A Reeb graph-based representation for non-sequential construction of topologically complex shapes
The shape data of many complex objects, such as anatomical structures, are available in the form of contour slices. To visualize these complex shapes, most existing methods focus on resolving the contours connectivities and generating surface models. These methods do not offer any way of abstracting characteristic features from these complex shapes. To address this problem, Morse theory has previously been proposed as a topological abstraction tool, and a Morse theory-based surface coding system has been developed to code complex shapes as a sequence of basic operators. The topological structure of the coded surface is, however, implicitly stored in the operators. Topological information is thus not readily available without evaluating the operators. This paper proposes a more versatile representation by incorporating a contour containment relation into the earlier representation. With the explicit topological information, non-sequential construction and modifications can be performed without violating topological integrity. Editing operators similar to the Euler operators are also proposed.
topology contours euler operators critical points reeb graph