# HLO: Half-kernel Laplacian Operator for Surface Smoothing

## TL;DR

This paper introduces the Half-kernel Laplacian Operator (HLO), a novel surface smoothing method that better preserves features and reduces artifacts compared to traditional Laplacian-based techniques, especially on noisy meshes.

## Contribution

The paper proposes the HLO, a new Laplacian operator that improves feature preservation and artifact reduction in surface smoothing, with a simple iterative approach and only one parameter.

## Key findings

- HLO outperforms traditional Laplacian smoothing in feature preservation.
- HLO significantly reduces shrinkage artifacts.
- HLO is effective on high-noise meshes.

## Abstract

This paper presents a simple yet effective method for feature-preserving surface smoothing. Through analyzing the differential property of surfaces, we show that the conventional discrete Laplacian operator with uniform weights is not applicable to feature points at which the surface is non-differentiable and the second order derivatives do not exist. To overcome this difficulty, we propose a Half-kernel Laplacian Operator (HLO) as an alternative to the conventional Laplacian. Given a vertex v, HLO first finds all pairs of its neighboring vertices and divides each pair into two subsets (called half windows); then computes the uniform Laplacians of all such subsets and subsequently projects the computed Laplacians to the full-window uniform Laplacian to alleviate flipping and degeneration. The half window with least regularization energy is then chosen for v. We develop an iterative approach to apply HLO for surface denoising. Our method is conceptually simple and easy to use because it has a single parameter, i.e., the number of iterations for updating vertices. We show that our method can preserve features better than the popular uniform Laplacian-based denoising and it significantly alleviates the shrinkage artifact. Extensive experimental results demonstrate that HLO is better than or comparable to state-of-the-art techniques both qualitatively and quantitatively and that it is particularly good at handling meshes with high noise. We will make our source code publicly available.