Beyond Angular Error: Global and Local Geometry of Failure in Monocular Surface-Normal Models

Per-pixel angular error measures how much a surface-normal prediction is wrong, but it hides the geometric form of that error. We show that failures of modern monocular surface-normal estimators exhibit reproducible structure at both global and local scales. Globally, we use continuous generalized bas-relief (GBR) fitting as an interpretable coordinate system for prediction–reference discrepancies. Across four architectures, matched pairs concentrate in a shared low-shear regime, while correspondence-destroying controls move the fitted transforms toward higher shear. The remaining relief coordinate $\lambda$ also transfers beyond the pixels used to estimate it: on 100 randomly sampled NYUv2 validation images, cross-fitted $\log \lambda$ correlates with angular error on disjoint held-out pixels across all four models ($r\in [-0.70,-0.60]$). Locally, we derive a prediction-only curl score from a single estimated normal field. On 500 NYUv2 images per model, it identifies the worst 10% of predictions with AUROC 0.77–0.91, outperforming generic normal roughness and improving selective risk–coverage. Together, these results show that normal-estimation errors are organized by reusable geometric structure, providing both a diagnostic view of model failure and a practical signal for test-time reliability.