New findings from the spiderweb algorithm: Toward a digital morse theory

Algorithms that tile isovalued surfaces should produce "correctly" tiled orientable manifold surfaces. Rigorous evaluation of different algorithms or case tables has been impossible up to now because of the lack of a clear and comprehensive theoretical framework. We propose the extension of Morse theory, as developed in the continuous domain, to apply to discretely sampled continuous domains (sampled Morse functions). We call this the digital domain, and thus formulate a Digital Morse Theory (DMT). We show that a discretely sampled continuous volume in which we are tiling a surface can have various classes of Morse criticalities. When the isosurface comes close (within a voxel length) of a criticality, this is what gives rise to certain apparent ambiguities in tiling surfaces. DMT provides a heuristic to correctly disambiguate tiling decisions. In addition, DMT gives insight to correctly simplifying a volume data set so as to produce an isosurface with a reduced number of tiles, and yet maintain a topology. Additionally, one can establish a hierarchical relationship between each criticality and its associated regions.