These methods iteratively remove elements (vertices, edges or triangles) from the mesh. When removing a triangle, it is replaced by a single vertex that is chosen to minimize some error metric. After updating the neighborhood this step is repeated until some stop criterion is reached. Other approaches remove vertices and retriangulate the resulting holes, thus successively simplifying the mesh. The removal of edges and their adjacent faces is also possible. As stated above, Progressive Meshes can be considered a special case of edge decimation.
Different proposals were made to bound the error of the simplified mesh. Some methods define an error volume surrounding the original mesh. Simplification stops when the simplified mesh is no longer completely contained in this error volume. In [11] the HAUSDORFF-distance is explicitly measured between regions of topological correspondence. The deviation spaces explained in section 4.1 are constructed in a similar way.
Another method worth mentioning is presented by LINDSTROM et al. in [12]. It is used for (and restricted to) the rendering of height fields. The mesh must be defined over a regular grid. Adjacent triangles are merged when the screen-space geometric error introduced by this operation is below a user-definable threshold. This error is measured as the projection to the screen of the vertical deviation between the simplified and the original mesh. HOPPE refers to this idea in [6].