If we are interested in large scale structures in the data we can remove small zone components that contribute only small scale topological and geometric features. More importantly, the zones of criticalities allow us to simplify the dataset in a way that does not change the essential topology of any level set. For example, as long as we retain the maximum point or set p, we can remove readings (vertices) from the zone and not change the topology of any iso-surface of the associated family. If we are interested in topology only, the criticality graph, the criticalities, and their values contain all the essential information to produce all the topologically distinct boundary iso-surfaces of the level sets. If we had only the boundaries of the zones we would have all the essential information needed to recover objects from the data. We could form a new data lattice containing only a very small percentage of the initial readings, and still have lost no essential topological information. For example, in three dimensions we can form, say, 4 by 4 by 4 cubes, keeping only the 8 data readings at the corners and removing the 60 readings between them, within a zone. This means that we can manage Level of Detail rendering, within zones, in a very simple fashion. This is different from Triangle Decimation, where we start with a rendered triangular mesh, and search around for almost co-planar tiles to cull (See [31]), or merge traingles in a manner that preserves the topology of the complex.
We can perform this topology-based simplification of the data, prior to constructing a tiling of any level set. We can reduce the tiling complexity of the resulting iso-surfaces, prior to actually constructing them. We believe this idea is quite new and unique, as most methods employ simplification only after constructing an iso-surface.