Learning Geometry-Aware Embedding Fields for Intrinsic Riemannian Mappings

Computing exponential and logarithmic maps on raw geome-tries is a fundamental yet challenging task in geometry processing, oftenhindered by the lack of explicit connectivity and the presence of noise.We present a novel learning-based framework that computes these mapsefficiently by learning a continuous geometric embedding field directlyfrom discrete point sets. Our key insight is to construct an embeddingfield that encapsulates the intrinsic local geometry governing exponen-tial and logarithmic maps. Specifically, we encode the input into a high-dimensional feature volume using a sparse Octree-based CNN. For anyarbitrary query point, we retrieve its corresponding embedding to mod-ulate two specialized triplane-based neural networks, which then predictthe maps in a single forward pass. By formulating the problem as learninga query-able latent field, our method bypasses the need for mesh con-nectivity while ensuring robustness against irregular sampling and noise.Our results demonstrate that the proposed framework achieves compet-itive accuracy and runtime over previous methods, while maintainingrobustness on challenging geometric structures.