prohibitive cost of manual labeling. To address this, we employ sparse Bayesian learning, to automatically prune irrelevant frequency components, leveraging the dominant frequency structure of heartbeat and respiration to identify the most informative frames to label. We show that the resulting greedy selection scheme admits approximation guarantees relative to the optimal scheme. Finally, spatial correlations between slices are incorporated into the Bayesian framework, improving predictive accuracy when labeled data are scarce. The second part turns to multi-agent localization of an unknown pollutant source under partial observation. We cast the problem as a continuous–discrete stochastic control system: between measurements, the value function evolves according to Hamilton–Jacobi–Bellman dynamics, and at observation times Gaussian Bayesian updates incorporate new data. Directly solving the high-dimensional HJB equation is computationally demanding, so we exploit structural properties of the problem to improve efficiency. For example, since the dominant uncertainty often aligns with the wind direction, the discretization of the posterior can be concentrated along this axis, reducing the number of grid points required. Similarly, permutation symmetry across agents allows a decomposition of the value function analogous to analysis of variance, enabling scalable approximations by retaining only single- and pairwise interaction terms. This approach mitigates the curse of dimensionality and provides flexibility, as objectives such as collision avoidance or patrolling trajectories can be incorporated seamlessly. This work combines Bayesian inference and stochastic differential equations to address challenges in cardiac imaging and autonomous sensing. By exploiting inherent problem structure, it renders otherwise intractable inference and control problems computationally feasible, advancing both the theory and application of uncertainty quantification and optimal decision-making.