# Efficient Minimization of New Quadric Metric for Simplifying Meshes with
Appearance Attributes

Hugues Hoppe
and
Steve Marschner.

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.

## Abstract

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.

## Availability

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.

Steve Marschner