Laplacian smoothing revisited

TL;DR

This paper revisits Laplacian smoothing, revealing its connection to the mean ratio quality measure, and introduces generalized convex functions for mesh quality that enable global optimization for untangling and smoothing.

Contribution

It provides a new understanding of Laplacian smoothing's regularization effect and proposes convex quality functions for polygons and polyhedra, facilitating improved mesh optimization.

Findings

Laplacian smoothing's effect is linked to the mean ratio quality measure.

Convex functions for mesh quality enable global optimization.

Comparison of smoothing methods shows effectiveness with topology modifications.

Abstract

We show that the driving force behind the regularizing effect of Laplacian smoothing on surface elements is the popular mean ratio quality measure. We use these insights to provide natural generalizations to polygons and polyhedra. The corresponding functions measuring the quality of meshes are easily seen to be convex and can be used for global optimization-based untangling and smoothing. Using a simple backtracking line-search we compare several smoothing methods with respect to the resulting mesh quality. We also discuss their effectiveness in combination with topology modification.