GreenCloud: Volumetric Gradient Filtering via Regularized Green's Functions

Abstract

Gradient-based optimization is a fundamental tool in geometry processing, but it is often hampered by geometric distortion arising from noisy or sparse gradients. Existing methods mitigate these issues by filtering (i.e., diffusing) gradients over a surface mesh, but they require explicit mesh connectivity and solving large linear systems, making them unsuitable for point-based representation. In this work, we introduce a gradient filtering method tailored for point-based geometry. Our method bypasses explicit connectivity by leveraging regularized Green's functions to directly compute the filtered gradient field from discrete spatial points. Additionally, our approach incorporates elastic deformation based on Green's function of linear elasticity (known as Kelvinlets), reproducing various elastic behaviors such as smoothness and volume preservation while improving robustness in affine transformations. We further accelerate computation using a hierarchical Barnes–Hut style approximation, enabling scalable optimization of one million points. Our method significantly improves convergence across a wide range of applications, including reconstruction, editing, stylization, and simplified optimization experiments with Gaussian splatting.