Deep learning over spatial data: From a 3D reconstructive perspective

The ability to create and interact with high-fidelity digital representations of the physical world is crucial for applications ranging from digital twins to autonomous navigation. Yet, the robust interpretation and reconstruction of 3D environments from sparse or noisy observations remains a significant challenge for artificial intelligence. This thesis addresses the fundamental problems in 3D reconstruction by exploring the evolving landscape of deep learning over spatial data, from structured parametric models to flexible implicit representations. Our research navigates the trade-offs between these paradigms to develop novel methods that enhance reconstruction fidelity, efficiency, and user control. The investigation begins by confronting the limitations of explicit parametric methods in handling complex topologies from unstructured point clouds. This analysis motivates a pivot towards implicit neural representations. Our work introduces key innovations in this area, including Seed-Net, an interactive framework that differentiably incorporates sparse user guidance to refine local geometric details in neural fields, and NeuLap, a geometry-aware training scheme that leverages a learned Laplacian prior to refine the convergence process and improve the reconstruction of sharp features from limited data. Building on these insights, the research path leads to the development of a general-purpose backbone for large-scale 3D learning: a Hierarchical Attention OctTree. This architecture introduces a novel attention propagation mechanism that efficiently captures multi-scale spatial context, demonstrating competitive performance and memory efficiency. Collectively, these contributions offer a suite of methods and a conceptual roadmap for 3D spatial learning. Through advancements in interactive reconstruction, prior guided optimization, and scalable deep learning architectures, this research contributes to the field of representing and reconstructing high-fidelity 3D spatial data, providing technical insights and solutions that are valuable for various future applications in fields such as digital twins, robotics, and creative design.