Still, the total rendering time is often dominated by the extraction time of the isosurface, that is the computation of the isosurface triangle vertices. Volumetric multiresolution methods allow a triangle reduction during the extraction phase through suitable approximations of the volume, thereby speeding up not only the rendering time but also the extraction time.
Ideally, a coarse isosurface should have the same topological structure as the original. The topological genus of the isosurface is one important property which is often neglected in multiresolution algorithms. This results in uncontrolled topological changes which can occur whenever the level-of-detail is changed. Topology control can be achieved by the fast computation and classification of hierarchical critical points and intervals.
Above you see isosurfaces of the CT scan of a lobster (courtesy of AVS) for varying thresholds. For this isovalue the topology of the isosurface is very complex due to the multitude of lobster shells. At coarse resolutions only the main structure of the lobster is left. The number of triangles differs by a factor of 2 from image to image.
Here you see isosurfaces of an algebraic function for varying thresholds without (upper row) and with (lower row) topology preservation. Note that without topology preservation delicate structure of the singularity in the center of the function gets lost for large error thresholds.
These are isosurfaces of the buckyball (courtesy of AVS) for varying thresholds again without (upper row) and with (lower row) topology preservation. The topological genus of this isosurface is much higher than in the previous example. As in the lobster case, for coarse resolutions only the overall shape of the isosurface is visible without topology preservation. The topology preserving method preserves the genus exactly, albeit at a slightly higher triangle count in this example.