A Comprehensive Guide to Mesh Simplification using Edge Collapse

TL;DR

This paper provides a detailed, implementation-oriented overview of mesh simplification through edge collapse, covering mathematical foundations, cost functions, algorithms, and practical safeguards for preserving mesh quality.

Contribution

It offers a comprehensive, step-by-step guide with reference implementations, bridging theoretical concepts and practical techniques for mesh simplification using edge collapse.

Findings

Includes reference implementations of QEM and Lindstrom-Turk cost functions

Details safeguards against mesh degeneracies and boundary issues

Provides a complete algorithmic outline for edge collapse processes

Abstract

Mesh simplification is the process of reducing the number of vertices, edges and triangles in a three-dimensional (3D) mesh while preserving the overall shape and salient features of the mesh. A popular strategy for this is edge collapse, where an edge connecting two vertices is merged into a single vertex. The edge to collapse is chosen based on a cost function that estimates the error introduced by this collapse. This paper presents a comprehensive, implementation-oriented guide to edge collapse for practitioners and researchers seeking both theoretical grounding and practical insight. We review and derive the underlying mathematics and provide reference implementations for foundational cost functions including Quadric Error Metrics (QEM) and Lindstrom-Turk's geometric criteria. We also explain the mathematics behind attribute-aware edge collapse in QEM variants and Hoppe's energy-based method used in progressive meshes. In addition to cost functions, we outline the complete edge collapse algorithm, including the specific sequence of operations and the data structures that are commonly used. To create a robust system, we also cover the necessary programmatic safeguards that prevent issues like mesh degeneracies, inverted normals, and improper handling of boundary conditions. The goal of this work is not only to consolidate established methods but also to bridge the gap between theory and practice, offering a clear, step-by-step guide for implementing mesh simplification pipelines based on edge collapse.