What is the best mesh simplification algorithm for your computational geometry project?

1 Quadric error metrics

One of the most widely used and influential algorithms for mesh simplification is based on quadric error metrics (QEM), proposed by Garland and Heckbert in 1997. The idea is to measure the distance between a vertex and a set of planes that approximate the local surface around it, and use this as a criterion for selecting and collapsing edges. QEM can produce high-quality simplifications with low distortion and smooth boundaries, and it can handle meshes with arbitrary topology and attributes. However, QEM can also introduce artifacts such as shape flattening, texture stretching, and feature loss, especially for very low-resolution meshes. QEM is also sensitive to the choice of weighting and scaling factors for the planes and the vertices.

2 Progressive meshes

Another influential algorithm for mesh simplification is based on progressive meshes (PM), proposed by Hoppe in 1996. The idea is to generate a sequence of meshes with decreasing levels of detail, starting from the original mesh and applying edge collapse operations in a greedy manner. Each operation is recorded as a vertex split operation that can be reversed to restore the original mesh. PM can produce smooth and continuous transitions between different levels of detail, and it can support selective refinement and adaptive rendering. However, PM can also suffer from quality degradation and feature loss, especially for meshes with sharp edges and corners. PM is also computationally expensive and memory intensive, as it requires storing the entire sequence of meshes and the vertex split operations.

3 Vertex clustering

A simpler and faster algorithm for mesh simplification is based on vertex clustering (VC), proposed by Rossignac and Borrel in 1993. The idea is to divide the bounding box of the mesh into a uniform grid of cells, and replace all the vertices that fall into the same cell with their centroid. The faces of the mesh are then updated to connect the new vertices, and any degenerate faces are removed. VC can achieve significant reduction in the number of vertices and faces, and it can handle meshes with any topology and attributes. However, VC can also produce poor simplifications with high distortion and jagged boundaries, and it can lose important features and details of the mesh. VC is also sensitive to the choice of the grid resolution and the placement of the cells.

4 Surface simplification using quadric error metrics with texture preservation

A more recent and advanced algorithm for mesh simplification is based on surface simplification using quadric error metrics with texture preservation (SSTP), proposed by Li et al. in 2015. The idea is to extend the QEM algorithm to incorporate texture information into the quadric error metrics, and use a texture-aware edge collapse strategy that minimizes the texture distortion. SSTP can produce high-quality simplifications with low geometric and texture errors, and it can preserve the features and details of the mesh. However, SSTP can also be slow and complex, as it requires computing and updating the texture quadrics and the texture distortion for each edge. SSTP is also dependent on the quality and resolution of the texture map and the texture coordinates.

5 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?