Digital Morse theory for anatomic modeling

We use a digital adaptation of classical Morse theory to organize (typically density based) interpolated geometry in sampled data. The heart of this new thinking is the recognition and cataloging (indexing) of the transition between families of iso-surfaces at criticalities, as the threshold is varied, into a hierarchical criticality graph, with associated zones. The methods work without any assumption that the underlying function is Morse, in the traditional sense. Using our combinatorial approach, we completely characterize all critical points and structures in multi-dimensional sampled functions. We give an efficient algorithm for computing the criticality graph and zones. We demonstrate that the criticality graph and associated zones have a number of important properties, including the following: Each zone has an associated real interval. Each level set boundary iso-surface (at any threshold) is completely contained within component of a zone. A zone component contains an iso-surface if and only if the iso-value lies within the range of the zone. Moreover, for any pair of iso-values in this range, the iso-surfaces within a zone component are topologically equivalent (homeomorphic). We use this software to build a graph of a representative region of the visible human project CT image database as a test case.