Key Takeaways
- The QEM edge collapse algorithm can be effective with the right cost function and implementation, but hard constraints can harm quality and simplification.
- Soft constraints, such as penalizing changes in normal direction, can improve the algorithm’s performance.
- The standard QEM formulation can be improved by detecting sharp or disconnected edges and adding virtual planes to the cost matrix.
- The topology of the input mesh can limit the effectiveness of edge collapses, and re-meshing algorithms or manual simplification may be necessary for complex meshes.
- The QEM edge collapse algorithm can produce good results, but may require adjustments to the cost function to preserve mesh silhouettes and textures.
Introduction to QEM Edge Collapse
The QEM edge collapse algorithm is a popular method for simplifying 3D meshes while preserving their overall shape and features. However, many implementations of this algorithm suffer from a major deficiency: they enforce hard constraints on when edges can be collapsed, which can harm both the quality and the ability to produce a well-simplified mesh. For example, preventing the flipping of normals of triangles if the angle changes by more than 90 degrees can limit the algorithm’s ability to simplify complex meshes. Instead, using a soft cost function that penalizes changes in the normal direction of adjacent triangles can improve the algorithm’s performance.
The Importance of Soft Constraints
Soft constraints are essential for achieving good results with the QEM edge collapse algorithm. By using a soft cost function, the algorithm can exactly control the number of triangles in the output mesh, which is useful for level of detail (LOD) generation. This approach allows the algorithm to produce the best possible quality and efficiency at each LOD level. Additionally, soft constraints enable the algorithm to temporarily flip triangles more than 90 degrees, which can be necessary for simplifying complex meshes with noisy or rough surfaces. This flexibility is particularly important for meshes with detailed features, such as those generated by 3D reconstruction algorithms.
Limitations of the Standard QEM Formulation
The standard QEM formulation has a significant limitation: it does not penalize edge collapses within the plane of planar surfaces. This makes it easy for flat geometry, such as buildings, to get "eaten away" during the simplification process. To address this issue, it is necessary to detect sharp or disconnected edges and add extra "virtual" planes to the cost matrix, which are perpendicular to the edge normal. This modification helps to prevent bad simplification in the edge normal direction, resulting in a more accurate and detailed output mesh.
Topology and Mesh Simplification
The topology of the input mesh can significantly limit the effectiveness of edge collapses. If the input mesh is a simple closed manifold, such as one generated by isosurface extraction of a signed distance field (SDF), the algorithm works well. However, if the mesh is built from many separate parts, the algorithm may not simplify as well due to the disjointed mesh connectivity. Edge collapses cannot usually remove holes in a mesh, which can limit the algorithm’s ability to simplify complex meshes. In such cases, re-meshing algorithms or manual simplification may be necessary to achieve good results.
Re-meshing and Manual Simplification
Re-meshing algorithms, such as converting the input mesh to an SDF and extracting a manifold isosurface at the desired resolution, can help to merge disjointed parts of the mesh and produce a better quality simplification. However, keeping textures consistent after re-meshing can be challenging. Manual simplification, where an artist creates simplified versions of meshes that do not simplify well with edge collapses, can also be effective. This approach allows the artist to manually omit building details and replace complex meshes with simpler ones, resulting in a more efficient and accurate output.
Example Results
The QEM edge collapse algorithm can produce good results, as demonstrated by the example of simplifying the Sponza mesh from 262k triangles to 10k triangles and 1k triangles. The 10k triangle mesh still looks reasonably good, while the 1k triangle mesh starts to show some degradation. The algorithm handles texture coordinates and normals reasonably well, but may require adjustments to the cost function to better preserve mesh silhouettes with fewer triangles. Overall, the QEM edge collapse algorithm is a powerful tool for simplifying 3D meshes, but its effectiveness depends on the choice of cost function, implementation, and input mesh topology.