Statistical Gradient Filtering for Geometry Optimization Under Limited Observations

摘要

Geometry optimization often encounters sparse gradients from limited observations, which can cause optimization to drift toward unnatural shapes. Prior work stabilizes this process by exploiting spatial structure encoded by the Laplacian operator within a single shape, enforcing spatial smoothness on gradients. However, such smoothness alone propagates gradients to unobserved regions without any prior knowledge of how shapes typically deform in a given domain. We introduce statistical gradient filtering, which leverages statistical structure across a shape collection by learning shape variations via principal component analysis (PCA) and guiding geometry updates along directions consistent with this learned prior. Unlike previous PCA-based methods that constrain solutions to a linear subspace or modify the objective with regularization terms, we filter gradients at each iteration to steer the optimization path toward plausible shapes, without restricting the solution space or altering the original objective. We validate our approach across a range of shape optimization tasks, demonstrating robust convergence even under challenging conditions.