Microsoft Research Technical Report MSR-TR-2000-64, June 2000.
This brief technical note expands on Hugues's earlier paper (from IEEE Visualization '99) about quadric error metrics for simplification. It shows a simple block matrix manipulation that you can use to compute optimal vertex positions more efficiently.
In an earlier paper we introduced a new quadric metric for simplifying triangle meshes using the edge collapse operation. The quadric measures both the geometric accuracy of the simplified mesh surface and the fidelity of appearance fields defined on the surface (such as normals or colors). The minimization of this quadric metric involves solving a linear system of size (3 + m) by (3 + m), where m is the number of distinct appearance attributes. The system has only O(m) nonzero entries, so it can be solved in O(m^2) time using traditional sparse solvers such as the method of conjugate gradients. In this short addendum, we show that the special structure of the sparsity permits the system to be solved in O(m) time.
This paper is available as a 39K PDF file. The canonical copy at MSR is here. If you don't already have it, you will need Adobe Acrobat Reader.