Towards Understanding 3D Vision: the Role of Gaussian Curvature

TL;DR

This paper explores the significance of Gaussian curvature in 3D surface modeling, demonstrating its invariance, compactness, and correlation with stereo reconstruction performance, proposing it as a geometric prior for future algorithms.

Contribution

It highlights Gaussian curvature as a fundamental geometric invariant and correlates it with stereo method performance, suggesting its use as a prior in 3D reconstruction.

Findings

Gaussian curvature is invariant under observer changes.

It provides a sparse, compact surface description.

Performance of stereo methods correlates with low Gaussian curvature.

Abstract

Recent advances in computer vision have predominantly relied on data-driven approaches that leverage deep learning and large-scale datasets. Deep neural networks have achieved remarkable success in tasks such as stereo matching and monocular depth reconstruction. However, these methods lack explicit models of 3D geometry that can be directly analyzed, transferred across modalities, or systematically modified for controlled experimentation. We investigate the role of Gaussian curvature in 3D surface modeling. Besides Gaussian curvature being an invariant quantity under change of observers or coordinate systems, we demonstrate using the Middlebury stereo dataset that it offers a sparse and compact description of 3D surfaces. Furthermore, we show a strong correlation between the performance rank of top state-of-the-art stereo and monocular methods and the low total absolute Gaussian curvature. We propose that this property can serve as a geometric prior to improve future 3D reconstruction algorithms.