Towards Efficient Deep Learning for 3D Geometric Understanding and Generation

3D geometric understanding and generation are fundamental problems in computer vision, with broad applications in augmented and virtual reality, human computer interaction, and autonomous systems, and they form a core component of learning world models. Recent advances in deep learning have led to substantial progress across many 3D tasks, yet achieving accuracy, geometric consistency, and computational efficiency simultaneously remains a key challenge. This dissertation addresses this challenge by developing efficient and principled learning frameworks that explicitly leverage geometric structure across multiple 3D representations. The thesis first studies explicit surface geometry and introduces an optimal transport based framework for learning diffeomorphic mesh deformations. By modeling surfaces as probability measures and employing sliced Wasserstein distance, this approach defines efficient and geometrically meaningful discrepancies between meshes, enabling accurate cortical surface reconstruction. Building on this perspective, the dissertation then investigates unsupervised shape correspondence by integrating functional maps with efficient optimal transport objectives, resulting in robust point to point correspondences under both near isometric and non isometric deformations. The focus then shifts from explicit surfaces to implicit and image conditioned geometry. A geometry guided diffusion framework is proposed for metric depth estimation, formulating depth prediction as an inverse problem constrained by stereo geometry and solved using pretrained diffusion models without retraining. Finally, the dissertation presents a hybrid framework for novel view synthesis that unifies deterministic rendering with masked autoregressive diffusion, operating on ray based representations to balance geometric fidelity and generative completion while significantly improving inference efficiency.